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+Scaling laws for single-agent reinforcement learning
+Jacob Hilton
+OpenAI
+jacob.hilton@gmail.com
+Jie Tang
+OpenAI
+jietang@openai.com
+John Schulman
+OpenAI
+joschu@openai.com
+Abstract
+Recent work has shown that, in generative modeling, cross-entropy loss improves
+smoothly with model size and training compute, following a power law plus
+constant scaling law. One challenge in extending these results to reinforcement
+learning is that the main performance objective of interest, mean episode return,
+need not vary smoothly. To overcome this, we introduce intrinsic performance,
+a monotonic function of the return defined as the minimum compute required to
+achieve the given return across a family of models of different sizes. We find that,
+across a range of environments, intrinsic performance scales as a power law in
+model size and environment interactions. Consequently, as in generative modeling,
+the optimal model size scales as a power law in the training compute budget.
+Furthermore, we study how this relationship varies with the environment and with
+other properties of the training setup. In particular, using a toy MNIST-based
+environment, we show that varying the “horizon length” of the task mostly changes
+the coefficient but not the exponent of this relationship.
+1
+Introduction
+Recent studies of how neural network performance varies with model size and training compute have
+found these relationships to be governed by smooth power laws [Kaplan et al., 2020, Henighan et al.,
+2020, Droppo and Elibol, 2021, Ghorbani et al., 2021]. These studies have focused primarily on
+generative modeling, in which the training objective is cross-entropy loss, and have found test loss to
+scale smoothly. In this work we seek to extend these results to reinforcement learning, in which there
+is generally no cross-entropy loss.
+In some reinforcement learning environments, there is still a performance metric that varies smoothly.
+For example, in competitive games, it is often possible to assign Elo ratings to players such that
+scaled differences in Elo ratings give approximate logit probabilities of victory. Recently it has been
+shown that, in the board games Hex [Jones, 2021], Connect Four and Pentago [Neumann and Gros,
+2022], the exponentiated Elo rating of a policy trained using AlphaZero [Silver et al., 2018] follows a
+power law in training compute (within a certain Elo range). We call metrics that follow such simple
+relationships natural performance metrics.
+However, in other reinforcement learning environments, there may be no obvious natural performance
+metric. For example, there may be no reason to expect the number of objects collected in a video
+game to vary smoothly, since crossing some threshold may require some challenging new capability.
+To overcome this difficulty, we introduce intrinsic performance, which is defined to be equal to
+training compute on the compute-efficient frontier of the tradeoff between model size and environment
+interactions. This causes the relationship between performance and training compute to follow a
+power law by definition, thereby making it possible to study the remaining relationships between
+performance, model size and environment interactions.
+We study these relationships across a range of environments: the easy and hard modes of environments
+from Procgen Benchmark [Cobbe et al., 2020]; a 1v1 version of Dota 2 [OpenAI et al., 2019]; and a toy
+environment based on MNIST [LeCun, 1998] for which we vary the “horizon length”. Across these
+arXiv:2301.13442v1  [cs.LG]  31 Jan 2023
+
+environments, we find intrinsic performance to scale as a power law in model size and environment
+interactions, in much the same way as the analogous quantities in generative modeling.
+One consequence of this scaling law is that, as in generative modeling, the optimal model size for
+a given training compute budget follows a power law. We study in detail how the coefficient and
+exponent of this relationship vary with properties of the training setup, including: the difficulty mode
+of environment, for Procgen; the horizon length of the task, for the MNIST-based environment; the
+period of training used to fit the power law; and whether the width or depth of the model is scaled.
+Contents
+1
+Introduction
+1
+2
+Scaling laws without cross-entropy loss
+3
+2.1
+Intrinsic performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+2.2
+The power law for intrinsic performance . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.3
+Optimal model size vs compute . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+3
+Experimental setup
+5
+4
+Results
+7
+4.1
+Optimal model size vs compute . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+4.2
+Effect of task horizon length . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+4.3
+Variability of exponents over training
+. . . . . . . . . . . . . . . . . . . . . . . .
+10
+4.4
+Scaling depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+4.5
+Natural performance metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+5
+Discussion
+13
+5.1
+Extrapolating sample efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+5.2
+Cost-efficient reinforcement learning . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+5.3
+Limitations
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+5.4
+Forecasting compute requirements . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+6
+Conclusion
+16
+A Curve-fitting methodology
+19
+B
+Hyperparameters
+21
+C Results in full
+24
+D Parameter and FLOP calculations
+27
+E
+Fitted constants
+28
+F
+Proof of the lemma
+32
+G Proof sketch of the proposition
+33
+2
+
+1014
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+5
+10
+15
+20
+25
+30
+Mean episode return
+StarPilot, hard
+(a) Using the usual metric of mean episode return.
+1014
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+Parameters
+104.3
+104.6
+104.9
+105.2
+105.5
+105.8
+106.1
+106.4
+106.7
+107.0
+StarPilot, hard
+(b) Using intrinsic performance instead.
+Figure 1: Learning curves as a function of total training compute for StarPilot, an environment from
+Procgen Benchmark, using CNNs of different widths. Mean ±1 sample standard deviation over three
+seeds shown.
+2
+Scaling laws without cross-entropy loss
+2.1
+Intrinsic performance
+In generative modeling, cross-entropy test loss scales smoothly with training compute, following a
+power law plus constant scaling law [Henighan et al., 2020]. However, in reinforcement learning
+(RL), there is generally no cross-entropy loss, and the usual objective of mean episode return need
+not scale so smoothly.
+For example, consider StarPilot, a side-scrolling shooter from Procgen Benchmark [Cobbe et al.,
+2020]. The agent receives a reward of 1 for destroying each enemy, and the episode continues until
+either the agent is destroyed, or the agent reaches the end of the level and obtains a bonus reward
+of 10. There is no reason to expect mean episode return in this game to scale smoothly. Indeed, it
+takes some ability with aiming and dodging to reach a mean episode return of 5 or 10, but not much
+additional skill to reach a mean episode return of 15 or 20. This irregular difficulty profile is reflected
+in the uneven shape of learning curves for this environment (see Figure 1(a)).
+It may be tempting to conclude that the scaling law methodology cannot be applied to such an
+environment. However, in generative modeling, there are smooth scaling laws that do not depend on
+test loss per se. For example, the model size that achieves the minimum test loss for a given compute
+budget scales as a power law with compute. In order to study such relationships in the context of
+RL, we would like a performance metric that behaves like test loss, i.e., some monotonic function
+of the return that scales as a power law with compute. We achieve this with our notion of intrinsic
+performance by simply using compute itself as our performance metric.
+Definition. A scalable model family is collection of models trained in a uniform way, parameterized
+by the model size and the total compute used in training. Given a scalable model family, the intrinsic
+performance of an arbitrary policy is the minimum compute required to train a model of any size in
+the family to reach the same return (averaged over random seeds).
+Another way of explaining this definition is to consider learning curves as a function of compute
+for a family of models of different sizes, as in Figure 1. The maximum performance over all model
+sizes defines the compute-efficient frontier. When using the usual metric of mean episode return (as
+in Figure 1(a)), the compute-efficient frontier need not follow any particular trend. However, when
+using intrinsic performance instead (as in Figure 1(b)), the efficient frontier is mapped onto the line
+3
+
+1014
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+5
+10
+15
+20
+25
+30
+Mean episode return
+StarPilot, hard
+(a) Using mean episode return.
+1014
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+Parameters
+104.3
+104.6
+104.9
+105.2
+105.5
+105.8
+106.1
+106.4
+106.7
+107.0
+Learning
+curve
+Power law
+fit
+Power law
+asymptote
+Efficient
+frontier
+Efficient
+points
+StarPilot, hard
+(b) Using intrinsic performance.
+Figure 2: Learning curves as a function of total training compute for StarPilot, together with their
+power law fits. The asymptotes show the E → ∞ limits of the power law fits, representing the
+predicted performance at convergence. The efficient points show where the power law fits are tangent
+to the efficient frontier. Mean over three seeds shown.
+y = x by definition. This reveals the regularity of the learning curves, which, as we shall see next,
+now follow a power law trend.
+We describe in detail how we compute intrinsic performance in Appendix A.
+2.2
+The power law for intrinsic performance
+Our main empirical result is that intrinsic performance I scales approximately as a power law with
+model parameters N and environment interactions E,
+I−β =
+�Nc
+N
+�αN
++
+�Ec
+E
+�αE
+,
+(1)
+where αN, αE, β, Nc and Ec are positive constants.
+This is essentially the same as the corresponding scaling law for language models [Kaplan et al.,
+2020, equation (1.6)], but with test loss replaced by I−β. Although it appears that we have introduced
+an additional exponent β, the intrinsic definition of I means that β is actually determined by αN and
+αE (see Lemma 1).
+The intuition behind this equation is that, when the number of interactions is not bottlenecked
+(E → ∞), I scales as a power law in N, and when model size is not bottlenecked (N → ∞), I
+scales as a power law in E.
+2.3
+Optimal model size vs compute
+An important implication of equation (1) is that the optimal model size for a given compute budget
+scales as a power law in that compute budget.
+More precisely, we assume that total training compute is proportional to NE (ignoring the compute
+required to run the environment, at least for now). Hence, for a given compute budget, there is a
+trade-off between N and E (the optimum of which defines a point on the compute-efficient frontier).
+What we will now show is that, under equation (1), the optimal value of N scales as a power law in
+the compute budget, with an exponent that we will specify.
+4
+
+Since training compute is proportional to NE, for convenience we choose units of compute such
+that training compute equals NE exactly (although in plots we will continue to display compute in
+FLOPs). This implies that I = NE along the compute-efficient frontier.
+Lemma 1. If I satisfies equation (1) and I = NE along the compute-efficient frontier, then the
+compute-efficient frontier is described by the equation
+αN
+�Nc
+N
+�αN
+= αE
+�Ec
+E
+�αE
+.
+(2)
+Moreover, once αN and αE are chosen, β and NcEc are determined:
+1
+β =
+1
+αN
++ 1
+αE
+and
+1
+NcEc
+=
+�
+1 + αN
+αE
+�
+1
+αN �
+1 + αE
+αN
+�
+1
+αE .
+For a proof, see Appendix F.
+Substituting equation (2) into equation (1), it follows that along the compute-efficient frontier,
+N = Nc
+�
+1 + αN
+αE
+�
+1
+αN C
+1
+1+ αN
+αE ,
+where C := NE. In other words, for a given compute budget C, the optimal model size N scales as
+N ∝ C
+1
+1+ αN
+αE .
+3
+Experimental setup
+We ran experiments using variety of RL environments:
+• Procgen Benchmark [Cobbe et al., 2020]: CoinRun, StarPilot and FruitBot in both easy
+and hard modes, separately varying CNN width and depth.
+• Dota 2 [OpenAI et al., 2019]: a 1v1 version of the game, varying LSTM size.
+• MNIST: an RL environment in which the agent has to correctly label a handwritten digit
+from MNIST [LeCun, 1998], using hyperparameters to artificially alter the “horizon length”
+of the task, varying CNN width.
+All our experiments used a variant of either the PPO algorithm [Schulman et al., 2017] or its close
+cousin PPG [Cobbe et al., 2021], along with the Adam optimization algorithm [Kingma and Ba,
+2014].
+The remainder of this section discusses further details of our experimental setup. Hyperparameters
+for all our experiments are given in Appendix B.
+3.1
+Procgen Benchmark
+For our Procgen Benchmark experiments, we used CoinRun, StarPilot and FruitBot. We chose these
+environments because they have lower-variance learning curves than other Procgen environments,
+and because CoinRun’s binary reward enabled us to study the scaling of natural performance metrics
+(see Section 4.5). We used both the easy and hard difficulty modes of these environments to see if
+this would have an effect on the scaling constants.
+We used PPG-EWMA [Hilton et al., 2021] with a fixed KL penalty objective [Cobbe et al., 2021],
+and trained for 200 million environment interactions.
+We used the CNN architecture from IMPALA [Espeholt et al., 2018] and conducted both width-
+scaling and depth-scaling experiments. For our width-scaling experiments, we varied the total number
+of parameters from
+1
+64 of the default to 8 times the default, rounding to integer numbers of channels.
+For our depth-scaling experiments, we varied the number of residual blocks per stack from 1 to 64,
+and used 1
+4 of the default width since the default number of residual blocks per stack was only 2.
+5
+
+3.2
+Dota 2
+For our Dota 2 experiments, we used a 1v1 version of the game to save computational expense.
+Following OpenAI et al. [2019], we used PPO, but we adjusted the asynchronous setup to ensure that
+training used only on-policy data with no data reuse. We used 8 parallel GPU workers and trained for
+between 13.6 billion and 82.6 billion environment interactions.
+We used an LSTM architecture and varied the width of the network, with the sizes of the embedding
+and hidden state varying from 8 to 4096.
+3.3
+MNIST
+Our MNIST environment samples a handwritten digit from the MNIST training set uniformly and
+independently random at each timestep, and provides an immediate reward of 1 for a correct label
+and 0 for an incorrect label. There are no episode boundaries, and so we measure mean training
+accuracy instead of mean episode return.
+The use of immediate rewards with no episode boundaries allows the horizon length of the task
+to be artificially controlled by varying the hyperparameters of our method advantage estimation,
+GAE [Schulman et al., 2015]. First, we set the GAE credit assignment parameter λ to 1, so that the
+algorithm assigns credit for each reward to all previous actions, instead of assigning more immediate
+credit. Then we vary the GAE discount rate γ, so that the algorithm discounts future rewards at this
+rate. In separate experiments, we set γ = 1 −
+2
+h+1 for different values of the “horizon length” h
+ranging from 1 to 256. (This equation is equivalent to saying that an exponentially-weighted moving
+average with decay parameter γ has the same center of mass as the interval [0, h − 1].)
+We used PPO-EWMA [Hilton et al., 2021] with rollouts of length 512 (twice as long as our maximum
+value of h), and trained for 225 environment interactions.
+We used a simple CNN architecture with ReLU activations and the following layers: a 5 × 5
+convolutional layer with 40 channels, 2×2 max pooling, a 3×3 convolutional layer with 80 channels,
+2 × 2 max pooling, and a dense layer with 1,000 channels. We scaled the width of this network by
+varying total number of parameters from
+1
+64 of the default to 8 times the default. We used separate
+policy and value function networks because we did not expect there to be much transfer between the
+two objectives, since the environment samples digits independently.
+3.4
+Learning rates
+Although we would not expect our qualitative results to change much, our quantitative results
+such as scaling exponents depend crucially on using well-tuned hyperparameters. By far the most
+important hyperparameter to tune in our setup is the Adam learning rate, whose optimal value can
+vary substantially with model size and compute budget.
+When varying model size, we found that a good heuristic is to keep the Adam learning rate propor-
+tional to the initialization scale. For our width-scaling experiments, this means keeping the Adam
+learning rate proportional to 1/
+√
+width, since we use Kaiming He initialization [He et al., 2015]. For
+our Procgen depth-scaling experiments, which use a residual network, it means keeping the Adam
+learning rate proportional to 1/
+�
+depth
+1
+L , where L is the number of layers per residual block (L = 2
+in our case), since we use an initialization similar to Fixup initialization [Zhang et al., 2019]. For
+Procgen and MNIST, we tuned the learning rate at one model size and followed this heuristic to select
+the learning rate for the other model sizes. For Dota 2, we tuned the learning rate separately for each
+model size, but this amounted to following approximately the same heuristic.
+When varying the compute budget for a given model size, it can actually be necessary to use separate
+training runs for each compute budget, each with its own learning rate schedule, rather than taking
+different snapshots at different points of the same training run [Hoffmann et al., 2022]. Unfortunately,
+due to the challenge of carefully tuning learning rate schedules for RL and the expense of multiplying
+the number of training runs, we took the latter approach. To mitigate the impact of this, we found a
+learning rate schedule that seemed to work well for a variety of compute budgets, which we explain
+in Appendix B.1. Nevertheless, the values of our scaling exponents should be considered uncertain
+because of this.
+6
+
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+αN
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+αE
+1
+1 + αN/αE
+= 0.8
+1
+1 + αN/αE
+= 0.7
+1
+1 + αN/αE
+= 0.6
+Procgen (width)
+CoinRun
+StarPilot
+FruitBot
+Easy, single seed
+Easy, mean return
+Hard, single seed
+Hard, mean return
+Dota 2
+1v1
+Reference
+αN/αE = const
+MNIST horizons
+1
+2
+4
+8
+16
+32
+64
+128
+192
+256
+αN vs αE
+Figure 3: Fitted values of αN and αE. For Procgen, we also show the values fitted using each of
+the 3 random seeds, to show the variation due to the choice of random seed. The dotted lines show
+contours for
+1
+1+αN/αE , the exponent for the scaling of optimal model size with compute.
+4
+Results
+Our main result is that our power law for intrinsic performance, equation (1), holds across envi-
+ronments and model sizes, at least after an initial transient period of training (which we discuss in
+more detail in Section 4.3). This result is supported by the closeness of the power law fit to our
+learning curves, as shown in Figure 2 for StarPilot and in Appendix C for all our environments. Our
+methodology for fitting this power law is described in Appendix A.
+It is interesting to study the sensitivity of the exponents αN and αE, which govern the scaling
+behavior of I with N and E (and determine the other exponents of interest). The fitted values of
+these exponents for the different environments are shown in Figure 3. The numerical values of all of
+the fitted constants may be found in Appendix E.
+Although our measurements of these exponents are uncertain, due to the limitations discussed in
+Section 5.3, we make a number of observations:
+• The primary determinant of αN and αE is the domain (Procgen, Dota 2, or MNIST), which
+we expect is a consequence of the fact that so many experimental details are shared within
+each domain.
+• Within MNIST, increasing the horizon seems to lower αE, but as we explain in Section 4.2,
+this effect is confounded by a measurement problem caused by under-training.
+• Within Procgen, the easy and hard modes of each Procgen game tend to have closer
+exponents to one another than to other Procgen games. We believe that this is because
+identifying visual features is a core part of Procgen, and the two modes of each game have
+very similar observation distributions.
+7
+
+10−7
+10−6
+10−5
+10−4
+10−3
+10−2
+Compute (PF-days)
+103
+104
+105
+106
+107
+Parameters
+Procgen (width)
+CoinRun
+StarPilot
+FruitBot
+Easy
+Hard
+Dota 2
+1v1
+Generative modeling
+Language
+(Hoffmann et al.)
+Language
+(Kaplan et al.)
+Image 32x32
+(Henighan et al.)
+MNIST horizons
+1
+2
+4
+8
+16
+32
+64
+128
+192
+256
+Optimal model size vs compute
+Figure 4: Optimal model size vs compute for all our environments. Note that the individual points,
+which correspond to the sizes of models that we trained, are themselves obtained from a power law
+best fit. Hence the fact that the lines pass through the points exactly is automatic and does not indicate
+goodness of fit.
+• The Procgen difficulty mode does not obviously have any particular effect on the scaling
+exponents. We hypothesize that humans tend to judge a task as easier when a near-perfect
+score can be achieved with less compute, even if it takes a lot of additional compute to eke
+out the final few points. Conversely, it does not seem to matter to the RL algorithm exactly
+how the score maps on to intrinsic performance (i.e., the compute required).
+4.1
+Optimal model size vs compute
+As explained in Section 2.3, our power law for intrinsic performance implies that, for a given compute
+budget, the optimal model size scales as a power law with exponent
+1
+1+αN/αE .
+Figure 4 shows these inferred relationships for our different environments, along with some generative
+modeling relationships taken from the literature. The full equations for these relationships are
+provided in Appendix E.
+The exponent
+1
+1+αN/αE varied between around 0.40 and 0.65 for Procgen and 0.66 and 0.80 for
+MNIST, and was around 0.76 for Dota 2. By comparison, the corresponding exponent for language
+modeling, which was carefully measured by Hoffmann et al. [2022], is around 0.50. Previous work
+by Kaplan et al. [2020] and Henighan et al. [2020] measured this exponent less carefully but using a
+methodology that more closely matches our own, and found an exponent of around 0.73 for language
+0.65 for 32x32 images.
+An intriguing conjecture, which is also suggested by theoretical considerations [Bahri et al., 2021],
+is that the exponent of this relationship would be around 0.5 in every domain if it were measured
+carefully enough (i.e., with optimal hyperparameters and enough random seeds). Given the limitations
+of our experiments, we consider our results to be inconclusive on this question.
+Nevertheless, it is clear that the scaling coefficient of this relationship varies significantly between
+domains. With the exception of our toy MNIST environment, the optimal model size for RL for
+8
+
+a given compute budget is consistently smaller than for generative modeling, in some cases by
+multiple orders of magnitude. We believe that this is because RL tasks have a longer horizon length
+than generative modeling in some sense, and explore this hypothesis with our MNIST environment
+in Section 4.2. Another possibility is that the arithmetic intensity (i.e., the number of FLOPs per
+parameter in a forward pass) of the architecture is a confounder, which we discuss in more depth in
+Section 4.4.
+4.2
+Effect of task horizon length
+As explained in Section 3.3, for our MNIST experiments, we artificially altered the “horizon length”
+of the task by setting the GAE credit assignment parameter λ to 1 and varying the GAE discount rate
+γ.
+The expected effect of varying γ in this context is given by the following theoretical result.
+Proposition 1. Consider an MDP with independent timesteps (by which we mean that each st is
+identically distributed and independent of st−1 and at−1, and episodes never terminate). Suppose we
+train a model with parameters θ on this MDP using Vanilla Policy Gradient,1 estimating advantages
+using GAE with γ = 1 −
+2
+h+1 and λ = 1, and working with separate policy and value function
+networks. Then the covariance matrix of the policy gradient is approximately
+Σθ + Πθ
+�
+h + 1
+h − 2
+�
+for some symmetric positive semi-definite matrices Σθ and Πθ that do not depend on h.
+For a proof sketch, see Appendix G.
+Intuitively, this result says that gradient variance may be decomposed into two pieces: one piece that
+is inherent to the task (the Σθ term), and one piece that comes from imperfect credit assignment (the
+Πθ term). For example, when h = 1 (i.e., γ = 0), credit is correctly assigned to the previous action
+only, and hence the second term vanishes. Ignoring the 1
+h term (since h ≥ 1), we may stylize this
+result as: gradient variance is an affine function of h (i.e., a linear function with an intercept).
+This can be directly translated into a statement about sample efficiency, since multiplying the gradient
+variance by some factor c can be exactly compensated for by multiplying the batch size by c, which
+multiplies the number of samples used by c. Hence in order to reach a given performance level,
+the number of environment interactions required should be an affine function of h. This affine
+function will come from integrating certain functionals of Σθ and Πθ over the course of training,
+and will therefore depend both on the model architecture and on the choice of performance level.
+To test this prediction, we looked at the number of environment interactions required to reach a
+1% failure rate (i.e., 99% training accuracy) on MNIST as a function of the horizon length h. Our
+results are shown in Figure 5, along with affine fits. As expected, the number of interactions closely
+follows an affine function of the horizon length, although the fit is less good for shorter horizons and
+larger models. At very short horizons, the number of interactions even decreases with the horizon
+length, suggesting a hyperparameter issue (perhaps a suboptimal learning rate schedule, or reward
+normalization implicitly decreasing the KL penalty and entropy bonus).
+The implication of this for our optimal model size vs compute scaling law is that once h becomes large
+enough, further increasing h should lead to a proportional increase the compute budget corresponding
+to each given optimal model size, without changing the scaling exponent of this relationship. This
+is because the intercept term of the affine function will eventually become dominated by the term
+involving h, and so the number of environment interactions required to reach a given performance
+level will eventually scale approximately proportionally to h. (For small values of h, however, the
+relationship between the two components of the covariance matrix of the policy gradient may have a
+more complex dependence on model size.)
+This effect is visible in Figure 4, where the main impact of increasing the horizon length is to shift
+the optimal model size vs compute curve to the right. The curve also gets shallower as the horizon
+1Vanilla Policy Gradient is a primitive version of PPO, explained here: https://spinningup.openai.
+com/en/latest/algorithms/vpg.html
+9
+
+0
+50
+100
+150
+200
+250
+Horizon length h
+1
+2
+3
+4
+5
+6
+Interactions
+×106
+Parameters
+104.8
+105.1
+105.4
+105.7
+106.0
+106.3
+106.6
+106.9
+107.2
+107.5
+Value
+Affine fit
+Interactions required to reach a 1% failure rate, MNIST
+Figure 5: Sample efficiency for MNIST as a function of the horizon length h, for all our model sizes.
+length is increased, but this effect is confounded by a measurement problem caused by under-training,
+which we explain in more detail in Section 4.3.
+Our MNIST environment is useful because our it allows us to vary the task horizon length in a fine-
+grained, quantifiable way by varying γ. But our analysis of this environment relies on the assumption
+of independent timesteps, which does not hold in most environments (and in particular removes the
+need for exploration). Nevertheless, our results are suggestive of a more general explanation for the
+large differences in optimal model size for a given compute budget between different environments:
+that different environments have different task horizon lengths in a more general sense. We speculate
+that, in this more general sense, task horizon length is influenced by how long rewards are delayed
+for relative to the actions the agent is currently learning (which may increase throughout training as
+the agent learns skills with feedback loops that are less and less tight), and that γ determines only an
+upper bound on the task horizon length.
+4.3
+Variability of exponents over training
+Although our power law for intrinsic performance holds across environments and model sizes, we
+only obtain a good fit by excluding an initial transient period of training. Put another way, the scaling
+constants vary over the course of training.
+This phenomenon is clearest with with our MNIST environment, since we were able to use many
+random seeds to reduce variance. Recall that in this environment, the agent observes a randomly
+sampled MNIST training set digit each timestep, and the horizon length of the task is artificially
+controlled using the GAE discount rate γ, as explained in Section 3.3. We fitted our power law to
+three different periods of training for this environment: an early period (216–219 interactions), a
+middle period (219–222 interactions), and a late period (222–225 interactions).
+Figure 6 shows the fitted values of αN and αE for these different periods of training. We found αE
+to be significantly lower during the early and middle periods of training, especially for the shorter
+horizon lengths.
+In order to accurately measure the scaling constants for optimal model size vs compute, it is best to
+use a period of training during which the learning curves reach the compute-efficient frontier, since
+otherwise the measurement is an extrapolation. As shown in Figure 7, this is always in the late period
+10
+
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+αN
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+αE
+1
+1 + αN/αE
+= 0.9
+1
+1 + αN/αE
+= 0.8
+1
+1 + αN/αE
+= 0.7
+MNIST periods
+Early
+Middle
+Late
+MNIST horizons
+1
+2
+4
+8
+16
+32
+64
+128
+192
+256
+αN vs αE, MNIST
+Figure 6: Fitted values of αN and αE for MNIST
+with different horizons, using different periods of
+training to fit the power laws. The horizon h is
+defined by γ = 1 −
+2
+h+1, where γ is the discount
+rate.
+1013
+1014
+1015
+1016
+1014
+1016
+MNIST, horizon 1, late period
+Parameters
+104.8
+105.1
+105.4
+105.7
+106.0
+106.3
+106.6
+106.9
+107.2
+107.5
+1013
+1014
+1015
+1016
+1013
+1014
+1015
+Intrinsic performance (FLOPs)
+MNIST, horizon 256, late period
+1012
+1013
+1014
+1015
+Compute (FLOPs)
+1012
+1013
+MNIST, horizon 1, middle period
+Learning
+curve
+Power law
+fit
+Efficient
+frontier
+Efficient
+points
+Figure 7: Learning curves as a function of total
+training compute for MNIST, using different hori-
+zons and different periods of training, together
+with their power law fits. Mean over the middle-
+performing 16 of 20 random seeds shown.
+of training, if at all. For this reason, we use the late period of training for all of our results on MNIST
+outside of this section.
+Figure 7 also shows that, for the longer horizon lengths, the learning curves of the larger models
+did not reach the compute-efficient frontier even during the late period of training. Hence our
+measurements of
+1
+1+αN/αE , the exponent for the scaling of optimal model size with compute, are
+likely underestimates for these longer horizon lengths.
+For our other environments, we found that it was enough to exclude only the first
+1
+64 of training
+in order for our power law for intrinsic performance to be a good fit around the compute-efficient
+frontier. This is similar to what is needed for the corresponding law for language [Kaplan et al., 2020,
+Figure 4, right]. Nevertheless, it is possible that the measurement problem identified in this section
+affects some of our other results.
+4.4
+Scaling depth
+Most of our experiments involved scaling the width of our networks, but for Procgen, we also tried
+scaling the depth, as explained in Section 3.1. We found that our power law for intrinsic performance
+still held, but with more noise than the width-scaling experiments, as a consequence of using fewer
+model sizes. The fitted values of αN and αE for the depth-scaling experiments lay in a similar region
+to the width-scaling experiments, but there were no clear relationships between the depth-scaling
+exponents for the different environments, nor between the width-scaling and depth-scaling exponents
+11
+
+10−5
+10−4
+10−3
+10−2
+Compute (PF-days)
+104
+105
+106
+Parameters
+Optimal model size vs compute, Procgen
+(a) Using parameters as the measure of model size.
+10−5
+10−4
+10−3
+10−2
+Compute (PF-days)
+106
+107
+108
+FLOPs per forward pass
+Generative modeling
+Language
+(Hoffmann et al.)
+Language
+(Kaplan et al.)
+Image 32x32
+(Henighan et al.)
+Procgen
+CoinRun
+StarPilot
+FruitBot
+Easy
+Hard
+Width
+Depth (*)
+Optimal model size vs compute, Procgen,
+arithmetic intensity-adjusted
+(b) Using FLOPs per forward pass instead of parameters.
+Figure 8: Comparison of optimal model size vs compute for our Procgen width- and depth-scaling
+experiments. (*) It is important to understand how parameters and FLOPs were counted to interpret
+the depth-scaling results. This is explained in detail in Appendix D.
+for a given environment. Plots of our results may be found in Appendix C, and the numerical values
+of the fitted constants may be found in Appendix E.
+The main difference between our width-scaling and depth-scaling results is that the optimal model
+size for a given compute budget was significantly smaller for our depth-scaling experiments, but
+this was an artifact of how we counted parameters and FLOPs. As explained in Appendix D, we
+only included the part of the network being scaled in our parameter and FLOP calculations, which
+meant excluding the final dense layer of the network for our depth-scaling experiments, but not our
+width-scaling experiments. If this layer had been included in our depth-scaling calculations, it would
+have accounted for between 16% and 90% of the parameters but only 2% or fewer of the FLOPs,
+depending on the depth.
+Interestingly, as shown in Figure 8, the optimal model size vs compute scaling laws for our width-
+and depth-scaling experiments become much more similar if we measure model size using FLOPs
+per forward pass rather than parameters. This is because excluding the final dense layer from the
+parameter and FLOP calculations significantly increases the arithmetic intensity (i.e., FLOPs per
+parameter in a forward pass) as calculated for the depth-scaling experiments. This suggests that,
+when comparing models with very different arithmetic intensities, FLOPs per forward pass may
+be a better measure of model size than parameters (or perhaps arithmetic intensity should even be
+considered as an additional independent variable).
+4.5
+Natural performance metrics
+Although in general there may be no obvious performance metric that scales smoothly with model
+parameters and environment interactions, motivating our use of intrinsic performance, there may still
+be such a metric in some environments. We call such metrics natural performance metrics, and we
+were able to find them in a couple of our environments:
+• CoinRun: In the CoinRun environment from Procgen Benchmark, the episode return is
+always either 10 or 0, corresponding to whether or the agent successfully collects the coin at
+the end of the level. We found the fail-to-success ratio F := 10−R
+R
+, where R is the mean
+episode return, to be a natural performance metric for CoinRun. This is similar to the failure
+rate 1 − R
+10, since R is close to 10 for most of training, but provides a slightly better fit
+early in training, since it does not have an upper bound of 1. Note that the logarithm of the
+12
+
+1014
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+10−2
+10−1
+Fail-to-success ratio
+Easy
+Hard
+Learning
+curves
+Power law
+fitted to
+I−β
+(arbitrary
+function
+of ratio)
+Fail-to-
+success
+ratio
+CoinRun, efficient frontier fits
+Figure 9: Comparison of the efficient frontier fits
+for CoinRun, using intrinsic performance and the
+fail-to-success ratio.
+1014
+1016
+1018
+1020
+Compute (FLOPs)
+−5
+0
+5
+10
+15
+20
+25
+TrueSkill
+Learning
+curves
+Power law
+fitted to
+I−β
+(arbitrary
+function
+of T)
+e−αT T
+Dota 2, efficient frontier fits
+Figure 10: Comparison of the efficient frontier
+fits for Dota 2, using intrinsic performance and
+exponentiated scaled TrueSkill.
+fail-to-success ratio can also be thought of as the logit function (inverse sigmoid) of the
+failure rate.
+• Dota 2: Dota 2 is a two-player game, and so the performance of a policy must be measured
+by comparing it to other policies. The standard method for this is the TrueSkill rating
+system,2 in which differences in rating between policies correspond to win probabilities
+when the policies are played against one another, similarly to the Elo rating system. We
+found TrueSkill to be a natural performance metric for Dota 2.
+Specifically, we found that our power law for intrinsic performance, equation (1), still roughly held
+with the left-hand side replaced by a suitable function of the natural performance metric. For CoinRun,
+we used the fail-to-success ratio directly, but discarded data from early in training where this ratio
+was above 0.5. For Dota 2, we used e−αT T , where T is TrueSkill and αT is a fitted constant, which
+was needed because the scale of T is arbitrary.
+Figures 9 and 10 compare the efficient frontier fits for intrinsic performance and for the natural
+performance metric, for CoinRun and Dota 2 respectively. The fits match closely, except for Dota 2 at
+higher levels of TrueSkill. We conjecture that Dota 2 has an analog of an irreducible loss [Henighan
+et al., 2020], representing the maximum attainable TrueSkill for the family of models we trained.
+We explored introducing an additional fitted constant T ∗ for this maximum attainable TrueSkill, and
+using either of the functional forms e−αT T − e−αT T ∗ and (T ∗ − T)αT . However, it was unclear
+to us which of these forms made the most theoretical sense, and we were unsure whether we could
+justify the extra degree of freedom given the lack of data at higher levels of TrueSkill.
+The fitted constants for all of these alternative power laws for both CoinRun and Dota 2 are given in
+Appendix E. Interestingly, for CoinRun, the values of the scaling exponent for the fail-to-success
+ratio F in terms of intrinsic performance I, corresponding to the slopes of the lines in Figure 9, are
+similar between the two difficulty modes: F ∝ I−0.40 in easy mode and F ∝ I−0.48 in hard mode.
+5
+Discussion
+5.1
+Extrapolating sample efficiency
+We may use our power law for intrinsic performance, equation (1), to extrapolate sample efficiency
+to unseen model sizes N and environment interactions E. For example, in Figure 11, we show the
+2https://en.wikipedia.org/wiki/TrueSkill
+13
+
+0.0
+0.5
+1.0
+1.5
+2.0
+Interactions
+×108
+5
+10
+15
+20
+25
+30
+Mean episode return
+Parameters
+104.3
+104.6
+104.9
+105.2
+105.5
+105.8
+106.1
+106.4
+106.7
+107.0
+Learning
+curve
+Power law fit
+Power law
+N → ∞
+limit
+Sample efficiency, StarPilot, hard
+Figure 11: Learning curves for StarPilot (hard
+mode, scaling width), together with their power
+law fits, and the N → ∞ limit of the power law.
+10−7
+10−6
+10−5
+10−4
+10−3
+10−2
+Compute (PF-days)
+103
+104
+105
+106
+107
+Parameters
+Procgen (width)
+CoinRun
+StarPilot
+FruitBot
+Easy
+Hard
+Dota 2
+1v1
+GM (various)
+MNIST horizons
+1–256
+Optimal model size vs compute, Ne = 105
+Figure 12: Optimal model size vs compute, taking
+into account a hypothetical compute cost per en-
+vironment interaction equal to that of a model of
+size Ne = 105. See Figure 4 for the full legend.
+extrapolated learning curve for StarPilot in the infinite-width limit. This reaches the final performance
+of our largest model in about half the number of environment interactions. Note, however, that
+without a natural performance metric, we cannot extrapolate to unseen performance levels.
+It is natural to ask how this extrapolated infinite-width limit compares to human sample efficiency. On
+StarPilot (slowed down to 3 frames per second), a human can reach a mean episode return of around
+20 after a few episodes, whereas the extrapolated infinitely-wide model takes 18 million interactions,
+around 10,000 times as many. This is not really a fair comparison though, because much of the
+challenge in Procgen is to learn to identify basic visual features, which humans are already able to do.
+For Dota 2, we crudely estimate that it would take a human around 50–500 hours of gameplay to
+reach the performance of the extrapolated infinitely-wide LSTM after 5 billion interactions, a factor
+of 100–1,000 in sample efficiency. This comparison may be fairer, because Dota 2 has a structured
+observation space and is more challenging than StarPilot, although it still draws on many pre-existing
+human intuitions. Of course, our models were all trained from scratch, and we should expect this
+factor to be smaller for models that have been pre-trained to learn useful representations.
+5.2
+Cost-efficient reinforcement learning
+In the reinforcement learning literature, sample efficiency is usually taken to be the primary metric
+of algorithmic progress. This can be thought of as focusing on the cost of running the environment,
+but not the algorithm. At the other extreme, we have so far focused on the computational cost of the
+algorithm, but not on the cost of the environment. However, it is straightforward to now take both
+into account. To do this, let Ne be the cost of the environment, measured in terms of the number of
+parameters in a model with the same cost per interaction. Thus the total cost of both the algorithm
+and the environment is proportional to (N + Ne) E.
+The cost-efficient frontier is now described by the following generalization of equation (2):
+�
+1 + Ne
+N
+�
+αN
+�Nc
+N
+�αN
+= αE
+�Ec
+E
+�αE
+.
+Substituting this into our power law given by equation (1), it follows that along the cost-efficient
+frontier,
+C =
+�
+1 + Ne
+N
+� �
+1
+1 + αN
+αE
+�
+1 + Ne
+N
+�
+�
+1
+αN +
+1
+αE � N
+Nc
+�1+ αN
+αE ,
+14
+
+where C := (N + Ne) E. Thus for a given budget C, the optimal model size N scales as the same
+power law in C as before once N ≫ Ne, and it is only efficient to take N ≪ Ne when C is very
+small. This validates and makes precise the rule-of-thumb that it is usually inefficient to use a model
+that is much cheaper to run than the environment, at least when training from scratch.
+To illustrate this relationship, Figure 12 shows the optimal model size vs compute relationship from
+Figure 4, but incorporating a fixed hypothetical compute cost associated with each environment
+interaction.
+5.3
+Limitations
+Our experiments have several limitations:
+• As explained in Section 3.4, we did not use separate training runs for each compute budget,
+each with their own learning rate schedule, which can be necessary to accurately measure
+scaling exponents [Hoffmann et al., 2022]. We tried to mitigate this by using a learning rate
+schedule that worked well for a variety of compute budgets, as explained in Appendix B.1,
+but this may not have been enough.
+• As explained in Section 4.3, the variability of exponents over training gives rise to a
+measurement problem. We mitigated this to some extent by excluding data from early in
+training when fitting our power law, but this does not fully correct for the fact that some of
+our models were under-trained relative to the compute-efficient frontier.
+• We did not carefully optimize the aspect ratios of our models, instead scaling width and
+depth separately. More generally, suboptimal hyperparameters or other problems with our
+training setups could have lead to errors in our measurements of scaling constants.
+• Learning curves in reinforcement learning are often very high-variance, adding significant
+noise to power law fits. We mitigated this to some extent by choosing environments with
+relatively low-variance learning curves and using multiple random seeds, but a lot of variance
+still remained.
+As a result of these limitations, we do not think conclusions that depend on the precise fitted values of
+our scaling constants can be drawn with confidence, although we consider our mitigations sufficient
+for more qualitative conclusions. We are excited for future work to fix these limitations, explore
+new domains, and more carefully disentangle the effects of the choice of algorithm, architecture and
+hyperparameters as well as properties of the environment.
+5.4
+Forecasting compute requirements
+The scaling of optimal model size with compute is a key input into the biological anchors framework
+for forecasting transformative artificial intelligence [Cotra, 2020]. In this framework, the human brain
+is used as a biological anchor for estimating the number of parameters in a transformative model, and
+optimal model size vs compute scaling laws are used to forecast the total compute required to train
+such a model. In this section we summarize the main implications of our work for this framework.
+Scaling exponents for reinforcement learning lie in a similar range to generative modeling. The
+exponent for the scaling of optimal model size with compute,
+1
+1+αN/αE , varied between around 0.4
+and 0.8 for our environments, a range that encompasses previous measurements of this exponent for
+generative modeling. However, as discussed in Section 5.3, we do not think our measurements of this
+exponent should be taken literally, due to the limitations of our experiments. The results of Hoffmann
+et al. [2022] and Bahri et al. [2021] suggest the possibility that this exponent would be around 0.5 in
+every domain if it were measured carefully enough, and we consider our results to be inconclusive on
+this question.
+Scaling coefficients for reinforcement learning vary by multiple orders of magnitude. The
+coefficient for the scaling of optimal model size with compute, Nc
+�
+1 + αN
+αE
+�
+1
+αN , varied substantially,
+enough that we do not think this variation is attributable only to the limitations of our experiments.
+For example, the scaling exponents for MNIST (with a horizon length of 1) and Dota 2 are very
+similar, but a model of the same size needs to be trained for around 2,000 times longer on Dota 2
+than on MNIST to be compute-efficient. By comparison, Henighan et al. [2020] found generative
+15
+
+modeling to require around 20 times as much training on 32x32 images than on language. Moreover,
+our analysis of the effect of the task horizon length gives a plausible mechanism for this variation.
+Arithmetic intensity may confound scaling coefficients. As discussed in Section 4.4, the coefficient
+for the scaling of optimal model size with compute can be affected by the arithmetic intensity (i.e.,
+the number of FLOPs per parameter in a forward pass) of the model. This alone does not explain the
+large variation in this coefficient between MNIST and Dota 2, for example, but it may explain some
+of the other variation. We hypothesize that, when comparing models with very different arithmetic
+intensities, due to parameter sharing or methods such as mixture of experts, it may be better to
+measure model size in FLOPs per forward pass rather than in parameters.
+Sample efficiency is an affine function of the task horizon length. We study the effect of the
+task horizon length using a toy MNIST-based environment in Section 4.2. Both theoretically (see
+Proposition 1) and empirically, the number of samples required to reach a given level of performance
+grows with the horizon length as an affine function (i.e., a linear function with an intercept) that
+depends on both the model size and the target performance level. However, our analysis makes a
+simplifying assumption of independent timesteps, which does not hold in most environments. In
+particular, we do not analyze the need for curricula and/or exploration to solve tasks for which it is
+challenging to obtain useful feedback. Instead, we simply assume that the algorithm pays attention to
+rewards over a longer time horizon, making credit assignment harder.
+This result validates and refines the analysis of Cotra [2020], who defined the “effective horizon
+length” as a quantity that scales linearly with training data requirements, incorporating not only the
+horizon length as we define it, but also reward sparsity, noise and so on. Our result specifically isolates
+the explicit horizon length, showing that training data requirements are a sum of two components,
+at least in our toy setting: one corresponding to a version of the task in which the horizon ends
+immediately, and another that is proportional to the horizon length. This implies that, for a given fixed
+task, continuing to increase the horizon length will eventually lead to a proportional increase in the
+compute budget corresponding to a given optimal model size, without changing the exponent of this
+scaling law. However, this will only happen once the first component has become negligible, and it is
+unclear whether there are realistic tasks of different horizon lengths for which this first component is
+negligible in practice.
+We are excited for future work to study other aspects of the “effective horizon length”, such as
+reward sparsity and noise, as well as studying the explicit horizon length in environments that are less
+artificial. It is not entirely clear how to quantify these properties in general, and they could potentially
+affect scaling exponents as well as scaling coefficients, if for example they change over the course of
+training.
+Measuring scaling exponents precisely is challenging. The biological anchors framework uses
+the scaling of optimal model size with compute to perform a substantial extrapolation, making it
+particularly sensitive to the exponent of this relationship. This makes it challenging to measure this
+exponent with sufficient precision. In addition to the challenges raised by Hoffmann et al. [2022]
+involving learning rate schedules, we hope that others will benefit from learning about the other
+challenges we faced, which are summarized in Section 5.3.
+6
+Conclusion
+We have shown how to extend scaling laws to single-agent reinforcement learning using the notion of
+intrinsic performance. Across a range of environments, intrinsic performance scales as a power law
+in model size and environment interactions, and hence the optimal model size scales as a power law in
+the training compute budget. We have studied how this relationship is affected by various properties
+of the training setup, including the horizon length of the task, and have discussed the implications of
+this for the biological anchors framework for forecasting transformative artificial intelligence.
+7
+Acknowledgments
+Thanks to Mira Murati, Karl Cobbe, Chris Hesse, David Farhi, Paul Christiano, Jared Kaplan, Long
+Ouyang and Ajeya Cotra for discussions, ideas, help, advice, support and inspiration that have greatly
+benefited this project.
+16
+
+References
+Y. Bahri, E. Dyer, J. Kaplan, J. Lee, and U. Sharma. Explaining neural scaling laws. arXiv preprint
+arXiv:2102.06701, 2021.
+K. Cobbe, C. Hesse, J. Hilton, and J. Schulman. Leveraging procedural generation to benchmark
+reinforcement learning. In International conference on machine learning, pages 2048–2056.
+PMLR, 2020.
+K. W. Cobbe, J. Hilton, O. Klimov, and J. Schulman. Phasic policy gradient. In International
+Conference on Machine Learning, pages 2020–2027. PMLR, 2021.
+A. Cotra. Forecasting transformative AI with biological anchors, 2020.
+J. Droppo and O. Elibol. Scaling laws for acoustic models. arXiv preprint arXiv:2106.09488, 2021.
+L. Espeholt, H. Soyer, R. Munos, K. Simonyan, V. Mnih, T. Ward, Y. Doron, V. Firoiu, T. Harley,
+I. Dunning, et al. Impala: Scalable distributed deep-RL with importance weighted actor-learner
+architectures. In International conference on machine learning, pages 1407–1416. PMLR, 2018.
+B. Ghorbani, O. Firat, M. Freitag, A. Bapna, M. Krikun, X. Garcia, C. Chelba, and C. Cherry. Scaling
+laws for neural machine translation. arXiv preprint arXiv:2109.07740, 2021.
+K. He, X. Zhang, S. Ren, and J. Sun. Delving deep into rectifiers: Surpassing human-level per-
+formance on imagenet classification. In Proceedings of the IEEE international conference on
+computer vision, pages 1026–1034, 2015.
+T. Henighan, J. Kaplan, M. Katz, M. Chen, C. Hesse, J. Jackson, H. Jun, T. B. Brown, P. Dhari-
+wal, S. Gray, et al.
+Scaling laws for autoregressive generative modeling.
+arXiv preprint
+arXiv:2010.14701, 2020.
+J. Hilton, K. Cobbe, and J. Schulman. Batch size-invariance for policy optimization. arXiv preprint
+arXiv:2110.00641, 2021.
+J. Hoffmann, S. Borgeaud, A. Mensch, E. Buchatskaya, T. Cai, E. Rutherford, D. d. L. Casas, L. A.
+Hendricks, J. Welbl, A. Clark, et al. Training compute-optimal large language models. arXiv
+preprint arXiv:2203.15556, 2022.
+A. L. Jones. Scaling scaling laws with board games. arXiv preprint arXiv:2104.03113, 2021.
+J. Kaplan, S. McCandlish, T. Henighan, T. B. Brown, B. Chess, R. Child, S. Gray, A. Radford, J. Wu,
+and D. Amodei. Scaling laws for neural language models. arXiv preprint arXiv:2001.08361, 2020.
+D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980,
+2014.
+Y. LeCun. The MNIST database of handwritten digits, 1998.
+O. Neumann and C. Gros. Scaling laws for a multi-agent reinforcement learning model. arXiv
+preprint arXiv:2210.00849, 2022.
+OpenAI, C. Berner, G. Brockman, B. Chan, V. Cheung, P. D˛ebiak, C. Dennison, D. Farhi, Q. Fis-
+cher, S. Hashme, C. Hesse, R. Józefowicz, S. Gray, C. Olsson, J. Pachocki, M. Petrov, H. P.
+de Oliveira Pinto, J. Raiman, T. Salimans, J. Schlatter, J. Schneider, S. Sidor, I. Sutskever, J. Tang,
+F. Wolski, and S. Zhang. Dota 2 with large scale deep reinforcement learning. arXiv preprint
+arXiv:1912.06680, 2019.
+J. Schulman, P. Moritz, S. Levine, M. Jordan, and P. Abbeel. High-dimensional continuous control
+using generalized advantage estimation. arXiv preprint arXiv:1506.02438, 2015.
+J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov. Proximal policy optimization
+algorithms. arXiv preprint arXiv:1707.06347, 2017.
+D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez, M. Lanctot, L. Sifre, D. Ku-
+maran, T. Graepel, et al. A general reinforcement learning algorithm that masters chess, shogi, and
+Go through self-play. Science, 362(6419):1140–1144, 2018.
+17
+
+H. Zhang, Y. N. Dauphin, and T. Ma. Fixup initialization: Residual learning without normalization.
+arXiv preprint arXiv:1901.09321, 2019.
+18
+
+A
+Curve-fitting methodology
+In this section we discuss our methodology for computing intrinsic performance and fitting the power
+law constants, which require some care. Code for our full procedure, along with its application to
+our experiments, may be found in this Colab notebook: https://colab.research.google.com/
+drive/1PzwZyXsi9jRdVCj1GJrS8JdOPBQ7LHZV.
+Recall that the intrinsic performance of a policy is the minimum compute required to train a model of
+any size in the same family to reach the same return (averaged over random seeds). The naive way to
+compute this would be to train models of many different sizes, and to take the best-performing model
+size for each possible compute budget. However, it may not be feasible to train models of enough
+different sizes to get a reasonable level of granularity, while using enough different random seeds
+sufficiently to reduce the high variance of learning curves.
+To cope with this, we compute intrinsic performance and fit the power law constants together. This
+allows us to make use of all the data from each learning curve, instead of just a single point from
+each one. We do this by jointly fitting the power law constants and a monotonic function f to
+f (R)−β =
+�Nc
+N
+�αN
++
+�Ec
+E
+�αE
+,
+where R is the mean episode return (or another performance metric such as TrueSkill), N is the
+number of model parameters, and E is the number of environment interactions. By also requiring the
+relationships between the constants from Lemma 1 to hold, this provides us both with the power law
+constants, and with the desired function f satisfying f (R) = I, where I is intrinsic performance.
+We perform this fit by using a black-box optimization algorithm such as CMA-ES to fit αN, αE
+and Nc, which determine β and Ec, with monotonic regression3 in the inner loop to fit f, using the
+squared error of the regression as the black-box loss function. We actually fit log (f) rather than f in
+order to obtain a good fit to I on a logarithmic scale, and we weight the data in proportion to 1
+E so
+that each interval is given equal weight on a logarithmic scale. In our Colab notebook, this routine is
+performed by the function fit_coeffs.
+This procedure seems to work well off-the-shelf, typically converging to a unique local minimum.
+However:
+• When there is a lack of data or the data is very noisy, the local minimum may not be a global
+minimum, and the procedure can diverge to a degenerate solution.
+• It is necessary to first smooth learning curves so that they are mostly monotonic, to prevent
+the monotonic regression from overfitting. In our Colab notebook, we use the function
+smooth, which uses standard errors to automatically choose smoothing parameters (although
+note that we used slightly different smoothing parameters for MNIST).
+• As discussed in Section 4.3, it is important to exclude data from early in training.
+Our full procedure is therefore as follows.
+• Smooth learning curves. Plot the smoothed curves on a logarithmic scale to check the
+monotonicity and fit, and adjust the smoothing parameters if necessary.
+• Exclude data from early in training, balancing the need for data against how much the early
+data skews the fit. Typically at least the first
+1
+64 of training should be excluded.
+• Fit the power law constants and f using the black-box optimization with monotonic regres-
+sion routine.
+• Plot the fit to check the routine did not diverge. If it did, re-run routine, or constrain the
+constants and re-run, or include more data in step 2. If none of these fixes the divergence,
+then it may be necessary to collect more data.
+• Check the fit is not overly skewed by data from early in training. If it is, exclude more data
+in step 2.
+3https://en.wikipedia.org/wiki/Isotonic_regression
+19
+
+This procedure led us to exclude the first 3 million environment interactions for Procgen, the first 2
+billion environment interactions for Dota 2, and the first 216, 219 or 222 environment interactions for
+MNIST depending on the period of training being considered, as discussed in Section 4.3.
+A.1
+Fitting to natural performance metrics
+As discussed in Section 4.5, as well as fitting our power law with I−β on the left-hand side, as in
+equation (1), we also fit it using various other expressions, such as e−αT T , where T is TrueSkill and
+αT is a fitted constant. When doing this, we adopt the convention that the constraints on β and Ec
+from Lemma 1 should continue to hold. This necessitates introducing an additional multiplier, and
+instead fitting
+Tce−αT T =
+�Nc
+N
+�αN
++
+�Ec
+E
+�αE
+for example, where Tc is a fitted constant. Doing this allows us to continue interpret the left-hand
+side of this equation as I−β.
+To fit equations of this form, we continue use the same black-box optimization method, and simply
+replace the monotonic regression by another method of fitting log (f). For example, we may fit
+f (T)−β = Tce−αT T
+by using linear regression to fit log (f). (Recall that β is already determined by αN and αE.)
+The function from our Colab notebook, fit_coeffs, provides options for fitting various functional
+forms for f, although it can sometimes be slow. (This is because it sometimes uses black-box
+optimization again in the inner loop for ease of implementation, even though this could be collapsed
+into the outer loop if speed were important.)
+20
+
+B
+Hyperparameters
+Our default hyperparameters for Procgen, Dota 2 and MNIST are given in Tables 1, 2 and 3
+respectively. We modified these defaults in two ways:
+• We adjusted the Adam step size as the model was scaled, as explained in Section 3.4.
+• For Procgen and MNIST, we incorporated a batch ramp and learning rate schedule, as
+explained in Section B.1.
+Table 1: Default PPG-EWMA hyperparameters for Procgen.
+Hyperparameter
+Value
+PPO
+Parallel environments
+1024
+Timesteps per rollout (T)
+256
+Minibatches per epoch
+8
+Adam step size (α)
+5 × 10−4
+Value function coefficient
+0.5
+Entropy coefficient
+0.01
+PPO clipping parameter (ϵ)
+Not used
+PPO KL penalty coefficient (β)
+1
+GAE discount rate (γ)
+0.999
+GAE bootstrapping parameter (λ)
+0.95
+Reward normalization?
+Yes
+Advantage normalization?
+Yes
+Total environment interactions
+200 million
+PPG
+Policy iterations per phase (Nπ)
+32
+Policy phase policy epochs (Eπ)
+1
+Policy phase value function epochs (EV )
+1
+Auxiliary phase epochs (Eaux)
+6
+Auxiliary phase minibatches per epoch
+16Nπ
+Auxiliary phase cloning coefficient (βclone)
+1
+PPG-EWMA
+Proximal policy EWMA decay rate (βprox)
+8
+9
+Batch ramp
+Initial batch size multiplier
+1
+32
+Table 2: PPO hyperparameters for Dota 2.
+Hyperparameter
+Value
+Parallel environments
+6144
+Timesteps per rollout (T)
+512
+Minibatches per epoch
+32
+Epochs (E)
+1
+Adam step size (α)
+10−4 to 10−3
+PPO clipping parameter (ϵ)
+0.2
+PPO KL penalty coefficient (β)
+Not used
+GAE bootstrapping parameter (λ)
+0.95
+Total environment interactions
+13.6–82.6 billion
+21
+
+Table 3: Default PPO-EWMA hyperparameters for MNIST in terms the horizon length h, which
+varied from 1 to 256.
+Hyperparameter
+Value
+PPO
+Parallel environments
+16
+Timesteps per rollout (T)
+512
+Minibatches per epoch
+8
+Epochs (E)
+1
+Adam step size (α)
+1 × 10−3
+Value function coefficient
+0.5
+Entropy coefficient
+0.01
+PPO clipping parameter (ϵ)
+Not used
+PPO KL penalty coefficient (β)
+1
+GAE discount rate (γ)
+1 −
+2
+h+1
+GAE bootstrapping parameter (λ)
+1
+Reward normalization?
+Yes
+Advantage normalization?
+Yes
+Total environment interactions
+225
+PPO-EWMA
+Proximal policy EWMA decay rate (βprox)
+8
+9
+Batch ramp
+Initial batch size multiplier
+√
+h
+64
+B.1
+Batch ramp and learning rate schedule
+As explained in Section 3.4, it was important to use a well-tuned learning rate schedule, and to use
+a schedule that works well for a variety of compute budgets. It was also important to use a batch
+ramp, i.e., to start with a small batch size and increase it over the course of training, because the
+critical batch size is smaller at the start of training, and we needed training to still be sample-efficient
+for small compute budgets. Without a batch ramp, we would have needed to adjust our power law,
+equation (1), in much the same way as the corresponding law for language [Kaplan et al., 2020,
+equation (1.6)], which uses Smin (S), the minimum number of optimization steps as estimated using
+a power law fit to the gradient noise scale.
+Note, however, that increasing the batch size has a very similar effect to lowering the learning rate.
+To simplify matters, we used PPO-EWMA and PPG-EWMA, which are batch size-invariant [Hilton
+et al., 2021], allowing us to have almost the same effect as increasing the batch size by instead
+lowering the learning rate and increasing the center of mass of the proximal policy EWMA. We then
+considered only the batch size schedule, whether implemented explicitly or implicitly via these other
+hyperparameters.
+To explore promising schedules, we implemented a greedy adaptive batch size algorithm, which tries
+doubling the batch size and switches if that performs better, or else backtracks and stays with the
+current batch size. We experimented with this on StarPilot’s easy difficulty setting, using model sizes
+spanning a factor of around 2048. We found our algorithm to fairly consistently choose a schedule
+that can be well-approximated by the power law
+B = max
+�
+Bmin, E0.84
+80
+�
+,
+where B is the batch size in interactions, E is the total number of interactions so far, and Bmin = 256
+was our initial batch size.
+Having fit this power law schedule on one Procgen environment, we tested it on several different
+Procgen environments, and found it to consistently outperform our usual fixed batch size both at the
+start and end of training. (Curiously, our schedule sometimes underperformed the fixed batch size in
+the middle of training. We believe this may be explained by the smaller initial batch size causing the
+entropy to fall too quickly at the start of training, highlighting a pitfall of the greedy approach.) In
+particular, we were able to use the same schedule on both the easy and hard difficulty settings. Our
+22
+
+usual fixed batch size, on the other hand, was larger for the hard setting, corresponding to the fact
+that it was tuned to longer training runs.
+The same schedule also worked well on our MNIST environment at every horizon length, although it
+was necessary to tune Bmin. Using too small a value for Bmin seemed to result in an instability which
+could not always be recovered from. We found the optimal Bmin to vary based on the horizon length
+h, and we took Bmin = 16
+√
+h (though taking Bmin to have the form A0 + A1h would probably have
+made more theoretical sense in hindsight, given the results of Section 4.2). If trying our schedule
+on other environments, we suggest tuning Bmin to ensure stability at the start of training, but it is
+probably less important to tune the power law constants.
+We used this batch size schedule for both our Procgen and MNIST experiments (although it would
+probably have been better to fully re-fit the schedule for MNIST). We implemented this using a batch
+size multiplier, explicitly reducing the batch size when the multiplier was less than 1, and changing
+the learning rate and center of mass of the proximal policy EWMA instead when the multiplier was
+greater than 1. With Procgen, for which we used PPG-EWMA, we also changed the number of policy
+iterations per phase, Nπ, in proportion to the batch size, since we thought the number of optimization
+steps per phase should remain constant, and we rounded the batch size multiplier to the nearest power
+of two, with minimum and maximum multipliers of
+1
+32 and 4 (corresponding to batch sizes of 1024
+and 131072 respectively).
+For Dota 2, we did not use a batch size schedule, since those experiments were carried out before we
+investigated batch size schedules.
+23
+
+C
+Results in full
+All the data from our experiments may be accessed using this Colab notebook: https://colab.
+research.google.com/drive/1PzwZyXsi9jRdVCj1GJrS8JdOPBQ7LHZV.
+This also includes
+code for analyzing this data, including model size and compute calculations, intrinsic performance
+and power law fitting, and generating all the plots in this paper.
+Figures 13, 14, 15 and 16 show learning curves as a function of total training compute, together with
+their power law fits, for all of our experiments. On the left of each figure we show mean episode
+return (or failure rate for CoinRun and MNIST, or TrueSkill for Dota 2), with error bars showing
+mean ±1 sample standard deviation over the random seeds. On the right of each figure, we show
+intrinsic performance, with error bars hidden for clarity.
+1015
+1017
+Compute (FLOPs)
+10−2
+10−1
+Failure rate
+CoinRun, easy
+1015
+1017
+Compute (FLOPs)
+10−1
+Failure rate
+CoinRun, hard
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+Intrinsic performance (FLOPs)
+CoinRun, easy
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+CoinRun, hard
+Parameters
+104.3
+104.6
+104.9
+105.2
+105.5
+105.8
+106.1
+106.4
+106.7
+107.0
+1015
+1017
+Compute (FLOPs)
+10
+20
+30
+40
+50
+60
+Mean episode return
+StarPilot, easy
+1015
+1017
+Compute (FLOPs)
+5
+10
+15
+20
+25
+30
+Mean episode return
+StarPilot, hard
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+StarPilot, easy
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+StarPilot, hard
+Learning
+curve
+Power law
+fit
+Power law
+asymptote
+Efficient
+frontier
+Efficient
+points
+1015
+1017
+Compute (FLOPs)
+5
+10
+15
+20
+25
+30
+Mean episode return
+FruitBot, easy
+1015
+1017
+Compute (FLOPs)
+0
+5
+10
+15
+20
+25
+Mean episode return
+FruitBot, hard
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+Intrinsic performance (FLOPs)
+FruitBot, easy
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+FruitBot, hard
+Procgen, width
+Figure 13: Learning curves as a function of total training compute for our Procgen width-scaling
+experiments, together with their power law fits. Left half: mean episode return or failure rate, mean
+±1 sample standard deviation over three seeds shown. Right half: intrinsic performance, mean only
+shown.
+24
+
+1015
+1017
+Compute (FLOPs)
+10−2
+10−1
+Failure rate
+CoinRun, easy
+1015
+1017
+Compute (FLOPs)
+10−1
+Failure rate
+CoinRun, hard
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+Intrinsic performance (FLOPs)
+CoinRun, easy
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+CoinRun, hard
+Parameters
+103.9
+104.1
+104.4
+104.6
+104.9
+105.2
+105.5
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+20
+30
+40
+50
+60
+Mean episode return
+StarPilot, easy
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+5
+10
+15
+20
+25
+Mean episode return
+StarPilot, hard
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+StarPilot, easy
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+StarPilot, hard
+Learning
+curve
+Power law
+fit
+Power law
+asymptote
+Efficient
+frontier
+Efficient
+points
+1015
+1017
+Compute (FLOPs)
+5
+10
+15
+20
+25
+30
+Mean episode return
+FruitBot, easy
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+0
+5
+10
+15
+20
+25
+Mean episode return
+FruitBot, hard
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+Intrinsic performance (FLOPs)
+FruitBot, easy
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+FruitBot, hard
+Procgen, depth
+Figure 14: Learning curves as a function of total training compute for our Procgen depth-scaling
+experiments, together with their power law fits. Left half: mean episode return or failure rate, mean
+±1 sample standard deviation over three seeds shown. Right half: intrinsic performance, mean only
+shown.
+1014
+1016
+1018
+1020
+Compute (FLOPs)
+−5
+0
+5
+10
+15
+20
+25
+TrueSkill
+1014
+1016
+1018
+1020
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+1017
+1018
+1019
+Intrinsic performance (FLOPs)
+Parameters
+102.7
+104.5
+105.1
+105.7
+106.3
+106.9
+108.1
+Learning
+curve
+Power law
+fit
+Power law
+asymptote
+Efficient
+frontier
+Efficient
+points
+Dota 2
+Figure 15: Learning curves as a function of total training compute for Dota 2, together with their
+power law fits. Only one random seed was used. Left: TrueSkill. Right: intrinsic performance.
+25
+
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 1
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 2
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 1
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 2
+Parameters
+104.8
+105.1
+105.4
+105.7
+106.0
+106.3
+106.6
+106.9
+107.2
+107.5
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 4
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 8
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 4
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 8
+Learning
+curve
+Power law
+fit
+Power law
+asymptote
+Efficient
+frontier
+Efficient
+points
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 16
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 32
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 16
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 32
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 64
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+10−2
+Failure rate
+Horizon 128
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 64
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 128
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+10−2
+Failure rate
+Horizon 192
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+10−2
+Failure rate
+Horizon 256
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+Intrinsic performance (FLOPs)
+Horizon 192
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+Intrinsic performance (FLOPs)
+Horizon 256
+MNIST, late period
+Figure 16: Learning curves as a function of total training compute for MNIST, together with their
+power law fits, for the late period of training (222–225 environment interactions). Left half: failure
+rate, mean ±1 sample standard deviation over the middle-performing 16 of 20 random seeds shown.
+Right: intrinsic performance, mean only shown.
+26
+
+D
+Parameter and FLOP calculations
+In counting parameters and FLOPs, we apply the following principles:
+• We only include the part of the network that is being scaled (ignoring things like embedding
+parameters), since we consider that to be the bottleneck.
+• We use round numbers (ignoring negligible contributions such as as biases and activations),
+for simplicity.
+• We include both rollout and optimization FLOPs (including any additional overhead of
+PPO-EWMA).
+• We treat an add-multiply as 2 FLOPs.
+For example, we treat the forward pass of a dense layer as taking 2 FLOPs per batch item per
+parameter, and a convolutional layer as taking 2houtwout FLOPs per batch item per parameter. We
+treat a backward pass as taking 2× the FLOPs of a forward pass.
+For the Procgen width-scaling experiments, we ignore the first convolution, since it scales as width
+(instead of as width squared), and has few parameters. Similarly, for the depth-scaling experiments,
+we ignore the final dense layer, since we only vary the number of convolutional layers. Unfortunately,
+as discussed in Section 4.4, the final dense layer contains many parameters, which skews our constants.
+In both cases, we include both the policy and value networks, which are separate with identical
+architectures. We use PPG-EWMA with 1 policy epoch and 6 auxiliary epochs, totaling 9 forward
+and 7 backward passes per interaction.
+For the Dota experiments, we ignore the embedding layer, considering only the LSTM. Since each
+interaction was used only once, we count 2 forward passes and 1 backward pass per interaction (1
+forward pass for the rollout, and 1 forward-backward pass for optimization).
+For the MNIST experiments, we ignore the first convolution, as for the Procgen width-scaling
+experiments. However, we only include the policy network, since the task of the value network is
+trivial (due to timesteps being independent). We use PPO-EWMA with 1 epoch, totaling 3 forward
+passes and 1 backward pass per interaction.
+The numerical results of these calculations are as follows.
+• Procgen, scaling width: for the width multiplier w = 2−3, 2−2.52−2, . . . , 22.5, we count
+1242112w2 parameters and 2652897280w2 FLOPs per interaction.
+• Procgen, scaling depth: for the number of residual blocks b = 1, 2, 4, . . . , 64, we count
+5184b + 1944 parameters and 61046784b + 81395712 FLOPs per interaction.
+• Dota 2: for the LSTM size s = 8, 64, 128, 256, 512, 1024, 4096, we count 8s2 parameters
+and 64s2 FLOPs per interaction.
+• MNIST: for the width multiplier w = 2−3, 2−2.52−2, . . . , 22.5, we count 3948800w2
+parameters and 95648000w2 FLOPs per interaction.
+Note that one of our modeling assumptions is that the number of FLOPs per interaction is proportional
+to the number of parameters, but this is not true for our Procgen depth-scaling experiments. In other
+words, the number of FLOPs per param-interact, which is used to convert compute from units of
+parameters × interactions to units of FLOPs, is not constant. However, this number differs by at most
+40% from the mean of this number over the different depths, and so we simply used the mean when
+doing this conversion.
+27
+
+E
+Fitted constants
+In this section we provide the constants αN, αE and Nc, together with the values of β and Ec derived
+using Lemma 1, for our fitted power laws for intrinsic performance I as given by equation (1). We
+also provide Imin and Imax, the minimum and maximum intrinsic performance obtained during the
+span of interaction counts considered; our model is not able to predict mean episode return outside
+this range. Recall that the units of I are parameters × interactions; the conversion to FLOPs may be
+performed using the values given in Appendix D.
+We also provide the derived equations for optimal model size N vs compute C in PF-days. By
+substituting equation (2) for the compute-efficient frontier into equation (1), these are given by
+N = Nc
+�
+1 + αN
+αE
+�
+1
+αN � C × 1015 × 24 × 3600
+FLOPs per param-interact
+�
+1
+1+ αN
+αE
+for
+Nmin ≤ N ≤ Nmax.
+We take Nmin and Nmax to be the minimum and maximum model sizes we tested whose power law
+fit intersects the compute-efficient frontier somewhere between Imin and Imax.
+For our comparison to generative modeling, we use these equations for optimal model size N vs
+compute C in PF-days:
+• Language [Hoffmann et al., 2022]: N = (
+C
+1.4×10−18 )0.5
+• Language [Kaplan et al., 2020]: N = (
+C
+3.3×10−13 )0.73
+• Image 32x32 [Henighan et al., 2020]: N = (
+C
+1.6×10−13 )0.65
+Further fitted constants, such as for single seeds, for different spans of interaction counts (see Section
+4.2), and fitted to natural performance metrics (see Section 4.5), may be found in this Colab notebook:
+https://colab.research.google.com/drive/1PzwZyXsi9jRdVCj1GJrS8JdOPBQ7LHZV.
+E.1
+Procgen, scaling width
+The fitted constants for our Procgen width-scaling experiments are as follows.
+Environment
+αN
+αE
+β
+Nc
+Ec
+Imin
+Imax
+CoinRun, easy
+0.542
+0.462
+0.249
+2.53 × 10−2
+2.49 × 100
+4.83 × 1010
+2.55 × 1014
+CoinRun, hard
+0.759
+0.576
+0.328
+1.55 × 10−1
+8.00 × 10−1
+6.07 × 1010
+3.45 × 1014
+StarPilot, easy
+0.318
+0.604
+0.208
+2.25 × 10−4
+2.02 × 102
+4.88 × 1010
+1.95 × 1015
+StarPilot, hard
+0.453
+0.533
+0.245
+4.55 × 10−3
+1.31 × 101
+5.43 × 1010
+1.09 × 1015
+FruitBot, easy
+0.527
+0.350
+0.210
+9.17 × 10−2
+4.46 × 10−1
+5.24 × 1010
+1.67 × 1014
+FruitBot, hard
+0.478
+0.346
+0.201
+1.14 × 10−1
+2.96 × 10−1
+6.00 × 1010
+7.26 × 1014
+These imply the following equations for optimal model size N vs compute C in PF-days.
+• CoinRun, easy: N = 4.615 × 106 × C0.4600 for 19408 ≤ N ≤ 310528
+• CoinRun, hard: N = 6.881 × 106 × C0.4315 for 43668 ≤ N ≤ 587092
+• StarPilot, easy: N = 6.383 × 107 × C0.6549 for 19408 ≤ N ≤ 4968448
+• StarPilot, hard: N = 1.668 × 107 × C0.5404 for 19408 ≤ N ≤ 1242112
+• FruitBot, easy: N = 2.243 × 106 × C0.3994 for 19408 ≤ N ≤ 174672
+• FruitBot, hard: N = 6.631 × 106 × C0.4201 for 43668 ≤ N ≤ 587092
+As discussed in Section 4.5, for CoinRun, we also fit power laws using the fail-to-success ratio F,
+excluding data for which F > 0.5. As explained in Section A.1, we replaced I−β with F
+Fc , where Fc
+is a fitted constant. The fitted constants for these power laws are as follows.
+28
+
+Difficulty
+αN
+αE
+β
+Nc
+Ec
+Imin
+Imax
+Easy
+0.899
+1.007
+0.475
+1.00 × 10−2
+2.33 × 101
+2.55 × 1010
+2.60 × 1014
+Hard
+0.833
+0.776
+0.402
+4.69 × 10−2
+3.80 × 100
+5.14 × 1011
+7.38 × 1014
+Difficulty
+Fc
+Easy
+3.88 × 104
+Hard
+2.52 × 104
+These imply the following relationships between I and F.
+• Easy: I = 4.57 × 109 × F −
+1
+0.475
+• Hard: I = 9.15 × 1010 × F −
+1
+0.402
+They also imply the following equations for optimal model size N vs compute C in PF-days.
+• Easy: N = 1.216 × 107 × C0.5285 for 19408 ≤ N ≤ 587092
+• Hard: N = 1.148 × 107 × C0.4822 for 77632 ≤ N ≤ 1242112
+E.2
+Procgen, scaling depth
+The fitted constants for our Procgen depth-scaling experiments are as follows.
+Environment
+αN
+αE
+β
+Nc
+Ec
+Imin
+Imax
+CoinRun, easy
+0.351
+0.469
+0.201
+2.64 × 10−4
+1.26 × 102
+5.43 × 109
+3.72 × 1013
+CoinRun, hard
+0.336
+0.581
+0.213
+1.02 × 10−4
+4.47 × 102
+6.58 × 109
+6.24 × 1013
+StarPilot, easy
+0.800
+0.821
+0.405
+9.65 × 10−3
+1.87 × 101
+1.70 × 1010
+5.52 × 1013
+StarPilot, hard
+0.380
+0.381
+0.190
+2.87 × 10−3
+9.11 × 100
+1.58 × 1010
+5.21 × 1013
+FruitBot, easy
+0.539
+0.564
+0.276
+2.92 × 10−3
+2.77 × 101
+9.58 × 109
+3.76 × 1013
+FruitBot, hard
+0.401
+0.463
+0.215
+1.23 × 10−3
+3.26 × 101
+1.34 × 1010
+4.64 × 1013
+These imply the following equations for optimal model size N vs compute C in PF-days. Note,
+however, that:
+• As discussed in Section 4.4, we exclude the final dense layer, which would have accounted
+for between 16% and 90% of the parameters, depending on the depth. This skews the
+leading constants here.
+• As discussed in Appendix D, we also ignored the variation in the number of FLOPs per
+param-interact between models of different depths, leading to errors of up to 40%.
+• CoinRun, easy: N = 1.390 × 106 × C0.5723 for 7128 ≤ N ≤ 43416
+• CoinRun, hard: N = 3.962 × 106 × C0.6337 for 7128 ≤ N ≤ 167832
+• StarPilot, easy: N = 2.202 × 106 × C0.5063 for 7128 ≤ N ≤ 167832
+• StarPilot, hard: N = 1.410 × 106 × C0.5007 for 7128 ≤ N ≤ 84888
+• FruitBot, easy: N = 1.172 × 106 × C0.5110 for 7128 ≤ N ≤ 84888
+• FruitBot, hard: N = 1.671 × 106 × C0.5359 for 7128 ≤ N ≤ 84888
+29
+
+E.3
+Dota 2
+As explained in Sections 4.5 and A.1, we fit power laws to I−β, Tce−αT T , Tc
+�
+e−αT T − eαT T ∗�
+and Tc (T ∗ − T)αT , where I is intrinsic performance, T is TrueSkill, and αT , Tc and T ∗ are fitted
+constants. The fitted constants for these different functional forms are as follows.
+Fit to
+αN
+αE
+β
+Nc
+Ec
+Imin
+Imax
+I−β
+0.186
+0.593
+0.141
+1.98 × 10−8
+1.04 × 106
+6.83 × 1011
+1.79 × 1018
+Tce−αT T
+0.180
+0.486
+0.131
+3.53 × 10−8
+3.33 × 105
+4.62 × 1011
+2.24 × 1017
+Tc(e−αT T − eαT T ∗)
+0.181
+0.560
+0.137
+2.07 × 10−8
+8.32 × 105
+6.31 × 1011
+1.77 × 1018
+Tc(T ∗ − T)αT
+0.183
+0.569
+0.138
+2.06 × 10−8
+8.82 × 1005
+6.71 × 1011
+1.23 × 1018
+Fit to
+αT
+Tc
+T ∗
+I−β
+-
+-
+-
+Tce−αT T
+0.0572
+2.16 × 10−2
+-
+Tc(e−αT T − eαT T ∗)
+0.0402
+2.40 × 10−2
+35.43
+Tc(T ∗ − T)αT
+2.84
+2.14 × 10−7
+54.01
+As discussed in Section 4.5, we have less confidence in the last two functional forms, which is
+reflected in the very different estimates for T ∗, which represents the maximum attainable TrueSkill
+for the family of models we trained.
+These imply the following relationships between I and T for the last three fits.
+• Tce−αT T :
+I = 4.93 × 1012 × 1.5462T
+• Tc(e−αT T − eαT T ∗):
+I = 6.49 × 1011 ×
+�
+1.0410−T − 1.0410−35.43�−
+1
+0.137
+• Tc(T ∗ − T)αT :
+I = 1.48 × 1048 × (54.01 − T)− 2.84
+0.138
+They also imply the following equations for optimal model size N vs compute C in PF-days.
+• I−β:
+N = 2.703 × 107 × C0.7617 for 512 ≤ N ≤ 2097152
+• Tce−αT T :
+N = 1.607 × 107 × C0.7302 for 512 ≤ N ≤ 524288
+• Tc(e−αT T − eαT T ∗):
+N = 2.305 × 107 × C0.7552 for 512 ≤ N ≤ 2097152
+• Tc(T ∗ − T)αT :
+N = 2.385 × 107 × C0.7567 for 512 ≤ N ≤ 2097152
+E.4
+MNIST
+The fitted constants for our MNIST experiments are as follows. As discussed in Section 4.3, these
+constants are for the late period of training (222–225 environment interactions). Recall also that the
+horizon h is such that the interval [0, h − 1] has the same center of mass as an exponentially-weighted
+moving average with decay parameter γ, i.e., γ = 1 −
+2
+h+1.
+30
+
+Horizon
+αN
+αE
+β
+Nc
+Ec
+Imin
+Imax
+1
+0.263
+1.050
+0.210
+9.79 × 10−6
+9.43 × 103
+1.79 × 1011
+1.00 × 1015
+2
+0.265
+0.979
+0.208
+1.32 × 10−5
+6.30 × 103
+1.87 × 1011
+9.66 × 1014
+4
+0.284
+0.791
+0.209
+4.21 × 10−5
+1.50 × 103
+1.94 × 1011
+4.19 × 1014
+8
+0.276
+0.826
+0.207
+2.83 × 10−5
+2.33 × 103
+1.80 × 1011
+6.24 × 1014
+16
+0.252
+0.830
+0.193
+1.59 × 10−5
+3.78 × 103
+1.62 × 1011
+7.69 × 1014
+32
+0.263
+0.856
+0.201
+1.73 × 10−5
+3.83 × 103
+1.59 × 1011
+7.47 × 1014
+64
+0.307
+0.736
+0.217
+7.27 × 10−5
+8.40 × 102
+1.64 × 1011
+4.16 × 1014
+128
+0.315
+0.769
+0.224
+6.27 × 10−5
+1.08 × 103
+1.45 × 1011
+3.64 × 1014
+192
+0.330
+0.688
+0.223
+1.22 × 10−4
+4.86 × 102
+1.33 × 1011
+2.08 × 1014
+256
+0.358
+0.681
+0.235
+2.11 × 10−4
+3.04 × 102
+1.33 × 1011
+1.53 × 1014
+These imply the following equations for optimal model size N vs compute C in PF-days.
+• Horizon 1:
+N = 1.586 × 1010 × C0.7999 for 61700 ≤ N ≤ 15795200
+• Horizon 2:
+N = 1.309 × 1010 × C0.7871 for 61700 ≤ N ≤ 15795200
+• Horizon 4:
+N = 5.507 × 109 × C0.7357 for 61700 ≤ N ≤ 3948800
+• Horizon 8:
+N = 6.406 × 109 × C0.7493 for 61700 ≤ N ≤ 7739648
+• Horizon 16: N = 7.787 × 109 × C0.7671 for 61700 ≤ N ≤ 7739648
+• Horizon 32: N = 7.535 × 109 × C0.7652 for 61700 ≤ N ≤ 7739648
+• Horizon 64: N = 2.746 × 109 × C0.7053 for 61700 ≤ N ≤ 3948800
+• Horizon 128: N = 2.681 × 109 × C0.7092 for 61700 ≤ N ≤ 3948800
+• Horizon 192: N = 1.376 × 109 × C0.6757 for 61700 ≤ N ≤ 987200
+• Horizon 256: N = 9.876 × 108 × C0.6553 for 61700 ≤ N ≤ 987200
+31
+
+F
+Proof of the lemma
+Proof of Lemma 1. We may write I (N, E) as a function of N and compute C := NE:
+I (N, C)−β =
+�Nc
+N
+�αN
++
+�EcN
+C
+�αE
+.
+The compute-efficient frontier is defined by the value of N that maximizes I (N, C) for each C.
+Equivalently, since β > 0, this value of N minimizes I (N, C)−β, and so it satisfies
+∂
+∂N
+�
+I (N, C)−β�
+= 0.
+Differentiating and multiplying through by N, this equation becomes
+−αN
+�Nc
+N
+�αN
++ αE
+�EcN
+C
+�αE
+= 0.
+Eliminating C, this is exactly equation (2), as required.
+By assumption, we also have I (N, E) = NE along the compute-efficient frontier. Substituting (2)
+into I (N, E), this equation becomes
+�
+1 + αN
+αE
+� �Nc
+N
+�αN
+= (NE)−β .
+(3)
+Thus both equations (2) and (3) are power law relationships between N and E that hold along the
+compute-efficient frontier, so we may simply equate exponents and constants. Equating exponents,
+αN
+αE
+= αN
+β − 1
+and hence
+1
+β =
+1
+αN
++ 1
+αE
+,
+as required. Equating constants,
+�αN
+αE
+�
+1
+αE N
+αN
+αE
+c
+E−1
+c
+=
+�
+1 + αN
+αE
+� 1
+β
+N
+αN
+β
+c
+,
+and hence
+1
+NcEc
+=
+�
+1 + αN
+αE
+�
+1
+αN +
+1
+αE � αE
+αN
+�
+1
+αE =
+�
+1 + αN
+αE
+�
+1
+αN �
+1 + αE
+αN
+�
+1
+αE ,
+as required.
+32
+
+G
+Proof sketch of the proposition
+A formal statement and proof of Proposition 1 would require a formal analysis of Vanilla Policy
+Gradient, which is beyond the scope of this work. Instead, we provide a proof sketch in which we
+make approximations informally.
+Proof sketch of Proposition 1. The horizon length h only affects the algorithm via GAE, which in
+the case λ = 1 produces the value function targets and advantage estimates
+ˆVt := rt + γrt+1 + · · · + γT −trT = rt + γ ˆVt+1
+and
+ˆAt := ˆVt − V (st) = rt − V (st) + γ ˆVt+1,
+where V is the value function. Since timesteps are independent, γ ˆVt+1 is independent of st and at,
+and so should be thought of as noise. The value function will quickly learn to incorporate the mean
+of this noise, and so
+V (st) ≈ V 0 (st) + E
+�
+γ ˆVt+1
+�
+,
+where V 0 (st) is the “immediate reward value function” that would have been obtained had we
+used the value function targets ˆV 0
+t := ˆVt − E
+�
+γ ˆVt+1
+�
+. Writing ϵ := γ ˆVt+1 − E
+�
+γ ˆVt+1
+�
+for the
+zero-mean component of γ ˆVt+1, we obtain
+ˆV 0
+t = rt + ϵ
+and
+ˆAt ≈ rt − V 0 (st) + ϵ.
+In other words, the entire impact of varying h is that it changes the variance of the noise term ϵ added
+to the value function targets and advantage estimates.
+Let us now analyze the policy gradient, which equals
+ˆEt
+�
+∇θρt (θ) ˆAt
+�
+≈ ˆEt
+�
+∇θρt (θ)
+�
+rt − V 0 (st) + ϵ
+��
+,
+where ρt (θ) :=
+πθ(at|st)
+πθold(at|st). Since ϵ is independent of st and at and E [ϵ] = 0, the covariance matrix
+of this decomposes as
+Σθ + ΦθVar [ϵ] ,
+where Σθ is the covariance matrix of ∇θρt (θ)
+�
+rt − V 0 (st)
+�
+, and Φθ := E
+�
+∇θρt (θ) ∇T
+θ ρt (θ)
+�
+is
+the uncentered covariance matrix of ∇θρt (θ).
+Note that V 0 (st) simply estimates E [rt], which does not depend on h. The variance of V 0 (st) does
+depend on h via the addition of ϵ to the value function targets, but this additional variance is small
+compared to the variance of ϵ itself. We may therefore treat Σθ as approximately independent of h.
+It remains to express Var [ϵ] in terms of h. We assume that T is large enough compared to h that we
+may take T → ∞. (In our experiments, we use rollouts of length 512 and h ≤ 256.) Thus
+Var [ϵ] = Var
+�
+γ ˆVt+1
+�
+=
+�
+γ2 + γ4 + γ6 + . . .
+�
+Var [rt]
+=
+γ2
+1 − γ2 Var [rt]
+= 1
+4
+�
+h + 1
+h − 2
+�
+Var [rt] .
+Hence the covariance matrix of the policy gradient is approximately
+Σθ + Πθ
+�
+h + 1
+h − 2
+�
+,
+where Σθ and Πθ := 1
+4Var [rt] Φθ are symmetric positive semi-definite matrices that do not depend
+on h, as required.
+33
+
diff --git a/-9FQT4oBgHgl3EQf7Ta5/content/tmp_files/load_file.txt b/-9FQT4oBgHgl3EQf7Ta5/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a7e96cfba93bef32a5475c8f9777e99fba2002a8
--- /dev/null
+++ b/-9FQT4oBgHgl3EQf7Ta5/content/tmp_files/load_file.txt
@@ -0,0 +1,1833 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf,len=1832
+page_content='Scaling laws for single-agent reinforcement learning  Jacob Hilton  OpenAI  jacob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='hilton@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='com  Jie Tang  OpenAI  jietang@openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='com  John Schulman  OpenAI  joschu@openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='com  Abstract  Recent work has shown that, in generative modeling, cross-entropy loss improves  smoothly with model size and training compute, following a power law plus  constant scaling law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' One challenge in extending these results to reinforcement  learning is that the main performance objective of interest, mean episode return,  need not vary smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' To overcome this, we introduce intrinsic performance,  a monotonic function of the return defined as the minimum compute required to  achieve the given return across a family of models of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We find that,  across a range of environments, intrinsic performance scales as a power law in  model size and environment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Consequently, as in generative modeling,  the optimal model size scales as a power law in the training compute budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Furthermore, we study how this relationship varies with the environment and with  other properties of the training setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In particular, using a toy MNIST-based  environment, we show that varying the “horizon length” of the task mostly changes  the coefficient but not the exponent of this relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  1  Introduction  Recent studies of how neural network performance varies with model size and training compute have  found these relationships to be governed by smooth power laws [Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2020, Henighan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=',  2020, Droppo and Elibol, 2021, Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' These studies have focused primarily on  generative modeling, in which the training objective is cross-entropy loss, and have found test loss to  scale smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In this work we seek to extend these results to reinforcement learning, in which there  is generally no cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  In some reinforcement learning environments, there is still a performance metric that varies smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For example, in competitive games, it is often possible to assign Elo ratings to players such that  scaled differences in Elo ratings give approximate logit probabilities of victory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Recently it has been  shown that, in the board games Hex [Jones, 2021], Connect Four and Pentago [Neumann and Gros,  2022], the exponentiated Elo rating of a policy trained using AlphaZero [Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2018] follows a  power law in training compute (within a certain Elo range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We call metrics that follow such simple  relationships natural performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  However, in other reinforcement learning environments, there may be no obvious natural performance  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For example, there may be no reason to expect the number of objects collected in a video  game to vary smoothly, since crossing some threshold may require some challenging new capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  To overcome this difficulty, we introduce intrinsic performance, which is defined to be equal to  training compute on the compute-efficient frontier of the tradeoff between model size and environment  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This causes the relationship between performance and training compute to follow a  power law by definition, thereby making it possible to study the remaining relationships between  performance, model size and environment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We study these relationships across a range of environments: the easy and hard modes of environments  from Procgen Benchmark [Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2020];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' a 1v1 version of Dota 2 [OpenAI et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2019];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' and a toy  environment based on MNIST [LeCun, 1998] for which we vary the “horizon length”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Across these  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='13442v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='LG]  31 Jan 2023  environments, we find intrinsic performance to scale as a power law in model size and environment  interactions, in much the same way as the analogous quantities in generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  One consequence of this scaling law is that, as in generative modeling, the optimal model size for  a given training compute budget follows a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We study in detail how the coefficient and  exponent of this relationship vary with properties of the training setup, including: the difficulty mode  of environment, for Procgen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' the horizon length of the task, for the MNIST-based environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' the  period of training used to fit the power law;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' and whether the width or depth of the model is scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Contents  1  Introduction  1  2  Scaling laws without cross-entropy loss  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  Intrinsic performance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  The power law for intrinsic performance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  Optimal model size vs compute .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='  12  5  Discussion  13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  Extrapolating sample efficiency .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='  13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  Cost-efficient reinforcement learning .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='  14  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  Limitations  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  15  6  Conclusion  16  A Curve-fitting methodology  19  B  Hyperparameters  21  C Results in full  24  D Parameter and FLOP calculations  27  E  Fitted constants  28  F  Proof of the lemma  32  G Proof sketch of the proposition  33  2  1014  1015  1016  1017  1018  Compute (FLOPs)  5  10  15  20  25  30  Mean episode return  StarPilot, hard  (a) Using the usual metric of mean episode return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  1014  1015  1016  1017  1018  Compute (FLOPs)  1014  1015  1016  1017  1018  Intrinsic performance (FLOPs)  Parameters  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='9  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='8  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0  StarPilot, hard  (b) Using intrinsic performance instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Figure 1: Learning curves as a function of total training compute for StarPilot, an environment from  Procgen Benchmark, using CNNs of different widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Mean ±1 sample standard deviation over three  seeds shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  2  Scaling laws without cross-entropy loss  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  Intrinsic performance  In generative modeling, cross-entropy test loss scales smoothly with training compute, following a  power law plus constant scaling law [Henighan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' However, in reinforcement learning  (RL), there is generally no cross-entropy loss, and the usual objective of mean episode return need  not scale so smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For example, consider StarPilot, a side-scrolling shooter from Procgen Benchmark [Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=',  2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The agent receives a reward of 1 for destroying each enemy, and the episode continues until  either the agent is destroyed, or the agent reaches the end of the level and obtains a bonus reward  of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' There is no reason to expect mean episode return in this game to scale smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Indeed, it  takes some ability with aiming and dodging to reach a mean episode return of 5 or 10, but not much  additional skill to reach a mean episode return of 15 or 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This irregular difficulty profile is reflected  in the uneven shape of learning curves for this environment (see Figure 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  It may be tempting to conclude that the scaling law methodology cannot be applied to such an  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' However, in generative modeling, there are smooth scaling laws that do not depend on  test loss per se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For example, the model size that achieves the minimum test loss for a given compute  budget scales as a power law with compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In order to study such relationships in the context of  RL, we would like a performance metric that behaves like test loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', some monotonic function  of the return that scales as a power law with compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We achieve this with our notion of intrinsic  performance by simply using compute itself as our performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' A scalable model family is collection of models trained in a uniform way, parameterized  by the model size and the total compute used in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Given a scalable model family, the intrinsic  performance of an arbitrary policy is the minimum compute required to train a model of any size in  the family to reach the same return (averaged over random seeds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Another way of explaining this definition is to consider learning curves as a function of compute  for a family of models of different sizes, as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The maximum performance over all model  sizes defines the compute-efficient frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' When using the usual metric of mean episode return (as  in Figure 1(a)), the compute-efficient frontier need not follow any particular trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' However, when  using intrinsic performance instead (as in Figure 1(b)), the efficient frontier is mapped onto the line  3  1014  1015  1016  1017  1018  Compute (FLOPs)  5  10  15  20  25  30  Mean episode return  StarPilot, hard  (a) Using mean episode return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  1014  1015  1016  1017  1018  Compute (FLOPs)  1014  1015  1016  1017  1018  Intrinsic performance (FLOPs)  Parameters  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='9  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='8  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0  Learning  curve  Power law  fit  Power law  asymptote  Efficient  frontier  Efficient  points  StarPilot, hard  (b) Using intrinsic performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Figure 2: Learning curves as a function of total training compute for StarPilot, together with their  power law fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The asymptotes show the E → ∞ limits of the power law fits, representing the  predicted performance at convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The efficient points show where the power law fits are tangent  to the efficient frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Mean over three seeds shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  y = x by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This reveals the regularity of the learning curves, which, as we shall see next,  now follow a power law trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We describe in detail how we compute intrinsic performance in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  The power law for intrinsic performance  Our main empirical result is that intrinsic performance I scales approximately as a power law with  model parameters N and environment interactions E,  I−β =  �Nc  N  �αN  +  �Ec  E  �αE  ,  (1)  where αN, αE, β, Nc and Ec are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  This is essentially the same as the corresponding scaling law for language models [Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=',  2020, equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6)], but with test loss replaced by I−β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Although it appears that we have introduced  an additional exponent β, the intrinsic definition of I means that β is actually determined by αN and  αE (see Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The intuition behind this equation is that, when the number of interactions is not bottlenecked  (E → ∞), I scales as a power law in N, and when model size is not bottlenecked (N → ∞), I  scales as a power law in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  Optimal model size vs compute  An important implication of equation (1) is that the optimal model size for a given compute budget  scales as a power law in that compute budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  More precisely, we assume that total training compute is proportional to NE (ignoring the compute  required to run the environment, at least for now).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hence, for a given compute budget, there is a  trade-off between N and E (the optimum of which defines a point on the compute-efficient frontier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  What we will now show is that, under equation (1), the optimal value of N scales as a power law in  the compute budget, with an exponent that we will specify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  4  Since training compute is proportional to NE, for convenience we choose units of compute such  that training compute equals NE exactly (although in plots we will continue to display compute in  FLOPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This implies that I = NE along the compute-efficient frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' If I satisfies equation (1) and I = NE along the compute-efficient frontier, then the  compute-efficient frontier is described by the equation  αN  �Nc  N  �αN  = αE  �Ec  E  �αE  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  (2)  Moreover, once αN and αE are chosen, β and NcEc are determined:  1  β =  1  αN  + 1  αE  and  1  NcEc  =  �  1 + αN  αE  �  1  αN �  1 + αE  αN  �  1  αE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For a proof, see Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Substituting equation (2) into equation (1), it follows that along the compute-efficient frontier,  N = Nc  �  1 + αN  αE  �  1  αN C  1  1+ αN  αE ,  where C := NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In other words, for a given compute budget C, the optimal model size N scales as  N ∝ C  1  1+ αN  αE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  3  Experimental setup  We ran experiments using variety of RL environments:  Procgen Benchmark [Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2020]: CoinRun, StarPilot and FruitBot in both easy  and hard modes, separately varying CNN width and depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Dota 2 [OpenAI et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2019]: a 1v1 version of the game, varying LSTM size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  MNIST: an RL environment in which the agent has to correctly label a handwritten digit  from MNIST [LeCun, 1998], using hyperparameters to artificially alter the “horizon length”  of the task, varying CNN width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  All our experiments used a variant of either the PPO algorithm [Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2017] or its close  cousin PPG [Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2021], along with the Adam optimization algorithm [Kingma and Ba,  2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The remainder of this section discusses further details of our experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hyperparameters  for all our experiments are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  Procgen Benchmark  For our Procgen Benchmark experiments, we used CoinRun, StarPilot and FruitBot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We chose these  environments because they have lower-variance learning curves than other Procgen environments,  and because CoinRun’s binary reward enabled us to study the scaling of natural performance metrics  (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We used both the easy and hard difficulty modes of these environments to see if  this would have an effect on the scaling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We used PPG-EWMA [Hilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2021] with a fixed KL penalty objective [Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2021],  and trained for 200 million environment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We used the CNN architecture from IMPALA [Espeholt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2018] and conducted both width-  scaling and depth-scaling experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For our width-scaling experiments, we varied the total number  of parameters from  1  64 of the default to 8 times the default, rounding to integer numbers of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For our depth-scaling experiments, we varied the number of residual blocks per stack from 1 to 64,  and used 1  4 of the default width since the default number of residual blocks per stack was only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  Dota 2  For our Dota 2 experiments, we used a 1v1 version of the game to save computational expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Following OpenAI et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' [2019], we used PPO, but we adjusted the asynchronous setup to ensure that  training used only on-policy data with no data reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We used 8 parallel GPU workers and trained for  between 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6 billion and 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6 billion environment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We used an LSTM architecture and varied the width of the network, with the sizes of the embedding  and hidden state varying from 8 to 4096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  MNIST  Our MNIST environment samples a handwritten digit from the MNIST training set uniformly and  independently random at each timestep, and provides an immediate reward of 1 for a correct label  and 0 for an incorrect label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' There are no episode boundaries, and so we measure mean training  accuracy instead of mean episode return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The use of immediate rewards with no episode boundaries allows the horizon length of the task  to be artificially controlled by varying the hyperparameters of our method advantage estimation,  GAE [Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' First, we set the GAE credit assignment parameter λ to 1, so that the  algorithm assigns credit for each reward to all previous actions, instead of assigning more immediate  credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Then we vary the GAE discount rate γ, so that the algorithm discounts future rewards at this  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In separate experiments, we set γ = 1 −  2  h+1 for different values of the “horizon length” h  ranging from 1 to 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' (This equation is equivalent to saying that an exponentially-weighted moving  average with decay parameter γ has the same center of mass as the interval [0, h − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=')  We used PPO-EWMA [Hilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2021] with rollouts of length 512 (twice as long as our maximum  value of h), and trained for 225 environment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We used a simple CNN architecture with ReLU activations and the following layers: a 5 × 5  convolutional layer with 40 channels, 2×2 max pooling, a 3×3 convolutional layer with 80 channels,  2 × 2 max pooling, and a dense layer with 1,000 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We scaled the width of this network by  varying total number of parameters from  1  64 of the default to 8 times the default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We used separate  policy and value function networks because we did not expect there to be much transfer between the  two objectives, since the environment samples digits independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  Learning rates  Although we would not expect our qualitative results to change much, our quantitative results  such as scaling exponents depend crucially on using well-tuned hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' By far the most  important hyperparameter to tune in our setup is the Adam learning rate, whose optimal value can  vary substantially with model size and compute budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  When varying model size, we found that a good heuristic is to keep the Adam learning rate propor-  tional to the initialization scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For our width-scaling experiments, this means keeping the Adam  learning rate proportional to 1/  √  width, since we use Kaiming He initialization [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For  our Procgen depth-scaling experiments, which use a residual network, it means keeping the Adam  learning rate proportional to 1/  �  depth  1  L , where L is the number of layers per residual block (L = 2  in our case), since we use an initialization similar to Fixup initialization [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For  Procgen and MNIST, we tuned the learning rate at one model size and followed this heuristic to select  the learning rate for the other model sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For Dota 2, we tuned the learning rate separately for each  model size, but this amounted to following approximately the same heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  When varying the compute budget for a given model size, it can actually be necessary to use separate  training runs for each compute budget, each with its own learning rate schedule, rather than taking  different snapshots at different points of the same training run [Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Unfortunately,  due to the challenge of carefully tuning learning rate schedules for RL and the expense of multiplying  the number of training runs, we took the latter approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' To mitigate the impact of this, we found a  learning rate schedule that seemed to work well for a variety of compute budgets, which we explain  in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Nevertheless, the values of our scaling exponents should be considered uncertain  because of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='8  αN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0  αE  1  1 + αN/αE  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='8  1  1 + αN/αE  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7  1  1 + αN/αE  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6  Procgen (width)  CoinRun  StarPilot  FruitBot  Easy, single seed  Easy, mean return  Hard, single seed  Hard, mean return  Dota 2  1v1  Reference  αN/αE = const  MNIST horizons  1  2  4  8  16  32  64  128  192  256  αN vs αE  Figure 3: Fitted values of αN and αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For Procgen, we also show the values fitted using each of  the 3 random seeds, to show the variation due to the choice of random seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The dotted lines show  contours for  1  1+αN/αE , the exponent for the scaling of optimal model size with compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  4  Results  Our main result is that our power law for intrinsic performance, equation (1), holds across envi-  ronments and model sizes, at least after an initial transient period of training (which we discuss in  more detail in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This result is supported by the closeness of the power law fit to our  learning curves, as shown in Figure 2 for StarPilot and in Appendix C for all our environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Our  methodology for fitting this power law is described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  It is interesting to study the sensitivity of the exponents αN and αE, which govern the scaling  behavior of I with N and E (and determine the other exponents of interest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The fitted values of  these exponents for the different environments are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The numerical values of all of  the fitted constants may be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Although our measurements of these exponents are uncertain, due to the limitations discussed in  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3, we make a number of observations:  The primary determinant of αN and αE is the domain (Procgen, Dota 2, or MNIST), which  we expect is a consequence of the fact that so many experimental details are shared within  each domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Within MNIST, increasing the horizon seems to lower αE, but as we explain in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2,  this effect is confounded by a measurement problem caused by under-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Within Procgen, the easy and hard modes of each Procgen game tend to have closer  exponents to one another than to other Procgen games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We believe that this is because  identifying visual features is a core part of Procgen, and the two modes of each game have  very similar observation distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  7  10−7  10−6  10−5  10−4  10−3  10−2  Compute (PF-days)  103  104  105  106  107  Parameters  Procgen (width)  CoinRun  StarPilot  FruitBot  Easy  Hard  Dota 2  1v1  Generative modeling  Language  (Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=')  Language  (Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=')  Image 32x32  (Henighan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=')  MNIST horizons  1  2  4  8  16  32  64  128  192  256  Optimal model size vs compute  Figure 4: Optimal model size vs compute for all our environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Note that the individual points,  which correspond to the sizes of models that we trained, are themselves obtained from a power law  best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hence the fact that the lines pass through the points exactly is automatic and does not indicate  goodness of fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The Procgen difficulty mode does not obviously have any particular effect on the scaling  exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We hypothesize that humans tend to judge a task as easier when a near-perfect  score can be achieved with less compute, even if it takes a lot of additional compute to eke  out the final few points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Conversely, it does not seem to matter to the RL algorithm exactly  how the score maps on to intrinsic performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', the compute required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  Optimal model size vs compute  As explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3, our power law for intrinsic performance implies that, for a given compute  budget, the optimal model size scales as a power law with exponent  1  1+αN/αE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Figure 4 shows these inferred relationships for our different environments, along with some generative  modeling relationships taken from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The full equations for these relationships are  provided in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The exponent  1  1+αN/αE varied between around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='40 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='65 for Procgen and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='66 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='80 for  MNIST, and was around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='76 for Dota 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' By comparison, the corresponding exponent for language  modeling, which was carefully measured by Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' [2022], is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Previous work  by Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' [2020] and Henighan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' [2020] measured this exponent less carefully but using a  methodology that more closely matches our own, and found an exponent of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='73 for language  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='65 for 32x32 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  An intriguing conjecture, which is also suggested by theoretical considerations [Bahri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2021],  is that the exponent of this relationship would be around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5 in every domain if it were measured  carefully enough (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', with optimal hyperparameters and enough random seeds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Given the limitations  of our experiments, we consider our results to be inconclusive on this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Nevertheless, it is clear that the scaling coefficient of this relationship varies significantly between  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' With the exception of our toy MNIST environment, the optimal model size for RL for  8  a given compute budget is consistently smaller than for generative modeling, in some cases by  multiple orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We believe that this is because RL tasks have a longer horizon length  than generative modeling in some sense, and explore this hypothesis with our MNIST environment  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Another possibility is that the arithmetic intensity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', the number of FLOPs per  parameter in a forward pass) of the architecture is a confounder, which we discuss in more depth in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  Effect of task horizon length  As explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3, for our MNIST experiments, we artificially altered the “horizon length”  of the task by setting the GAE credit assignment parameter λ to 1 and varying the GAE discount rate  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The expected effect of varying γ in this context is given by the following theoretical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Consider an MDP with independent timesteps (by which we mean that each st is  identically distributed and independent of st−1 and at−1, and episodes never terminate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Suppose we  train a model with parameters θ on this MDP using Vanilla Policy Gradient,1 estimating advantages  using GAE with γ = 1 −  2  h+1 and λ = 1, and working with separate policy and value function  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Then the covariance matrix of the policy gradient is approximately  Σθ + Πθ  �  h + 1  h − 2  �  for some symmetric positive semi-definite matrices Σθ and Πθ that do not depend on h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For a proof sketch, see Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Intuitively, this result says that gradient variance may be decomposed into two pieces: one piece that  is inherent to the task (the Σθ term), and one piece that comes from imperfect credit assignment (the  Πθ term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For example, when h = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', γ = 0), credit is correctly assigned to the previous action  only, and hence the second term vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Ignoring the 1  h term (since h ≥ 1), we may stylize this  result as: gradient variance is an affine function of h (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', a linear function with an intercept).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  This can be directly translated into a statement about sample efficiency, since multiplying the gradient  variance by some factor c can be exactly compensated for by multiplying the batch size by c, which  multiplies the number of samples used by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hence in order to reach a given performance level,  the number of environment interactions required should be an affine function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This affine  function will come from integrating certain functionals of Σθ and Πθ over the course of training,  and will therefore depend both on the model architecture and on the choice of performance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  To test this prediction, we looked at the number of environment interactions required to reach a  1% failure rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 99% training accuracy) on MNIST as a function of the horizon length h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Our  results are shown in Figure 5, along with affine fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' As expected, the number of interactions closely  follows an affine function of the horizon length, although the fit is less good for shorter horizons and  larger models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' At very short horizons, the number of interactions even decreases with the horizon  length, suggesting a hyperparameter issue (perhaps a suboptimal learning rate schedule, or reward  normalization implicitly decreasing the KL penalty and entropy bonus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The implication of this for our optimal model size vs compute scaling law is that once h becomes large  enough, further increasing h should lead to a proportional increase the compute budget corresponding  to each given optimal model size, without changing the scaling exponent of this relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This  is because the intercept term of the affine function will eventually become dominated by the term  involving h, and so the number of environment interactions required to reach a given performance  level will eventually scale approximately proportionally to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' (For small values of h, however, the  relationship between the two components of the covariance matrix of the policy gradient may have a  more complex dependence on model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=')  This effect is visible in Figure 4, where the main impact of increasing the horizon length is to shift  the optimal model size vs compute curve to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The curve also gets shallower as the horizon  1Vanilla Policy Gradient is a primitive version of PPO, explained here: https://spinningup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  com/en/latest/algorithms/vpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='html  9  0  50  100  150  200  250  Horizon length h  1  2  3  4  5  6  Interactions  ×106  Parameters  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='8  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='9  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  Value  Affine fit  Interactions required to reach a 1% failure rate, MNIST  Figure 5: Sample efficiency for MNIST as a function of the horizon length h, for all our model sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  length is increased, but this effect is confounded by a measurement problem caused by under-training,  which we explain in more detail in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Our MNIST environment is useful because our it allows us to vary the task horizon length in a fine-  grained, quantifiable way by varying γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' But our analysis of this environment relies on the assumption  of independent timesteps, which does not hold in most environments (and in particular removes the  need for exploration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Nevertheless, our results are suggestive of a more general explanation for the  large differences in optimal model size for a given compute budget between different environments:  that different environments have different task horizon lengths in a more general sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We speculate  that, in this more general sense, task horizon length is influenced by how long rewards are delayed  for relative to the actions the agent is currently learning (which may increase throughout training as  the agent learns skills with feedback loops that are less and less tight), and that γ determines only an  upper bound on the task horizon length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  Variability of exponents over training  Although our power law for intrinsic performance holds across environments and model sizes, we  only obtain a good fit by excluding an initial transient period of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Put another way, the scaling  constants vary over the course of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  This phenomenon is clearest with with our MNIST environment, since we were able to use many  random seeds to reduce variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Recall that in this environment, the agent observes a randomly  sampled MNIST training set digit each timestep, and the horizon length of the task is artificially  controlled using the GAE discount rate γ, as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We fitted our power law to  three different periods of training for this environment: an early period (216–219 interactions), a  middle period (219–222 interactions), and a late period (222–225 interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Figure 6 shows the fitted values of αN and αE for these different periods of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We found αE  to be significantly lower during the early and middle periods of training, especially for the shorter  horizon lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  In order to accurately measure the scaling constants for optimal model size vs compute, it is best to  use a period of training during which the learning curves reach the compute-efficient frontier, since  otherwise the measurement is an extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' As shown in Figure 7, this is always in the late period  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6  αN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0  αE  1  1 + αN/αE  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='9  1  1 + αN/αE  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='8  1  1 + αN/αE  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7  MNIST periods  Early  Middle  Late  MNIST horizons  1  2  4  8  16  32  64  128  192  256  αN vs αE, MNIST  Figure 6: Fitted values of αN and αE for MNIST  with different horizons, using different periods of  training to fit the power laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The horizon h is  defined by γ = 1 −  2  h+1, where γ is the discount  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  1013  1014  1015  1016  1014  1016  MNIST, horizon 1, late period  Parameters  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='8  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='9  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  1013  1014  1015  1016  1013  1014  1015  Intrinsic performance (FLOPs)  MNIST, horizon 256, late period  1012  1013  1014  1015  Compute (FLOPs)  1012  1013  MNIST, horizon 1, middle period  Learning  curve  Power law  fit  Efficient  frontier  Efficient  points  Figure 7: Learning curves as a function of total  training compute for MNIST, using different hori-  zons and different periods of training, together  with their power law fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Mean over the middle-  performing 16 of 20 random seeds shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  of training, if at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For this reason, we use the late period of training for all of our results on MNIST  outside of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Figure 7 also shows that, for the longer horizon lengths, the learning curves of the larger models  did not reach the compute-efficient frontier even during the late period of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hence our  measurements of  1  1+αN/αE , the exponent for the scaling of optimal model size with compute, are  likely underestimates for these longer horizon lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For our other environments, we found that it was enough to exclude only the first  1  64 of training  in order for our power law for intrinsic performance to be a good fit around the compute-efficient  frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This is similar to what is needed for the corresponding law for language [Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2020,  Figure 4, right].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Nevertheless, it is possible that the measurement problem identified in this section  affects some of our other results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  Scaling depth  Most of our experiments involved scaling the width of our networks, but for Procgen, we also tried  scaling the depth, as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We found that our power law for intrinsic performance  still held, but with more noise than the width-scaling experiments, as a consequence of using fewer  model sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The fitted values of αN and αE for the depth-scaling experiments lay in a similar region  to the width-scaling experiments, but there were no clear relationships between the depth-scaling  exponents for the different environments, nor between the width-scaling and depth-scaling exponents  11  10−5  10−4  10−3  10−2  Compute (PF-days)  104  105  106  Parameters  Optimal model size vs compute, Procgen  (a) Using parameters as the measure of model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  10−5  10−4  10−3  10−2  Compute (PF-days)  106  107  108  FLOPs per forward pass  Generative modeling  Language  (Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=')  Language  (Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=')  Image 32x32  (Henighan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=')  Procgen  CoinRun  StarPilot  FruitBot  Easy  Hard  Width  Depth (*)  Optimal model size vs compute, Procgen,  arithmetic intensity-adjusted  (b) Using FLOPs per forward pass instead of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Figure 8: Comparison of optimal model size vs compute for our Procgen width- and depth-scaling  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' (*) It is important to understand how parameters and FLOPs were counted to interpret  the depth-scaling results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This is explained in detail in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  for a given environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Plots of our results may be found in Appendix C, and the numerical values  of the fitted constants may be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The main difference between our width-scaling and depth-scaling results is that the optimal model  size for a given compute budget was significantly smaller for our depth-scaling experiments, but  this was an artifact of how we counted parameters and FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' As explained in Appendix D, we  only included the part of the network being scaled in our parameter and FLOP calculations, which  meant excluding the final dense layer of the network for our depth-scaling experiments, but not our  width-scaling experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' If this layer had been included in our depth-scaling calculations, it would  have accounted for between 16% and 90% of the parameters but only 2% or fewer of the FLOPs,  depending on the depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Interestingly, as shown in Figure 8, the optimal model size vs compute scaling laws for our width-  and depth-scaling experiments become much more similar if we measure model size using FLOPs  per forward pass rather than parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This is because excluding the final dense layer from the  parameter and FLOP calculations significantly increases the arithmetic intensity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', FLOPs per  parameter in a forward pass) as calculated for the depth-scaling experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This suggests that,  when comparing models with very different arithmetic intensities, FLOPs per forward pass may  be a better measure of model size than parameters (or perhaps arithmetic intensity should even be  considered as an additional independent variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  Natural performance metrics  Although in general there may be no obvious performance metric that scales smoothly with model  parameters and environment interactions, motivating our use of intrinsic performance, there may still  be such a metric in some environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We call such metrics natural performance metrics, and we  were able to find them in a couple of our environments:  CoinRun: In the CoinRun environment from Procgen Benchmark, the episode return is  always either 10 or 0, corresponding to whether or the agent successfully collects the coin at  the end of the level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We found the fail-to-success ratio F := 10−R  R  , where R is the mean  episode return, to be a natural performance metric for CoinRun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This is similar to the failure  rate 1 − R  10, since R is close to 10 for most of training, but provides a slightly better fit  early in training, since it does not have an upper bound of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Note that the logarithm of the  12  1014  1015  1016  1017  1018  Compute (FLOPs)  10−2  10−1  Fail-to-success ratio  Easy  Hard  Learning  curves  Power law  fitted to  I−β  (arbitrary  function  of ratio)  Fail-to-  success  ratio  CoinRun, efficient frontier fits  Figure 9: Comparison of the efficient frontier fits  for CoinRun, using intrinsic performance and the  fail-to-success ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  1014  1016  1018  1020  Compute (FLOPs)  −5  0  5  10  15  20  25  TrueSkill  Learning  curves  Power law  fitted to  I−β  (arbitrary  function  of T)  e−αT T  Dota 2, efficient frontier fits  Figure 10: Comparison of the efficient frontier  fits for Dota 2, using intrinsic performance and  exponentiated scaled TrueSkill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  fail-to-success ratio can also be thought of as the logit function (inverse sigmoid) of the  failure rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Dota 2: Dota 2 is a two-player game, and so the performance of a policy must be measured  by comparing it to other policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The standard method for this is the TrueSkill rating  system,2 in which differences in rating between policies correspond to win probabilities  when the policies are played against one another, similarly to the Elo rating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We  found TrueSkill to be a natural performance metric for Dota 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Specifically, we found that our power law for intrinsic performance, equation (1), still roughly held  with the left-hand side replaced by a suitable function of the natural performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For CoinRun,  we used the fail-to-success ratio directly, but discarded data from early in training where this ratio  was above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For Dota 2, we used e−αT T , where T is TrueSkill and αT is a fitted constant, which  was needed because the scale of T is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Figures 9 and 10 compare the efficient frontier fits for intrinsic performance and for the natural  performance metric, for CoinRun and Dota 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The fits match closely, except for Dota 2 at  higher levels of TrueSkill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We conjecture that Dota 2 has an analog of an irreducible loss [Henighan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2020], representing the maximum attainable TrueSkill for the family of models we trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We explored introducing an additional fitted constant T ∗ for this maximum attainable TrueSkill, and  using either of the functional forms e−αT T − e−αT T ∗ and (T ∗ − T)αT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' However, it was unclear  to us which of these forms made the most theoretical sense, and we were unsure whether we could  justify the extra degree of freedom given the lack of data at higher levels of TrueSkill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The fitted constants for all of these alternative power laws for both CoinRun and Dota 2 are given in  Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Interestingly, for CoinRun, the values of the scaling exponent for the fail-to-success  ratio F in terms of intrinsic performance I, corresponding to the slopes of the lines in Figure 9, are  similar between the two difficulty modes: F ∝ I−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='40 in easy mode and F ∝ I−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='48 in hard mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  5  Discussion  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  Extrapolating sample efficiency  We may use our power law for intrinsic performance, equation (1), to extrapolate sample efficiency  to unseen model sizes N and environment interactions E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For example, in Figure 11, we show the  2https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='org/wiki/TrueSkill  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0  Interactions  ×108  5  10  15  20  25  30  Mean episode return  Parameters  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='9  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='8  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0  Learning  curve  Power law fit  Power law  N → ∞  limit  Sample efficiency, StarPilot, hard  Figure 11: Learning curves for StarPilot (hard  mode, scaling width), together with their power  law fits, and the N → ∞ limit of the power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  10−7  10−6  10−5  10−4  10−3  10−2  Compute (PF-days)  103  104  105  106  107  Parameters  Procgen (width)  CoinRun  StarPilot  FruitBot  Easy  Hard  Dota 2  1v1  GM (various)  MNIST horizons  1–256  Optimal model size vs compute, Ne = 105  Figure 12: Optimal model size vs compute, taking  into account a hypothetical compute cost per en-  vironment interaction equal to that of a model of  size Ne = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' See Figure 4 for the full legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  extrapolated learning curve for StarPilot in the infinite-width limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This reaches the final performance  of our largest model in about half the number of environment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Note, however, that  without a natural performance metric, we cannot extrapolate to unseen performance levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  It is natural to ask how this extrapolated infinite-width limit compares to human sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' On  StarPilot (slowed down to 3 frames per second), a human can reach a mean episode return of around  20 after a few episodes, whereas the extrapolated infinitely-wide model takes 18 million interactions,  around 10,000 times as many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This is not really a fair comparison though, because much of the  challenge in Procgen is to learn to identify basic visual features, which humans are already able to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For Dota 2, we crudely estimate that it would take a human around 50–500 hours of gameplay to  reach the performance of the extrapolated infinitely-wide LSTM after 5 billion interactions, a factor  of 100–1,000 in sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This comparison may be fairer, because Dota 2 has a structured  observation space and is more challenging than StarPilot, although it still draws on many pre-existing  human intuitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Of course, our models were all trained from scratch, and we should expect this  factor to be smaller for models that have been pre-trained to learn useful representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  Cost-efficient reinforcement learning  In the reinforcement learning literature, sample efficiency is usually taken to be the primary metric  of algorithmic progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This can be thought of as focusing on the cost of running the environment,  but not the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' At the other extreme, we have so far focused on the computational cost of the  algorithm, but not on the cost of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' However, it is straightforward to now take both  into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' To do this, let Ne be the cost of the environment, measured in terms of the number of  parameters in a model with the same cost per interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Thus the total cost of both the algorithm  and the environment is proportional to (N + Ne) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The cost-efficient frontier is now described by the following generalization of equation (2):  �  1 + Ne  N  �  αN  �Nc  N  �αN  = αE  �Ec  E  �αE  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Substituting this into our power law given by equation (1), it follows that along the cost-efficient  frontier,  C =  �  1 + Ne  N  � �  1  1 + αN  αE  �  1 + Ne  N  �  �  1  αN +  1  αE � N  Nc  �1+ αN  αE ,  14  where C := (N + Ne) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Thus for a given budget C, the optimal model size N scales as the same  power law in C as before once N ≫ Ne, and it is only efficient to take N ≪ Ne when C is very  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This validates and makes precise the rule-of-thumb that it is usually inefficient to use a model  that is much cheaper to run than the environment, at least when training from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  To illustrate this relationship, Figure 12 shows the optimal model size vs compute relationship from  Figure 4, but incorporating a fixed hypothetical compute cost associated with each environment  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  Limitations  Our experiments have several limitations:  As explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4, we did not use separate training runs for each compute budget,  each with their own learning rate schedule, which can be necessary to accurately measure  scaling exponents [Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We tried to mitigate this by using a learning rate  schedule that worked well for a variety of compute budgets, as explained in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1,  but this may not have been enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  As explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3, the variability of exponents over training gives rise to a  measurement problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We mitigated this to some extent by excluding data from early in  training when fitting our power law, but this does not fully correct for the fact that some of  our models were under-trained relative to the compute-efficient frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We did not carefully optimize the aspect ratios of our models, instead scaling width and  depth separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' More generally, suboptimal hyperparameters or other problems with our  training setups could have lead to errors in our measurements of scaling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Learning curves in reinforcement learning are often very high-variance, adding significant  noise to power law fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We mitigated this to some extent by choosing environments with  relatively low-variance learning curves and using multiple random seeds, but a lot of variance  still remained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  As a result of these limitations, we do not think conclusions that depend on the precise fitted values of  our scaling constants can be drawn with confidence, although we consider our mitigations sufficient  for more qualitative conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We are excited for future work to fix these limitations, explore  new domains, and more carefully disentangle the effects of the choice of algorithm, architecture and  hyperparameters as well as properties of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  Forecasting compute requirements  The scaling of optimal model size with compute is a key input into the biological anchors framework  for forecasting transformative artificial intelligence [Cotra, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In this framework, the human brain  is used as a biological anchor for estimating the number of parameters in a transformative model, and  optimal model size vs compute scaling laws are used to forecast the total compute required to train  such a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In this section we summarize the main implications of our work for this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Scaling exponents for reinforcement learning lie in a similar range to generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The  exponent for the scaling of optimal model size with compute,  1  1+αN/αE , varied between around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='8 for our environments, a range that encompasses previous measurements of this exponent for  generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' However, as discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3, we do not think our measurements of this  exponent should be taken literally, due to the limitations of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The results of Hoffmann  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' [2022] and Bahri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' [2021] suggest the possibility that this exponent would be around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5 in  every domain if it were measured carefully enough, and we consider our results to be inconclusive on  this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Scaling coefficients for reinforcement learning vary by multiple orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The  coefficient for the scaling of optimal model size with compute, Nc  �  1 + αN  αE  �  1  αN , varied substantially,  enough that we do not think this variation is attributable only to the limitations of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For example, the scaling exponents for MNIST (with a horizon length of 1) and Dota 2 are very  similar, but a model of the same size needs to be trained for around 2,000 times longer on Dota 2  than on MNIST to be compute-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' By comparison, Henighan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' [2020] found generative  15  modeling to require around 20 times as much training on 32x32 images than on language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Moreover,  our analysis of the effect of the task horizon length gives a plausible mechanism for this variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Arithmetic intensity may confound scaling coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4, the coefficient  for the scaling of optimal model size with compute can be affected by the arithmetic intensity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=',  the number of FLOPs per parameter in a forward pass) of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This alone does not explain the  large variation in this coefficient between MNIST and Dota 2, for example, but it may explain some  of the other variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We hypothesize that, when comparing models with very different arithmetic  intensities, due to parameter sharing or methods such as mixture of experts, it may be better to  measure model size in FLOPs per forward pass rather than in parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Sample efficiency is an affine function of the task horizon length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We study the effect of the  task horizon length using a toy MNIST-based environment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Both theoretically (see  Proposition 1) and empirically, the number of samples required to reach a given level of performance  grows with the horizon length as an affine function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', a linear function with an intercept) that  depends on both the model size and the target performance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' However, our analysis makes a  simplifying assumption of independent timesteps, which does not hold in most environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In  particular, we do not analyze the need for curricula and/or exploration to solve tasks for which it is  challenging to obtain useful feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Instead, we simply assume that the algorithm pays attention to  rewards over a longer time horizon, making credit assignment harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  This result validates and refines the analysis of Cotra [2020], who defined the “effective horizon  length” as a quantity that scales linearly with training data requirements, incorporating not only the  horizon length as we define it, but also reward sparsity, noise and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Our result specifically isolates  the explicit horizon length, showing that training data requirements are a sum of two components,  at least in our toy setting: one corresponding to a version of the task in which the horizon ends  immediately, and another that is proportional to the horizon length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This implies that, for a given fixed  task, continuing to increase the horizon length will eventually lead to a proportional increase in the  compute budget corresponding to a given optimal model size, without changing the exponent of this  scaling law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' However, this will only happen once the first component has become negligible, and it is  unclear whether there are realistic tasks of different horizon lengths for which this first component is  negligible in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We are excited for future work to study other aspects of the “effective horizon length”, such as  reward sparsity and noise, as well as studying the explicit horizon length in environments that are less  artificial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' It is not entirely clear how to quantify these properties in general, and they could potentially  affect scaling exponents as well as scaling coefficients, if for example they change over the course of  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Measuring scaling exponents precisely is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The biological anchors framework uses  the scaling of optimal model size with compute to perform a substantial extrapolation, making it  particularly sensitive to the exponent of this relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This makes it challenging to measure this  exponent with sufficient precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In addition to the challenges raised by Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' [2022]  involving learning rate schedules, we hope that others will benefit from learning about the other  challenges we faced, which are summarized in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  6  Conclusion  We have shown how to extend scaling laws to single-agent reinforcement learning using the notion of  intrinsic performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Across a range of environments, intrinsic performance scales as a power law  in model size and environment interactions, and hence the optimal model size scales as a power law in  the training compute budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We have studied how this relationship is affected by various properties  of the training setup, including the horizon length of the task, and have discussed the implications of  this for the biological anchors framework for forecasting transformative artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  7  Acknowledgments  Thanks to Mira Murati, Karl Cobbe, Chris Hesse, David Farhi, Paul Christiano, Jared Kaplan, Long  Ouyang and Ajeya Cotra for discussions, ideas, help, advice, support and inspiration that have greatly  benefited this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  16  References  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Bahri, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Dyer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Kaplan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Lee, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Sharma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Explaining neural scaling laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' arXiv preprint  arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='06701, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Cobbe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hesse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hilton, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Schulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Leveraging procedural generation to benchmark  reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In International conference on machine learning, pages 2048–2056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Cobbe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hilton, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Klimov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Schulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Phasic policy gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In International  Conference on Machine Learning, pages 2020–2027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Cotra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Forecasting transformative AI with biological anchors, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Droppo and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Elibol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Scaling laws for acoustic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='09488, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Espeholt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Soyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Munos, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Simonyan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Mnih, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Ward, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Doron, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Firoiu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Harley,  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Dunning, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Impala: Scalable distributed deep-RL with importance weighted actor-learner  architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In International conference on machine learning, pages 1407–1416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Ghorbani, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Firat, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Freitag, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Bapna, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Krikun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Garcia, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Chelba, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Cherry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Scaling  laws for neural machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='07740, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Ren, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Delving deep into rectifiers: Surpassing human-level per-  formance on imagenet classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In Proceedings of the IEEE international conference on  computer vision, pages 1026–1034, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Henighan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Kaplan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Katz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hesse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Jackson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Jun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Brown, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Dhari-  wal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Gray, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Scaling laws for autoregressive generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  arXiv preprint  arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='14701, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hilton, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Cobbe, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Schulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Batch size-invariance for policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' arXiv preprint  arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='00641, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hoffmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Borgeaud, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Mensch, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Buchatskaya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Cai, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Rutherford, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Casas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Hendricks, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Welbl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Clark, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Training compute-optimal large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' arXiv  preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='15556, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Jones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Scaling scaling laws with board games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='03113, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Kaplan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' McCandlish, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Henighan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Brown, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Chess, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Child, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Gray, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Radford, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Wu,  and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Amodei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Scaling laws for neural language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' arXiv preprint arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='08361, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Kingma and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Adam: A method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6980,  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' LeCun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The MNIST database of handwritten digits, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Neumann and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Gros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Scaling laws for a multi-agent reinforcement learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' arXiv  preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='00849, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  OpenAI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Berner, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Brockman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Chan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Cheung, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' D˛ebiak, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Dennison, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Farhi, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Fis-  cher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hashme, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hesse, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Józefowicz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Gray, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Olsson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Pachocki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Petrov, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  de Oliveira Pinto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Raiman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Salimans, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Schlatter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Schneider, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Sidor, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Sutskever, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Tang,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Wolski, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Dota 2 with large scale deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' arXiv preprint  arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='06680, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Schulman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Moritz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Levine, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Jordan, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Abbeel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' High-dimensional continuous control  using generalized advantage estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' arXiv preprint arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='02438, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Schulman, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Wolski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Dhariwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Radford, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Klimov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Proximal policy optimization  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' arXiv preprint arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='06347, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Silver, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Hubert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Schrittwieser, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Antonoglou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Lai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Guez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Lanctot, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Sifre, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Ku-  maran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Graepel, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' A general reinforcement learning algorithm that masters chess, shogi, and  Go through self-play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Science, 362(6419):1140–1144, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  17  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Dauphin, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Fixup initialization: Residual learning without normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  arXiv preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='09321, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  18  A  Curve-fitting methodology  In this section we discuss our methodology for computing intrinsic performance and fitting the power  law constants, which require some care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Code for our full procedure, along with its application to  our experiments, may be found in this Colab notebook: https://colab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='com/  drive/1PzwZyXsi9jRdVCj1GJrS8JdOPBQ7LHZV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Recall that the intrinsic performance of a policy is the minimum compute required to train a model of  any size in the same family to reach the same return (averaged over random seeds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The naive way to  compute this would be to train models of many different sizes, and to take the best-performing model  size for each possible compute budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' However, it may not be feasible to train models of enough  different sizes to get a reasonable level of granularity, while using enough different random seeds  sufficiently to reduce the high variance of learning curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  To cope with this, we compute intrinsic performance and fit the power law constants together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This  allows us to make use of all the data from each learning curve, instead of just a single point from  each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We do this by jointly fitting the power law constants and a monotonic function f to  f (R)−β =  �Nc  N  �αN  +  �Ec  E  �αE  ,  where R is the mean episode return (or another performance metric such as TrueSkill), N is the  number of model parameters, and E is the number of environment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' By also requiring the  relationships between the constants from Lemma 1 to hold, this provides us both with the power law  constants, and with the desired function f satisfying f (R) = I, where I is intrinsic performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We perform this fit by using a black-box optimization algorithm such as CMA-ES to fit αN, αE  and Nc, which determine β and Ec, with monotonic regression3 in the inner loop to fit f, using the  squared error of the regression as the black-box loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We actually fit log (f) rather than f in  order to obtain a good fit to I on a logarithmic scale, and we weight the data in proportion to 1  E so  that each interval is given equal weight on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In our Colab notebook, this routine is  performed by the function fit_coeffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  This procedure seems to work well off-the-shelf, typically converging to a unique local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  However:  When there is a lack of data or the data is very noisy, the local minimum may not be a global  minimum, and the procedure can diverge to a degenerate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  It is necessary to first smooth learning curves so that they are mostly monotonic, to prevent  the monotonic regression from overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In our Colab notebook, we use the function  smooth, which uses standard errors to automatically choose smoothing parameters (although  note that we used slightly different smoothing parameters for MNIST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3, it is important to exclude data from early in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Our full procedure is therefore as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Smooth learning curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Plot the smoothed curves on a logarithmic scale to check the  monotonicity and fit, and adjust the smoothing parameters if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Exclude data from early in training, balancing the need for data against how much the early  data skews the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Typically at least the first  1  64 of training should be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Fit the power law constants and f using the black-box optimization with monotonic regres-  sion routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Plot the fit to check the routine did not diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' If it did, re-run routine, or constrain the  constants and re-run, or include more data in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' If none of these fixes the divergence,  then it may be necessary to collect more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Check the fit is not overly skewed by data from early in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' If it is, exclude more data  in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  3https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='org/wiki/Isotonic_regression  19  This procedure led us to exclude the first 3 million environment interactions for Procgen, the first 2  billion environment interactions for Dota 2, and the first 216, 219 or 222 environment interactions for  MNIST depending on the period of training being considered, as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  Fitting to natural performance metrics  As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5, as well as fitting our power law with I−β on the left-hand side, as in  equation (1), we also fit it using various other expressions, such as e−αT T , where T is TrueSkill and  αT is a fitted constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' When doing this, we adopt the convention that the constraints on β and Ec  from Lemma 1 should continue to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This necessitates introducing an additional multiplier, and  instead fitting  Tce−αT T =  �Nc  N  �αN  +  �Ec  E  �αE  for example, where Tc is a fitted constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Doing this allows us to continue interpret the left-hand  side of this equation as I−β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  To fit equations of this form, we continue use the same black-box optimization method, and simply  replace the monotonic regression by another method of fitting log (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' For example, we may fit  f (T)−β = Tce−αT T  by using linear regression to fit log (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' (Recall that β is already determined by αN and αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=')  The function from our Colab notebook, fit_coeffs, provides options for fitting various functional  forms for f, although it can sometimes be slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' (This is because it sometimes uses black-box  optimization again in the inner loop for ease of implementation, even though this could be collapsed  into the outer loop if speed were important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=')  20  B  Hyperparameters  Our default hyperparameters for Procgen, Dota 2 and MNIST are given in Tables 1, 2 and 3  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We modified these defaults in two ways:  We adjusted the Adam step size as the model was scaled, as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For Procgen and MNIST, we incorporated a batch ramp and learning rate schedule, as  explained in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Table 1: Default PPG-EWMA hyperparameters for Procgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Hyperparameter  Value  PPO  Parallel environments  1024  Timesteps per rollout (T)  256  Minibatches per epoch  8  Adam step size (α)  5 × 10−4  Value function coefficient  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  Entropy coefficient  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='01  PPO clipping parameter (ϵ)  Not used  PPO KL penalty coefficient (β)  1  GAE discount rate (γ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='999  GAE bootstrapping parameter (λ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='95  Reward normalization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Yes  Advantage normalization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Yes  Total environment interactions  200 million  PPG  Policy iterations per phase (Nπ)  32  Policy phase policy epochs (Eπ)  1  Policy phase value function epochs (EV )  1  Auxiliary phase epochs (Eaux)  6  Auxiliary phase minibatches per epoch  16Nπ  Auxiliary phase cloning coefficient (βclone)  1  PPG-EWMA  Proximal policy EWMA decay rate (βprox)  8  9  Batch ramp  Initial batch size multiplier  1  32  Table 2: PPO hyperparameters for Dota 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Hyperparameter  Value  Parallel environments  6144  Timesteps per rollout (T)  512  Minibatches per epoch  32  Epochs (E)  1  Adam step size (α)  10−4 to 10−3  PPO clipping parameter (ϵ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  PPO KL penalty coefficient (β)  Not used  GAE bootstrapping parameter (λ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='95  Total environment interactions  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6–82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6 billion  21  Table 3: Default PPO-EWMA hyperparameters for MNIST in terms the horizon length h, which  varied from 1 to 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Hyperparameter  Value  PPO  Parallel environments  16  Timesteps per rollout (T)  512  Minibatches per epoch  8  Epochs (E)  1  Adam step size (α)  1 × 10−3  Value function coefficient  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  Entropy coefficient  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='01  PPO clipping parameter (ϵ)  Not used  PPO KL penalty coefficient (β)  1  GAE discount rate (γ)  1 −  2  h+1  GAE bootstrapping parameter (λ)  1  Reward normalization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Yes  Advantage normalization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Yes  Total environment interactions  225  PPO-EWMA  Proximal policy EWMA decay rate (βprox)  8  9  Batch ramp  Initial batch size multiplier  √  h  64  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  Batch ramp and learning rate schedule  As explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4, it was important to use a well-tuned learning rate schedule, and to use  a schedule that works well for a variety of compute budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' It was also important to use a batch  ramp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', to start with a small batch size and increase it over the course of training, because the  critical batch size is smaller at the start of training, and we needed training to still be sample-efficient  for small compute budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Without a batch ramp, we would have needed to adjust our power law,  equation (1), in much the same way as the corresponding law for language [Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2020,  equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6)], which uses Smin (S), the minimum number of optimization steps as estimated using  a power law fit to the gradient noise scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Note, however, that increasing the batch size has a very similar effect to lowering the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  To simplify matters, we used PPO-EWMA and PPG-EWMA, which are batch size-invariant [Hilton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2021], allowing us to have almost the same effect as increasing the batch size by instead  lowering the learning rate and increasing the center of mass of the proximal policy EWMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We then  considered only the batch size schedule, whether implemented explicitly or implicitly via these other  hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  To explore promising schedules, we implemented a greedy adaptive batch size algorithm, which tries  doubling the batch size and switches if that performs better, or else backtracks and stays with the  current batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We experimented with this on StarPilot’s easy difficulty setting, using model sizes  spanning a factor of around 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We found our algorithm to fairly consistently choose a schedule  that can be well-approximated by the power law  B = max  �  Bmin, E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='84  80  �  ,  where B is the batch size in interactions, E is the total number of interactions so far, and Bmin = 256  was our initial batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Having fit this power law schedule on one Procgen environment, we tested it on several different  Procgen environments, and found it to consistently outperform our usual fixed batch size both at the  start and end of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' (Curiously, our schedule sometimes underperformed the fixed batch size in  the middle of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We believe this may be explained by the smaller initial batch size causing the  entropy to fall too quickly at the start of training, highlighting a pitfall of the greedy approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=') In  particular, we were able to use the same schedule on both the easy and hard difficulty settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Our  22  usual fixed batch size, on the other hand, was larger for the hard setting, corresponding to the fact  that it was tuned to longer training runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The same schedule also worked well on our MNIST environment at every horizon length, although it  was necessary to tune Bmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Using too small a value for Bmin seemed to result in an instability which  could not always be recovered from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We found the optimal Bmin to vary based on the horizon length  h, and we took Bmin = 16  √  h (though taking Bmin to have the form A0 + A1h would probably have  made more theoretical sense in hindsight, given the results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' If trying our schedule  on other environments, we suggest tuning Bmin to ensure stability at the start of training, but it is  probably less important to tune the power law constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We used this batch size schedule for both our Procgen and MNIST experiments (although it would  probably have been better to fully re-fit the schedule for MNIST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We implemented this using a batch  size multiplier, explicitly reducing the batch size when the multiplier was less than 1, and changing  the learning rate and center of mass of the proximal policy EWMA instead when the multiplier was  greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' With Procgen, for which we used PPG-EWMA, we also changed the number of policy  iterations per phase, Nπ, in proportion to the batch size, since we thought the number of optimization  steps per phase should remain constant, and we rounded the batch size multiplier to the nearest power  of two, with minimum and maximum multipliers of  1  32 and 4 (corresponding to batch sizes of 1024  and 131072 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For Dota 2, we did not use a batch size schedule, since those experiments were carried out before we  investigated batch size schedules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  23  C  Results in full  All the data from our experiments may be accessed using this Colab notebook: https://colab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='com/drive/1PzwZyXsi9jRdVCj1GJrS8JdOPBQ7LHZV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  This also includes  code for analyzing this data, including model size and compute calculations, intrinsic performance  and power law fitting, and generating all the plots in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Figures 13, 14, 15 and 16 show learning curves as a function of total training compute, together with  their power law fits, for all of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' On the left of each figure we show mean episode  return (or failure rate for CoinRun and MNIST, or TrueSkill for Dota 2), with error bars showing  mean ±1 sample standard deviation over the random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' On the right of each figure, we show  intrinsic performance, with error bars hidden for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  1015  1017  Compute (FLOPs)  10−2  10−1  Failure rate  CoinRun, easy  1015  1017  Compute (FLOPs)  10−1  Failure rate  CoinRun, hard  1015  1017  Compute (FLOPs)  1014  1015  1016  1017  Intrinsic performance (FLOPs)  CoinRun, easy  1015  1017  Compute (FLOPs)  1014  1015  1016  1017  1018  Intrinsic performance (FLOPs)  CoinRun, hard  Parameters  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='9  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='8  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0  1015  1017  Compute (FLOPs)  10  20  30  40  50  60  Mean episode return  StarPilot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' easy  1015  1017  Compute (FLOPs)  5  10  15  20  25  30  Mean episode return  StarPilot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' hard  1015  1017  Compute (FLOPs)  1014  1015  1016  1017  1018  Intrinsic performance (FLOPs)  StarPilot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' easy  1015  1017  Compute (FLOPs)  1014  1015  1016  1017  1018  Intrinsic performance (FLOPs)  StarPilot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' hard  Learning  curve  Power law  fit  Power law  asymptote  Efficient  frontier  Efficient  points  1015  1017  Compute (FLOPs)  5  10  15  20  25  30  Mean episode return  FruitBot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' easy  1015  1017  Compute (FLOPs)  0  5  10  15  20  25  Mean episode return  FruitBot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' hard  1015  1017  Compute (FLOPs)  1014  1015  1016  1017  Intrinsic performance (FLOPs)  FruitBot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' easy  1015  1017  Compute (FLOPs)  1014  1015  1016  1017  1018  Intrinsic performance (FLOPs)  FruitBot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' hard  Procgen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' width  Figure 13: Learning curves as a function of total training compute for our Procgen width-scaling  experiments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' together with their power law fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Left half: mean episode return or failure rate, mean  ±1 sample standard deviation over three seeds shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Right half: intrinsic performance, mean only  shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='1  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='9  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content=' easy  1015  1016  1017  1018  Compute (FLOPs)  0  5  10  15  20  25  Mean episode return  FruitBot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' hard  1015  1017  Compute (FLOPs)  1014  1015  1016  1017  Intrinsic performance (FLOPs)  FruitBot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' easy  1015  1016  1017  1018  Compute (FLOPs)  1015  1016  1017  1018  Intrinsic performance (FLOPs)  FruitBot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' hard  Procgen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' depth  Figure 14: Learning curves as a function of total training compute for our Procgen depth-scaling  experiments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' together with their power law fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Left half: mean episode return or failure rate, mean  ±1 sample standard deviation over three seeds shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='1  Learning  curve  Power law  fit  Power law  asymptote  Efficient  frontier  Efficient  points  Dota 2  Figure 15: Learning curves as a function of total training compute for Dota 2, together with their  power law fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='2  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='1016  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='Failure rate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='Horizon 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='Failure rate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='Horizon 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1013  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='1016  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='1016  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='1013  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='1016  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='Horizon 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='Learning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='curve  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='fit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='Failure rate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='Horizon 16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
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+page_content='Intrinsic performance (FLOPs)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='Horizon 256  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='MNIST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' late period  Figure 16: Learning curves as a function of total training compute for MNIST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' together with their  power law fits,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' for the late period of training (222–225 environment interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Left half: failure  rate, mean ±1 sample standard deviation over the middle-performing 16 of 20 random seeds shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Right: intrinsic performance, mean only shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  26  D  Parameter and FLOP calculations  In counting parameters and FLOPs, we apply the following principles:  We only include the part of the network that is being scaled (ignoring things like embedding  parameters), since we consider that to be the bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We use round numbers (ignoring negligible contributions such as as biases and activations),  for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We include both rollout and optimization FLOPs (including any additional overhead of  PPO-EWMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We treat an add-multiply as 2 FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For example, we treat the forward pass of a dense layer as taking 2 FLOPs per batch item per  parameter, and a convolutional layer as taking 2houtwout FLOPs per batch item per parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We  treat a backward pass as taking 2× the FLOPs of a forward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For the Procgen width-scaling experiments, we ignore the first convolution, since it scales as width  (instead of as width squared), and has few parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Similarly, for the depth-scaling experiments,  we ignore the final dense layer, since we only vary the number of convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Unfortunately,  as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4, the final dense layer contains many parameters, which skews our constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  In both cases, we include both the policy and value networks, which are separate with identical  architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We use PPG-EWMA with 1 policy epoch and 6 auxiliary epochs, totaling 9 forward  and 7 backward passes per interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For the Dota experiments, we ignore the embedding layer, considering only the LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Since each  interaction was used only once, we count 2 forward passes and 1 backward pass per interaction (1  forward pass for the rollout, and 1 forward-backward pass for optimization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For the MNIST experiments, we ignore the first convolution, as for the Procgen width-scaling  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' However, we only include the policy network, since the task of the value network is  trivial (due to timesteps being independent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We use PPO-EWMA with 1 epoch, totaling 3 forward  passes and 1 backward pass per interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The numerical results of these calculations are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Procgen, scaling width: for the width multiplier w = 2−3, 2−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='52−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' , 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5, we count  1242112w2 parameters and 2652897280w2 FLOPs per interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Procgen, scaling depth: for the number of residual blocks b = 1, 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' , 64, we count  5184b + 1944 parameters and 61046784b + 81395712 FLOPs per interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Dota 2: for the LSTM size s = 8, 64, 128, 256, 512, 1024, 4096, we count 8s2 parameters  and 64s2 FLOPs per interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  MNIST: for the width multiplier w = 2−3, 2−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='52−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' , 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5, we count 3948800w2  parameters and 95648000w2 FLOPs per interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Note that one of our modeling assumptions is that the number of FLOPs per interaction is proportional  to the number of parameters, but this is not true for our Procgen depth-scaling experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' In other  words, the number of FLOPs per param-interact, which is used to convert compute from units of  parameters × interactions to units of FLOPs, is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' However, this number differs by at most  40% from the mean of this number over the different depths, and so we simply used the mean when  doing this conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  27  E  Fitted constants  In this section we provide the constants αN, αE and Nc, together with the values of β and Ec derived  using Lemma 1, for our fitted power laws for intrinsic performance I as given by equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We  also provide Imin and Imax, the minimum and maximum intrinsic performance obtained during the  span of interaction counts considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' our model is not able to predict mean episode return outside  this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Recall that the units of I are parameters × interactions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' the conversion to FLOPs may be  performed using the values given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We also provide the derived equations for optimal model size N vs compute C in PF-days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' By  substituting equation (2) for the compute-efficient frontier into equation (1), these are given by  N = Nc  �  1 + αN  αE  �  1  αN � C × 1015 × 24 × 3600  FLOPs per param-interact  �  1  1+ αN  αE  for  Nmin ≤ N ≤ Nmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  We take Nmin and Nmax to be the minimum and maximum model sizes we tested whose power law  fit intersects the compute-efficient frontier somewhere between Imin and Imax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  For our comparison to generative modeling, we use these equations for optimal model size N vs  compute C in PF-days:  Language [Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2022]: N = (  C  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4×10−18 )0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5  Language [Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2020]: N = (  C  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3×10−13 )0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='73  Image 32x32 [Henighan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', 2020]: N = (  C  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6×10−13 )0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='65  Further fitted constants, such as for single seeds, for different spans of interaction counts (see Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2), and fitted to natural performance metrics (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5), may be found in this Colab notebook:  https://colab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='com/drive/1PzwZyXsi9jRdVCj1GJrS8JdOPBQ7LHZV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1  Procgen, scaling width  The fitted constants for our Procgen width-scaling experiments are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Environment  αN  αE  β  Nc  Ec  Imin  Imax  CoinRun, easy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='542  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='462  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='249  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='53 × 10−2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='49 × 100  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='83 × 1010  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='55 × 1014  CoinRun, hard  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='759  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='576  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='328  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='55 × 10−1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='00 × 10−1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='07 × 1010  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='45 × 1014  StarPilot, easy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='318  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='604  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='208  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='25 × 10−4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='02 × 102  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='88 × 1010  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='95 × 1015  StarPilot, hard  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='453  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='533  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='245  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='55 × 10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='31 × 101  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='43 × 1010  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='09 × 1015  FruitBot, easy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='527  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='350  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='210  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='17 × 10−2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='46 × 10−1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='24 × 1010  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='67 × 1014  FruitBot, hard  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='478  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='346  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='201  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='14 × 10−1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='96 × 10−1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='00 × 1010  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='26 × 1014  These imply the following equations for optimal model size N vs compute C in PF-days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  CoinRun, easy: N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='615 × 106 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4600 for 19408 ≤ N ≤ 310528  CoinRun, hard: N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='881 × 106 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4315 for 43668 ≤ N ≤ 587092  StarPilot, easy: N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='383 × 107 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6549 for 19408 ≤ N ≤ 4968448  StarPilot, hard: N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='668 × 107 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5404 for 19408 ≤ N ≤ 1242112  FruitBot, easy: N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='243 × 106 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3994 for 19408 ≤ N ≤ 174672  FruitBot, hard: N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='631 × 106 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4201 for 43668 ≤ N ≤ 587092  As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5, for CoinRun, we also fit power laws using the fail-to-success ratio F,  excluding data for which F > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' As explained in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1, we replaced I−β with F  Fc , where Fc  is a fitted constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The fitted constants for these power laws are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  28  Difficulty  αN  αE  β  Nc  Ec  Imin  Imax  Easy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='899  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='475  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='00 × 10−2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='33 × 101  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='55 × 1010  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='60 × 1014  Hard  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='833  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='776  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='402  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='69 × 10−2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='80 × 100  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='14 × 1011  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='38 × 1014  Difficulty  Fc  Easy  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='88 × 104  Hard  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='52 × 104  These imply the following relationships between I and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Easy: I = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='57 × 109 × F −  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='475  Hard: I = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='15 × 1010 × F −  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='402  They also imply the following equations for optimal model size N vs compute C in PF-days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Easy: N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='216 × 107 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5285 for 19408 ≤ N ≤ 587092  Hard: N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='148 × 107 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4822 for 77632 ≤ N ≤ 1242112  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='2  Procgen, scaling depth  The fitted constants for our Procgen depth-scaling experiments are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Environment  αN  αE  β  Nc  Ec  Imin  Imax  CoinRun, easy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='351  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='469  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='201  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='64 × 10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='26 × 102  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='43 × 109  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='72 × 1013  CoinRun, hard  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='336  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='581  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='213  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='02 × 10−4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='47 × 102  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='58 × 109  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='24 × 1013  StarPilot, easy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='821  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='405  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='65 × 10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='87 × 101  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='70 × 1010  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='52 × 1013  StarPilot, hard  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='380  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='381  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='190  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='87 × 10−3  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='11 × 100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='58 × 1010  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='21 × 1013  FruitBot, easy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='539  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='564  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='276  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='92 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='77 × 101  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='58 × 109  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='76 × 1013  FruitBot, hard  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='401  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='463  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='215  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='23 × 10−3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='26 × 101  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='34 × 1010  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='64 × 1013  These imply the following equations for optimal model size N vs compute C in PF-days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Note,  however, that:  As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4, we exclude the final dense layer, which would have accounted  for between 16% and 90% of the parameters, depending on the depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' This skews the  leading constants here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  As discussed in Appendix D, we also ignored the variation in the number of FLOPs per  param-interact between models of different depths, leading to errors of up to 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  CoinRun, easy: N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='390 × 106 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5723 for 7128 ≤ N ≤ 43416  CoinRun, hard: N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='962 × 106 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6337 for 7128 ≤ N ≤ 167832  StarPilot, easy: N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='202 × 106 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5063 for 7128 ≤ N ≤ 167832  StarPilot, hard: N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='410 × 106 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5007 for 7128 ≤ N ≤ 84888  FruitBot, easy: N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='172 × 106 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5110 for 7128 ≤ N ≤ 84888  FruitBot, hard: N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='671 × 106 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5359 for 7128 ≤ N ≤ 84888  29  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3  Dota 2  As explained in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='1, we fit power laws to I−β, Tce−αT T , Tc  �  e−αT T − eαT T ∗�  and Tc (T ∗ − T)αT , where I is intrinsic performance, T is TrueSkill, and αT , Tc and T ∗ are fitted  constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The fitted constants for these different functional forms are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Fit to  αN  αE  β  Nc  Ec  Imin  Imax  I−β  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='186  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='593  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='141  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='98 × 10−8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='04 × 106  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='83 × 1011  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='79 × 1018  Tce−αT T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='180  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='486  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='131  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='53 × 10−8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='33 × 105  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='62 × 1011  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='24 × 1017  Tc(e−αT T − eαT T ∗)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='181  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='560  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='137  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='07 × 10−8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='32 × 105  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='31 × 1011  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='77 × 1018  Tc(T ∗ − T)αT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='183  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='569  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='138  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='06 × 10−8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='82 × 1005  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='71 × 1011  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='23 × 1018  Fit to  αT  Tc  T ∗  I−β Tce−αT T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0572  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='16 × 10−2 Tc(e−αT T − eαT T ∗)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0402  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='40 × 10−2  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='43  Tc(T ∗ − T)αT  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='14 × 10−7  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='01  As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5, we have less confidence in the last two functional forms, which is  reflected in the very different estimates for T ∗, which represents the maximum attainable TrueSkill  for the family of models we trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  These imply the following relationships between I and T for the last three fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Tce−αT T :  I = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='93 × 1012 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='5462T  Tc(e−αT T − eαT T ∗):  I = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='49 × 1011 ×  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0410−T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='0410−35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='43�−  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='137  Tc(T ∗ − T)αT :  I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='48 × 1048 × (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='01 − T)− 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='138  They also imply the following equations for optimal model size N vs compute C in PF-days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  I−β:  N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='703 × 107 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7617 for 512 ≤ N ≤ 2097152  Tce−αT T :  N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='607 × 107 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7302 for 512 ≤ N ≤ 524288  Tc(e−αT T − eαT T ∗):  N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='305 × 107 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7552 for 512 ≤ N ≤ 2097152  Tc(T ∗ − T)αT :  N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='385 × 107 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7567 for 512 ≤ N ≤ 2097152  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='4  MNIST  The fitted constants for our MNIST experiments are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='3, these  constants are for the late period of training (222–225 environment interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Recall also that the  horizon h is such that the interval [0, h − 1] has the same center of mass as an exponentially-weighted  moving average with decay parameter γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=', γ = 1 −  2  h+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  30  Horizon  αN  αE  β  Nc  Ec  Imin  Imax  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='263  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='210  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='79 × 10−6  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='43 × 103  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='79 × 1011  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='00 × 1015  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='265  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='979  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='208  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='32 × 10−5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='30 × 103  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='87 × 1011  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='66 × 1014  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='284  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='791  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='209  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='21 × 10−5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='50 × 103  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='94 × 1011  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='19 × 1014  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='276  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='826  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='207  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='83 × 10−5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='33 × 103  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='80 × 1011  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='24 × 1014  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='252  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='830  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='193  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='59 × 10−5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='78 × 103  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='62 × 1011  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='69 × 1014  32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='263  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='856  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='201  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='73 × 10−5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='83 × 103  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='59 × 1011  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='47 × 1014  64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='307  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='736  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='217  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='27 × 10−5  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='40 × 102  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='64 × 1011  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='16 × 1014  128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='315  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='769  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='224  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='27 × 10−5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='08 × 103  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='45 × 1011  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='64 × 1014  192  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='330  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='688  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='223  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='22 × 10−4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='86 × 102  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='33 × 1011  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='08 × 1014  256  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='358  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='681  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='235  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='11 × 10−4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='04 × 102  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='33 × 1011  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='53 × 1014  These imply the following equations for optimal model size N vs compute C in PF-days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Horizon 1:  N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='586 × 1010 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7999 for 61700 ≤ N ≤ 15795200  Horizon 2:  N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='309 × 1010 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7871 for 61700 ≤ N ≤ 15795200  Horizon 4:  N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='507 × 109 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7357 for 61700 ≤ N ≤ 3948800  Horizon 8:  N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='406 × 109 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7493 for 61700 ≤ N ≤ 7739648  Horizon 16: N = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='787 × 109 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7671 for 61700 ≤ N ≤ 7739648  Horizon 32: N = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='535 × 109 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7652 for 61700 ≤ N ≤ 7739648  Horizon 64: N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='746 × 109 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7053 for 61700 ≤ N ≤ 3948800  Horizon 128: N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='681 × 109 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='7092 for 61700 ≤ N ≤ 3948800  Horizon 192: N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='376 × 109 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6757 for 61700 ≤ N ≤ 987200  Horizon 256: N = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='876 × 108 × C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='6553 for 61700 ≤ N ≤ 987200  31  F  Proof of the lemma  Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We may write I (N, E) as a function of N and compute C := NE:  I (N, C)−β =  �Nc  N  �αN  +  �EcN  C  �αE  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  The compute-efficient frontier is defined by the value of N that maximizes I (N, C) for each C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Equivalently, since β > 0, this value of N minimizes I (N, C)−β, and so it satisfies  ∂  ∂N  �  I (N, C)−β�  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Differentiating and multiplying through by N, this equation becomes  −αN  �Nc  N  �αN  + αE  �EcN  C  �αE  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Eliminating C, this is exactly equation (2), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  By assumption, we also have I (N, E) = NE along the compute-efficient frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Substituting (2)  into I (N, E), this equation becomes  �  1 + αN  αE  � �Nc  N  �αN  = (NE)−β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  (3)  Thus both equations (2) and (3) are power law relationships between N and E that hold along the  compute-efficient frontier, so we may simply equate exponents and constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Equating exponents,  αN  αE  = αN  β − 1  and hence  1  β =  1  αN  + 1  αE  ,  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Equating constants,  �αN  αE  �  1  αE N  αN  αE  c  E−1  c  =  �  1 + αN  αE  � 1  β  N  αN  β  c  ,  and hence  1  NcEc  =  �  1 + αN  αE  �  1  αN +  1  αE � αE  αN  �  1  αE =  �  1 + αN  αE  �  1  αN �  1 + αE  αN  �  1  αE ,  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  32  G  Proof sketch of the proposition  A formal statement and proof of Proposition 1 would require a formal analysis of Vanilla Policy  Gradient, which is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Instead, we provide a proof sketch in which we  make approximations informally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Proof sketch of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The horizon length h only affects the algorithm via GAE, which in  the case λ = 1 produces the value function targets and advantage estimates  ˆVt := rt + γrt+1 + · · · + γT −trT = rt + γ ˆVt+1  and  ˆAt := ˆVt − V (st) = rt − V (st) + γ ˆVt+1,  where V is the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Since timesteps are independent, γ ˆVt+1 is independent of st and at,  and so should be thought of as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The value function will quickly learn to incorporate the mean  of this noise, and so  V (st) ≈ V 0 (st) + E  �  γ ˆVt+1  �  ,  where V 0 (st) is the “immediate reward value function” that would have been obtained had we  used the value function targets ˆV 0  t := ˆVt − E  �  γ ˆVt+1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Writing ϵ := γ ˆVt+1 − E  �  γ ˆVt+1  �  for the  zero-mean component of γ ˆVt+1, we obtain  ˆV 0  t = rt + ϵ  and  ˆAt ≈ rt − V 0 (st) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  In other words, the entire impact of varying h is that it changes the variance of the noise term ϵ added  to the value function targets and advantage estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Let us now analyze the policy gradient, which equals  ˆEt  �  ∇θρt (θ) ˆAt  �  ≈ ˆEt  �  ∇θρt (θ)  �  rt − V 0 (st) + ϵ  ��  ,  where ρt (θ) :=  πθ(at|st)  πθold(at|st).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' Since ϵ is independent of st and at and E [ϵ] = 0, the covariance matrix  of this decomposes as  Σθ + ΦθVar [ϵ] ,  where Σθ is the covariance matrix of ∇θρt (θ)  �  rt − V 0 (st)  �  , and Φθ := E  �  ∇θρt (θ) ∇T  θ ρt (θ)  �  is  the uncentered covariance matrix of ∇θρt (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Note that V 0 (st) simply estimates E [rt], which does not depend on h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' The variance of V 0 (st) does  depend on h via the addition of ϵ to the value function targets, but this additional variance is small  compared to the variance of ϵ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We may therefore treat Σθ as approximately independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  It remains to express Var [ϵ] in terms of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' We assume that T is large enough compared to h that we  may take T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' (In our experiments, we use rollouts of length 512 and h ≤ 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=') Thus  Var [ϵ] = Var  �  γ ˆVt+1  �  =  �  γ2 + γ4 + γ6 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  �  Var [rt]  =  γ2  1 − γ2 Var [rt]  = 1  4  �  h + 1  h − 2  �  Var [rt] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  Hence the covariance matrix of the policy gradient is approximately  Σθ + Πθ  �  h + 1  h − 2  �  ,  where Σθ and Πθ := 1  4Var [rt] Φθ are symmetric positive semi-definite matrices that do not depend  on h, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
+page_content='  33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FQT4oBgHgl3EQf7Ta5/content/2301.13442v1.pdf'}
diff --git a/-NFLT4oBgHgl3EQfuy_h/content/tmp_files/2301.12157v1.pdf.txt b/-NFLT4oBgHgl3EQfuy_h/content/tmp_files/2301.12157v1.pdf.txt
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+Simple Realistic Model of Spin Reorientation in 4f-3d
+Compounds
+Alexander Moskvin*, Evgenii Vasinovich, Anton Shadrin
+Ural Federal University, Ekaterinburg, Russia
+Abstract: Spin reorientation is an important phenomenon of rare-earth perovskites, orthoferrites
+and orthochromites. In this study, we consider a simple but realistic microscopic theory of the
+spontaneous spin-reorientation transitions induced by the 4f-3d interaction, more specifically, the
+interaction of the main Kramers doublet or non-Kramers quasi-doublet of the 4f ion with an effective
+magnetic field induced by the 3d sublattice. The obtained results indicate that the cause of both the
+temperature and the character of the spin-reorientation transition is a competition between the second
+and fourth order spin anisotropy of the 3d sublattice, the crystal field for 4f ions, and the 4f-3d
+interaction.
+Keywords: 4f-3d interaction; (quasi)doublets; spin reorientation
+1 Introduction
+Rare-earth orthorhombic perovskites, orthoferrites RFeO3 and orthochromites RCrO3 (where
+R is a rare-earth ion and yttrium), exhibit many important features such as weak ferro- and
+antiferromagnetism, magnetization reversal, anomalous circular magnetooptics, and the phenomenon
+of the spontaneous spin reorientation. The spin reorientation (SR) is one of their unique properties
+that have attracted a lot of attention back in the 70s of the last century [1, 2], though their exact
+microscopic origin is still a challenge to theorists and experimentalists.
+The revival of interest in the mechanism of the spontaneous spin reorientation and
+magnetic compensation in rare-earth perovskites in recent years is related with the discovery
+of the magnetoelectric and the exchange bias effect, which can have a direct application in
+magnetoelectronics. Along with the emergence of new experimental studies (see, e.g., Refs. [3, 4]),
+there also appeared theoretical works claiming to modify the mean-field theory of the spontaneous
+spin-reorientation transitions [5] or to scrutinize the microscopic mechanism responsible for
+spin reorientations and magnetization reversal [6]. In fact, these results are not directly related
+to the microscopic theory of the spontaneous spin reorientation in rare-earth orthoferrites and
+orthochromites. For instance, the authors of the most recent paper [6] did not take into account
+*alexander.moskvin@urfu.ru
+1
+arXiv:2301.12157v1  [cond-mat.str-el]  28 Jan 2023
+
+a number of interactions, such as the fourth-order anisotropy for the 3𝑑 sublattice of orthoferrites
+and the crystal field for 𝑅-ions, which play a fundamental role in determining the spontaneous
+spin reorientation. The spin anisotropy of the second order in the 3𝑑 sublattice of orthorhombic
+orthoferrites and orthochromites is generally not reduced to an effective uniaxial form as adopted in
+Ref. [6]. Furthermore, the density functional theory does not allow in principle to give an adequate
+description of such effects of higher orders of perturbation theory as spin anisotropy or antisymmetric
+exchange [7].
+In this paper, we present the results of a simple but realistic microscopic model of the spontaneous
+spin reorientation in rare-earth orthoferrites and orthochromites, which takes into account all the main
+relevant interactions. This model was developed back in the 80s of the last century [8], but has not
+been published until now.
+2 Model formulation
+The most popular examples of systems with the spontaneous SR transitions are magnets based
+on 3𝑑 and 4𝑓 elements such as rare-earth orthoferrites RFeO3, orthochromites RCrO3, intermetallic
+compounds RCo5, RFe2 etc. In all cases, an important cause of the spontaneous SR is the 4𝑓 − 3𝑑
+interaction. Usually this interaction is taken into account by introducing an effective field of the
+magnetically ordered 3𝑑 sublattice acting on the 4𝑓 ions.
+To consider the contribution of the rare-earth sublattice to the free energy at low temperatures, we
+are developing a model which takes into account either the well isolated lower Kramers doublet of the
+4𝑓 ions (with an odd number of the 4𝑓 electrons) or the well isolated two lower Stark sublevels with
+close energies that form a quasi-doublet.
+Within the framework of such “single-doublet” approximation we consider the spontaneous SR
+transition in orthorhombic weak ferromagnets RFeO3 and RCrO3, where the free energy per ion can
+be represented as follows
+Φ(𝜃) = 𝐾1 cos 2𝜃 + 𝐾2 cos 4𝜃 − 𝑘𝑇 ln 2 cosh ∆(𝜃)
+2𝑘𝑇 ,
+(1)
+where 𝐾1, 𝐾2 are the first and second anisotropy constants of the 3𝑑 sublattice, which are temperature
+independent (at least in the SR region), 𝜃 is the orientation angle of the antiferromagnetic, or N´eel
+vector G of the 3𝑑 sublattice (e.g. in the 𝑎𝑐 plane), and ∆(𝜃) is the lower doublet (quasi-doublet)
+splitting of the 4𝑓 ion in a magnetic field induced by the 3𝑑 sublattice.
+Theoretical estimations [8–10] of the different contributions to the first constants of the magnetic
+anisotropy for orthoferrites RFeO3 point to a competition of several main mechanisms with relatively
+regular (Dzyaloshinskii-Moriya (DM) coupling, magnetodipole interaction) or irregular (single-ion
+anisotropy, SIA) dependence on the type of R-ion. For instance, the microscopic theory predicts an
+unexpectedly strong increase in values of the constant 𝐾1(𝑎𝑐) for LuFeO3 as compared with YFeO3.
+The SIA contribution to 𝐾1(𝑎𝑐) partially compensates for the large contribution of the DM interaction
+in YFeO3, whereas in LuFeO3, they add up. This result is confirmed by experimental data on the
+2
+
+measurement of the threshold field 𝐻𝑆𝑅 of spin reorientation Γ4 → Γ2 (𝐺𝑥 → 𝐺𝑧) in the orthoferrite
+Lu0.5Y0.5FeO3, in which 𝐻𝑆𝑅 = 15 T as compared to 𝐻𝑆𝑅 = 7.5 T in YFeO3 [10]. Thus, one can
+estimate 𝐾1(𝑎𝑐) in LuFeO3 as around three times as much as 𝐾1(𝑎𝑐) in YFeO3.
+Let us pay attention to recent works on the determination of the parameters of the spin Hamiltonian
+in YFeO3 from measurements of the spin-wave spectrum by the inelastic neutron scattering [11,
+12] and terahertz absorption spectroscopy [13]. However, these authors started with a simplified
+spin-Hamiltonian that took into account only Heisenberg exchange, DM interaction, and single-
+ion anisotropy. Obviously, disregarding the magnetic dipole and exchange-relativistic anisotropy, the
+“single-ion anisotropy” constants found by the authors are some effective quantities that are not directly
+related to the SIA.
+Unfortunately, despite numerous, including fairly recent, studies of the magnetic anisotropy of
+orthoferrites, we do not have reliable experimental data on the magnitude of the contributions of
+various anisotropy mechanisms.
+As shown by theoretical calculations [8,9,14] the constants 𝐾2 of the fourth order spin anisotropy
+rather smoothly decrease in absolute value, changing by no more than two times on going from La to
+Lu. But the most interesting was the conclusion about the different signs of these constants, positive
+for the 𝑎𝑐 and 𝑏𝑐 planes and negative for the 𝑎𝑏 plane, thus indicating a different character of spin-
+reorientation transitions in the corresponding planes, i.e., second-order transitions in the 𝑎𝑐 and 𝑏𝑐
+planes and first-order transitions in the 𝑎𝑏 plane [2]. Indeed, all currently known spin-reorientation
+transitions of the Γ4 −Γ2 (𝐺𝑥 −𝐺𝑧) type in orthoferrites RFeO3 (R = Pr, Nd, Sm, Tb, Ho, Er, Tm, Yb)
+are smooth, with two characteristic temperatures of the second-order phase transitions to be a start
+and finish of the spin reorientation, and the only known jump-like first order SR transition for these
+crystals is the SR transition Γ4 − Γ1 (𝐺𝑥 − 𝐺𝑦) in the 𝑎𝑏 plane in DyFeO3 [2]. A unique example
+that confirms the conclusions about the sign of the second anisotropy constant is a mixed orthoferrite
+Ho0.5Dy0.5FeO3 [2] in which two spin-reorientation transitions 𝐺𝑥 − 𝐺𝑦 (𝑇 = 46 K) and 𝐺𝑦 − 𝐺𝑧
+(18 ÷ 24 K) are realized through one phase transition of the first order in the 𝑎𝑏 plane and two phase
+transitions of the second order in the 𝑏𝑐 plane, respectively.
+The splitting value ∆(𝜃) for the Kramers doublet in a magnetic field H has the well-known form
+∆(𝜃) = 𝜇𝐵
+[︀
+(𝑔𝑥𝑥𝐻𝑥 + 𝑔𝑥𝑦𝐻𝑦)2 + (𝑔𝑥𝑦𝐻𝑥 + 𝑔𝑦𝑦𝐻𝑦)2 + 𝑔2
+𝑧𝑧𝐻2
+𝑧
+]︀1/2 ,
+(2)
+where it is taken into account that for the 4𝑓 ions in RFeO3 the ˆ𝑔-tensor (with the local symmetry 𝐶𝑠)
+has the form
+ˆ𝑔 =
+⎛
+⎜
+⎝
+𝑔𝑥𝑥
+𝑔𝑥𝑦
+0
+𝑔𝑥𝑦
+𝑔𝑦𝑦
+0
+0
+0
+𝑔𝑧𝑧
+⎞
+⎟
+⎠ .
+(3)
+The effective field H for the SR transition 𝐺𝑥 → 𝐺𝑧 in the 𝑎𝑐 plane can be represented as follows
+𝐻𝑥 = 𝐻(0)
+𝑥 cos 𝜃, 𝐻𝑦 = 𝐻(0)
+𝑦
+cos 𝜃, 𝐻𝑧 = 𝐻(0)
+𝑧
+sin 𝜃,
+(4)
+3
+
+so in the absence of an external magnetic field, for ∆(𝜃) we have the rather simple expression:
+∆(𝜃) =
+(︂∆2
+𝑎 − ∆2
+𝑐
+2
+cos 2𝜃 + ∆2
+𝑎 + ∆2
+𝑐
+2
+)︂1/2
+,
+(5)
+where ∆𝑎,𝑐 are the doublet splitting for the cases of 𝜃 = 0 (𝐺𝑧-phase) and 𝜃 = 𝜋/2 (𝐺𝑥-phase)
+respectively. The dependence ∆(𝜃) from (5) is also valid in the case of quasi-doublet.
+A contribution of splitting ∆ to the free energy Φ(𝜃) for the rare-earth sublattice is usually
+considered in the “high-temperature” approximation, when 𝑘𝑇 ≫ ∆ and the influence of the 4𝑓
+sublattice are reduced only to renormalization of the first anisotropy constant 𝐾1:
+𝐾*
+1 = 𝐾1
+(︂
+1 − 1
+𝜏
+)︂
+,
+(6)
+where 𝜏 = 𝑇/𝑇𝑆𝑅 is the reduced temperature and 𝑇𝑆𝑅 = (∆2
+𝑎 − ∆2
+𝑐)/16𝑘𝐾1 is the characteristic
+transition temperature.
+Below we will consider a specific situation when 𝐾1 > 0 and ∆𝑎 > ∆𝑐, i.e. when the configuration
+𝐺𝑥 (𝜃 = 𝜋/2) is realized at high temperatures and a decrease in temperature can lead to the spin
+reorientation 𝐺𝑥 → 𝐺𝑧 or 𝐺𝑥 → 𝐺𝑥𝑧 (transition to an angular spin structure). The type of the phase
+transition of the spin reorientation in the “high-temperature” approximation is determined by the sign
+of the second constant 𝐾2: at 𝐾2 < 0 it will be realized by one first-order phase transition at 𝑇 = 𝑇𝑆𝑅,
+i.e. 𝜏 = 1, or at 𝐾2 > 0 by two second-order phase transitions at 𝜏𝑠 = (1 + 𝛾)−1 and 𝜏𝑓 = (1 − 𝛾)−1,
+where 𝜏𝑠 and 𝜏𝑓 are the reduced temperatures of the beginning and end of the SR phase transition and
+𝛾 = 4𝐾2/𝐾1.
+3 Analysis of the “single-doublet” model
+A behavior of a system described by the free energy (1) can be analyzed rigorously. The condition
+𝜕Φ/𝜕𝜃 = 0 reduces in this case to two equations:
+sin 2𝜃 = 0,
+(7)
+𝛼𝜇 + 𝛽𝜇3 = tanh 𝜇
+𝜏 ;
+(8)
+where the following notations are introduced:
+𝛼 = 1 − 𝛾 ∆2
+𝑎 + ∆2
+𝑐
+∆2
+𝑎 − ∆2
+𝑐
+, 𝛽 =
+2𝛾
+𝜇2
+𝑓 − 𝜇2
+𝑠
+, 𝜇 = ∆(𝜃)
+2𝑘𝑇𝑆𝑅
+, 𝜇𝑠 =
+∆𝑐
+2𝑘𝑇𝑆𝑅
+, 𝜇𝑓 =
+∆𝑎
+2𝑘𝑇𝑆𝑅
+.
+(9)
+This corresponds to three possible magnetic configurations:
+• The configuration 𝐺𝑥: 𝜃 = ±𝜋/2, stable at tanh 𝜇𝑠/𝜏 ≤ 𝛼𝜇𝑠 + 𝛽𝜇3
+𝑠 .
+• The configuration 𝐺𝑧: 𝜃 = 0, 𝜋, stable at tanh 𝜇𝑓/𝜏 ≥ 𝛼𝜇𝑓 + 𝛽𝜇3
+𝑓 .
+4
+
+• The angular configuration 𝐺𝑥𝑧: the temperature dependence of 𝜃(𝜏) is determined by solving
+the equation (8) (see Figure 1), the state is stable at 𝜕𝜇/𝜕𝜏 ≤ 0.
+The peculiar 𝜇-𝜏 phase diagram which represents solutions of the master equation (8) given a fixed
+value of the 𝛼 parameter and different value of the 𝛽 parameter is shown in Figure 1, where areas
+with different character of the SR transition are highlighted in different colors. For the solutions in
+the FO region, the SR goes through one first-order phase transition, in the SO region we arrive at one
+or two second-order phase transitions, in the MO1,2 regions we arrive at a “mixture” of the first and
+second-order phase transitions. All the lines 𝜇(𝜏) on the right side converge to
+√︀
+|𝛼/𝛽| at 𝜏 → ∞; on
+the left side, when 𝜏 → 0 the branch point 𝜇 =
+3
+2𝛼 is obtained at 𝛽 = − 4
+27𝛼3, and the point 𝜇 = 1/𝛼
+at 𝛽 = 0; all the solutions, where 𝜇 can reach zero, converge to 𝜏 = 1/𝛼.
+0
+1/α
+τ
+1/α
+3
+2 α
+μ
+α / β1
+α / β2
+α / β3
+FO
+MO1
+MO2
+SO
+Fig. 1: (Color online) The peculiar 𝜇-𝜏 phase diagram which represents solutions of the master
+equation (8) given a fixed value of the 𝛼 parameter and different value of the 𝛽 parameter (see text for
+detail).
+The character of the SR transition will be determined by the form of the solution of the equation
+(8) in the region 𝜇𝑠 ≤ 𝜇 ≤ 𝜇𝑓. Let us analyze this equation starting with the simplest case 𝐾2 = 0,
+i.e. 𝛼 = 1, 𝛽 = 0. In this case, the main equation transforms into the molecular field equation well
+known in the basic theory of ferromagnetism:
+𝜇 = tanh 𝜇
+𝜏 = 𝐵 1
+2
+(︁𝜇
+𝜏
+)︁
+,
+(10)
+where 𝐵1/2(𝑥) is the Brillouin function. The equation has only one non-trivial solution at 0 ≤ 𝜏 ≤ 1,
+0 ≤ 𝜇 ≤ 1, and the function 𝜇(𝜏) has the usual “Weiss” form. Thus, with the absence of the
+cubic anisotropy (𝐾2 = 0) in the “single-doublet” model the SR will be realized either through two
+second-order phase transitions at 𝜇𝑓 ≤ 1 (the complete spin-reorientation 𝐺𝑥 → 𝐺𝑧), or through one
+second-order phase transition at 𝜇𝑓 > 1, but in this case the SR will be incomplete, i.e. it will end
+with a transition to the angular spin structure 𝐺𝑥𝑧. The spin reorientation will begin at a temperature
+5
+
+𝑇𝑠 ≤ 𝑇𝑆𝑅 and 𝑇𝑠 is equal to 𝑇𝑆𝑅 only in the case 𝜇𝑠 = 0 (∆𝑐 = 0), which can be realized in the general
+case only for Ising ions (e.g. Dy3+ in DyFeO3). For this type of ions, the temperature dependence of
+the “order parameter” 𝜇 (in fact the splitting ∆(𝜃) of the doublet) in a close range of 𝑇𝑆𝑅 will be very
+sharp: 𝜇(𝑇) ∼ (𝑇 − 𝑇𝑆𝑅)−1/2. Nevertheless, the SR will be continuous and the temperature range of
+the SR ∆𝑇 = 𝑇𝑠 − 𝑇𝑓 at 𝜇 ≪ 1 can theoretically reach arbitrarily small values.
+Thus, the results of the rigorous analysis of the “single-doublet” model are fundamentally different
+from the conclusions of the simplified model (the “high-temperature” approximation), according to
+which for 𝐾2 = 0 the spin reorientation always occurs as the first-order phase transition at 𝑇 = 𝑇𝑆𝑅.
+For a positive second anisotropy constant (𝐾2 > 0, 𝛽 > 0), the main equation (8) has one non-
+trivial solution in the region 0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 𝜇0 at 𝛼 > 0, and one in the region 0 ≤ 𝜏 ≤ ∞,
+√︀
+|𝛼/𝛽| ≤ 𝜇 ≤ 𝜇0 at 𝛼 ≤ 0, where 𝜇0 is determined from the solution of the equation 𝛼𝜇0 +𝛽𝜇3
+0 = 1.
+The situation in this case is very similar to the previous one, i.e. the beginning of the SR will always
+be a second-order phase transition, and the reorientation will be complete (𝐺𝑥 → 𝐺𝑧) or incomplete
+(𝐺𝑥 → 𝐺𝑥𝑧). Note that under the condition (𝜇2
+𝑓 − 𝜇2
+𝑠)/(𝜇2
+𝑓 + 𝜇2
+𝑠) ≥ 𝛾, i.e. 𝛼 ≤ 0, the width of the
+reorientation region becomes very large, even if 𝜇𝑠 differs slightly from 𝜇𝑓.
+For Ising ions at ∆𝑐 = 0, the SR beginning temperature is determined in exactly the same way as
+in the “high-temperature” approximation 𝑇𝑠 = 𝑇𝑆𝑅/(1 + 𝛾).
+For a negative second anisotropy constant (𝐾2 < 0, 𝛽 < 0), the several fundamentally different
+solutions of the main equation (8) are possible. For 𝐾*
+2 ≥ 𝐾2, where 𝐾*
+2 is determined from the
+condition 𝛽 = − 1
+3𝛼3, i.e.
+2𝛾
+𝜇2
+𝑓 − 𝜇2
+𝑠
+= −1
+3
+(︃
+1 − 𝛾 𝜇2
+𝑓 + 𝜇2
+𝑠
+𝜇2
+𝑓 − 𝜇2
+𝑠
+)︃3
+,
+(11)
+there is one non-trivial solution of the equation (8) in the region 1/𝛼 ≤ 𝜏 < ∞, 𝜇 ≤
+√︀
+𝛼/𝛽, but
+here 𝜇(𝑇) decreases with decreasing temperature, i.e. 𝜕𝜇/𝜕𝜏 > 0. This solution is unstable and there
+is no fundamental possibility for a smooth rotation of spins, the SR is always realized through the
+first-order phase transition.
+In the intermediate range of values 𝐾2 (𝐾*
+2 < 𝐾2 < 0 or − 1
+3𝛼3 < 𝛽 < 0) the main equation
+has two non-trivial solutions, and for one of them 𝜕𝜇/𝜕𝜏 > 0 (corresponding to bigger values of 𝜇),
+and for the second 𝜕𝜇/𝜕𝜏 < 0 (corresponding to smaller values of 𝜇). It is convenient to consider
+separately three areas of variation 𝛽.
+1. − 4
+27𝛼3 < 𝛽 < 0:
+a) the first solution: 0 ≤ 𝜏 < ∞, 𝜇> ≤ 𝜇 <
+√︀
+|𝛼/𝛽|,
+b) the second solution: 0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 𝜇<,
+where 𝜇>, 𝜇< are the bigger and smaller positive solution of the equation 𝛼𝜇 + 𝛽𝜇3 = 1.
+2. 𝛽 = − 4
+27𝛼3:
+a) the first solution: 0 ≤ 𝜏 < ∞, 3/(2𝛼) ≤ 𝜇 <
+√︀
+|𝛼/𝛽|,
+b) the second solution: 0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 3/(2𝛼),
+moreover, in this case we have a branch point of the main equation solution at 𝜏 = 0, 𝜇 = 1.
+3. − 1
+3𝛼3 < 𝛽 < − 4
+27𝛼3:
+a) the first solution: 𝜏0 ≤ 𝜏 < ∞, 𝜇0 ≤ 𝜇 <
+√︀
+|𝛼/𝛽|,
+6
+
+b) the second solution: 𝜏0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 𝜇0,
+where the quantities 𝜇0, 𝜏0 correspond to the branch points of the main equation solutions.
+Illustrations of typical (a,b) and unconventional (c,d) SR transitions predicted by simple
+(quasi)doublet model are shown in Figure 2. The Figure 2a, built with 𝐾1 = 1, 𝛾 = 0.05, ∆𝑎 =
+30.84, ∆𝑐 = 14.82, which corresponds to 𝑇𝑆𝑅 = 45.73, 𝜇𝑠 = 0.162, 𝜇𝑓 = 0.337, 𝜏𝑠 = 1.04, 𝜏𝑓 =
+0.91, describes a typical smooth SR transition with two second-order phase transitions 𝐺𝑥 − 𝐺𝑥𝑧 at
+the beginning (𝜏𝑠) and 𝐺𝑥𝑧 − 𝐺𝑧 at the end (𝜏𝑓) of the spin reorientation.
+The Figure 2b, built with 𝐾1 = 1, 𝛾 = −0.1, ∆𝑎 = 33.19, ∆𝑐 = 27.1, which corresponds to
+𝑇𝑆𝑅 = 22.95, 𝜇𝑠 = 0.59, 𝜇𝑓 = 0.72, 𝜏𝑠 = 0.762, 𝜏𝑓 = 0.93, describes an abrupt first-order SR
+transition. For 𝜏 > 𝜏𝑓 there is the 𝐺𝑥-phase, which can remain stable up to 𝜏𝑠 when cooled. For 𝜏 < 𝜏𝑠
+there is the 𝐺𝑧-phase, which can remain stable up to 𝜏𝑓 when heated. The point 𝐴 marks a phase
+transition point when the phases 𝐺𝑥 and 𝐺𝑧 have equal energies.
+The Figure 2c, built with 𝐾1 = 1, 𝛾 = −0.222, ∆𝑎 = 6.72, ∆𝑐 = 1.63, which corresponds
+to 𝑇𝑆𝑅 = 2.65, 𝜇𝑠 = 0.307, 𝜇𝑓 = 1.266, 𝜏𝑠 = 0.778, 𝜏𝑓 = 0.523 and the Figure 2d, built with
+𝐾1 = 1, 𝛾 = −0.25, ∆𝑎 = 6.71, ∆𝑐 = 2.02, which corresponds to 𝑇𝑆𝑅 = 2.56, 𝜇𝑠 = 0.396, 𝜇𝑓 =
+1.31, 𝜏𝑠 = 0.73, 𝜏𝑓 = 0.545 describe unconventional "mixed"SR transitions. At 𝜏𝑠 there is the smooth
+second-order phase transition 𝐺𝑥 − 𝐺𝑥𝑧. At 𝜏 ≤ 𝜏𝑓 we have two stable phases 𝐺𝑧 and 𝐺𝑥𝑧: at those
+temperatures the sharp first-order phase transition 𝐺𝑥𝑧 − 𝐺𝑧 can happen, or the system could stay in
+the angular 𝐺𝑥𝑧-phase.
+(a)
+τ
+τs
+τf
+μ
+μs
+μf
+(c)
+τ
+τs
+τf
+μ
+μs
+μf
+(d)
+τ
+τs
+τf
+μ
+μs
+μf
+θ
+Φ
+θ
+Φ
+θ
+Φ
+τ > τf
+τ < τs
+A
+(b)
+τ
+τs
+τf
+μ
+μs
+μf
+A
+Fig. 2: Illustrations of typical (a,b) and unconventional (c,d) SR transitions predicted by simple
+(quasi)doublet model (see text for detail). The arrows indicate the direction of the antiferromagnetic
+vector G in the 𝑎𝑐 plane. The insets in panel (b) show the 𝜃-dependence of the free energy.
+7
+
+Thus, there are not only the smooth and abrupt SR transitions, a characteristic feature of the range
+of intermediate values 𝐾2 is the fundamental possibility of the existence of “mixed” SR transitions, in
+which the spins first smoothly rotate through a certain angle and then jump to the position with 𝜃 = 0.
+For this, it is sufficient that 𝜇𝑓 corresponds to a point on the upper branch of solutions, and 𝜇𝑠 to a point
+on the lower branch of solutions at 𝜏𝑓 < 𝜏𝑠. In this case, the spin reorientation begins with the single
+second-order transition 𝐺𝑥 → 𝐺𝑥𝑧 and then ends with the first-order phase transition 𝐺𝑥𝑧 → 𝐺𝑧.
+In contrast to the “high-temperature” approximation, the “single-doublet” model claims the nature
+of the phase transition is determined not simply by the sign of the second anisotropy constant, but
+also it depends on the ratio between 𝐾1, 𝐾2 and the doublet splitting in both phases. Nevertheless,
+if we apply the simplified model to describe the SR transition, we have to renormalize both the first
+and the second anisotropy constant, giving the last one sometimes a rather complicated temperature
+dependence, in particular with a change in sign when considering transitions of the “mixed” type.
+Of course, in this case Fe sublattice alone is not enough to provide the value of the effective second
+constant.
+4 Conclusion
+The model of the spin-reorientation transitions induced by the 4𝑓 − 3𝑑 interaction in rare-earth
+orthoferrites and orthochromites has been investigated. It is shown that both the temperature and
+the character of the spin-reorientation transition following from the solution of the transcendental
+equation (8) are the result of competition between the second and fourth order spin anisotropy of
+the 3𝑑 sublattice, the crystal field for 4f ions, and the 4𝑓 − 3𝑑 interaction. At variance with the
+“high-temperature” approximation, the “single-doublet” model, along with typical smooth and abrupt
+SR transitions, predicts the appearance of mixed-type SR transitions, with an initial second-order
+transition and a final abrupt first-order transition.
+Funding: The research was supported by the Ministry of Education and Science of the Russian
+Federation, project № FEUZ-2020-0054, and by Russian Science Foundation, project № 22-22-00682.
+References
+[1] Belov, K.P.; Zvezdin, A.K.; Kadomtseva, A.M.; Levitin, R.Z. Spin-reorientation transitions in
+rare-earth magnets. Sov. Phys. Usp. 1976, 19, 574.
+[2] Belov, K.P.; Zvezdin, A.K.; Kadomtseva, A.M.; Levitin, R.Z. Orientational Transitions in Rare-
+Earth Magnetics; Nauka: Moscow, Russia, 1979. (In Russian)
+[3] Singh, A.; Rajput, S.; Padmanabhan, B.; Anas, M.; Damay, F.; Kumar, C.M.N.; Eguchi, G.; Jain,
+A.; Yusuf, S.M.; Maitra, T.; Malik V.K. Successive spin reorientations and rare earth ordering
+in Nd0.5Dy0.5FeO3: Experimental and ab initio investigations. Phys. Rev. B 2020, 102, 144432.
+8
+
+[4] Hoogeboom, G.R.; Kuschel, T.; Bauer, G.E.W.; Mostovoy, M.V.; Kimel, A.V.; van Wees, B.J.
+Magnetic order of Dy3+ and Fe3+ moments in antiferromagnetic DyFeO3 probed by spin Hall
+magnetoresistance and spin Seebeck effect. Phys. Rev. B 2021, 103, 134406.
+[5] Tsymbal, L.T.; Bazaliy, Y.B.; Derkachenko, V.N.; Kamenev, V.I.; Kakazei, G.N.; Palomares, F.J.;
+Wigen, P.E. Magnetic and structural properties of spin-reorientation transitions in orthoferrites.
+J. Appl. Phys. 2007, 101, 123919–123926.
+[6] Sasani, A.; I˜niguez, J.; Bousquet, E. Magnetic phase diagram of rare-earth orthorhombic
+perovskite oxides. Phys. Rev. B 2021, 104, 064431.
+[7] Moskvin, A.S. Dzyaloshinskii–Moriya Coupling in 3d Insulators. Condens. Matter 2019, 4, 84.
+[8] Moskvin, A.S. Antisymmetric Exchange and Magnetic Anisotropy in Weak Ferromagnets. D.
+Sc. Thesis, Lomonosov Moscow State University, Moscow, Russia, 1984. (In Russian)
+[9] Moskvin,
+A.
+Structure–Property
+Relationships
+for
+Weak
+Ferromagnetic
+Perovskites.
+Magnetochemistry 2021, 7, 111.
+[10] Kadomtseva, A.M.; Agafonov, A.P.; Lukina, M.M.; Milov, V.N.; Moskvin, A.S.; Semenov, V.A.;
+Sinitsyn, E.V. Nature of the Magnetic Anisotropy and Magnetostriction of Orthoferrites and
+Orthochromites. JETP 1981, 81, 700–706.
+[11] Hahn, S.E.; Podlesnyak, A.A.; Ehlers, G.; Granroth, G.E.; Fishman, R.S.; Kolesnikov, A.I.;
+Pomjakushina, E.; Conder, K. Inelastic neutron scattering studies of YFeO3. Phys. Rev. B 2014,
+89, 014420.
+[12] Park, K.; Sim, H.; Leiner, J.C.; Yoshida, Y.; Jeong, J.; Yano, S.; Gardner, J.; Bourges, P.; Klicpera,
+M.; Sechovsk´y, V.; Boehm, M.; Park, J.-G. Low-energy spin dynamics of orthoferrites AFeO3
+(A = Y, La, Bi). J. Phys. Condens. Matter 2018, 30, 235802.
+[13] Amelin, K.; Nagel, U.; Fishman, R.S.; Yoshida, Y.; Sim, H.; Park, K.; Park, J.-G.; R˜o˜om, T.
+Terahertz absorption spectroscopy study of spin waves in orthoferrite YFeO3 in a magnetic field.
+Phys. Rev. B 2018, 98, 174417.
+[14] Moskvin, A.S.; Bostrem, I.G. Cubic Anisotropy of Rare-Earth Orthoferrites. Sov. Phys. Solid St.
+1979, 21, 628.
+9
+
diff --git a/-NFLT4oBgHgl3EQfuy_h/content/tmp_files/load_file.txt b/-NFLT4oBgHgl3EQfuy_h/content/tmp_files/load_file.txt
new file mode 100644
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+++ b/-NFLT4oBgHgl3EQfuy_h/content/tmp_files/load_file.txt
@@ -0,0 +1,410 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf,len=409
+page_content='Simple Realistic Model of Spin Reorientation in 4f-3d  Compounds  Alexander Moskvin*, Evgenii Vasinovich, Anton Shadrin  Ural Federal University, Ekaterinburg, Russia  Abstract: Spin reorientation is an important phenomenon of rare-earth perovskites, orthoferrites  and orthochromites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' In this study, we consider a simple but realistic microscopic theory of the  spontaneous spin-reorientation transitions induced by the 4f-3d interaction, more specifically, the  interaction of the main Kramers doublet or non-Kramers quasi-doublet of the 4f ion with an effective  magnetic field induced by the 3d sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The obtained results indicate that the cause of both the  temperature and the character of the spin-reorientation transition is a competition between the second  and fourth order spin anisotropy of the 3d sublattice, the crystal field for 4f ions, and the 4f-3d  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Keywords: 4f-3d interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' (quasi)doublets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' spin reorientation  1 Introduction  Rare-earth orthorhombic perovskites, orthoferrites RFeO3 and orthochromites RCrO3 (where  R is a rare-earth ion and yttrium), exhibit many important features such as weak ferro- and  antiferromagnetism, magnetization reversal, anomalous circular magnetooptics, and the phenomenon  of the spontaneous spin reorientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The spin reorientation (SR) is one of their unique properties  that have attracted a lot of attention back in the 70s of the last century [1, 2], though their exact  microscopic origin is still a challenge to theorists and experimentalists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The revival of interest in the mechanism of the spontaneous spin reorientation and  magnetic compensation in rare-earth perovskites in recent years is related with the discovery  of the magnetoelectric and the exchange bias effect, which can have a direct application in  magnetoelectronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Along with the emergence of new experimental studies (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' [3, 4]),  there also appeared theoretical works claiming to modify the mean-field theory of the spontaneous  spin-reorientation transitions [5] or to scrutinize the microscopic mechanism responsible for  spin reorientations and magnetization reversal [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' In fact, these results are not directly related  to the microscopic theory of the spontaneous spin reorientation in rare-earth orthoferrites and  orthochromites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For instance, the authors of the most recent paper [6] did not take into account  alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='moskvin@urfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='ru  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='12157v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='str-el]  28 Jan 2023  a number of interactions, such as the fourth-order anisotropy for the 3𝑑 sublattice of orthoferrites  and the crystal field for 𝑅-ions, which play a fundamental role in determining the spontaneous  spin reorientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The spin anisotropy of the second order in the 3𝑑 sublattice of orthorhombic  orthoferrites and orthochromites is generally not reduced to an effective uniaxial form as adopted in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Furthermore, the density functional theory does not allow in principle to give an adequate  description of such effects of higher orders of perturbation theory as spin anisotropy or antisymmetric  exchange [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  In this paper, we present the results of a simple but realistic microscopic model of the spontaneous  spin reorientation in rare-earth orthoferrites and orthochromites, which takes into account all the main  relevant interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' This model was developed back in the 80s of the last century [8], but has not  been published until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  2 Model formulation  The most popular examples of systems with the spontaneous SR transitions are magnets based  on 3𝑑 and 4𝑓 elements such as rare-earth orthoferrites RFeO3, orthochromites RCrO3, intermetallic  compounds RCo5, RFe2 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' In all cases, an important cause of the spontaneous SR is the 4𝑓 − 3𝑑  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Usually this interaction is taken into account by introducing an effective field of the  magnetically ordered 3𝑑 sublattice acting on the 4𝑓 ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  To consider the contribution of the rare-earth sublattice to the free energy at low temperatures, we  are developing a model which takes into account either the well isolated lower Kramers doublet of the  4𝑓 ions (with an odd number of the 4𝑓 electrons) or the well isolated two lower Stark sublevels with  close energies that form a quasi-doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Within the framework of such “single-doublet” approximation we consider the spontaneous SR  transition in orthorhombic weak ferromagnets RFeO3 and RCrO3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' where the free energy per ion can  be represented as follows  Φ(𝜃) = 𝐾1 cos 2𝜃 + 𝐾2 cos 4𝜃 − 𝑘𝑇 ln 2 cosh ∆(𝜃)  2𝑘𝑇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  (1)  where 𝐾1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝐾2 are the first and second anisotropy constants of the 3𝑑 sublattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' which are temperature  independent (at least in the SR region),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝜃 is the orientation angle of the antiferromagnetic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' or N´eel  vector G of the 3𝑑 sublattice (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' in the 𝑎𝑐 plane), and ∆(𝜃) is the lower doublet (quasi-doublet)  splitting of the 4𝑓 ion in a magnetic field induced by the 3𝑑 sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Theoretical estimations [8–10] of the different contributions to the first constants of the magnetic  anisotropy for orthoferrites RFeO3 point to a competition of several main mechanisms with relatively  regular (Dzyaloshinskii-Moriya (DM) coupling, magnetodipole interaction) or irregular (single-ion  anisotropy, SIA) dependence on the type of R-ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For instance, the microscopic theory predicts an  unexpectedly strong increase in values of the constant 𝐾1(𝑎𝑐) for LuFeO3 as compared with YFeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The SIA contribution to 𝐾1(𝑎𝑐) partially compensates for the large contribution of the DM interaction  in YFeO3, whereas in LuFeO3, they add up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' This result is confirmed by experimental data on the  2  measurement of the threshold field 𝐻𝑆𝑅 of spin reorientation Γ4 → Γ2 (𝐺𝑥 → 𝐺𝑧) in the orthoferrite  Lu0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5FeO3, in which 𝐻𝑆𝑅 = 15 T as compared to 𝐻𝑆𝑅 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5 T in YFeO3 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Thus, one can  estimate 𝐾1(𝑎𝑐) in LuFeO3 as around three times as much as 𝐾1(𝑎𝑐) in YFeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Let us pay attention to recent works on the determination of the parameters of the spin Hamiltonian  in YFeO3 from measurements of the spin-wave spectrum by the inelastic neutron scattering [11,  12] and terahertz absorption spectroscopy [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' However, these authors started with a simplified  spin-Hamiltonian that took into account only Heisenberg exchange, DM interaction, and single-  ion anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Obviously, disregarding the magnetic dipole and exchange-relativistic anisotropy, the  “single-ion anisotropy” constants found by the authors are some effective quantities that are not directly  related to the SIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Unfortunately, despite numerous, including fairly recent, studies of the magnetic anisotropy of  orthoferrites, we do not have reliable experimental data on the magnitude of the contributions of  various anisotropy mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  As shown by theoretical calculations [8,9,14] the constants 𝐾2 of the fourth order spin anisotropy  rather smoothly decrease in absolute value, changing by no more than two times on going from La to  Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' But the most interesting was the conclusion about the different signs of these constants, positive  for the 𝑎𝑐 and 𝑏𝑐 planes and negative for the 𝑎𝑏 plane, thus indicating a different character of spin-  reorientation transitions in the corresponding planes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=', second-order transitions in the 𝑎𝑐 and 𝑏𝑐  planes and first-order transitions in the 𝑎𝑏 plane [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Indeed, all currently known spin-reorientation  transitions of the Γ4 −Γ2 (𝐺𝑥 −𝐺𝑧) type in orthoferrites RFeO3 (R = Pr, Nd, Sm, Tb, Ho, Er, Tm, Yb)  are smooth, with two characteristic temperatures of the second-order phase transitions to be a start  and finish of the spin reorientation, and the only known jump-like first order SR transition for these  crystals is the SR transition Γ4 − Γ1 (𝐺𝑥 − 𝐺𝑦) in the 𝑎𝑏 plane in DyFeO3 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' A unique example  that confirms the conclusions about the sign of the second anisotropy constant is a mixed orthoferrite  Ho0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5Dy0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5FeO3 [2] in which two spin-reorientation transitions 𝐺𝑥 − 𝐺𝑦 (𝑇 = 46 K) and 𝐺𝑦 − 𝐺𝑧  (18 ÷ 24 K) are realized through one phase transition of the first order in the 𝑎𝑏 plane and two phase  transitions of the second order in the 𝑏𝑐 plane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The splitting value ∆(𝜃) for the Kramers doublet in a magnetic field H has the well-known form  ∆(𝜃) = 𝜇𝐵  [︀  (𝑔𝑥𝑥𝐻𝑥 + 𝑔𝑥𝑦𝐻𝑦)2 + (𝑔𝑥𝑦𝐻𝑥 + 𝑔𝑦𝑦𝐻𝑦)2 + 𝑔2  𝑧𝑧𝐻2  𝑧  ]︀1/2 ,  (2)  where it is taken into account that for the 4𝑓 ions in RFeO3 the ˆ𝑔-tensor (with the local symmetry 𝐶𝑠)  has the form  ˆ𝑔 =  ⎛  ⎜  ⎝  𝑔𝑥𝑥  𝑔𝑥𝑦  0  𝑔𝑥𝑦  𝑔𝑦𝑦  0  0  0  𝑔𝑧𝑧  ⎞  ⎟  ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  (3)  The effective field H for the SR transition 𝐺𝑥 → 𝐺𝑧 in the 𝑎𝑐 plane can be represented as follows  𝐻𝑥 = 𝐻(0)  𝑥 cos 𝜃, 𝐻𝑦 = 𝐻(0)  𝑦  cos 𝜃, 𝐻𝑧 = 𝐻(0)  𝑧  sin 𝜃,  (4)  3  so in the absence of an external magnetic field, for ∆(𝜃) we have the rather simple expression:  ∆(𝜃) =  (︂∆2  𝑎 − ∆2  𝑐  2  cos 2𝜃 + ∆2  𝑎 + ∆2  𝑐  2  )︂1/2  ,  (5)  where ∆𝑎,𝑐 are the doublet splitting for the cases of 𝜃 = 0 (𝐺𝑧-phase) and 𝜃 = 𝜋/2 (𝐺𝑥-phase)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The dependence ∆(𝜃) from (5) is also valid in the case of quasi-doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  A contribution of splitting ∆ to the free energy Φ(𝜃) for the rare-earth sublattice is usually  considered in the “high-temperature” approximation, when 𝑘𝑇 ≫ ∆ and the influence of the 4𝑓  sublattice are reduced only to renormalization of the first anisotropy constant 𝐾1:  𝐾*  1 = 𝐾1  (︂  1 − 1  𝜏  )︂  ,  (6)  where 𝜏 = 𝑇/𝑇𝑆𝑅 is the reduced temperature and 𝑇𝑆𝑅 = (∆2  𝑎 − ∆2  𝑐)/16𝑘𝐾1 is the characteristic  transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Below we will consider a specific situation when 𝐾1 > 0 and ∆𝑎 > ∆𝑐, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' when the configuration  𝐺𝑥 (𝜃 = 𝜋/2) is realized at high temperatures and a decrease in temperature can lead to the spin  reorientation 𝐺𝑥 → 𝐺𝑧 or 𝐺𝑥 → 𝐺𝑥𝑧 (transition to an angular spin structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The type of the phase  transition of the spin reorientation in the “high-temperature” approximation is determined by the sign  of the second constant 𝐾2: at 𝐾2 < 0 it will be realized by one first-order phase transition at 𝑇 = 𝑇𝑆𝑅,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝜏 = 1, or at 𝐾2 > 0 by two second-order phase transitions at 𝜏𝑠 = (1 + 𝛾)−1 and 𝜏𝑓 = (1 − 𝛾)−1,  where 𝜏𝑠 and 𝜏𝑓 are the reduced temperatures of the beginning and end of the SR phase transition and  𝛾 = 4𝐾2/𝐾1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  3 Analysis of the “single-doublet” model  A behavior of a system described by the free energy (1) can be analyzed rigorously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The condition  𝜕Φ/𝜕𝜃 = 0 reduces in this case to two equations:  sin 2𝜃 = 0,  (7)  𝛼𝜇 + 𝛽𝜇3 = tanh 𝜇  𝜏 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  (8)  where the following notations are introduced:  𝛼 = 1 − 𝛾 ∆2  𝑎 + ∆2  𝑐  ∆2  𝑎 − ∆2  𝑐  , 𝛽 =  2𝛾  𝜇2  𝑓 − 𝜇2  𝑠  , 𝜇 = ∆(𝜃)  2𝑘𝑇𝑆𝑅  , 𝜇𝑠 =  ∆𝑐  2𝑘𝑇𝑆𝑅  , 𝜇𝑓 =  ∆𝑎  2𝑘𝑇𝑆𝑅  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  (9)  This corresponds to three possible magnetic configurations:  The configuration 𝐺𝑥: 𝜃 = ±𝜋/2, stable at tanh 𝜇𝑠/𝜏 ≤ 𝛼𝜇𝑠 + 𝛽𝜇3  𝑠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The configuration 𝐺𝑧: 𝜃 = 0, 𝜋, stable at tanh 𝜇𝑓/𝜏 ≥ 𝛼𝜇𝑓 + 𝛽𝜇3  𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  4  The angular configuration 𝐺𝑥𝑧: the temperature dependence of 𝜃(𝜏) is determined by solving  the equation (8) (see Figure 1), the state is stable at 𝜕𝜇/𝜕𝜏 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The peculiar 𝜇-𝜏 phase diagram which represents solutions of the master equation (8) given a fixed  value of the 𝛼 parameter and different value of the 𝛽 parameter is shown in Figure 1, where areas  with different character of the SR transition are highlighted in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For the solutions in  the FO region, the SR goes through one first-order phase transition, in the SO region we arrive at one  or two second-order phase transitions, in the MO1,2 regions we arrive at a “mixture” of the first and  second-order phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' All the lines 𝜇(𝜏) on the right side converge to  √︀  |𝛼/𝛽| at 𝜏 → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' on  the left side, when 𝜏 → 0 the branch point 𝜇 =  3  2𝛼 is obtained at 𝛽 = − 4  27𝛼3, and the point 𝜇 = 1/𝛼  at 𝛽 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' all the solutions, where 𝜇 can reach zero, converge to 𝜏 = 1/𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  0  1/α  τ  1/α  3  2 α  μ  \uf603α / β1\uf604  \uf603α / β2\uf604  \uf603α / β3\uf604  FO  MO1  MO2  SO  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 1: (Color online) The peculiar 𝜇-𝜏 phase diagram which represents solutions of the master  equation (8) given a fixed value of the 𝛼 parameter and different value of the 𝛽 parameter (see text for  detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The character of the SR transition will be determined by the form of the solution of the equation  (8) in the region 𝜇𝑠 ≤ 𝜇 ≤ 𝜇𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Let us analyze this equation starting with the simplest case 𝐾2 = 0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝛼 = 1, 𝛽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' In this case, the main equation transforms into the molecular field equation well  known in the basic theory of ferromagnetism:  𝜇 = tanh 𝜇  𝜏 = 𝐵 1  2  (︁𝜇  𝜏  )︁  ,  (10)  where 𝐵1/2(𝑥) is the Brillouin function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The equation has only one non-trivial solution at 0 ≤ 𝜏 ≤ 1,  0 ≤ 𝜇 ≤ 1, and the function 𝜇(𝜏) has the usual “Weiss” form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Thus, with the absence of the  cubic anisotropy (𝐾2 = 0) in the “single-doublet” model the SR will be realized either through two  second-order phase transitions at 𝜇𝑓 ≤ 1 (the complete spin-reorientation 𝐺𝑥 → 𝐺𝑧), or through one  second-order phase transition at 𝜇𝑓 > 1, but in this case the SR will be incomplete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' it will end  with a transition to the angular spin structure 𝐺𝑥𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The spin reorientation will begin at a temperature  5  𝑇𝑠 ≤ 𝑇𝑆𝑅 and 𝑇𝑠 is equal to 𝑇𝑆𝑅 only in the case 𝜇𝑠 = 0 (∆𝑐 = 0), which can be realized in the general  case only for Ising ions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Dy3+ in DyFeO3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For this type of ions, the temperature dependence of  the “order parameter” 𝜇 (in fact the splitting ∆(𝜃) of the doublet) in a close range of 𝑇𝑆𝑅 will be very  sharp: 𝜇(𝑇) ∼ (𝑇 − 𝑇𝑆𝑅)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Nevertheless, the SR will be continuous and the temperature range of  the SR ∆𝑇 = 𝑇𝑠 − 𝑇𝑓 at 𝜇 ≪ 1 can theoretically reach arbitrarily small values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Thus, the results of the rigorous analysis of the “single-doublet” model are fundamentally different  from the conclusions of the simplified model (the “high-temperature” approximation), according to  which for 𝐾2 = 0 the spin reorientation always occurs as the first-order phase transition at 𝑇 = 𝑇𝑆𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  For a positive second anisotropy constant (𝐾2 > 0, 𝛽 > 0), the main equation (8) has one non-  trivial solution in the region 0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 𝜇0 at 𝛼 > 0, and one in the region 0 ≤ 𝜏 ≤ ∞,  √︀  |𝛼/𝛽| ≤ 𝜇 ≤ 𝜇0 at 𝛼 ≤ 0, where 𝜇0 is determined from the solution of the equation 𝛼𝜇0 +𝛽𝜇3  0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The situation in this case is very similar to the previous one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' the beginning of the SR will always  be a second-order phase transition, and the reorientation will be complete (𝐺𝑥 → 𝐺𝑧) or incomplete  (𝐺𝑥 → 𝐺𝑥𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Note that under the condition (𝜇2  𝑓 − 𝜇2  𝑠)/(𝜇2  𝑓 + 𝜇2  𝑠) ≥ 𝛾, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝛼 ≤ 0, the width of the  reorientation region becomes very large, even if 𝜇𝑠 differs slightly from 𝜇𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  For Ising ions at ∆𝑐 = 0, the SR beginning temperature is determined in exactly the same way as  in the “high-temperature” approximation 𝑇𝑠 = 𝑇𝑆𝑅/(1 + 𝛾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  For a negative second anisotropy constant (𝐾2 < 0, 𝛽 < 0), the several fundamentally different  solutions of the main equation (8) are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For 𝐾*  2 ≥ 𝐾2, where 𝐾*  2 is determined from the  condition 𝛽 = − 1  3𝛼3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  2𝛾  𝜇2  𝑓 − 𝜇2  𝑠  = −1  3  (︃  1 − 𝛾 𝜇2  𝑓 + 𝜇2  𝑠  𝜇2  𝑓 − 𝜇2  𝑠  )︃3  ,  (11)  there is one non-trivial solution of the equation (8) in the region 1/𝛼 ≤ 𝜏 < ∞, 𝜇 ≤  √︀  𝛼/𝛽, but  here 𝜇(𝑇) decreases with decreasing temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝜕𝜇/𝜕𝜏 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' This solution is unstable and there  is no fundamental possibility for a smooth rotation of spins, the SR is always realized through the  first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  In the intermediate range of values 𝐾2 (𝐾*  2 < 𝐾2 < 0 or − 1  3𝛼3 < 𝛽 < 0) the main equation  has two non-trivial solutions, and for one of them 𝜕𝜇/𝜕𝜏 > 0 (corresponding to bigger values of 𝜇),  and for the second 𝜕𝜇/𝜕𝜏 < 0 (corresponding to smaller values of 𝜇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' It is convenient to consider  separately three areas of variation 𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' − 4  27𝛼3 < 𝛽 < 0:  a) the first solution: 0 ≤ 𝜏 < ∞, 𝜇> ≤ 𝜇 <  √︀  |𝛼/𝛽|,  b) the second solution: 0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 𝜇<,  where 𝜇>, 𝜇< are the bigger and smaller positive solution of the equation 𝛼𝜇 + 𝛽𝜇3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝛽 = − 4  27𝛼3:  a) the first solution: 0 ≤ 𝜏 < ∞, 3/(2𝛼) ≤ 𝜇 <  √︀  |𝛼/𝛽|,  b) the second solution: 0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 3/(2𝛼),  moreover, in this case we have a branch point of the main equation solution at 𝜏 = 0, 𝜇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' − 1  3𝛼3 < 𝛽 < − 4  27𝛼3:  a) the first solution: 𝜏0 ≤ 𝜏 < ∞, 𝜇0 ≤ 𝜇 <  √︀  |𝛼/𝛽|,  6  b) the second solution: 𝜏0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 𝜇0,  where the quantities 𝜇0, 𝜏0 correspond to the branch points of the main equation solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Illustrations of typical (a,b) and unconventional (c,d) SR transitions predicted by simple  (quasi)doublet model are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The Figure 2a, built with 𝐾1 = 1, 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='05, ∆𝑎 =  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='84, ∆𝑐 = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='82, which corresponds to 𝑇𝑆𝑅 = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='73, 𝜇𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='162, 𝜇𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='337, 𝜏𝑠 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='04, 𝜏𝑓 =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='91, describes a typical smooth SR transition with two second-order phase transitions 𝐺𝑥 − 𝐺𝑥𝑧 at  the beginning (𝜏𝑠) and 𝐺𝑥𝑧 − 𝐺𝑧 at the end (𝜏𝑓) of the spin reorientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The Figure 2b, built with 𝐾1 = 1, 𝛾 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='1, ∆𝑎 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='19, ∆𝑐 = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='1, which corresponds to  𝑇𝑆𝑅 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='95, 𝜇𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='59, 𝜇𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='72, 𝜏𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='762, 𝜏𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='93, describes an abrupt first-order SR  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For 𝜏 > 𝜏𝑓 there is the 𝐺𝑥-phase, which can remain stable up to 𝜏𝑠 when cooled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For 𝜏 < 𝜏𝑠  there is the 𝐺𝑧-phase, which can remain stable up to 𝜏𝑓 when heated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The point 𝐴 marks a phase  transition point when the phases 𝐺𝑥 and 𝐺𝑧 have equal energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The Figure 2c, built with 𝐾1 = 1, 𝛾 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='222, ∆𝑎 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='72, ∆𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='63, which corresponds  to 𝑇𝑆𝑅 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='65, 𝜇𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='307, 𝜇𝑓 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='266, 𝜏𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='778, 𝜏𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='523 and the Figure 2d, built with  𝐾1 = 1, 𝛾 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='25, ∆𝑎 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='71, ∆𝑐 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='02, which corresponds to 𝑇𝑆𝑅 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='56, 𝜇𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='396, 𝜇𝑓 =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='31, 𝜏𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='73, 𝜏𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='545 describe unconventional "mixed"SR transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' At 𝜏𝑠 there is the smooth  second-order phase transition 𝐺𝑥 − 𝐺𝑥𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' At 𝜏 ≤ 𝜏𝑓 we have two stable phases 𝐺𝑧 and 𝐺𝑥𝑧: at those  temperatures the sharp first-order phase transition 𝐺𝑥𝑧 − 𝐺𝑧 can happen, or the system could stay in  the angular 𝐺𝑥𝑧-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  (a)  τ  τs  τf  μ  μs  μf  (c)  τ  τs  τf  μ  μs  μf  (d)  τ  τs  τf  μ  μs  μf  θ  Φ  θ  Φ  θ  Φ  τ > τf  τ < τs  A  (b)  τ  τs  τf  μ  μs  μf  A  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 2: Illustrations of typical (a,b) and unconventional (c,d) SR transitions predicted by simple  (quasi)doublet model (see text for detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The arrows indicate the direction of the antiferromagnetic  vector G in the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The insets in panel (b) show the 𝜃-dependence of the free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  7  Thus, there are not only the smooth and abrupt SR transitions, a characteristic feature of the range  of intermediate values 𝐾2 is the fundamental possibility of the existence of “mixed” SR transitions, in  which the spins first smoothly rotate through a certain angle and then jump to the position with 𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  For this, it is sufficient that 𝜇𝑓 corresponds to a point on the upper branch of solutions, and 𝜇𝑠 to a point  on the lower branch of solutions at 𝜏𝑓 < 𝜏𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' In this case, the spin reorientation begins with the single  second-order transition 𝐺𝑥 → 𝐺𝑥𝑧 and then ends with the first-order phase transition 𝐺𝑥𝑧 → 𝐺𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  In contrast to the “high-temperature” approximation, the “single-doublet” model claims the nature  of the phase transition is determined not simply by the sign of the second anisotropy constant, but  also it depends on the ratio between 𝐾1, 𝐾2 and the doublet splitting in both phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Nevertheless,  if we apply the simplified model to describe the SR transition, we have to renormalize both the first  and the second anisotropy constant, giving the last one sometimes a rather complicated temperature  dependence, in particular with a change in sign when considering transitions of the “mixed” type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Of course, in this case Fe sublattice alone is not enough to provide the value of the effective second  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  4 Conclusion  The model of the spin-reorientation transitions induced by the 4𝑓 − 3𝑑 interaction in rare-earth  orthoferrites and orthochromites has been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' It is shown that both the temperature and  the character of the spin-reorientation transition following from the solution of the transcendental  equation (8) are the result of competition between the second and fourth order spin anisotropy of  the 3𝑑 sublattice, the crystal field for 4f ions, and the 4𝑓 − 3𝑑 interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' At variance with the  “high-temperature” approximation, the “single-doublet” model, along with typical smooth and abrupt  SR transitions, predicts the appearance of mixed-type SR transitions, with an initial second-order  transition and a final abrupt first-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Funding: The research was supported by the Ministry of Education and Science of the Russian  Federation, project № FEUZ-2020-0054, and by Russian Science Foundation, project № 22-22-00682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  References  [1] Belov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Zvezdin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Kadomtseva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Levitin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Spin-reorientation transitions in  rare-earth magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Usp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 1976, 19, 574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [2] Belov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Zvezdin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Kadomtseva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Levitin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Orientational Transitions in Rare-  Earth Magnetics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Nauka: Moscow, Russia, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' (In Russian)  [3] Singh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Rajput, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Padmanabhan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Anas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Damay, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Kumar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Eguchi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Jain,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Yusuf, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Maitra, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Malik V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Successive spin reorientations and rare earth ordering  in Nd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5Dy0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5FeO3: Experimental and ab initio investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' B 2020, 102, 144432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  8  [4] Hoogeboom, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Kuschel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Bauer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Mostovoy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Kimel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' van Wees, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Magnetic order of Dy3+ and Fe3+ moments in antiferromagnetic DyFeO3 probed by spin Hall  magnetoresistance and spin Seebeck effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' B 2021, 103, 134406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [5] Tsymbal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Bazaliy, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Derkachenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Kamenev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Kakazei, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Palomares, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Wigen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' 2007, 101, 123919–123926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [6] Sasani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' I˜niguez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Bousquet, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Magnetic phase diagram of rare-earth orthorhombic  perovskite oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' B 2021, 104, 064431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [7] Moskvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Dzyaloshinskii–Moriya Coupling in 3d Insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Matter 2019, 4, 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [8] Moskvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Antisymmetric Exchange and Magnetic Anisotropy in Weak Ferromagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Thesis, Lomonosov Moscow State University, Moscow, Russia, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' (In Russian)  [9] Moskvin,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Structure–Property  Relationships  for  Weak  Ferromagnetic  Perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Magnetochemistry 2021, 7, 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [10] Kadomtseva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Agafonov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Lukina, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Milov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Semenov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Sinitsyn, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Nature of the Magnetic Anisotropy and Magnetostriction of Orthoferrites and  Orthochromites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' JETP 1981, 81, 700–706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [11] Hahn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Podlesnyak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Ehlers, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Pomjakushina, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Conder, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Inelastic neutron scattering studies of YFeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' B 2014,  89, 014420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [12] Park, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Sim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Leiner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Yoshida, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Jeong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Yano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Gardner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Bourges, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Klicpera,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Sechovsk´y, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Boehm, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Low-energy spin dynamics of orthoferrites AFeO3  (A = Y, La, Bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Matter 2018, 30, 235802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [13] Amelin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Nagel, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' R˜o˜om, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Terahertz absorption spectroscopy study of spin waves in orthoferrite YFeO3 in a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' B 2018, 98, 174417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [14] Moskvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Bostrem, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Cubic Anisotropy of Rare-Earth Orthoferrites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Solid St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  1979, 21, 628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+Ion filling of a one-dimensional nanofluidic channel in the interaction
+confinement regime
+Paul Robin,1 Adrien Delahais,1 Lyd´eric Bocquet,1 and Nikita Kavokine2, 3, a)
+1)Laboratoire de Physique de l’´Ecole Normale Sup´erieure, ENS, Universit´e PSL, CNRS, Sorbonne Universit´e,
+Universit´e Paris Cit´e, Paris, France
+2)Department of Molecular Spectroscopy, Max Planck Institute for Polymer Research, Ackermannweg 10,
+55128 Mainz, Germany
+3)Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010,
+USA
+(Dated: 12 January 2023)
+Ion transport measurements are widely used as an indirect probe for various properties of confined electrolytes.
+It is generally assumed that the ion concentration in a nanoscale channel is equal to the ion concentration
+in the macroscopic reservoirs it connects to, with deviations arising only in the presence of surface charges
+on the channel walls.
+Here, we show that this assumption may break down even in a neutral channel,
+due to electrostatic correlations between the ions arising in the regime of interaction confinement, where
+Coulomb interactions are reinforced due to the presence of the channel walls. We focus on a one-dimensional
+channel geometry, where an exact evaluation of the electrolyte’s partition function is possible with a transfer
+operator approach. Our exact solution reveals that in nanometre-scale channels, the ion concentration is
+generally lower than in the reservoirs, and depends continuously on the bulk salt concentration, in contrast to
+conventional mean-field theory that predicts an abrupt filling transition. We develop a modified mean-field
+theory taking into account the presence of ion pairs that agrees quantitatively with the exact solution and
+provides predictions for experimentally-relevant observables such as the ionic conductivity. Our results will
+guide the interpretation of nanoscale ion transport measurements.
+I.
+INTRODUCTION
+A channel connects two reservoirs filled with a salt so-
+lution at concentration cout. What is the salt concentra-
+tion cin inside the channel? The straightforward answer
+cin = cout is challenged as soon as the channel’s dimen-
+sions are at the nanometre scale1. A deviation typically
+occurs because of the presence of a surface charge density
+Σ on the channel walls. Indeed, a sufficiently long chan-
+nel must remain electrically neutral2, which results in an
+imbalance of the concentrations c±
+in of the positive and
+negative ions. In a cylindrical channel of radius R that
+is smaller than the electrolyte’s Debye length, the con-
+centrations are given by the famous Donnan equilibrium
+result3:
+c±
+in =
+�
+c2
+out + (2Σ/R)2 ± 2Σ/R.
+(1)
+Eq. (1) is widely used to infer a channel’s surface charge
+from measurements of its conductivity at different salt
+concentrations.
+For sufficiently small surface charges
+(2Σ/R ≪ cout), Eq. (1) predicts cin = cout even at ex-
+treme nanoscales.
+Importantly, this prediction under-
+lies the method for extracting confined ion mobilities
+from transport measurements, which has been applied
+down to 7-˚A-wide two-dimensional channels4. Yet, phys-
+ically, cin = cout stems from the assumption that the
+electrolyte solutions, both in the reservoirs and in the
+a)Electronic mail: nikita.kavokine@mpip-mainz.mpg.de
+channel, behave as ideal gases of non-interacting ions.
+While such a description is valid in the bulk reservoirs
+at reasonable salt concentrations5, it must be challenged
+in the nanometre-scale channel which is subject to in-
+teraction confinement6 – a reinforcement of the effective
+Coulomb interactions between the ions due to the dielec-
+tric contrast between the solvent (water) and the channel
+wall3,6–14.
+Due to interaction confinement, ions face a self-energy
+barrier Es when entering the channel7,8.
+It was first
+noted by Parsegian7 that this should result in ion exclu-
+sion: the salt concentration within the channel is then
+given by an Arrhenius scaling cin = coute−Es/kBT under
+the assumption of non-interacting ions. However, the re-
+sult becomes more subtle as the confinement-reinforced
+ionic interactions are taken into account.
+Within a
+mean-field description of a spherical nanopore, Dres-
+ner15 predicted an abrupt filling transition, where cin
+was a discontinuous function of cout.
+Later, Palmeri
+and coworkers16,17 recovered a similar transition using a
+three-dimensional model of a cylindrical channel, treated
+within the variational field theory formalism of Netz and
+Orland18.
+While this approach could be applied to a
+realistic geometry, it took into account electrostatic cor-
+relations only approximately.
+An exact treatment of electrostatic correlations is pos-
+sible upon simplification of the geometry to a purely
+one-dimensional model, with the channel wall being
+taken into account by introducing an effective confined
+Coulomb interaction.
+The 1D electrolyte can then be
+mapped onto an Ising or 1D Coulomb-gas-type model;
+the transfer matrix solution of such models was used, for
+arXiv:2301.04622v1  [cond-mat.soft]  11 Jan 2023
+
+2
+Effective interaction
+Self-energy
+B
+A
+FIG. 1.
+Ion filling in the interaction confinement regime.
+A. Schematic of the ion filling problem: a cylindrical
+nanochannel (radius R ∼ 1 nm) is connected to macroscopic reservoirs of aqueous electrolyte. The salt concentration inside
+the channel, cin, may differ from that in the reservoirs, cout. B. Physics of interaction confinement. When a charged species
+enters a nanochannel, the dielectric contrast between water (ϵw ∼ 80) and walls (ϵm ∼ 2) constraints the electric field lines to
+remain within the channel. This process can be interpreted in terms of image charges inside the channel walls, and results in
+an electrostatic self-energy barrier for ions to enter the channel, and reinforced interactions between ions.
+example, to discuss the capacitance of nanoporous sys-
+tems19–21. The lattice models may be taken to the con-
+tinuum limit, and the resulting path integral solutions
+have been used to discuss various ion-exchange phase
+transitions that arise in the presence of fixed discrete
+charges inside the channel9,22,23 and the ionic Coulomb
+blockade phenomenon13.
+Such models are particularly
+rich theoretically, as they support a mapping to non-
+Hermitian quantum mechanics24.
+Nevertheless, to our
+knowledge, the fundamental problem of ion filling in
+an uncharged channel has not been tackled within this
+framework.
+In this paper, we treat the ion-filling problem in the
+interaction confinement regime using an exactly-solvable
+one-dimensional model.
+We find that the value of cin
+is strongly affected by the formation of Bjerrum pairs
+– pairs of oppositely charged ions – within the channel,
+which preclude the occurence of an abrupt filling transi-
+tion. This is in contrast to the prediction of Palmeri and
+coworkers16,17, and to the result of conventional mean-
+field theory. We then build on our exact results to pro-
+pose a modified mean-field model that accounts for the
+relevant physical ingredients, and, particularly, for the
+presence of ion pairs.
+The paper is organized as follows.
+In Section II,
+we present the one-dimensional model and its solution
+within a path-integral formalism. The reader interested
+only in the physical outcomes may skip directly to Sec-
+tion III, where we discuss the model’s prediction for the
+ion concentration within the channel, compare it to the
+mean-field solution, and interpret it in terms of tightly
+bound Bjerrum pairs. In Section IV, we establish a mod-
+ified mean-field theory, based on the notion of phantom
+pairs, that reproduces our exact solution. The mean-field
+theory allows us to determine the number of unpaired
+ions and produces experimentally relevant predictions for
+a nanochannel’s ionic conductance. Section V establishes
+our conclusions.
+II.
+1D COULOMB GAS MODEL
+A.
+Confined interaction
+We consider a cylindrical channel of radius R and
+length L, connected to macroscopic reservoirs (Fig. 1A).
+We first assume for simplicity that the channel is filled
+with water that has isotropic dielectric permittivity ϵw =
+80, and that it is embedded in an insulating medium
+with much lower permittivity ϵm (for a lipid membrane7,
+ϵm ∼ 2).
+The effective Coulomb interaction V (x) be-
+tween two monovalent ions separated by a distance x
+on the channel axis can be computed exactly by solving
+Poisson’s equation8,12,13. A simple approximate expres-
+sion can be obtained for x ∼ R (ref.3):
+V (x) ≈
+e2α
+2πϵ0ϵwRe−|x|/(αR),
+(2)
+where α is a numerical coefficient that depends on the
+ratio ϵw/ϵm (α = 6.3 for ϵw/ϵm = 40). The reinforce-
+ment of electrostatic interactions compared to the usual
+e2/4πϵ0ϵwr Coulomb interaction that ions experience in
+bulk water can be interpreted in terms of images charges
+within the channel walls (Fig. 1B). Two confined ions
+interact not only with each other, but also with their
+respective image charges.
+We introduce the parameters ξ ≡ αR and xT
+≡
+2πϵ0ϵwR2kBT/e2: both have the dimension of a length.
+
+3
+With these notations,
+V (x) = kBT ξ
+xT
+e−|x|/ξ.
+(3)
+The effects of ion valence and of anisotropic dielectric
+response of confined water can be taken into account by
+adjusting ξ and xT 13. Formally, the expression in Eq. (2)
+is valid for any channel radius.
+Yet, it is only physi-
+cally relevant if at x ∼ R the interaction is significant
+compared to kBT, which restricts in practice the appli-
+cability of Eq. (2) to R ≲ 2 nm. In such extreme 1D
+confinement, we may neglect the ions’ degrees of free-
+dom perpendicular to the channel axis and assume that
+they are constrained to move in one dimension. The par-
+tition function of such a 1D electrolyte may be computed
+exactly, as detailed in the next section.
+B.
+Path integral formalism
+Here, we detail the analytical solution for the partition
+function of a 1D Coulomb gas-like system that was first
+introduced in ref.13. We set kBT = 1 until the end of Sec.
+II. We start from a lattice model, in order to rigorously
+establish a path integral description in the continuum
+limit.
+Our computation is inspired by the original solution of the 1D Coulomb gas model by Lenard and Edwards25, and
+subsequent studies by Demery, Dean and coworkers19,21,26,27, as well as Shklovskii and coworkers22,23. We consider
+a one-dimensional lattice with sites 1, . . . , M as a model for the nanochannel of radius R and length L. Each lattice
+site i carries a spin Si, which takes the values {0, 1, −1}, corresponding respectively to no ion, a positive ion, or a
+negative ion occupying the site. We model the surface charge distribution as an extra fixed charge qi added at each
+lattice site. The spins interact with the Hamiltonian
+H({Si}) =
+ξ
+2xT
+M
+�
+i,j=1
+(Si + qi)(Sj + qj)e−|i−j|/ξ ≡
+1
+2xT
+(S + q)T C(S + q).
+(4)
+The system is in contact with a particle reservoir at concentration cout. Here the parameters ξ and xT are dimension-
+less, expressed in number of lattice sites.
+The grand partition function is given by
+Ξ =
+�
+S1,...,SM
+z
+�
+i |Si|e−
+1
+2xT (S+q)T C(S+q),
+(5)
+with z = coutπR2L/M the fugacity. The matrix C can be analytically inverted:
+C−1 =
+1
+2ξ sinh(1/ξ) ·
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+e1/ξ
+−1
+0
+0
+. . .
+0
+0
+−1
+2 cosh(1/ξ) −1
+0
+. . .
+0
+0
+...
+...
+... ...
+...
+...
+...
+... ... ...
+...
+...
+...
+... ...
+...
+...
+0
+0
+. . .
+0
+−1 2 cosh(1/ξ)
+−1
+0
+0
+. . . . . .
+0
+−1
+e1/ξ
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(6)
+Hence we can carry out a Hubbard-Stratonovich transformation, that is rewrite the partition function as a gaussian
+integral, introducing the integration variable ϕ:
+Ξ =
+�
+xM
+T
+(2π)Mdet(C) ·
+�
+S1,...,SM
+z
+�
+i |Si|
+�
+dϕe− xT
+2 ϕT C−1ϕ+i(S+q)T ϕ,
+(7)
+with det(C) =
+e1/ξ
+2 sinh(1/ξ) ·
+�
+ξ(1 − e−2/ξ)
+�M. After performing the sum over the spins, which is now decoupled, we
+obtain
+Ξ =
+�
+xM
+T
+(2π)Mdet(C) ·
+�
+dϕ1 . . . dϕM
+M
+�
+j=1
+(1 + 2z cos ϕj)
+M
+�
+j=1
+eiqjϕj . . .
+. . . exp
+�
+�−
+xT
+4ξ sinh(1/ξ)
+�
+�
+M−1
+�
+j=1
+(ϕj+1 − ϕj)2 + 2(cosh(1/ξ) − 1)
+M−1
+�
+j=2
+ϕ2
+j + (e1/ξ − 1)(ϕ2
+1 + ϕ2
+M)
+�
+�
+�
+� .
+(8)
+
+4
+We now take a continuum limit of the lattice model. We call a the physical lattice spacing and let ˜ξ = aξ, ˜xT = axT
+and ˜z = Mz/L. We then let a → 0 and M → ∞ while keeping the physical length of the system L = aM constant.
+We then drop the tilde sign to lighten the notation and obtain
+Ξ =
+�
+dϕ(0)e−xT ϕ(0)2/4ξ
+�
+[dϕ]e−S[ϕ]
+�
+dϕ(L)e−xT ϕ(L)2/4ξ
+(9)
+with
+S[ϕ] =
+� L
+0
+dx
+�
+xT
+4
+�dϕ
+dx
+�2
++ xT
+4ξ2 ϕ(x)2 − iq(x)ϕ(x) − 2z cos ϕ(x)
+�
+≡
+� L
+0
+L(ϕ, ˙ϕ).
+(10)
+q(x) is the one-dimensional density corresponding to the surface charge, and z ≡ πR2cout. At this point ξ and xT
+have the dimension of length. The path integral measure is defined as
+[dϕ] =
+lim
+a→0
+M→∞
+L=aM
+�
+�
+M
+�
+j=1
+� xT
+4πadϕj
+�
+� .
+(11)
+We now define the propagator P(ϕ, x|ϕ0, 0), or simply P(ϕ, x), as
+P(ϕ, x) =
+�
+dϕ(x)δ(ϕ(x) − ϕ)
+�
+[dϕ]e−
+� x
+0 L(ϕ, ˙ϕ)
+�
+dϕ(0)δ(ϕ(0) − ϕ0).
+(12)
+Considering an infinitesimal displacement ∆x,
+P(ϕ, x) =
+� xT
+4π∆x
+�
+d(∆ϕ)P(ϕ − ∆ϕ, x − ∆x) . . .
+. . . exp
+�
+−
+� x
+x−∆x
+dx′
+�
+xT
+4
+�∆ϕ
+∆x
+�2
++ xT
+4ξ2 ϕ2 − iq(x)ϕ − 2z cos ϕ
+��
+.
+(13)
+Expanding the propagator as P(ϕ − ∆ϕ, x − ∆x) = P(ϕ, x) − ∆x∂P/∂x − ∆ϕ∂P/∂ϕ + (1/2)(∆ϕ2)∂2P/∂ϕ2, and
+carrying out the gaussian integrals, we obtain
+P(ϕ, x) =
+�
+P(ϕ, x) − ∆x∂P
+∂x + O(∆x2)
+� �
+1 − ∆x
+� xT
+4ξ2 ϕ2 − iq(x)ϕ − 2z cos ϕ
+�
++ O(∆x2)
+�
++ ∆x
+xT
+∂2P
+∂x2 (1 + O(∆x)).
+(14)
+P(ϕ, x) thus solves the partial differential equation
+∂P
+∂x = 1
+xT
+∂2P
+∂ϕ2 +
+�
+iqϕ − xT
+4ξ2 ϕ2 + 2z cos ϕ
+�
+P,
+(15)
+with initial condition P(ϕ, 0) = δ(ϕ − ϕ0), which is the equivalent of a Schr¨odinger equation for the path integral
+representation (9). The partition function can thus be computed as
+Ξ =
+�
+dϕ(L)e−xT ϕ2/4ξP(ϕ, L|f0),
+(16)
+where P(ϕ, L|f0) is the solution of (15) with initial condition P(ϕ, 0) = f0(ϕ) ≡ e−xT ϕ2/4ξ.
+C.
+Transfer operator
+We introduce the Fourier transform of P with respect to ϕ:
+˜P(k, x) =
+1
+√
+2π
+�
+dϕe−ikϕP(ϕ, x).
+(17)
+
+5
+Then ˜P(k, x) satisfies
+∂ ˜P
+∂x = − k2
+xT
+˜P − q ∂ ˜P
+∂k + xT
+4ξ2
+∂2 ˜P
+∂k2 + z
+�
+˜P(k + 1, x) + ˜P(k − 1, x)
+�
+.
+(18)
+From now on, we restrict ourselves to an uncharged channel (q = 0). We then define the operator T such that
+[T ( ˜P)](k) = − k2
+xT
+˜P + xT
+4ξ2
+∂2 ˜P
+∂k2 + z
+�
+˜P(k + 1, x) + ˜P(k − 1, x)
+�
+,
+(19)
+which plays the role of a functional transfer matrix. Recalling eq. (16), the partition function then reads
+Ξ = ⟨f0|eLT |f0⟩
+(20)
+with f0(k) = e−ξk2/xT and ⟨f(k)|g(k)⟩ ≡
+�
+dkf ∗(k)g(k).
+Now, in the limit L → ∞, we may consider the largest eigenvalue λ of the operator T , and the associated eigen-
+function χ:
+[T (χ)](k) = λχ(k).
+(21)
+Then, up to an exponentially small correction,
+Ξ = |⟨f0|χ⟩|2⟨χ|χ⟩eλL.
+(22)
+D.
+Ion concentration
+Our aim is to compute the salt concentration cin in the nanoscale channel given a salt concentration cout in the
+reservoir. At the level of the lattice model, the probability to find, say, a positive ion at position k, can be computed
+by replacing a factor (1 + 2z cos ϕk) by zeiϕk in Eq. (8). In the continuum limit, we obtain the positive (negative) ion
+linear density at position x by inserting the operator zeiϕ (ze−iϕ) at position x:
+πR2⟨c±
+in(x)⟩ = 1
+Ξ
+�
+dϕ(0)dϕ(x)dϕ(L)e−xT ϕ(0)2/4ξP(ϕ(x), x|ϕ(0), 0)ze±iϕ(x)P(ϕ(L), L|ϕ(x), x)e−xT ϕ(L)2/4ξ,
+(23)
+Upon Fourier-transformation, the insertion of eiϕ amounts to a shift by unity. Introducing the operator,
+SQ : f �→ (g : k �→ f(k − Q)),
+(24)
+the concentrations are given by
+⟨c±
+in(x)⟩ =
+z
+πR2
+⟨f0|exT S±1e(L−x)T |f0⟩
+Ξ
+= cout
+⟨f0|exT S±1e(L−x)T |f0⟩
+Ξ
+,
+(25)
+since z = coutπR2. In the thermodynamic limit, and using Eq. (22) for the partition function, we obtain
+⟨c±
+in⟩ = cout
+⟨χ(k)|χ(k ∓ 1)⟩
+⟨χ(k)|χ(k)⟩
+.
+(26)
+Eq. (26) is the main result of our exact computation. In practice, the function χ(k) is determined numerically, by
+finite-difference integration of Eq. (18).
+III.
+PHYSICS OF ION FILLING
+A.
+Debye-H¨uckel solution
+We now go back to the ion filling problem (Fig. 1A)
+and present first a one-dimensional mean-field solution.
+Typically, the mean-field solution of an electrolyte prob-
+lem is obtained by solving the Poisson-Boltzmann equa-
+tion28,29. For the conventional Poisson-Boltzmann equa-
+tion to apply, we would need to consider the full three-
+dimensional geometry of our problem, and the effective
+interaction of Eq. (3) would be introduced implicitly
+through the boundary conditions at the channel walls15.
+
+6
+B
+A
+C
+Concentration
+Distance
+Anions
+Cations
+Debye cloud
+10-4
+10-2
+100
+Reservoir concentration (M)
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Channel conc./Res. conc.
+Exact solution
+Series expansion
+Poisson-Boltzmann
+Debye-Hückel
+10-4
+10-2
+100
+Reservoir concentration (M)
+10-3
+10-2
+10-1
+100
+Channel conc./Res. conc.
+Exact solution
+Series expansion
+Poisson-Boltzmann
+Debye-Hückel
+Bulk
+Self-energy
+barrier
+Bulk
+Self-energy
+barrier
+Ion pairs
+Weak interactions:
+Es = 0.5 kBT
+Strong interactions:
+Es = 6 kBT
+FIG. 2. Comparing mean-field approximations with the exact Coulomb gas solution. A. Schematic description
+of the mean-field approaches.
+The chemical potential of confined ions is determined by solving the (linear or nonlinear)
+Poisson-Boltzmann equation around a given ion, interacting with an oppositely charged Debye cloud.
+B. Dependence of
+the channel salt concentration cin on the reservoir salt concentration cout, in a weakly-interacting case (R = 1 nm, ξ = 7 nm,
+xT = 7 nm, Es = 0.5 kBT). We plot four different predictions for the ratio cin/cout: the exact field-theoretical solution (Eq. (26),
+blue circles), its low concentration expansion (Eq. (47), black line), the mean-field predictions from solving the full Poisson-
+Boltzmann equation (Eq. (40), orange curve) or from its Debye-H¨uckel linearization (Eq. (36), yellow line). The two mean-field
+predictions are indistinguishable.
+In all cases, the naive estimate cin = cout is recovered for high enough concentrations.
+In the dilute limit, the concentration inside the channel is well approximated by the Arrhenius scaling cin = coute−Es/kBT .
+C. Dependence of the channel salt concentration cin on the reservoir salt concentration cout, in a strongly-interacting case
+(R = 1 nm, ξ = 7 nm, xT = 0.6 nm, Es = 6 kBT). The color code is the same as in B. Here, the mean-field predictions strongly
+deviate from the exact solution, with the Debye-H¨uckel model predicting an abrupt filling transition. This discrepancy is due
+to the formation of Bjerrum pairs at intermediate concentrations, as evidenced by the scaling cin ∝ c2
+out in the exact solution.
+In order to obtain a mean-field solution directly in the
+1D geometry, we need to introduce a modified Poisson’s
+equation for the electrostatic potential Φ whose Green’s
+function coincides with Eq. (3):
+� d2
+dx2 − 1
+ξ2
+�
+φ = −2πR2 c+ − c−
+xT
+,
+(27)
+with φ ≡ eΦ/kBT the dimensionless potential.
+Im-
+posing that the ions follow a Boltzmann distribution
+(c± = cine∓φ, where cin is understood as the average con-
+centration inside the channel), we obtain the analogue of
+the Poisson-Boltzmann equation in our 1D geometry:
+� d2
+dx2 − 1
+ξ2
+�
+φ = 2πR2 cin
+xT
+sinh φ.
+(28)
+In order to proceed analytically, we make a Debye-
+H¨uckel-type approximation and linearize Eq. (28) with
+respect to φ. Then, the potential around an ion placed
+in the channel at x = 0 is given by
+φ(x) = ξeff
+xT
+e−|x|/ξeff,
+(29)
+with
+ξ2
+eff =
+ξ2
+1 + 4πR2cinξ2/xT
+.
+(30)
+The chemical potential inside the channel is the sum of
+an ideal gas entropic part and of an excess part due to
+interactions:
+µin = µent + µex,
+(31)
+with
+µent = kBT log coutΛ3,
+(32)
+Λ being the De Broglie thermal wavelength of the ions.
+µex can be obtained via a Debye charging process30:
+µex
+kBT =
+� 1
+0
+φλ(0)dλ, φλ(0) =
+λξ/xT
+�
+1 + 4λπR2cinξ2/xT
+.
+(33)
+We determine cin by imposing equality of the chemical
+potentials between the channel and the reservoir:
+µout = kBT log coutΛ3 = µin,
+(34)
+which yields
+cin = coute−µex/kBT .
+(35)
+Evaluating analytically the integral in Eq. (33), we obtain
+an implicit equation for cin.
+With the notation ˆcin ≡
+πR2cin,
+cin = cout exp
+�
+− ξ
+2xT
+×
+x2
+T
+6ξ2ˆc2
+inξ2
+�
+1 − 3
+2(1 + 4ˆcinξ2/xT )1/2
++1
+2(1 + 4ˆcinξ2/xT )3/2
+��
+.
+(36)
+
+7
+In Fig. 2B and C, we plot the ratio cin/cout as a func-
+tion of cout, as obtained by numerically solving Eq. (36).
+We fix ξ = 7 nm (which corresponds to a channel with
+R ≈ 1 nm and strong dielectric contrast), and vary
+xT to set the ionic interaction strength.
+The interac-
+tion strength may be quantified through the self-energy
+barrier, Es = kBT × ξ/(2xT ). The limiting behavior of
+cin/cout may be understood directly from Eq. (36). When
+cin is small, Eq. (36) reduces to the Arrhenius scaling
+cin = coute−Es/kBT : this results typically holds for bio-
+logical ion channels which may contain either 0 or 1 ion at
+any given time, and the effect of inter-ionic interactions
+is negligible. When cin is large, we recover cin = cout. In-
+deed, the excess term in the chemical potential vanishes
+at high concentrations, which is then dominated by the
+entropic term. The fact that µex → 0 as cin → ∞ is non-
+trivial: it can be seen, physically, as resulting from the
+Coulomb potential of each ion being perfectly screened
+by the other ions. At small values of Es, Eq. (36) has
+a single solution for all values of cout, which interpolates
+smoothly between the two limiting regimes.
+However,
+for Es ≳ 5kBT, it has three solutions in a certain range
+of cout, pointing to a pseudo-first-order phase transition
+between a low-concentration and a high-concentration
+phase, similar to the one predicted by Dresner15 and
+Palmeri et al.16. The transition occurs at ˆcin ∼ xT /ξ2: as
+per Eq. (30), this corresponds to the concentration where
+the effect of the screening cloud on an ion’s Coulomb po-
+tential becomes significant.
+B.
+Full Poisson-Boltzmann solution
+The physical content of the mean-field solution pre-
+sented above is similar to the one of Dresner, based on
+a linearized Poisson-Boltzmann equation15. The differ-
+ence in geometry, and the fact that he foregoes the use
+of the Debye charging process, do not seem to play a sig-
+nificant qualitative role. The solution of Palmeri et al.16
+takes ionic correlations into account to some extent, yet
+it still involves a Debye-H¨uckel-type linear equation for
+the mean-field interaction potential between the ions.
+One may ask whether the same phenomenology per-
+sists if one does not linearize the Poisson-Boltzmann
+equation. The full Poisson-Boltzmann equation cannot
+be solved analytically, but supports the following inte-
+gral form:
+�dφ
+dx
+�2
+− 1
+ξ2 φ2 = 4πR2 cin
+xT
+(cosh φ − 1) ,
+(37)
+where we have used the fact that φ should vanish at x →
+∞. For x → 0, the solution of Eq. (37) should reduce
+to the unscreened potential in Eq. (3) up to an additive
+constant, so that
+1
+x2
+T
+− 1
+ξ2 φ2(0) = 4πR2 cin
+xT
+(cosh φ(0) − 1) .
+(38)
+Once again, one may express the excess chemical po-
+tential of the confined ions through a Debye charging
+process:
+µex
+kBT =
+� 1
+0
+φλ(0)dλ,
+λ2
+x2
+T
+− 1
+ξ2 φ2
+λ(0) = 4πR2 λcin
+xT
+(cosh φλ(0) − 1) .
+(39)
+This result is the analogue of Eq. (33), with φλ(0) now
+being the solution of an implicit non-linear equation, so
+that µex must be determined numerically. As before, the
+concentration inside the channel is then given by:
+cin = coute−µex/kBT .
+(40)
+The prediction of the full Poisson-Boltzmann equation
+is shown in Fig. 2B and C: we find cin to be a smooth
+function of cout for all values of parameters, in contrast to
+the linearized solution. We may not, however, unambigu-
+ously conclude that the filling transition is an artifact of
+linearization, since the non-linear solution still involves a
+mean-field approximation and is not guaranteed to yield
+the correct result.
+Interestingly, the “physically-motivated” mean-field
+solution in Eq. (28) differs from the mean-field limit of
+our exact solution. It is obtained by taking the saddle-
+point approximation in the path-integral expression of
+the partition function (Eq. (9)).
+The Euler-Lagrange
+equation for the minimizer ϕ(x) of the action S[ϕ] in
+Eq. (10) is, upon identifying φ = −iϕ,
+� d2
+dx2 − 1
+ξ2
+�
+φ = 2πR2 cout
+xT
+sinh φ.
+(41)
+This is Eq. (28) with cin replaced with cout, and corre-
+sponds to a first order treatment of interactions. Indeed,
+if the ions are non-interacting, cin = cout. By solving the
+mean-field equation, we determine how the ions’ chemi-
+cal potential is affected by Debye screening, which then
+results in value of cin that is different from cout. Within
+a straightforward interaction expansion procedure, one
+should determine the effect of screening assuming the ze-
+roth order value for the ion concentration inside the chan-
+nel, which is cout: this corresponds to Eq. (41). Eq. (28)
+contains an additional self-consistency condition, as it
+assumes the actual value cin for the ion concentration,
+which is not known until Eq. (28) is solved. One may
+draw a loose condensed matter physics analogy, where
+Eq. (41) resembles the Born approximation for impu-
+rity scattering, while Eq. (28) is analogous to the self-
+consistent Born approximation.31
+C.
+Exact solution
+We now turn to the exact solution obtained in Sec.
+II to unambiguously solve the ion filling problem. We
+
+8
+determine cin according to Eq. (26):
+⟨c±
+in⟩ = cout
+⟨χ(k)|χ(k ∓ 1)⟩
+⟨χ(k)|χ(k)⟩
+,
+(42)
+where χ(k) is the highest eigenfunction of the trans-
+fer operator in Eq. (19), determined in practice by nu-
+merical integration. The exact results for cin, with the
+same parameter values as for the mean-field solution,
+are shown in Fig. 2 B and C. When interactions are
+weak (small values of Es, Fig. 2B), the exact and mean-
+field solutions are in good agreement. Notably, all so-
+lutions smoothly interpolate between the bulk scaling
+cin = cout at high concentration, and the Arrhenius scal-
+ing cin = coute−Es/kBT at low concentration. Conversely,
+in the strongly-interacting case (large Es, Fig. 2C), the
+exact result yields a much larger ion concentration that
+the mean-field solutions for intermediate values of cout.
+In this intermediate regime, cin remains a smooth func-
+tion of cout, and obeys the scaling cin ∝ c2
+out.
+Such a scaling is the signature of the formation of
+tightly bound Bjerrum pairs of positive and negative ions
+– strongly-correlated configurations that are not taken
+into account by mean-field solutions. Indeed, let us as-
+sume that the channel contains an ideal gas of ion pairs at
+concentration cin. We further assume that in a pair, the
+distance between the two ions is uniformly distributed
+in the interval [−xT /2, xT /2], and the binding energy of
+a pair is kBTξ/xT = 2Es.
+Then, the grand partition
+function reads
+Ξ =
+�
+N
+(ze−βEs)2N 1
+N!
+N
+�
+i=1
+L
+� xT /2
+−xT /2
+dx e2βEs
+(43)
+=
+�
+N
+(z2LxT )N
+N!
+= ez2LxT ,
+(44)
+where we recall that z = πR2cout and β ≡ 1/(kBT).
+Using that
+πR2cin = 1
+L
+∂ log Ξ
+∂(βµ) = z
+L
+∂ log Ξ
+∂z
+,
+(45)
+we obtain
+cin = 2z2xT
+πR2
+= 2πR2xT c2
+out.
+(46)
+We recover indeed the quadratic scaling.
+We may check that the prefactor in Eq. (46) is the
+correct one by evaluating analytically the expression in
+Eq. (26) in the low concentration limit zT ≡ zxT ≪ 1.
+An analytical expansion of the function χ(k) in powers
+of zT was derived in ref.13. Substituting it into Eq. (26),
+we obtain
+πR2cin = z(e−βEs + 2zT − 13
+2 z2
+T e−βEs
+−7z3
+T + O(z4
+T ) + O(e−2βEs)).
+(47)
+The first term in the expansion corresponds to cin =
+coute−βEs.
+At the lowest salt concentrations, forming
+Bjerrum pairs is too entropically unfavorable, and the
+concentration inside the channel is controlled by the self-
+energy barrier.
+However, as the salt concentration in-
+creases, there is no abrupt transition to a highly-screened
+concentrated phase inside the channel; instead, the chan-
+nel is progressively filled by Bjerrum pairs. This corre-
+sponds to the quadratic term in the expansion, with the
+prefactor agreeing indeed with Eq. (46).1 The expansion
+in Eq. (47) reproduces quite well the low-concentration
+behavior of the exact solution as shown in Fig. 2B and
+C. However, it fails at high concentrations, where it does
+not recover cin = cout.
+Our exact analysis of the ion statistics in a nanoscale
+channel has revealed that Bjerrum pairs are a crucial in-
+gredient of the filling process. We now develop a modified
+mean-field theory that accounts the presence of Bjerrum
+pairs and compare it to the exact solution.
+IV.
+PAIR-ENHANCED MEAN-FIELD THEORY
+A.
+Debye-H¨uckel-Bjerrum theory
+The traditional mean-field treatment of electrolytes is
+incapable of taking Bjerrum pairs into account, as it nat-
+urally neglects any strong ion-ion correlations – pairing
+being a fundamentally discrete phenomenon.
+An idea
+proposed by Bjerrum to amend the Debye-H¨uckel theory
+was to introduce ion pairs as a separate species encapsu-
+lating all “strong” ion-ion correlations32. More precisely,
+any two oppositely charged ions that are closer than some
+minimum distance can be considered as a single neutral
+entity – a Bjerrum pair. The remaining “free” ions should
+then only experience weak interactions with each other,
+and can be treated at the mean-field level. Importantly,
+this last remark justifies the Debye-H¨uckel linearization,
+as all non-linear effects are assumed to be hidden in the
+definition of ion pairs.
+As before, we consider that pairs behave like particles
+of an ideal gas, and that their maximum extension is
+given by xT . Defining cp
+in the concentration pairs inside
+the channel, the chemical potential of pairs is given by:
+µp
+in = kBT log
+cp
+inΛ6
+2πxT R2 ,
+(48)
+where the geometrical factor inside the logarithm ac-
+counts for the internal degrees of freedom of a pair. The
+chemical potential only has an entropic term, because
+the binding energy of the pair exactly compensates the
+self-energy of the two separate ions. The chemical equi-
+librium between free ions and pairs inside the channel
+1 This justifies a posteriori our choice of [−xT /2, xT /2] as the
+interval in which a paired-up ion is allowed to move.
+
+9
+Concentration
+Distance
+Anions
+Cations
+B
+A
+C
+Debye cloud
+Bjerrum pair
+Well-defined
+pair
+Phantom pair
+10-4
+10-2
+100
+Reservoir concentration (M)
+10-3
+10-2
+10-1
+100
+101
+102
+Channel conc./Res. conc.
+Exact solution
+Debye-Hückel-Bjerrum mean-field
+Phantom pair mean-field
+Strong interactions:
+Es = 6 kBT
+FIG. 3. Pair-enhanced mean-field theory. A. Treatment of ion pairing in mean-field approaches. Top panel: Mean-field
+theories inevitably underestimate ion-ion correlations. To circumvent this problem, two ions that are distant by less than xT
+are considered to form an ion pair, which is treated as a separate chemical species. Bottom panel: schematic representation
+of ion distribution around a fixed positive ion. The distribution is very peaked close to the central ion, due to the formation
+of an ion pair, and then relaxes smoothly to the mean value cin. B. Evolution of channel concentration cin as function of
+reservoir concentration cout, in a strongly-interacting cacse (R = 1 nm, ξ = 7 nm, xT = 0.6 nm, Es = 6 kBT). We plot the ratio
+cin/cout obtained from three different models taking Bjerrum pairs into account: the exact field-theoretical solution (Eq. (26),
+blue circles), the Debye-H¨uckel-Bjerrum mean-field theory (Eq. (51), red line) and our modified mean-field theory based on the
+notion of phantom pairs (Eq. (55), orange line), which reproduces the exact solution quantitatively for all values of parameters.
+At high concentration, the Debye-H¨uckel-Bjerrum prediction fails due to the uncontrolled proliferation of Bjerrum pairs. C.
+Formation of phantom pairs inside the nanochannel. At low concentration (top panel), pairs are well-separated and ions forming
+a pair are tightly bound to each other. At high concentration (bottom panel), ionic interactions are weakened as a result of
+Debye screening, and two quasi-non-interacting ions may find themselves within a distance xT of each other without actually
+binding: this is a phantom pair.
+can be written as:
+µ+
+in + µ−
+in = 2µin = µp
+in,
+(49)
+where µ+
+in and µ−
+in are the chemical potentials of cations
+and anions, respectively.
+We then obtain, using the
+Debye-H¨uckel solution for µin (equations (31) to (33)):
+cp
+in = 2πR2xT c2
+out,
+(50)
+which is the result obtained in the previous section. The
+average concentration in free ions cf
+in is not modified com-
+pared to the Debye-H¨uckel solution, and is therefore the
+solution of the self-consistent Eq. (36).
+One can then
+compute the total concentration inside the channel as
+cin = cf
+in + cp
+in, or, explicitly
+cin = coute−µex(cf
+in)/kBT + 2πR2xT c2
+out.
+(51)
+In other words, the only impact of pairs in Bjerrum’s
+computation is to add a quadratic term 2πR2xT c2
+out to
+the Debye-H¨uckel result, matching with the expansion
+(47) of the exact solution up order 2 in the bulk concen-
+tration. We compare the two predictions on Fig. 3B. The
+Debye-H¨uckel-Bjerrum solution is found to match the ex-
+act one quite well at low and intermediate concentrations.
+This result is, however, unphysical for cout ≳ 1/πR2xT :
+cin is found to grow much faster than the bulk concen-
+tration. One solution would be to consider higher-order
+terms in the mean-field treatment through the inclusion
+of triplets, quadruplets, etc. of ions, and all possible in-
+teractions between these entities.
+Truncating the sum
+at any finite order, however, would not yield a solution
+valid in the entire range of concentrations, nor is it guar-
+anteed to converge to the exact solution. This approach
+is also unsatisfactory as it would not yield a closed-form
+expression for cin and would not allow for qualitative un-
+derstanding of the underlying physics.
+Instead, we develop a different method that, through
+physics-driven arguments, prevents the divergence of cin
+at high bulk concentrations and reproduces quantita-
+tively the exact solution.
+B.
+Phantom pairs
+Eq. (51) overestimates the number of Bjerrum pairs in
+the channel because it fails to account for the presence
+of Bjerrum pairs in the reservoir. The electrolyte in the
+reservoir is treated as an ideal gas : the ions are non-
+interacting and they cannot form actual tightly-bound
+pairs. Nevertheless, we have defined any two oppositely
+charged ions that find themselves in a cylinder of radius
+R and length xT to be a separate chemical species. Such
+configurations may arise in the reservoir simply out of
+statistical chance: we dub them phantom pairs. For our
+
+10
+mean-field theory to be consistent, these phantom pairs
+need to be taken into account.
+Let cp
+out be the concentration of phantom pairs in the
+reservoir.
+The chemical equilibrium between phantom
+pairs and free ions imposes
+cp
+out = 2πR2xT (cf
+out)2.
+(52)
+In addition, one has cf
+out + cp
+out = cout, since an ion must
+either be free or part of a pair. Imposing this condition
+yields:
+cf
+out =
+√
+1 + 8πcoutxT R2 − 1
+4xT πR2
+.
+(53)
+We use this result to control the proliferation of pairs in
+the channel: we now equilibrate the free ions inside the
+nanochannel with only the free ions in the reservoir:
+cf
+in = cf
+oute−µex(cf
+in)/kBT ,
+(54)
+which corresponds to Eq. (35) with cout replaced by cf
+out.
+Eq. (54) is again a self-consistent equation, this time on
+the concentration of free ions cf
+in, that must be solved
+numerically. Lastly, equilibrating pairs with free ions in-
+side the channel (or, equivalently, pairs inside with pairs
+outside), we obtain:
+cin = cf
+in + 2πR2xT (cf
+out)2,
+(55)
+where the second term corresponds again to Bjerrum
+pairs. Eqs. (53) to (55) constitute the main result of our
+modified mean-field theory. Note that µex may be deter-
+mined at the Debye-H¨uckel level (Eq. (33)), or by solving
+the full Poisson-Boltzmann equation (Eq. (39)). In what
+follows, we will only discuss the latter, as it offers greater
+accuracy; however, the Debye-H¨uckel prediction provides
+reasonable results even in the case of strong interactions,
+and yields for a convenient analytical expression for µex
+as function of cf
+in.
+The
+prediction
+of
+our
+phantom
+pair
+Poisson-
+Boltzmann model is compared to the exact solution (26)
+in Fig. 3B. The two solutions are found to be in near
+perfect agreement for all values of parameters, even in
+strong coupling limit Es ≫ kBT.
+In the next two sections, we use our modified mean-
+field model to predict the conductance of a nanochannel,
+first in the case of a neutral channel, and then in presence
+of a surface charge.
+C.
+Conductance
+One strength of our modified mean-field model is that
+it offers insight into the physical properties of the con-
+fined system beyond the value of the ionic concentra-
+tion. In particular, the decomposition of the electrolyte
+into free ions and bound pairs allows us to estimate the
+channel’s conductance. Tightly bound Bjerrum pairs are
+electrically neutral, so that they do not contribute to the
+ionic current to first order in applied electric field: it
+would then be straightforward to assume that the chan-
+nel’s conductance is proportional to the concentration
+of free ions. However, the reasoning needs to be more
+subtle, since the channel, in the same way as the reser-
+voir, may contain non-interacting phantom pairs.
+In-
+deed, we have decomposed the confined electrolyte into
+tightly bound pairs, that have no ionic atmosphere, and
+free ions that are dressed by a Debye screening cloud.
+As the concentration increases, the interaction between
+dressed ions becomes weak, and two of them may find
+themselves within a distance xT without actually bind-
+ing. Such a phantom pair is expected to still contribute
+to the conductance. The concentration of phantom pairs
+in the channel is obtained by imposing their chemical
+equilibrium with the free ions treated as an ideal gas.
+Thus, we estimate the channel’s conductance as:
+G = 2 e2D
+kBT
+πR2
+L
+�
+cf
+in + 2xT πR2(cf
+in)2�
+,
+(56)
+where D is the diffusion coefficient of ions; the second
+term corresponds to the contribution of phantom pairs.
+In Fig. 4A, we compare this result to the Ohm’s law
+prediction where pairs are neglected and one assumes
+cin = cout. Ohm’s law is found to greatly overestimate
+the conductance at low concentration. In the dilute limit,
+we instead recover the Arrhenius scaling, where one as-
+sumes cin = coute−Es/kBT .
+Finally, we stress that Eq. (56) only accounts for the
+electrophoresis of free ions, and is therefore only valid
+in the limit of weak external electric fields.
+Stronger
+voltage drops will result in the breaking of ion pairs,
+causing a conductivity increase in a process known as
+the second Wien effect. This phenomenon is described in
+refs.13,14, and has been used to create solid-state voltage-
+gated nanochannels33.
+D.
+Effect of a surface charge
+Up till now, we have restricted ourselves to channels
+with uncharged walls.
+However, in most experimen-
+tally relevant situations, the channel walls bear a sur-
+face charge density Σ, which strongly impacts nanofluidic
+transport. While introducing a surface charge is tedious
+within the exact framework, we may readily assess the
+effect of surface charge in the interaction confinement
+regime using our pair-enhanced mean-field theory.
+In the limit where the channel’s radius is smaller than
+the Debye length, we assume that the presence of the
+surface charge amounts to a homogeneous Donnan po-
+tential drop VD inside the channel, which we do not need
+to determine explicitly. Then, the chemical potential of
+ions inside the channel reads:
+µ±
+in = µex ± eVD + kBT log c±
+inΛ3.
+(57)
+
+11
+B
+A
+10-3
+10-2
+10-1
+100
+101
+Reservoir concentration (M)
+10-6
+10-4
+10-2
+100
+Channel conductance (nS)
+Ohm law
+Phantom pair mean-field
+Arrhenius model
+Actual surface charge
+10-3 C/m2
+Apparent surface charge
+10-2 C/m2
+10-3
+10-2
+10-1
+100
+101
+Reservoir concentration (M)
+10-2
+10-1
+100
+101
+Conductance (nS)
+Donnan equilibrium
+Phantom pair mean-field
+FIG. 4. Channel conductance in the pair-enhanced mean-field model. A. Conductance of a nanochannel (R = 1 nm,
+ξ = 7 nm, xT = 0.7 nm, Es = 10 kBT) as function of the reservoir concentration. The red line corresponds to the prediction of
+the phantom pair mean-field model (Eq. (56)) for T = 300 K, D = 10−9 m2/s and L = 100 nm. The Ohm’s law bulk prediction
+(cin = cout, blue line) and the Arrhenius model (cin = coute−Es/kBT , yellow line) are also represented for comparison. B.
+Conductance of a nanochannel with a weak surface charge Σ = 10−3 C/m2. We represented the predictions of the conventional
+Donnan equilibrium (Eq. (1), blue line) and of the phantom pair mean-field theory (equations (56) and (59), red line). Because
+interaction confinement results in a lower ion concentration in the channel, the usual formula Σ ∼ Rc∗/2, where c∗ is the reservoir
+concentration for which conductance levels off overestimates the surface charge by one order of magnitude, as indicated on the
+plot.
+Note that the concentration in free anions c−
+in and cations
+c+
+in are now distinct, so that µex is defined as a function of
+the average free ion concentration cf
+in = (c+
+in+c−
+in)/2. In a
+channel that is sufficiently long for local electroneutrality
+to hold,
+c+
+in − c−
+in + 2Σ/R = 0.
+(58)
+Imposing chemical equilibrium with the reservoir, we ob-
+tain a modified version of the Donnan result (Eq. (1)):
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+cin = cf
+in + cp
+in
+cf
+in =
+��
+cf
+oute−βµex(cf
+in)�2
++
+� 2Σ
+R
+�2,
+cp
+in = 2πR2xT (cf
+out)2,
+(59)
+with cf
+out given by Eq. (53).
+One can again obtain the channel’s conductance
+through Eq. (56), which we compare to the Donnan /
+Ohm’s law result in Fig. 4B. Importantly, the Donnan
+result predicts that conductance becomes independent of
+concentration for cout ∼ 2Σ/R (see Eq. (1)). In practice,
+this result is commonly used to estimate experimentally
+the surface charge as Σ ∼ Rc∗/2, where c∗ is the reser-
+voir concentration for which conductance levels off. In
+contrast, in the interaction confinement regime, we pre-
+dict that the transition occurs instead at cf
+in ∼ 2Σ/R –
+corresponding to a higher reservoir concentration, due to
+the self-energy barrier. In this case, Donnan’s prediction
+overestimates the surface charge by typically one order
+of magnitude, as shown in Fig. 4B.
+Finally, let us stress that we considered here a charge
+homogeneously distributed along the channel’s surface.
+This assumption is relevant in the case of conducting
+wall materials, such as systems where the charge is im-
+posed via a gating electrode connected to the chan-
+nel walls.
+This situation, however, may be different
+in experimentally-available devices, where the surface
+charge generally consists in localized charged groups and
+defects on the channel walls. In this case, the physics be-
+come more involved as ions may form bound pairs with
+the fixed surface charges.
+Some of these physics have
+been revealed by the exact computations of Shklovskii
+and coworkers9,22; a technically simpler approach to
+these physics using our pair-enhanced mean-field theory
+would be possible, but extends beyond the scope of the
+present work.
+V.
+DISCUSSION AND PERSPECTIVES
+We have determined the salt concentration inside a
+nanometric channel connected to reservoirs filled with
+electrolyte.
+In the case of a fully 1D geometry, corre-
+sponding to a nanotube of radius R ∼ 1nm, we devel-
+oped an exact field-theoretical solution that allowed us
+to compute channel concentration cin as function of the
+reservoir concentration cout. This solution clears up the
+ambiguities of pre-existing mean-field theories, and con-
+tradicts the naive expectation cin = cout. In particular,
+the concentration inside the nanochannel is found to be
+always lower than in the bulk, as the confinement of elec-
+trostatic interactions creates an energy barrier for ions to
+
+12
+enter the channel.
+Yet, we found that cin is in fact higher than the predic-
+tion of the mean-field Debye-H¨uckel theory, as ion pairing
+is counterbalances to some extent the energy cost of in-
+teraction confinement. Such strong ion-ion correlations
+cannot be directly accounted for in a mean-field theory,
+and the filling transition that emerges in Debye-H¨uckel
+theory appears to be an artefact of linearization. To over-
+come this issue, one can add Bjerrum pairs as a separate
+chemical species within the Debye-H¨uckel model. Care-
+fully accounting for the statistical formation of unbound
+phantom pairs, we obtain a modified mean-field theory
+that reproduces the result of the exact computation with
+nearly-perfect accuracy, and that can be extended to ac-
+count for a non-zero surface charge on the channel wall.
+Despite the concurring results, the two original for-
+malisms developed in this work serve distinct purposes.
+The field-theoretical solution plays the role of a touch-
+stone model, owing to its exact treatment of all many-
+body interactions. Modeling electrolytes is a notoriously
+hard problem in statistical physics, and simplified models
+often lack a lack a reference solution for benchmarking
+their approximations. This difficulty is lifted in the 1D
+geometry: thanks to the existence of the exact solution,
+we have been able to build a quantitatively precise mean-
+field model, adding step-by-step the qualitative ingredi-
+ents necessary to reproduce the exact result.
+Moreover, the field theory formalism gives access to the
+entire statistics of the system, including finite-size effects
+which elude any mean-field treatment.
+The latter are
+expected to be relevant in many experimental situations,
+as a substantial amount of current works focuses on short
+pores, where the length of the channel is comparable to
+its radius. For instance, one can expect shorter channels
+to deviate from electroneutrality2 – something entirely
+impossible in the limit of infinitely long channels.
+On the other hand, our modified mean-field formalism
+has the advantage of mathematical simplicity, allowing
+for convenient physical interpretations. The simple dis-
+tinction between free ions and Bjerrum pairs can be used
+to straightforwardly estimate the channel’s conductance.
+The influence of ion-ion correlations on conductivity is
+of particular importance as conductance measurements
+underpin many nanofluidic experiments. In contrast, the
+exact solution does not provide any such insight on trans-
+port properties, as it is limited to thermal equilibrium.
+Furthermore, the mean-field model may easily be
+adapted to other geometries, whereas an exact treatment
+is only possible in the strictly 1D case. Extensions of our
+results to 2D nanochannels would be of significant in-
+terest. In particular, 2D nanochannels can be made out
+of various materials with different electronic properties,
+which directly impact the confined ionic interactions6.
+Therefore, 2D nanochannels could serve as a platform
+for exploring the impact of wall metallicity on the ion
+filling problem.
+Both our exact and mean-field solutions can be ex-
+pected to fail at very high concentrations. Indeed, our
+work relies on a simplified picture of electrolytes, where
+all steric effects are discarded. We considered point-like
+ions with no short-distance repulsion; therefore, no effect
+like saturation or layering can be accounted for.
+Sim-
+ilarly, we neglected any interaction with the solvent –
+for example, we did not consider the decrement in rela-
+tive permittivity at high salt concentrations34. However,
+since all electrostatic interactions are screened in the
+limit of high concentrations, such considerations should
+not impact the conclusions of the present work: partic-
+ularly, we would still expect that cin = cout at high con-
+centration.
+Lastly, let us briefly recall our results for the ion filling
+problem. In channels larger than a few nanometers, the
+conventional mean-field picture is valid, so that in ab-
+sence of any surface charge the salt concentration inside
+the channel equals that of the reservoirs: cin = cout. For
+nanometre-scale confinement and low concentrations, in-
+teraction confinement amounts to a finite energy barrier
+for ions to enter the channel: cin = coute−Es/kBT . As
+concentration increases, more ions are able to overcome
+the barrier by forming Bjerrum pairs, neutralizing the
+electrostatic cost of confinement, at the price of entropy:
+cin ∝ c2
+out. Only at high concentrations can one recover
+the intuitive estimate cin = cout, as intense screening can-
+cels out all electrostatic interactions. Overall, interaction
+confinement has a significant impact on the properties
+of nanofluidic systems, and the assumption cin = cout
+should be questioned any time the system’s size reaches
+the nanometre scale.
+ACKNOWLEDGMENTS
+N.K. acknowledges support from a Humboldt fellow-
+ship.
+L.B. acknowledges funding from the EU H2020
+Framework Programme/ERC Advanced Grant agree-
+ment number 785911-Shadoks.
+The Flatiron Institute
+is a division of the Simons Foundation.
+DATA AVAILABILITY STATEMENT
+The data that support the findings of this study are
+available from the corresponding author upon reasonable
+request.
+1R. B. Schoch, J. Han, and P. Renaud, “Transport phenomena in
+nanofluidics,” Reviews of Modern Physics 80, 839–883 (2008).
+2A. Levy, J. P. de Souza,
+and M. Z. Bazant, “Breakdown of
+electroneutrality in nanopores,” Journal of Colloid and Interface
+Science 579, 162–176 (2020).
+3N. Kavokine, R. R. Netz,
+and L. Bocquet, “Fluids at the
+nanoscale: From continuum to subcontinuum transport,” An-
+nual Review of Fluid Mechanics 53, 377–410 (2021).
+4A. Esfandiar, B. Radha, F. C. Wang, Q. Yang, S. Hu, S. Garaj,
+R. R. Nair, A. K. Geim, and K. Gopinadhan, “Size effect in ion
+transport through angstrom-scale slits,” Science 358, 511–513
+(2017).
+
+13
+5Y. Avni, R. M. Adar, D. Andelman, and H. Orland, “Conductiv-
+ity of concentrated electrolytes,” Physical Review Letters 128,
+098002 (2022).
+6N. Kavokine, P. Robin,
+and L. Bocquet, “Interaction confine-
+ment and electronic screening in two-dimensional nanofluidic
+channels,” The Journal of Chemical Physics 157, 114703 (2022).
+7A. Parsegian, “Energy of an ion crossing a low dielectric mem-
+brane: Solutions to four relevant electrostatic problems,” Nature
+221, 844–846 (1969).
+8S. Teber, “Translocation energy of ions in nano-channels of cell
+membranes,” Journal of Statistical Mechanics: Theory and Ex-
+periment 2005, P07001–P07001 (2005).
+9J. Zhang, A. Kamenev,
+and B. I. Shklovskii, “Conductance of
+ion channels and nanopores with charged walls: A toy model,”
+Physical Review Letters 95, 148101 (2005).
+10Y. Levin, “Electrostatics of ions inside the nanopores and trans-
+membrane channels,” Europhysics Letters (EPL) 76, 163–169
+(2006).
+11S. Kondrat and A. Kornyshev, “Superionic state in double-layer
+capacitors with nanoporous electrodes,” Journal of Physics: Con-
+densed Matter 23, 022201 (2011).
+12P. Loche, C. Ayaz, A. Schlaich, Y. Uematsu,
+and R. R. Netz,
+“Giant axial dielectric response in water-fille nanotubes an ef-
+fective electrostatic ion-ion interactions from a tensorial dielec-
+tric model,” Journal of Physical Chemistry B 123, 10850–10857
+(2019).
+13N. Kavokine, S. Marbach, A. Siria,
+and L. Bocquet, “Ionic
+Coulomb blockade as a fractional Wien effect,” Nature Nanotech-
+nology 14, 573–578 (2019).
+14P. Robin, N. Kavokine, and L. Bocquet, “Modeling of emergent
+memory and voltage spiking in ionic transport through angstrom-
+scale slits,” Science 373, 687–691 (2021).
+15L. Dresner, “Ion exclusion from neutral and slightly charged
+pores,” Desalination 15, 39–57 (1974).
+16S. Buyukdagli, M. Manghi, and J. Palmeri, “Ionic capillary evap-
+oration in weakly charged nanopores,” Physical Review Letters
+105, 158103 (2010).
+17S. Buyukdagli, M. Manghi,
+and J. Palmeri, “Variational ap-
+proach for electrolyte solutions:
+From dielectric interfaces to
+charged nanopores,” Physical Review E 81, 041601 (2010).
+18R. R. Netz and H. Orland, “Variational charge renormalization in
+charged systems,” The European Physical Journal E 11, 301–311
+(2003).
+19V. D´emery, D. S. Dean, T. C. Hammant, R. R. Horgan,
+and
+R. Podgornik, “The one-dimensional Coulomb lattice fluid ca-
+pacitor,” The Journal of Chemical Physics 137, 064901 (2012).
+20A. A. Lee, S. Kondrat, and A. A. Kornyshev, “Single-file charge
+storage in conducting nanopores,” Physical Review Letters 113,
+1–5 (2014).
+21V. D´emery, R. Monsarrat, D. S. Dean, and R. Podgornik, “Phase
+diagram of a bulk 1d lattice Coulomb gas,” EPL (Europhysics
+Letters) 113, 18008 (2016).
+22J. Zhang, A. Kamenev, and B. I. Shklovskii, “Ion exchange phase
+transitions in water-filled channels with charged walls,” Physical
+Review E 73, 051205 (2006).
+23A. Kamenev, J. Zhang, A. Larkin,
+and B. Shklovskii, “Trans-
+port in one-dimensional Coulomb gases: From ion channels to
+nanopores,” Physica A 359, 129–161 (2006).
+24T. Gulden and A. Kamenev, “Dynamics of ion channels via non-
+hermitian quantum mechanics,” Entropy 23, 125 (2021).
+25S. F. Edwards and A. Lenard, “Exact statistical mechanics of
+a one-dimensional system with coulomb forces. ii. the method
+of functional integration,” Journal of Mathematical Physics 3,
+778–792 (1962).
+26V. D´emery, D. S. Dean, T. C. Hammant, R. R. Horgan,
+and
+R. Podgornik, “Overscreening in a 1d lattice coulomb gas model
+of ionic liquids,” EPL (Europhysics Letters) 97, 28004 (2012).
+27D. S. Dean, R. R. Horgan, and D. Sentenac, “Boundary effects in
+the one-dimensional Coulomb gas,” Journal of Statistical Physics
+90, 899–926 (1998).
+28D. Andelman, “Electrostatic properties of membranes:
+The
+Poisson-Boltzmann theory,” (1995).
+29C. Herrero and L. Joly, “Poisson-Boltzmann formulary,” arXiv
+preprint arXiv:2105.00720 (2021).
+30D. Frenkel and B. Smit, “Understanding molecular simulation,”
+(Academic Press, 2002) Chap. 7.
+31H. Bruus and K. Flensberg, “Many-body quantum theory in
+condensed matter physics,”
+(Oxford University Press, 2016)
+Chap. 12.
+32Y. Levin, “Electrostatic correlations: from plasma to biology,”
+Reports on progress in physics 65, 1577 (2002).
+33P. Robin, T. Emmerich, A. Ismail, A. Nigu`es, Y. You, G.-H.
+Nam, A. Keerthi, A. Siria, A. Geim, B. Radha, and L. Bocquet,
+“Long-term memory and synapse-like dynamics of ionic carriers
+in two-dimensional nanofluidic channels,” Science (in press).
+34A. Levy, D. Andelman,
+and H. Orland, “Dipolar Poisson-
+Boltzmann approach to ionic solutions: A mean field and loop ex-
+pansion analysis,” The Journal of Chemical Physics 139, 164909
+(2013).
+
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+page_content='1 Lyd´eric Bocquet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='1 and Nikita Kavokine2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' a)  1)Laboratoire de Physique de l’´Ecole Normale Sup´erieure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' ENS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Universit´e PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Sorbonne Universit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Universit´e Paris Cit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' France  2)Department of Molecular Spectroscopy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Max Planck Institute for Polymer Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Ackermannweg 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  55128 Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Germany  3)Center for Computational Quantum Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Flatiron Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 162 5th Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' NY 10010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  USA  (Dated: 12 January 2023)  Ion transport measurements are widely used as an indirect probe for various properties of confined electrolytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  It is generally assumed that the ion concentration in a nanoscale channel is equal to the ion concentration  in the macroscopic reservoirs it connects to, with deviations arising only in the presence of surface charges  on the channel walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Here, we show that this assumption may break down even in a neutral channel,  due to electrostatic correlations between the ions arising in the regime of interaction confinement, where  Coulomb interactions are reinforced due to the presence of the channel walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We focus on a one-dimensional  channel geometry, where an exact evaluation of the electrolyte’s partition function is possible with a transfer  operator approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Our exact solution reveals that in nanometre-scale channels, the ion concentration is  generally lower than in the reservoirs, and depends continuously on the bulk salt concentration, in contrast to  conventional mean-field theory that predicts an abrupt filling transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We develop a modified mean-field  theory taking into account the presence of ion pairs that agrees quantitatively with the exact solution and  provides predictions for experimentally-relevant observables such as the ionic conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Our results will  guide the interpretation of nanoscale ion transport measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  INTRODUCTION  A channel connects two reservoirs filled with a salt so-  lution at concentration cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' What is the salt concentra-  tion cin inside the channel?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The straightforward answer  cin = cout is challenged as soon as the channel’s dimen-  sions are at the nanometre scale1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' A deviation typically  occurs because of the presence of a surface charge density  Σ on the channel walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Indeed, a sufficiently long chan-  nel must remain electrically neutral2, which results in an  imbalance of the concentrations c±  in of the positive and  negative ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In a cylindrical channel of radius R that  is smaller than the electrolyte’s Debye length, the con-  centrations are given by the famous Donnan equilibrium  result3:  c±  in =  �  c2  out + (2Σ/R)2 ± 2Σ/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (1)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (1) is widely used to infer a channel’s surface charge  from measurements of its conductivity at different salt  concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  For sufficiently small surface charges  (2Σ/R ≪ cout), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (1) predicts cin = cout even at ex-  treme nanoscales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Importantly, this prediction under-  lies the method for extracting confined ion mobilities  from transport measurements, which has been applied  down to 7-˚A-wide two-dimensional channels4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Yet, phys-  ically, cin = cout stems from the assumption that the  electrolyte solutions, both in the reservoirs and in the  a)Electronic mail: nikita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='kavokine@mpip-mainz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='de  channel, behave as ideal gases of non-interacting ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  While such a description is valid in the bulk reservoirs  at reasonable salt concentrations5, it must be challenged  in the nanometre-scale channel which is subject to in-  teraction confinement6 – a reinforcement of the effective  Coulomb interactions between the ions due to the dielec-  tric contrast between the solvent (water) and the channel  wall3,6–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Due to interaction confinement, ions face a self-energy  barrier Es when entering the channel7,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  It was first  noted by Parsegian7 that this should result in ion exclu-  sion: the salt concentration within the channel is then  given by an Arrhenius scaling cin = coute−Es/kBT under  the assumption of non-interacting ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' However, the re-  sult becomes more subtle as the confinement-reinforced  ionic interactions are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Within a  mean-field description of a spherical nanopore, Dres-  ner15 predicted an abrupt filling transition, where cin  was a discontinuous function of cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Later, Palmeri  and coworkers16,17 recovered a similar transition using a  three-dimensional model of a cylindrical channel, treated  within the variational field theory formalism of Netz and  Orland18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  While this approach could be applied to a  realistic geometry, it took into account electrostatic cor-  relations only approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  An exact treatment of electrostatic correlations is pos-  sible upon simplification of the geometry to a purely  one-dimensional model, with the channel wall being  taken into account by introducing an effective confined  Coulomb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The 1D electrolyte can then be  mapped onto an Ising or 1D Coulomb-gas-type model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  the transfer matrix solution of such models was used, for  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='04622v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='soft]  11 Jan 2023  2  Effective interaction  Self-energy  B  A  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Ion filling in the interaction confinement regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Schematic of the ion filling problem: a cylindrical  nanochannel (radius R ∼ 1 nm) is connected to macroscopic reservoirs of aqueous electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The salt concentration inside  the channel, cin, may differ from that in the reservoirs, cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Physics of interaction confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' When a charged species  enters a nanochannel, the dielectric contrast between water (ϵw ∼ 80) and walls (ϵm ∼ 2) constraints the electric field lines to  remain within the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' This process can be interpreted in terms of image charges inside the channel walls, and results in  an electrostatic self-energy barrier for ions to enter the channel, and reinforced interactions between ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  example, to discuss the capacitance of nanoporous sys-  tems19–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The lattice models may be taken to the con-  tinuum limit, and the resulting path integral solutions  have been used to discuss various ion-exchange phase  transitions that arise in the presence of fixed discrete  charges inside the channel9,22,23 and the ionic Coulomb  blockade phenomenon13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Such models are particularly  rich theoretically, as they support a mapping to non-  Hermitian quantum mechanics24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Nevertheless, to our  knowledge, the fundamental problem of ion filling in  an uncharged channel has not been tackled within this  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  In this paper, we treat the ion-filling problem in the  interaction confinement regime using an exactly-solvable  one-dimensional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  We find that the value of cin  is strongly affected by the formation of Bjerrum pairs  – pairs of oppositely charged ions – within the channel,  which preclude the occurence of an abrupt filling transi-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' This is in contrast to the prediction of Palmeri and  coworkers16,17, and to the result of conventional mean-  field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We then build on our exact results to pro-  pose a modified mean-field model that accounts for the  relevant physical ingredients, and, particularly, for the  presence of ion pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  In Section II,  we present the one-dimensional model and its solution  within a path-integral formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The reader interested  only in the physical outcomes may skip directly to Sec-  tion III, where we discuss the model’s prediction for the  ion concentration within the channel, compare it to the  mean-field solution, and interpret it in terms of tightly  bound Bjerrum pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In Section IV, we establish a mod-  ified mean-field theory, based on the notion of phantom  pairs, that reproduces our exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The mean-field  theory allows us to determine the number of unpaired  ions and produces experimentally relevant predictions for  a nanochannel’s ionic conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Section V establishes  our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  1D COULOMB GAS MODEL  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Confined interaction  We consider a cylindrical channel of radius R and  length L, connected to macroscopic reservoirs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  We first assume for simplicity that the channel is filled  with water that has isotropic dielectric permittivity ϵw =  80, and that it is embedded in an insulating medium  with much lower permittivity ϵm (for a lipid membrane7,  ϵm ∼ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The effective Coulomb interaction V (x) be-  tween two monovalent ions separated by a distance x  on the channel axis can be computed exactly by solving  Poisson’s equation8,12,13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' A simple approximate expres-  sion can be obtained for x ∼ R (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='3):  V (x) ≈  e2α  2πϵ0ϵwRe−|x|/(αR),  (2)  where α is a numerical coefficient that depends on the  ratio ϵw/ϵm (α = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='3 for ϵw/ϵm = 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The reinforce-  ment of electrostatic interactions compared to the usual  e2/4πϵ0ϵwr Coulomb interaction that ions experience in  bulk water can be interpreted in terms of images charges  within the channel walls (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Two confined ions  interact not only with each other, but also with their  respective image charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  We introduce the parameters ξ ≡ αR and xT  ≡  2πϵ0ϵwR2kBT/e2: both have the dimension of a length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  3  With these notations,  V (x) = kBT ξ  xT  e−|x|/ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (3)  The effects of ion valence and of anisotropic dielectric  response of confined water can be taken into account by  adjusting ξ and xT 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Formally, the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (2)  is valid for any channel radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Yet, it is only physi-  cally relevant if at x ∼ R the interaction is significant  compared to kBT, which restricts in practice the appli-  cability of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (2) to R ≲ 2 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In such extreme 1D  confinement, we may neglect the ions’ degrees of free-  dom perpendicular to the channel axis and assume that  they are constrained to move in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The par-  tition function of such a 1D electrolyte may be computed  exactly, as detailed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Path integral formalism  Here, we detail the analytical solution for the partition  function of a 1D Coulomb gas-like system that was first  introduced in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We set kBT = 1 until the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We start from a lattice model, in order to rigorously  establish a path integral description in the continuum  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Our computation is inspired by the original solution of the 1D Coulomb gas model by Lenard and Edwards25, and  subsequent studies by Demery, Dean and coworkers19,21,26,27, as well as Shklovskii and coworkers22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We consider  a one-dimensional lattice with sites 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' , M as a model for the nanochannel of radius R and length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Each lattice  site i carries a spin Si, which takes the values {0, 1, −1}, corresponding respectively to no ion, a positive ion, or a  negative ion occupying the site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We model the surface charge distribution as an extra fixed charge qi added at each  lattice site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The spins interact with the Hamiltonian  H({Si}) =  ξ  2xT  M  �  i,j=1  (Si + qi)(Sj + qj)e−|i−j|/ξ ≡  1  2xT  (S + q)T C(S + q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (4)  The system is in contact with a particle reservoir at concentration cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Here the parameters ξ and xT are dimension-  less, expressed in number of lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The grand partition function is given by  Ξ =  �  S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=',SM  z  �  i |Si|e−  1  2xT (S+q)T C(S+q),  (5)  with z = coutπR2L/M the fugacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The matrix C can be analytically inverted:  C−1 =  1  2ξ sinh(1/ξ) ·  �  �  �  �  �  �  �  �  �  �  �  �  �  e1/ξ  −1  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  0  0  −1  2 cosh(1/ξ) −1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  0  −1 2 cosh(1/ξ)  −1  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  0  −1  e1/ξ  �  �  �  �  �  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (6)  Hence we can carry out a Hubbard-Stratonovich transformation, that is rewrite the partition function as a gaussian  integral, introducing the integration variable ϕ:  Ξ =  �  xM  T  (2π)Mdet(C) ·  �  S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=',SM  z  �  i |Si|  �  dϕe− xT  2 ϕT C−1ϕ+i(S+q)T ϕ,  (7)  with det(C) =  e1/ξ  2 sinh(1/ξ) ·  �  ξ(1 − e−2/ξ)  �M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' After performing the sum over the spins, which is now decoupled, we  obtain  Ξ =  �  xM  T  (2π)Mdet(C) ·  �  dϕ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' dϕM  M  �  j=1  (1 + 2z cos ϕj)  M  �  j=1  eiqjϕj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' exp  �  �−  xT  4ξ sinh(1/ξ)  �  �  M−1  �  j=1  (ϕj+1 − ϕj)2 + 2(cosh(1/ξ) − 1)  M−1  �  j=2  ϕ2  j + (e1/ξ − 1)(ϕ2  1 + ϕ2  M)  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (8)  4  We now take a continuum limit of the lattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We call a the physical lattice spacing and let ˜ξ = aξ, ˜xT = axT  and ˜z = Mz/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We then let a → 0 and M → ∞ while keeping the physical length of the system L = aM constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  We then drop the tilde sign to lighten the notation and obtain  Ξ =  �  dϕ(0)e−xT ϕ(0)2/4ξ  �  [dϕ]e−S[ϕ]  �  dϕ(L)e−xT ϕ(L)2/4ξ  (9)  with  S[ϕ] =  � L  0  dx  �  xT  4  �dϕ  dx  �2  + xT  4ξ2 ϕ(x)2 − iq(x)ϕ(x) − 2z cos ϕ(x)  �  ≡  � L  0  L(ϕ, ˙ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (10)  q(x) is the one-dimensional density corresponding to the surface charge, and z ≡ πR2cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' At this point ξ and xT  have the dimension of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The path integral measure is defined as  [dϕ] =  lim  a→0  M→∞  L=aM  �  �  M  �  j=1  � xT  4πadϕj  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (11)  We now define the propagator P(ϕ, x|ϕ0, 0), or simply P(ϕ, x), as  P(ϕ, x) =  �  dϕ(x)δ(ϕ(x) − ϕ)  �  [dϕ]e−  � x  0 L(ϕ, ˙ϕ)  �  dϕ(0)δ(ϕ(0) − ϕ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (12)  Considering an infinitesimal displacement ∆x,  P(ϕ, x) =  � xT  4π∆x  �  d(∆ϕ)P(ϕ − ∆ϕ, x − ∆x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' exp  �  −  � x  x−∆x  dx′  �  xT  4  �∆ϕ  ∆x  �2  + xT  4ξ2 ϕ2 − iq(x)ϕ − 2z cos ϕ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (13)  Expanding the propagator as P(ϕ − ∆ϕ, x − ∆x) = P(ϕ, x) − ∆x∂P/∂x − ∆ϕ∂P/∂ϕ + (1/2)(∆ϕ2)∂2P/∂ϕ2, and  carrying out the gaussian integrals, we obtain  P(ϕ, x) =  �  P(ϕ, x) − ∆x∂P  ∂x + O(∆x2)  � �  1 − ∆x  � xT  4ξ2 ϕ2 − iq(x)ϕ − 2z cos ϕ  �  + O(∆x2)  �  + ∆x  xT  ∂2P  ∂x2 (1 + O(∆x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (14)  P(ϕ, x) thus solves the partial differential equation  ∂P  ∂x = 1  xT  ∂2P  ∂ϕ2 +  �  iqϕ − xT  4ξ2 ϕ2 + 2z cos ϕ  �  P,  (15)  with initial condition P(ϕ, 0) = δ(ϕ − ϕ0), which is the equivalent of a Schr¨odinger equation for the path integral  representation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The partition function can thus be computed as  Ξ =  �  dϕ(L)e−xT ϕ2/4ξP(ϕ, L|f0),  (16)  where P(ϕ, L|f0) is the solution of (15) with initial condition P(ϕ, 0) = f0(ϕ) ≡ e−xT ϕ2/4ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Transfer operator  We introduce the Fourier transform of P with respect to ϕ:  ˜P(k, x) =  1  √  2π  �  dϕe−ikϕP(ϕ, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (17)  5  Then ˜P(k, x) satisfies  ∂ ˜P  ∂x = − k2  xT  ˜P − q ∂ ˜P  ∂k + xT  4ξ2  ∂2 ˜P  ∂k2 + z  �  ˜P(k + 1, x) + ˜P(k − 1, x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (18)  From now on, we restrict ourselves to an uncharged channel (q = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We then define the operator T such that  [T ( ˜P)](k) = − k2  xT  ˜P + xT  4ξ2  ∂2 ˜P  ∂k2 + z  �  ˜P(k + 1, x) + ˜P(k − 1, x)  �  ,  (19)  which plays the role of a functional transfer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Recalling eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (16), the partition function then reads  Ξ = ⟨f0|eLT |f0⟩  (20)  with f0(k) = e−ξk2/xT and ⟨f(k)|g(k)⟩ ≡  �  dkf ∗(k)g(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Now, in the limit L → ∞, we may consider the largest eigenvalue λ of the operator T , and the associated eigen-  function χ:  [T (χ)](k) = λχ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (21)  Then, up to an exponentially small correction,  Ξ = |⟨f0|χ⟩|2⟨χ|χ⟩eλL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (22)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Ion concentration  Our aim is to compute the salt concentration cin in the nanoscale channel given a salt concentration cout in the  reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' At the level of the lattice model, the probability to find, say, a positive ion at position k, can be computed  by replacing a factor (1 + 2z cos ϕk) by zeiϕk in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In the continuum limit, we obtain the positive (negative) ion  linear density at position x by inserting the operator zeiϕ (ze−iϕ) at position x:  πR2⟨c±  in(x)⟩ = 1  Ξ  �  dϕ(0)dϕ(x)dϕ(L)e−xT ϕ(0)2/4ξP(ϕ(x), x|ϕ(0), 0)ze±iϕ(x)P(ϕ(L), L|ϕ(x), x)e−xT ϕ(L)2/4ξ,  (23)  Upon Fourier-transformation, the insertion of eiϕ amounts to a shift by unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Introducing the operator,  SQ : f �→ (g : k �→ f(k − Q)),  (24)  the concentrations are given by  ⟨c±  in(x)⟩ =  z  πR2  ⟨f0|exT S±1e(L−x)T |f0⟩  Ξ  = cout  ⟨f0|exT S±1e(L−x)T |f0⟩  Ξ  ,  (25)  since z = coutπR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In the thermodynamic limit, and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (22) for the partition function, we obtain  ⟨c±  in⟩ = cout  ⟨χ(k)|χ(k ∓ 1)⟩  ⟨χ(k)|χ(k)⟩  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (26)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (26) is the main result of our exact computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In practice, the function χ(k) is determined numerically, by  finite-difference integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  PHYSICS OF ION FILLING  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Debye-H¨uckel solution  We now go back to the ion filling problem (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 1A)  and present first a one-dimensional mean-field solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Typically, the mean-field solution of an electrolyte prob-  lem is obtained by solving the Poisson-Boltzmann equa-  tion28,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' For the conventional Poisson-Boltzmann equa-  tion to apply, we would need to consider the full three-  dimensional geometry of our problem, and the effective  interaction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (3) would be introduced implicitly  through the boundary conditions at the channel walls15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  6  B  A  C  Concentration  Distance  Anions  Cations  Debye cloud  10-4  10-2  100  Reservoir concentration (M)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='9  1  Channel conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='/Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Exact solution  Series expansion  Poisson-Boltzmann  Debye-Hückel  10-4  10-2  100  Reservoir concentration (M)  10-3  10-2  10-1  100  Channel conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='/Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Exact solution  Series expansion  Poisson-Boltzmann  Debye-Hückel  Bulk  Self-energy  barrier  Bulk  Self-energy  barrier  Ion pairs  Weak interactions:  Es = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='5 kBT  Strong interactions:  Es = 6 kBT  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Comparing mean-field approximations with the exact Coulomb gas solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Schematic description  of the mean-field approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The chemical potential of confined ions is determined by solving the (linear or nonlinear)  Poisson-Boltzmann equation around a given ion, interacting with an oppositely charged Debye cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Dependence of  the channel salt concentration cin on the reservoir salt concentration cout, in a weakly-interacting case (R = 1 nm, ξ = 7 nm,  xT = 7 nm, Es = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='5 kBT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We plot four different predictions for the ratio cin/cout: the exact field-theoretical solution (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (26),  blue circles), its low concentration expansion (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (47), black line), the mean-field predictions from solving the full Poisson-  Boltzmann equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (40), orange curve) or from its Debye-H¨uckel linearization (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (36), yellow line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The two mean-field  predictions are indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  In all cases, the naive estimate cin = cout is recovered for high enough concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  In the dilute limit, the concentration inside the channel is well approximated by the Arrhenius scaling cin = coute−Es/kBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Dependence of the channel salt concentration cin on the reservoir salt concentration cout, in a strongly-interacting case  (R = 1 nm, ξ = 7 nm, xT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='6 nm, Es = 6 kBT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The color code is the same as in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Here, the mean-field predictions strongly  deviate from the exact solution, with the Debye-H¨uckel model predicting an abrupt filling transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' This discrepancy is due  to the formation of Bjerrum pairs at intermediate concentrations, as evidenced by the scaling cin ∝ c2  out in the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  In order to obtain a mean-field solution directly in the  1D geometry, we need to introduce a modified Poisson’s  equation for the electrostatic potential Φ whose Green’s  function coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (3):  � d2  dx2 − 1  ξ2  �  φ = −2πR2 c+ − c−  xT  ,  (27)  with φ ≡ eΦ/kBT the dimensionless potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Im-  posing that the ions follow a Boltzmann distribution  (c± = cine∓φ, where cin is understood as the average con-  centration inside the channel), we obtain the analogue of  the Poisson-Boltzmann equation in our 1D geometry:  � d2  dx2 − 1  ξ2  �  φ = 2πR2 cin  xT  sinh φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (28)  In order to proceed analytically, we make a Debye-  H¨uckel-type approximation and linearize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (28) with  respect to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Then, the potential around an ion placed  in the channel at x = 0 is given by  φ(x) = ξeff  xT  e−|x|/ξeff,  (29)  with  ξ2  eff =  ξ2  1 + 4πR2cinξ2/xT  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (30)  The chemical potential inside the channel is the sum of  an ideal gas entropic part and of an excess part due to  interactions:  µin = µent + µex,  (31)  with  µent = kBT log coutΛ3,  (32)  Λ being the De Broglie thermal wavelength of the ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  µex can be obtained via a Debye charging process30:  µex  kBT =  � 1  0  φλ(0)dλ, φλ(0) =  λξ/xT  �  1 + 4λπR2cinξ2/xT  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (33)  We determine cin by imposing equality of the chemical  potentials between the channel and the reservoir:  µout = kBT log coutΛ3 = µin,  (34)  which yields  cin = coute−µex/kBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (35)  Evaluating analytically the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (33), we obtain  an implicit equation for cin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  With the notation ˆcin ≡  πR2cin,  cin = cout exp  �  − ξ  2xT  ×  x2  T  6ξ2ˆc2  inξ2  �  1 − 3  2(1 + 4ˆcinξ2/xT )1/2  +1  2(1 + 4ˆcinξ2/xT )3/2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (36)  7  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 2B and C, we plot the ratio cin/cout as a func-  tion of cout, as obtained by numerically solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  We fix ξ = 7 nm (which corresponds to a channel with  R ≈ 1 nm and strong dielectric contrast), and vary  xT to set the ionic interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The interac-  tion strength may be quantified through the self-energy  barrier, Es = kBT × ξ/(2xT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The limiting behavior of  cin/cout may be understood directly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' When  cin is small, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (36) reduces to the Arrhenius scaling  cin = coute−Es/kBT : this results typically holds for bio-  logical ion channels which may contain either 0 or 1 ion at  any given time, and the effect of inter-ionic interactions  is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' When cin is large, we recover cin = cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In-  deed, the excess term in the chemical potential vanishes  at high concentrations, which is then dominated by the  entropic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The fact that µex → 0 as cin → ∞ is non-  trivial: it can be seen, physically, as resulting from the  Coulomb potential of each ion being perfectly screened  by the other ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' At small values of Es, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (36) has  a single solution for all values of cout, which interpolates  smoothly between the two limiting regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  However,  for Es ≳ 5kBT, it has three solutions in a certain range  of cout, pointing to a pseudo-first-order phase transition  between a low-concentration and a high-concentration  phase, similar to the one predicted by Dresner15 and  Palmeri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The transition occurs at ˆcin ∼ xT /ξ2: as  per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (30), this corresponds to the concentration where  the effect of the screening cloud on an ion’s Coulomb po-  tential becomes significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Full Poisson-Boltzmann solution  The physical content of the mean-field solution pre-  sented above is similar to the one of Dresner, based on  a linearized Poisson-Boltzmann equation15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The differ-  ence in geometry, and the fact that he foregoes the use  of the Debye charging process, do not seem to play a sig-  nificant qualitative role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The solution of Palmeri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='16  takes ionic correlations into account to some extent, yet  it still involves a Debye-H¨uckel-type linear equation for  the mean-field interaction potential between the ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  One may ask whether the same phenomenology per-  sists if one does not linearize the Poisson-Boltzmann  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The full Poisson-Boltzmann equation cannot  be solved analytically, but supports the following inte-  gral form:  �dφ  dx  �2  − 1  ξ2 φ2 = 4πR2 cin  xT  (cosh φ − 1) ,  (37)  where we have used the fact that φ should vanish at x →  ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' For x → 0, the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (37) should reduce  to the unscreened potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (3) up to an additive  constant, so that  1  x2  T  − 1  ξ2 φ2(0) = 4πR2 cin  xT  (cosh φ(0) − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (38)  Once again, one may express the excess chemical po-  tential of the confined ions through a Debye charging  process:  µex  kBT =  � 1  0  φλ(0)dλ,  λ2  x2  T  − 1  ξ2 φ2  λ(0) = 4πR2 λcin  xT  (cosh φλ(0) − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (39)  This result is the analogue of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (33), with φλ(0) now  being the solution of an implicit non-linear equation, so  that µex must be determined numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' As before, the  concentration inside the channel is then given by:  cin = coute−µex/kBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (40)  The prediction of the full Poisson-Boltzmann equation  is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 2B and C: we find cin to be a smooth  function of cout for all values of parameters, in contrast to  the linearized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We may not, however, unambigu-  ously conclude that the filling transition is an artifact of  linearization, since the non-linear solution still involves a  mean-field approximation and is not guaranteed to yield  the correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Interestingly, the “physically-motivated” mean-field  solution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (28) differs from the mean-field limit of  our exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' It is obtained by taking the saddle-  point approximation in the path-integral expression of  the partition function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The Euler-Lagrange  equation for the minimizer ϕ(x) of the action S[ϕ] in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (10) is, upon identifying φ = −iϕ,  � d2  dx2 − 1  ξ2  �  φ = 2πR2 cout  xT  sinh φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (41)  This is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (28) with cin replaced with cout, and corre-  sponds to a first order treatment of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Indeed,  if the ions are non-interacting, cin = cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' By solving the  mean-field equation, we determine how the ions’ chemi-  cal potential is affected by Debye screening, which then  results in value of cin that is different from cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Within  a straightforward interaction expansion procedure, one  should determine the effect of screening assuming the ze-  roth order value for the ion concentration inside the chan-  nel, which is cout: this corresponds to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (28)  contains an additional self-consistency condition, as it  assumes the actual value cin for the ion concentration,  which is not known until Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (28) is solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' One may  draw a loose condensed matter physics analogy, where  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (41) resembles the Born approximation for impu-  rity scattering, while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (28) is analogous to the self-  consistent Born approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='31  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Exact solution  We now turn to the exact solution obtained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  II to unambiguously solve the ion filling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We  8  determine cin according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (26):  ⟨c±  in⟩ = cout  ⟨χ(k)|χ(k ∓ 1)⟩  ⟨χ(k)|χ(k)⟩  ,  (42)  where χ(k) is the highest eigenfunction of the trans-  fer operator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (19), determined in practice by nu-  merical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The exact results for cin, with the  same parameter values as for the mean-field solution,  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 2 B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' When interactions are  weak (small values of Es, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 2B), the exact and mean-  field solutions are in good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Notably, all so-  lutions smoothly interpolate between the bulk scaling  cin = cout at high concentration, and the Arrhenius scal-  ing cin = coute−Es/kBT at low concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Conversely,  in the strongly-interacting case (large Es, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 2C), the  exact result yields a much larger ion concentration that  the mean-field solutions for intermediate values of cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  In this intermediate regime, cin remains a smooth func-  tion of cout, and obeys the scaling cin ∝ c2  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Such a scaling is the signature of the formation of  tightly bound Bjerrum pairs of positive and negative ions  – strongly-correlated configurations that are not taken  into account by mean-field solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Indeed, let us as-  sume that the channel contains an ideal gas of ion pairs at  concentration cin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We further assume that in a pair, the  distance between the two ions is uniformly distributed  in the interval [−xT /2, xT /2], and the binding energy of  a pair is kBTξ/xT = 2Es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Then, the grand partition  function reads  Ξ =  �  N  (ze−βEs)2N 1  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  N  �  i=1  L  � xT /2  −xT /2  dx e2βEs  (43)  =  �  N  (z2LxT )N  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  = ez2LxT ,  (44)  where we recall that z = πR2cout and β ≡ 1/(kBT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Using that  πR2cin = 1  L  ∂ log Ξ  ∂(βµ) = z  L  ∂ log Ξ  ∂z  ,  (45)  we obtain  cin = 2z2xT  πR2  = 2πR2xT c2  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (46)  We recover indeed the quadratic scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  We may check that the prefactor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (46) is the  correct one by evaluating analytically the expression in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (26) in the low concentration limit zT ≡ zxT ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  An analytical expansion of the function χ(k) in powers  of zT was derived in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Substituting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (26),  we obtain  πR2cin = z(e−βEs + 2zT − 13  2 z2  T e−βEs  −7z3  T + O(z4  T ) + O(e−2βEs)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (47)  The first term in the expansion corresponds to cin =  coute−βEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  At the lowest salt concentrations, forming  Bjerrum pairs is too entropically unfavorable, and the  concentration inside the channel is controlled by the self-  energy barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  However, as the salt concentration in-  creases, there is no abrupt transition to a highly-screened  concentrated phase inside the channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' instead, the chan-  nel is progressively filled by Bjerrum pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' This corre-  sponds to the quadratic term in the expansion, with the  prefactor agreeing indeed with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='1 The expansion  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (47) reproduces quite well the low-concentration  behavior of the exact solution as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 2B and  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' However, it fails at high concentrations, where it does  not recover cin = cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Our exact analysis of the ion statistics in a nanoscale  channel has revealed that Bjerrum pairs are a crucial in-  gredient of the filling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We now develop a modified  mean-field theory that accounts the presence of Bjerrum  pairs and compare it to the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  PAIR-ENHANCED MEAN-FIELD THEORY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Debye-H¨uckel-Bjerrum theory  The traditional mean-field treatment of electrolytes is  incapable of taking Bjerrum pairs into account, as it nat-  urally neglects any strong ion-ion correlations – pairing  being a fundamentally discrete phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  An idea  proposed by Bjerrum to amend the Debye-H¨uckel theory  was to introduce ion pairs as a separate species encapsu-  lating all “strong” ion-ion correlations32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' More precisely,  any two oppositely charged ions that are closer than some  minimum distance can be considered as a single neutral  entity – a Bjerrum pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The remaining “free” ions should  then only experience weak interactions with each other,  and can be treated at the mean-field level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Importantly,  this last remark justifies the Debye-H¨uckel linearization,  as all non-linear effects are assumed to be hidden in the  definition of ion pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  As before, we consider that pairs behave like particles  of an ideal gas, and that their maximum extension is  given by xT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Defining cp  in the concentration pairs inside  the channel, the chemical potential of pairs is given by:  µp  in = kBT log  cp  inΛ6  2πxT R2 ,  (48)  where the geometrical factor inside the logarithm ac-  counts for the internal degrees of freedom of a pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The  chemical potential only has an entropic term, because  the binding energy of the pair exactly compensates the  self-energy of the two separate ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The chemical equi-  librium between free ions and pairs inside the channel  1 This justifies a posteriori our choice of [−xT /2, xT /2] as the  interval in which a paired-up ion is allowed to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  9  Concentration  Distance  Anions  Cations  B  A  C  Debye cloud  Bjerrum pair  Well-defined  pair  Phantom pair  10-4  10-2  100  Reservoir concentration (M)  10-3  10-2  10-1  100  101  102  Channel conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='/Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Exact solution  Debye-Hückel-Bjerrum mean-field  Phantom pair mean-field  Strong interactions:  Es = 6 kBT  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Pair-enhanced mean-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Treatment of ion pairing in mean-field approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Top panel: Mean-field  theories inevitably underestimate ion-ion correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' To circumvent this problem, two ions that are distant by less than xT  are considered to form an ion pair, which is treated as a separate chemical species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Bottom panel: schematic representation  of ion distribution around a fixed positive ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The distribution is very peaked close to the central ion, due to the formation  of an ion pair, and then relaxes smoothly to the mean value cin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Evolution of channel concentration cin as function of  reservoir concentration cout, in a strongly-interacting cacse (R = 1 nm, ξ = 7 nm, xT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='6 nm, Es = 6 kBT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We plot the ratio  cin/cout obtained from three different models taking Bjerrum pairs into account: the exact field-theoretical solution (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (26),  blue circles), the Debye-H¨uckel-Bjerrum mean-field theory (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (51), red line) and our modified mean-field theory based on the  notion of phantom pairs (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (55), orange line), which reproduces the exact solution quantitatively for all values of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  At high concentration, the Debye-H¨uckel-Bjerrum prediction fails due to the uncontrolled proliferation of Bjerrum pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Formation of phantom pairs inside the nanochannel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' At low concentration (top panel), pairs are well-separated and ions forming  a pair are tightly bound to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' At high concentration (bottom panel), ionic interactions are weakened as a result of  Debye screening, and two quasi-non-interacting ions may find themselves within a distance xT of each other without actually  binding: this is a phantom pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  can be written as:  µ+  in + µ−  in = 2µin = µp  in,  (49)  where µ+  in and µ−  in are the chemical potentials of cations  and anions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  We then obtain, using the  Debye-H¨uckel solution for µin (equations (31) to (33)):  cp  in = 2πR2xT c2  out,  (50)  which is the result obtained in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The  average concentration in free ions cf  in is not modified com-  pared to the Debye-H¨uckel solution, and is therefore the  solution of the self-consistent Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  One can then  compute the total concentration inside the channel as  cin = cf  in + cp  in, or, explicitly  cin = coute−µex(cf  in)/kBT + 2πR2xT c2  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (51)  In other words, the only impact of pairs in Bjerrum’s  computation is to add a quadratic term 2πR2xT c2  out to  the Debye-H¨uckel result, matching with the expansion  (47) of the exact solution up order 2 in the bulk concen-  tration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We compare the two predictions on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The  Debye-H¨uckel-Bjerrum solution is found to match the ex-  act one quite well at low and intermediate concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  This result is, however, unphysical for cout ≳ 1/πR2xT :  cin is found to grow much faster than the bulk concen-  tration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' One solution would be to consider higher-order  terms in the mean-field treatment through the inclusion  of triplets, quadruplets, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' of ions, and all possible in-  teractions between these entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Truncating the sum  at any finite order, however, would not yield a solution  valid in the entire range of concentrations, nor is it guar-  anteed to converge to the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' This approach  is also unsatisfactory as it would not yield a closed-form  expression for cin and would not allow for qualitative un-  derstanding of the underlying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Instead, we develop a different method that, through  physics-driven arguments, prevents the divergence of cin  at high bulk concentrations and reproduces quantita-  tively the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Phantom pairs  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (51) overestimates the number of Bjerrum pairs in  the channel because it fails to account for the presence  of Bjerrum pairs in the reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The electrolyte in the  reservoir is treated as an ideal gas : the ions are non-  interacting and they cannot form actual tightly-bound  pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Nevertheless, we have defined any two oppositely  charged ions that find themselves in a cylinder of radius  R and length xT to be a separate chemical species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Such  configurations may arise in the reservoir simply out of  statistical chance: we dub them phantom pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' For our  10  mean-field theory to be consistent, these phantom pairs  need to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Let cp  out be the concentration of phantom pairs in the  reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The chemical equilibrium between phantom  pairs and free ions imposes  cp  out = 2πR2xT (cf  out)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (52)  In addition, one has cf  out + cp  out = cout, since an ion must  either be free or part of a pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Imposing this condition  yields:  cf  out =  √  1 + 8πcoutxT R2 − 1  4xT πR2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (53)  We use this result to control the proliferation of pairs in  the channel: we now equilibrate the free ions inside the  nanochannel with only the free ions in the reservoir:  cf  in = cf  oute−µex(cf  in)/kBT ,  (54)  which corresponds to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (35) with cout replaced by cf  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (54) is again a self-consistent equation, this time on  the concentration of free ions cf  in, that must be solved  numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Lastly, equilibrating pairs with free ions in-  side the channel (or, equivalently, pairs inside with pairs  outside), we obtain:  cin = cf  in + 2πR2xT (cf  out)2,  (55)  where the second term corresponds again to Bjerrum  pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (53) to (55) constitute the main result of our  modified mean-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Note that µex may be deter-  mined at the Debye-H¨uckel level (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (33)), or by solving  the full Poisson-Boltzmann equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (39)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In what  follows, we will only discuss the latter, as it offers greater  accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' however, the Debye-H¨uckel prediction provides  reasonable results even in the case of strong interactions,  and yields for a convenient analytical expression for µex  as function of cf  in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The  prediction  of  our  phantom  pair  Poisson-  Boltzmann model is compared to the exact solution (26)  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The two solutions are found to be in near  perfect agreement for all values of parameters, even in  strong coupling limit Es ≫ kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  In the next two sections, we use our modified mean-  field model to predict the conductance of a nanochannel,  first in the case of a neutral channel, and then in presence  of a surface charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Conductance  One strength of our modified mean-field model is that  it offers insight into the physical properties of the con-  fined system beyond the value of the ionic concentra-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In particular, the decomposition of the electrolyte  into free ions and bound pairs allows us to estimate the  channel’s conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Tightly bound Bjerrum pairs are  electrically neutral, so that they do not contribute to the  ionic current to first order in applied electric field: it  would then be straightforward to assume that the chan-  nel’s conductance is proportional to the concentration  of free ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' However, the reasoning needs to be more  subtle, since the channel, in the same way as the reser-  voir, may contain non-interacting phantom pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  In-  deed, we have decomposed the confined electrolyte into  tightly bound pairs, that have no ionic atmosphere, and  free ions that are dressed by a Debye screening cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  As the concentration increases, the interaction between  dressed ions becomes weak, and two of them may find  themselves within a distance xT without actually bind-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Such a phantom pair is expected to still contribute  to the conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The concentration of phantom pairs  in the channel is obtained by imposing their chemical  equilibrium with the free ions treated as an ideal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Thus, we estimate the channel’s conductance as:  G = 2 e2D  kBT  πR2  L  �  cf  in + 2xT πR2(cf  in)2�  ,  (56)  where D is the diffusion coefficient of ions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' the second  term corresponds to the contribution of phantom pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 4A, we compare this result to the Ohm’s law  prediction where pairs are neglected and one assumes  cin = cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Ohm’s law is found to greatly overestimate  the conductance at low concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In the dilute limit,  we instead recover the Arrhenius scaling, where one as-  sumes cin = coute−Es/kBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Finally, we stress that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (56) only accounts for the  electrophoresis of free ions, and is therefore only valid  in the limit of weak external electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Stronger  voltage drops will result in the breaking of ion pairs,  causing a conductivity increase in a process known as  the second Wien effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' This phenomenon is described in  refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='13,14, and has been used to create solid-state voltage-  gated nanochannels33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Effect of a surface charge  Up till now, we have restricted ourselves to channels  with uncharged walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  However, in most experimen-  tally relevant situations, the channel walls bear a sur-  face charge density Σ, which strongly impacts nanofluidic  transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' While introducing a surface charge is tedious  within the exact framework, we may readily assess the  effect of surface charge in the interaction confinement  regime using our pair-enhanced mean-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  In the limit where the channel’s radius is smaller than  the Debye length, we assume that the presence of the  surface charge amounts to a homogeneous Donnan po-  tential drop VD inside the channel, which we do not need  to determine explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Then, the chemical potential of  ions inside the channel reads:  µ±  in = µex ± eVD + kBT log c±  inΛ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (57)  11  B  A  10-3  10-2  10-1  100  101  Reservoir concentration (M)  10-6  10-4  10-2  100  Channel conductance (nS)  Ohm law  Phantom pair mean-field  Arrhenius model  Actual surface charge  10-3 C/m2  Apparent surface charge  10-2 C/m2  10-3  10-2  10-1  100  101  Reservoir concentration (M)  10-2  10-1  100  101  Conductance (nS)  Donnan equilibrium  Phantom pair mean-field  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Channel conductance in the pair-enhanced mean-field model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Conductance of a nanochannel (R = 1 nm,  ξ = 7 nm, xT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='7 nm, Es = 10 kBT) as function of the reservoir concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The red line corresponds to the prediction of  the phantom pair mean-field model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (56)) for T = 300 K, D = 10−9 m2/s and L = 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The Ohm’s law bulk prediction  (cin = cout, blue line) and the Arrhenius model (cin = coute−Es/kBT , yellow line) are also represented for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Conductance of a nanochannel with a weak surface charge Σ = 10−3 C/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We represented the predictions of the conventional  Donnan equilibrium (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (1), blue line) and of the phantom pair mean-field theory (equations (56) and (59), red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Because  interaction confinement results in a lower ion concentration in the channel, the usual formula Σ ∼ Rc∗/2, where c∗ is the reservoir  concentration for which conductance levels off overestimates the surface charge by one order of magnitude, as indicated on the  plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Note that the concentration in free anions c−  in and cations  c+  in are now distinct, so that µex is defined as a function of  the average free ion concentration cf  in = (c+  in+c−  in)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In a  channel that is sufficiently long for local electroneutrality  to hold,  c+  in − c−  in + 2Σ/R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  (58)  Imposing chemical equilibrium with the reservoir, we ob-  tain a modified version of the Donnan result (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (1)):  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  cin = cf  in + cp  in  cf  in =  ��  cf  oute−βµex(cf  in)�2  +  � 2Σ  R  �2,  cp  in = 2πR2xT (cf  out)2,  (59)  with cf  out given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  One can again obtain the channel’s conductance  through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (56), which we compare to the Donnan /  Ohm’s law result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Importantly, the Donnan  result predicts that conductance becomes independent of  concentration for cout ∼ 2Σ/R (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In practice,  this result is commonly used to estimate experimentally  the surface charge as Σ ∼ Rc∗/2, where c∗ is the reser-  voir concentration for which conductance levels off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In  contrast, in the interaction confinement regime, we pre-  dict that the transition occurs instead at cf  in ∼ 2Σ/R –  corresponding to a higher reservoir concentration, due to  the self-energy barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In this case, Donnan’s prediction  overestimates the surface charge by typically one order  of magnitude, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Finally, let us stress that we considered here a charge  homogeneously distributed along the channel’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  This assumption is relevant in the case of conducting  wall materials, such as systems where the charge is im-  posed via a gating electrode connected to the chan-  nel walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  This situation, however, may be different  in experimentally-available devices, where the surface  charge generally consists in localized charged groups and  defects on the channel walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In this case, the physics be-  come more involved as ions may form bound pairs with  the fixed surface charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Some of these physics have  been revealed by the exact computations of Shklovskii  and coworkers9,22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' a technically simpler approach to  these physics using our pair-enhanced mean-field theory  would be possible, but extends beyond the scope of the  present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  DISCUSSION AND PERSPECTIVES  We have determined the salt concentration inside a  nanometric channel connected to reservoirs filled with  electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  In the case of a fully 1D geometry, corre-  sponding to a nanotube of radius R ∼ 1nm, we devel-  oped an exact field-theoretical solution that allowed us  to compute channel concentration cin as function of the  reservoir concentration cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' This solution clears up the  ambiguities of pre-existing mean-field theories, and con-  tradicts the naive expectation cin = cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In particular,  the concentration inside the nanochannel is found to be  always lower than in the bulk, as the confinement of elec-  trostatic interactions creates an energy barrier for ions to  12  enter the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Yet, we found that cin is in fact higher than the predic-  tion of the mean-field Debye-H¨uckel theory, as ion pairing  is counterbalances to some extent the energy cost of in-  teraction confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Such strong ion-ion correlations  cannot be directly accounted for in a mean-field theory,  and the filling transition that emerges in Debye-H¨uckel  theory appears to be an artefact of linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' To over-  come this issue, one can add Bjerrum pairs as a separate  chemical species within the Debye-H¨uckel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Care-  fully accounting for the statistical formation of unbound  phantom pairs, we obtain a modified mean-field theory  that reproduces the result of the exact computation with  nearly-perfect accuracy, and that can be extended to ac-  count for a non-zero surface charge on the channel wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Despite the concurring results, the two original for-  malisms developed in this work serve distinct purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The field-theoretical solution plays the role of a touch-  stone model, owing to its exact treatment of all many-  body interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Modeling electrolytes is a notoriously  hard problem in statistical physics, and simplified models  often lack a lack a reference solution for benchmarking  their approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' This difficulty is lifted in the 1D  geometry: thanks to the existence of the exact solution,  we have been able to build a quantitatively precise mean-  field model, adding step-by-step the qualitative ingredi-  ents necessary to reproduce the exact result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Moreover, the field theory formalism gives access to the  entire statistics of the system, including finite-size effects  which elude any mean-field treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The latter are  expected to be relevant in many experimental situations,  as a substantial amount of current works focuses on short  pores, where the length of the channel is comparable to  its radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' For instance, one can expect shorter channels  to deviate from electroneutrality2 – something entirely  impossible in the limit of infinitely long channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  On the other hand, our modified mean-field formalism  has the advantage of mathematical simplicity, allowing  for convenient physical interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' The simple dis-  tinction between free ions and Bjerrum pairs can be used  to straightforwardly estimate the channel’s conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The influence of ion-ion correlations on conductivity is  of particular importance as conductance measurements  underpin many nanofluidic experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In contrast, the  exact solution does not provide any such insight on trans-  port properties, as it is limited to thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Furthermore, the mean-field model may easily be  adapted to other geometries, whereas an exact treatment  is only possible in the strictly 1D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Extensions of our  results to 2D nanochannels would be of significant in-  terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In particular, 2D nanochannels can be made out  of various materials with different electronic properties,  which directly impact the confined ionic interactions6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Therefore, 2D nanochannels could serve as a platform  for exploring the impact of wall metallicity on the ion  filling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Both our exact and mean-field solutions can be ex-  pected to fail at very high concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Indeed, our  work relies on a simplified picture of electrolytes, where  all steric effects are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' We considered point-like  ions with no short-distance repulsion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' therefore, no effect  like saturation or layering can be accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Sim-  ilarly, we neglected any interaction with the solvent –  for example, we did not consider the decrement in rela-  tive permittivity at high salt concentrations34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' However,  since all electrostatic interactions are screened in the  limit of high concentrations, such considerations should  not impact the conclusions of the present work: partic-  ularly, we would still expect that cin = cout at high con-  centration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Lastly, let us briefly recall our results for the ion filling  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' In channels larger than a few nanometers, the  conventional mean-field picture is valid, so that in ab-  sence of any surface charge the salt concentration inside  the channel equals that of the reservoirs: cin = cout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' For  nanometre-scale confinement and low concentrations, in-  teraction confinement amounts to a finite energy barrier  for ions to enter the channel: cin = coute−Es/kBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' As  concentration increases, more ions are able to overcome  the barrier by forming Bjerrum pairs, neutralizing the  electrostatic cost of confinement, at the price of entropy:  cin ∝ c2  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Only at high concentrations can one recover  the intuitive estimate cin = cout, as intense screening can-  cels out all electrostatic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Overall, interaction  confinement has a significant impact on the properties  of nanofluidic systems, and the assumption cin = cout  should be questioned any time the system’s size reaches  the nanometre scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' acknowledges support from a Humboldt fellow-  ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' acknowledges funding from the EU H2020  Framework Programme/ERC Advanced Grant agree-  ment number 785911-Shadoks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  The Flatiron Institute  is a division of the Simons Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  DATA AVAILABILITY STATEMENT  The data that support the findings of this study are  available from the corresponding author upon reasonable  request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  1R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Schoch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Han, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Renaud, “Transport phenomena in  nanofluidics,” Reviews of Modern Physics 80, 839–883 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Levy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' de Souza,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Bazant, “Breakdown of  electroneutrality in nanopores,” Journal of Colloid and Interface  Science 579, 162–176 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Kavokine, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Netz,  and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Bocquet, “Fluids at the  nanoscale: From continuum to subcontinuum transport,” An-  nual Review of Fluid Mechanics 53, 377–410 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  4A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Esfandiar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Radha, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Hu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Garaj,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Nair, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Geim, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Gopinadhan, “Size effect in ion  transport through angstrom-scale slits,” Science 358, 511–513  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  13  5Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Avni, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Adar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Andelman, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Orland, “Conductiv-  ity of concentrated electrolytes,” Physical Review Letters 128,  098002 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  6N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Kavokine, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Robin,  and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Bocquet, “Interaction confine-  ment and electronic screening in two-dimensional nanofluidic  channels,” The Journal of Chemical Physics 157, 114703 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  7A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Parsegian, “Energy of an ion crossing a low dielectric mem-  brane: Solutions to four relevant electrostatic problems,” Nature  221, 844–846 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  8S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Teber, “Translocation energy of ions in nano-channels of cell  membranes,” Journal of Statistical Mechanics: Theory and Ex-  periment 2005, P07001–P07001 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  9J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Kamenev,  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Shklovskii, “Conductance of  ion channels and nanopores with charged walls: A toy model,”  Physical Review Letters 95, 148101 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  10Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Levin, “Electrostatics of ions inside the nanopores and trans-  membrane channels,” Europhysics Letters (EPL) 76, 163–169  (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  11S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Kondrat and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Kornyshev, “Superionic state in double-layer  capacitors with nanoporous electrodes,” Journal of Physics: Con-  densed Matter 23, 022201 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  12P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Loche, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Ayaz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Schlaich, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Uematsu,  and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Netz,  “Giant axial dielectric response in water-fille nanotubes an ef-  fective electrostatic ion-ion interactions from a tensorial dielec-  tric model,” Journal of Physical Chemistry B 123, 10850–10857  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  13N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Kavokine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Marbach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Siria,  and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Bocquet, “Ionic  Coulomb blockade as a fractional Wien effect,” Nature Nanotech-  nology 14, 573–578 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  14P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Robin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Kavokine, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Bocquet, “Modeling of emergent  memory and voltage spiking in ionic transport through angstrom-  scale slits,” Science 373, 687–691 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  15L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Dresner, “Ion exclusion from neutral and slightly charged  pores,” Desalination 15, 39–57 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  16S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Buyukdagli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Manghi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Palmeri, “Ionic capillary evap-  oration in weakly charged nanopores,” Physical Review Letters  105, 158103 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  17S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Buyukdagli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Manghi,  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Palmeri, “Variational ap-  proach for electrolyte solutions:  From dielectric interfaces to  charged nanopores,” Physical Review E 81, 041601 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  18R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Netz and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Orland, “Variational charge renormalization in  charged systems,” The European Physical Journal E 11, 301–311  (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  19V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' D´emery, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Dean, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Hammant, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Horgan,  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Podgornik, “The one-dimensional Coulomb lattice fluid ca-  pacitor,” The Journal of Chemical Physics 137, 064901 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  20A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Kondrat, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Kornyshev, “Single-file charge  storage in conducting nanopores,” Physical Review Letters 113,  1–5 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  21V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' D´emery, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Monsarrat, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Dean, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Podgornik, “Phase  diagram of a bulk 1d lattice Coulomb gas,” EPL (Europhysics  Letters) 113, 18008 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  22J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Kamenev, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Shklovskii, “Ion exchange phase  transitions in water-filled channels with charged walls,” Physical  Review E 73, 051205 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  23A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Kamenev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Larkin,  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Shklovskii, “Trans-  port in one-dimensional Coulomb gases: From ion channels to  nanopores,” Physica A 359, 129–161 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  24T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Gulden and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Kamenev, “Dynamics of ion channels via non-  hermitian quantum mechanics,” Entropy 23, 125 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  25S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Edwards and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Lenard, “Exact statistical mechanics of  a one-dimensional system with coulomb forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' the method  of functional integration,” Journal of Mathematical Physics 3,  778–792 (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  26V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' D´emery, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Dean, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Hammant, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Horgan,  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Podgornik, “Overscreening in a 1d lattice coulomb gas model  of ionic liquids,” EPL (Europhysics Letters) 97, 28004 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  27D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Dean, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Horgan, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Sentenac, “Boundary effects in  the one-dimensional Coulomb gas,” Journal of Statistical Physics  90, 899–926 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  28D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Andelman, “Electrostatic properties of membranes:  The  Poisson-Boltzmann theory,” (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  29C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Herrero and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Joly, “Poisson-Boltzmann formulary,” arXiv  preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='00720 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  30D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Frenkel and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Smit, “Understanding molecular simulation,”  (Academic Press, 2002) Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  31H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Bruus and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Flensberg, “Many-body quantum theory in  condensed matter physics,”  (Oxford University Press, 2016)  Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  32Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Levin, “Electrostatic correlations: from plasma to biology,”  Reports on progress in physics 65, 1577 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  33P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Robin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Emmerich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Ismail, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Nigu`es, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' You, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  Nam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Keerthi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Siria, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Geim, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Radha, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Bocquet,  “Long-term memory and synapse-like dynamics of ionic carriers  in two-dimensional nanofluidic channels,” Science (in press).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content='  34A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Levy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Andelman,  and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
+page_content=' Orland, “Dipolar Poisson-  Boltzmann approach to ionic solutions: A mean field and loop ex-  pansion analysis,” The Journal of Chemical Physics 139, 164909  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E3T4oBgHgl3EQfnArs/content/2301.04622v1.pdf'}
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+arXiv:2301.00262v1  [math.PR]  31 Dec 2022
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+KOHEI SUZUKI
+Abstract. In this article, we show 1-Bakry–Émery lower Ricci curvature
+bound BE1(0, ∞) of a Dirichlet form on the configuration space whose in-
+variant measure is sineβ ensemble for any β > 0.
+As a particular case of
+β = 2, our result proves BE1(0, ∞) for a Dirichlet form related to the unlablled
+Dyson Brownian motion. We prove furthermore several functional inequalities
+including the integral Bochner inequality, the local Poincaré and the local log-
+Sobolev inequalities as well as the log-Harnack and the dimension-free Harnack
+inequalities, the Lipschitz contraction property and the L∞-to-Lipschitz regu-
+larisation property of the semigroup with the L2-transportation-type extended
+distance. At the end of the article, we provide a sufficient condition for the
+synthetic lower Ricci curvature bound in the case of general invariant measures
+beyond sineβ.
+Contents
+1.
+Introduction
+1
+2.
+Notation and Preliminaries
+4
+3.
+Curvature bound for finite-particle systems
+11
+4.
+Curvature bound for infinite-particle systems
+15
+5.
+Dimension-free/log Harnack inequalities and Lipschitz regularisation
+25
+6.
+Generalisation
+27
+Appendix A.
+29
+References
+31
+1. Introduction
+The objective of this article is to reveal the structure of lower curvature bound be-
+hind an infinite particle system of diffusions with logarithmic interactions. Such an
+interacting particle system is realised as a continuous-time strong Markov process
+having continuous paths (called a diffusion process) taking values in the configu-
+ration space Υ = Υ(R) over R and having the sineβ (β > 0) ensemble µ as an
+Date: 31/12/2022.
+Key words and phrases.
+Dyson Brownian motion, sine beta ensemble, Ricci curvature bound.
+Department of Mathematical Science, Durham University
+E-mail: kohei.suzuki@durham.ac.uk .
+1
+
+2
+K. SUZUKI
+invariant measure. We study a corresponding Dirichlet form (EΥ,µ, D(EΥ,µ)) with
+square field ΓΥ (Prop. 4.15) whose invariant measure is sineβ ensemble µ on the
+configuration space Υ. The case of β = 2 is particularly related to the diffusion
+process called (unlabelled) Dyson Brownian motion (cf. [Spo87, KT10, Osa13]). The
+labelled interacting diffusions can be phrased formally as the following infinitely
+many stochastic differential equation with logarithmic interaction (see [Tsa16] for a
+rigorous construction):
+dXk
+t = β
+2 lim
+r→∞
+�
+i̸=k:|Xk
+t −Xi
+t|<r
+1
+Xk
+t − Xi
+t
+dt + dBk
+t ,
+k ∈ N ,
+whereby {Bk
+· }k∈N are infinitely many independent Brownian motions on R.
+The main result of this article is to show that the aforementioned Dirichlet
+form (EΥ,µ, D(EΥ,µ)) satisfies the non-negative lower Ricci curvature bound BE1(0, ∞)
+in the sense of Bakry–Émery.
+We prove, furthermore, several related functional
+inequalities including the integral Bochner inequality with respect to the integral
+Γ2-operator (ΓΥ,µ
+2
+, D(ΓΥ,µ
+2
+)) (see (4.22)), the local Poincaré, the local log-Sobolev
+inequalities as well as the dimension-free Harnack inequality, the log-Harnack in-
+equality, the Lipschitz contraction and the L∞-to-Lipschitz regularisation property
+with respect to the L2-transportation-type extended distance ¯dΥ on Υ (see (2.13)).
+The main results are summarised in the following.
+Theorem. Let β > 0 and µ be the sineβ ensemble. The form (EΥ,µ, D(EΥ,µ)) satis-
+fies the following:
+• (Thm. 4.17) 1-Bakry–Émery estimate BE1(0, ∞): for u ∈ D(EΥ,µ), t > 0,
+ΓΥ�
+T Υ,µ
+t
+u
+� 1
+2 ≤ T Υ,µ
+t
+�
+ΓΥ(u)
+1
+2�
+;
+• (Cor. 4.18) Integral Bochner inequality: for every (u, ϕ) ∈ D(ΓΥ,µ
+2
+)
+ΓΥ,µ
+2
+(u, ϕ) ≥ 0 ;
+• (Cor. 4.18) Local Poincaré inequality: for u ∈ D(EΥ,µ), t > 0,
+T Υ,µ
+t
+u2 − (T Υ,µ
+t
+u)2 ≤ 2tT Υ,µ
+t
+ΓΥ(u) ,
+T Υ,µ
+t
+u2 − (T Υ,µ
+t
+u)2 ≥ 2tΓΥ(T Υ,µ
+t
+u) ;
+• (Cor. 4.18) Local log-Sobolev inequality: for non-negative u ∈ D(EΥ,µ), t > 0,
+T Υ,µ
+t
+u log u − T Υ,µ
+t
+u log T Υ,µ
+t
+u ≤ tT Υ,µ
+t
+�ΓΥ(u)
+u
+�
+,
+T Υ,µ
+t
+u log u − T Υ,µ
+t
+u log T Υ,µ
+t
+u ≥ tΓΥ(T Υ,µ
+t
+u)
+T Υ,µ
+t
+u
+.
+• (Thm. 5.1) Log-Harnack inequality: for every non-negative u ∈ L∞(Υ, µ),
+t > 0, there exists Ω ⊂ Υ so that µ(Ω) = 1 and
+T Υ,µ
+t
+(log u)(γ) ≤ log(T Υ,µ
+t
+u)(η) + ¯dΥ(γ, η)2 ,
+∀γ, η ∈ Ω ;
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+3
+• (Thm. 5.1) Dimension-free Harnack inequality: for every non-negative u ∈
+L∞(Υ, µ), t > 0 and α > 1 there exists Ω ⊂ Υ so that µ(Ω) = 1 and
+(T Υ,µ
+t
+u)α(γ) ≤ T Υ,µ
+t
+uα(η) exp
+�
+α
+2(α − 1)
+¯dΥ(γ, η)2�
+,
+∀γ, η ∈ Ω ;
+• (Thm. 5.1) Lipschitz contraction: for every u ∈ Lip(¯dΥ, µ) and t > 0
+T Υ,µ
+t
+u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ
+t
+u
+and the following Lipschitz contraction holds:
+Lip¯dΥ( ˜T Υ,µ
+t
+u) ≤ Lip¯dΥ(u) ;
+• (Thm. 5.1) L∞(Υ, µ)-to-Lip(¯dΥ, µ) regularisation by semigroup: For u ∈
+L∞(µ) and t > 0,
+T Υ,µ
+t
+u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ
+t
+u
+and the following estimate holds:
+Lip¯dΥ( ˜T Υ,µ
+t
+u) ≤
+1
+√
+2t∥u∥L∞(µ)
+∀u ∈ Lipb(¯dΥ, µ) .
+At the end of this article, Theorem will be extended to general point processes
+beyond the sineβ ensemble, see Thm. 6.2.
+Comparison with Literature. To the best knowledge of the author, this is the
+first article addressing lower Ricci curvature bound in the setting of interacting and
+infinite particle systems of diffusion processes. In the case of non-interacting case
+where the invariant measure is the Poisson measure, it has been studied in [EH15]
+in the case of Riemannian manifolds and in [DS22] in the case of general diffusion
+spaces. In the case of finite particle systems, a variable Ricci curvature bound has
+been addressed in [VG20] in the case of Coulomb-type potentials where the curvature
+bound depends on the number of particles.
+Outline of the article. After preparing the notation and the preliminaries in
+Section 2, we discuss in Section 3 the synthetic lower Ricci curvature bound for
+Dirichlet forms (EΥ(Br),µη
+r, D(EΥ(Br),µη
+r)) on the configuration space Υ(Br) over the
+closed metric ball Br with radius r > 0 centred at 0, whose invariant measure is
+the projected regular conditional probability µη
+r on Υ(Br) conditioned at η on the
+compliment Bc
+r ⊂ R. The key point of the proof is that the logarithm of the Radon–
+Nikodým density Ψη
+r := − log(dµη
+r/ dπmr) with respect the Poisson measure πmr on
+Υ(Br) with the intensity mr being the Lebesgue measure restricted on Br is geodesi-
+cally convex in (Υ(Br), ¯dΥ) due to the following DLR (Dobrushin–Lanford–Ruelle)
+equation (⋆) proven in [DHLM20, Thm.1.1] with the number-rigidity ([Gho15] for
+sine2; [NR18] and [DHLM20] for sineβ): for µ-a.e. η, there exists a unique k =
+
+4
+K. SUZUKI
+k(η) ∈ N0 so that µη
+r(Υl(Br)) > 0 if and only if l = k where Υl(Br) := {γ ∈
+Υ(Br) : γ(Br) = l}, and for γ = �k
+i=1 δxi ∈ Υ(Br)
+dµη
+r = 1
+Zη
+r e−Ψk,η
+r
+dm⊙k
+r
+,
+(⋆)
+Ψk,η
+r (γ) := − log
+� k
+�
+i<j
+|xi − xj|β
+k
+�
+i=1
+lim
+R→∞
+�
+y∈ηBcr ,|y|≤R
+���1 − xi
+y
+���
+β
+�
+,
+where m⊙k
+r
+is the k-symmetric product measure of the Lebesgue measure mr re-
+stricted on Br ⊂ R and Zη
+r is the normalising constant.
+In Section 4, we prove BE1(0, ∞) for (EΥ,µ, D(EΥ,µ)) in the following steps: we
+first construct the truncated form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)) on Υ whose gradient operator is
+truncated up to configurations on Br (Prop. 4.7). We then identify it with the super-
+position Dirichlet form ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+)) lifted from (EΥ(Br),µη
+r, D(EΥ(Br),µη
+r)) with re-
+spect to the conditioning η (Thm. 4.11). By this identification, we can lift BE(0, ∞)
+from the form (EΥ(Br),µη
+r, D(EΥ(Br),µη
+r)) onto the truncated form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)).
+Showing the monotonicity of the form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)) with respect to r and pass-
+ing to the limit r → ∞, we prove BE1(0, ∞) for the limit form (EΥ,µ, D(EΥ,µ))
+(Thm. 4.17). As a consequence of BE1(0, ∞), we obtain local Poincaré and local
+log-Sobolev inequalities (Cor. 4.18).
+In Section 5, we prove the log-Harnack inequality, the dimension-free Harnack
+inequality, the Lipschitz contraction and L∞(µ)-to-Lip(¯dΥ) properties (Thm. 5.1).
+Our proof strategy is to lift the corresponding functional inequalities from the space
+of finitely many configurations.
+In Section 6, we extend Theorem to the case of general point processes beyond
+sineβ (Thm. 6.2).
+Acknowledgement. A large part of the current work has been completed while
+the author was at Bielefeld University. He gratefully acknowledges funding by the
+Alexander von Humboldt Stiftung to support his stay.
+Data Availability Statement. Data sharing not applicable to this article as no
+datasets were generated or analysed during the current study.
+2. Notation and Preliminaries
+2.1. Numbers, Tensors, Function Spaces. We write N := {1, 2, 3, . . .}, N0 =
+{0, 1, 2, . . .}, N := N∪{+∞} and N0 := N0 ∪{+∞}. The uppercase letter N is used
+for N ∈ N0, while the lowercase letter n is used for n ∈ N0. We shall adhere to the
+following conventions:
+• the superscript □×N (the subscript □×N) denotes (N-fold) product objects;
+• the superscript □⊗N (the subscript □⊗N) denotes (N-fold) tensor objects;
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+5
+• the superscript □⊙N (the subscript □⊙N) denotes (N-fold) symmetric tensor
+objects;
+Let (X, τ) be a topological space with σ-finite Borel measure ν. We use the following
+symbols:
+(a) Lp(ν) (1 ≤ p ≤ ∞) for the space of ν-equivalence classes of functions u
+with |u|p ν-integrable when 1 ≤ p < ∞, and with u ν-essentially bounded
+when p = ∞. The Lp(ν)-norm is denoted by ∥u∥p
+Lp(ν) := ∥u∥p
+p :=
+�
+X |u|p dν
+for 1 ≤ p < ∞, and ∥u∥L∞(ν) := ∥u∥∞ = esssupX u. In the case of p = 2, the
+inner-product is denoted by (u, v)L2(ν) :=
+�
+X uv dν;
+(b) Lp
+s(ν⊗n) := {u ∈ Lp(ν⊗n) : u is symmetric} where u is said to be symmetric
+if and only if u(x1, . . . , xk) = u(xσ(1), . . . , xσ(k)) for any element σ ∈ S(k) in
+the k-symmetric group.
+(c) Cb(X) for the space of τ-continuous bounded functions on X; if X is locally
+compact, C0(X) denotes the space of τ-continuous and compactly supported
+functions on X; C∞
+0 (R) for the space of compactly supported smooth func-
+tions on R;
+(d) We write 1A for the indicator function on A, i.e., 1A(x) = 1 if x ∈ A, and
+1A(x) = 0 otherwise.
+2.2. Dirichlet forms. We refer the reader to [MR90, BH91] for this subsection.
+Throughout this paper, a Hilbert space always means a Hilbert space with inner
+product (·, ·)H taking value in R.
+Dirichlet forms.
+Given a bilinear form (Q, D(Q)) on a Hilbert space H, we write
+Q(u) := Q(u, u) ,
+Qα(u, v) := Q(u, v) + α(u, v)H , α > 0 .
+Let (X, Σ, ν) be a σ-finite measure space. A symmetric Dirichlet form on L2(ν) is
+a non-negative definite densely defined closed symmetric bilinear form (Q, D(Q))
+on L2(ν) satisfying the Markov property
+u0 := 0 ∨ u ∧ 1 ∈ D(Q)
+and
+Q(u0) ≤ Q(u) ,
+u ∈ D(Q) .
+Throughout this article, Dirichlet form always means symmetric Dirichlet form. If
+not otherwise stated, D(Q) is always regarded as a Hilbert space with norm
+∥ · ∥D(Q) := Q1( · )1/2 :=
+�
+Q( · ) + ∥ · ∥2
+L2(ν) .
+In order to distinguish Dirichlet forms defined in different base spaces with different
+reference measures, we often use the notation QX,ν to specify the base space X and
+the reference measure ν.
+Square field.
+A Dirichlet form (Q, D(Q)) admits square field Γ if there exists a
+dense subspace H ⊂ D(Q) ∩ L∞(ν) having the following property: for any u ∈ H,
+
+6
+K. SUZUKI
+there exists v ∈ L1(ν) so that
+2Q(uh, u) − Q(h, u2) =
+�
+X
+hv dν
+∀h ∈ D(Q) ∩ L∞(ν) .
+Such v is denoted by Γ(u). The square field Γ can be uniquely extended as an
+operator on D(Q) × D(Q) → L1(ν) ([BH91, Thm. I.4.1.3]).
+Semigroups and generators.
+We refer the reader to [MR90, Chap. I, Sec. 2] for
+the following contents.
+Let (Q, D(Q)) be a symmetric closed form on a Hilbert
+space H. The infinitesimal generator (A, D(A)) corresponding to (Q, D(Q)) is the
+unique densely defined closed operator on H satisfying the following integration-by-
+parts formula:
+−(u, Av)H = Q(u, v)
+∀u ∈ D(Q), v ∈ D(A) .
+The resolvent operator {Rα}α≥0 is the unique bounded linear operator on H satis-
+fying
+Qα(Rαu, v) = (u, v)H
+∀u ∈ H
+v ∈ D(Q) .
+The semigroup {Tt}t≥0 is the unique bounded linear operator on H satisfying
+Gαu =
+� ∞
+0
+e−αtTtu dt
+u ∈ H .
+Locality.
+Let (Q, D(Q)) be a Dirihclet form on L2(ν). It is called local ([BH91,
+Def. 5.1.2]) if for any F, G ∈ C∞
+c (R) and any u ∈ D(Q),
+supp[F] ∩ supp[G] = ∅ =⇒ Q(F0 ◦ u, G0 ◦ u) = 0 ,
+where F0(x) := F(x) − F(0) and G0(x) := G(x) − G(0).
+2.3. Metric space. Let X be any non-empty set. A function d: X ×2 → [0, ∞] is
+an extended distance if it is symmetric and satisfying the triangle inequality, and it
+does not vanish outside the diagonal in X ×2, i.e. d(x, y) = 0 iff x = y; a distance
+if it is finite. Let x0 ∈ X and r ∈ [0, ∞). We write Br(x0) := {dx0 ≤ r}, where
+dx0 := d(x0, ·).
+Lipschitz algebras.
+A function f : X → R is d-Lipschitz if there exists a con-
+stant L > 0 so that
+��u(x) − u(y)
+�� ≤ L d(x, y) ,
+x, y ∈ X .
+(2.1)
+The smallest constant L so that (2.1) holds is the (global) Lipschitz constant of u,
+denoted by Lipd(u). For any non-empty A ⊂ X we write Lip(A, d), resp. Lipb(A, d)
+for the family of all finite, resp. bounded, d-Lipschitz functions on A. For simplic-
+ity of notation, further let Lip(d) := Lip(X, d), resp. Lipb(d) := Lipb(X, d). Set also
+Lip1(d) := {u ∈ Lip(d) : Lipd(u) ≤ 1} and Lip1
+b(d) := Lip1(d) ∩ Lipb(d). For a given
+measure ν, we set
+Lip(d, ν) := {u ∈ Lip(d) : u is ν-measurable} ,
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+7
+as well as Lipb(d, ν) and Lip1
+b(d, ν) denoting the corresponding subspaces of ν-
+measurable functions respectively.
+Geodesical convexity.
+A metric space (X, d) is called a geodesic space if for any
+x0, x1 ∈ X there exists a constant speed geodesic ω : [0, 1] → X connecting x0 and x1:
+ω0 = x0 ,
+ω1 = x1 ,
+d(ωt, ωs) = |t − s|d(ω0, ω1)
+∀t, s ∈ [0, 1] .
+For a function U : X → R ∪ {+∞}, define D(U) := {x ∈ X : U(x) < ∞}. We say
+that U is K-geodesically convex for K ∈ R if for any x0, x1 ∈ D(U) there exists a
+constant speed geodesic ω : [0, 1] → X with ω0 = x0 and ω1 = x1 and
+U(ωt) ≤ (1 − t)U(ω0) + tU(ω1) − K
+2 t(1 − t)d2(ω0, ω1)
+∀t ∈ [0, 1] .
+When K = 0, we say that U is geodesically convex.
+2.4. Cheeger energies. A complete separable geodesic metric space (X, d) equipped
+with fully supported Radon measure ν with finite total mass ν(X) < ∞ is called a
+metric measure space in this article. Let (X, d, ν) be a metric measure space. For
+u ∈ Lip(d), the slope |Ddu|(x) is defined as
+|Ddu|(x) :=
+
+
+
+
+
+lim sup
+y→x
+|u(x) − u(y)|
+d(x, y)
+if x is not isolated;
+0
+otherwise .
+The slope is universally measurable, see [AGS14a, Lem. 2.6]. The Cheeger energy
+Chd,ν : L2(ν) → R ∪ {+∞} is defined as the L2(ν)-lower semi-continuous envelope
+of
+�
+X |Ddu|2 dν:
+Chd,ν(u) := inf
+�
+lim inf
+n→∞
+�
+X
+|Ddun|2 dν : un ∈ Lip(d) ∩ L2(ν)
+L2
+−→ u
+�
+.
+The domain is denoted by W 1,2(X, d, ν) := {u ∈ L2(ν) : Chd,ν(u) < ∞}.
+The
+Cheeger energy Chd,ν can be expressed by the following integration, see [AGS14a,
+Thm. 4.5] : there exists a measurable function |∇u|∗ ∈ L2(ν) so that |∇u|∗ ≤ |Ddu|
+ν-a.e. for every u ∈ Lip(d) and
+Chd,ν(u) =
+�
+X
+|∇u|2
+∗ dν
+∀u ∈ W 1,2(X, d, ν) ,
+where |∇u|∗ is called minimal relaxed slope.
+2.5. Riemannian Curvature-dimension condition. Let (X, d, ν) be a metric
+measure space. The following definition is an equivalent characterisation of RCD(K, ∞)
+by [AGS15, Cor. 4.18]. We say that (X, d, ν) satisfies the Riemannian Curvature-
+Dimension Condition RCD(K, ∞) for K ∈ R if
+(i) Chd,ν is quadratic, i.e., Chd,ν(u + v) + Chd,ν(u − v) = 2Chd,ν(u) + 2Chd,ν(v);
+(ii) Sobolev-to-Lipschitz property holds, i.e., every u ∈ W 1,2(X, d, ν) with |∇u|∗ ≤
+1 has a d-Lipschitz ν-representative ˜u satisfying Lip(˜u) ≤ 1;
+
+8
+K. SUZUKI
+(iii) Chd,ν satisfies BE2(K, ∞), i.e., |∇Ttu|2
+∗ ≤ e−2KtTt|∇u|2
+∗ for every u ∈ W 1,2(X, d, ν)
+and t > 0.
+In this case, the Cheeger energy Chd,ν is a local Dirichlet form ([AGS14b, §4.3]). We
+note that, while [AGS15, Cor. 4.18] is stated in terms of the minimal weak upper
+gradient denoted by |∇ · |w, it is identical to the minimal relaxed slope |∇ · |∗ due to
+[AGS14a, Thm. 6.2].
+2.6. Configuration spaces. A configuration on a locally compact Polish space X
+is any N0-valued Radon measure γ on X, which can be expressed by γ = �N
+ii δxi
+for N ∈ ¯N, where δx denotes the Dirac measure at x, i.e., δx(A) = 1 if and only if
+x ∈ A. The configuration space Υ = Υ(X) is the space of all configurations over X.
+The space Υ is equipped with the vague topology, i.e., the topology generated by
+the duality of the space C0(X) of continuous functions with compact support. We
+write the restriction γA := γ ⇂A for a Polish subspace A ⊂ X and the corresponding
+restriction map is denoted by
+prA : Υ −→ Υ(A): γ �−→ γA .
+(2.2)
+The N-particle configuration space is denoted by
+ΥN := {γ ∈ Υ : γ(X) = N} ,
+N ∈ N0 .
+Let Sk be the k-symmetric group. It can be readily seen that the k-particle config-
+uration space Υk is isomorphic to the quotient space X×k/Sk:
+Υk ∼= X⊙k := X×k/Sk ,
+k ∈ N0 .
+(2.3)
+The associated projection map from X×k to the quotient space X×k/Sk is denoted
+by Pr. For η ∈ Υ and r > 0, we set
+Υη
+r := {γ ∈ Υ : γBcr = ηBcr} .
+(2.4)
+Conditional probability.
+Let µ be a Borel probability measure on Υ. Let
+µ(· | prBcr(·) = ηBcr)
+denote the regular conditional probability of µ conditioned at η ∈ Υ with respect
+to the σ-field generated by the projection map γ ∈ Υ �→ prr(γ) = γBr ∈ Υ(Br) (see
+e.g., [DS21a, Def. 3.32] for the precise definition). Let µη
+r be the probability measure
+on Υ(Br) defined as
+µη
+r := (prr)#µ(· | prBcr(·) = ηBcr) ,
+(2.5)
+and its restriction on Υk(Br) is denoted by µk,η
+r
+:= µη
+r|Υk(Br).
+Note: The conditional probability µ(· | prBcr(·) = ηBcr) is a probability measure on
+the whole space Υ whose support is Υη
+r = {γ ∈ Υ : γBcr = ηBcr}. We may project the
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+9
+conditional probability to the probability measure µη
+r on Υ(Br) as in (2.5) without
+loss of information in the sense that
+prr : Υη
+r → Υ(Br) is a bi-measure-preserving bijection .
+(2.6)
+Namely, the projection map prr is bijective with the inverse map pr−1
+r
+defined
+as pr−1
+r (γ) := γ + η, and both prr and pr−1
+r
+are measure-preserving between the
+two measures µ(· | prBcr(·) = ηBcr) and µη
+r.
+For a measurable function u: Υ → R, r > 0 and for η ∈ Υ, we set
+uη
+r(γ) := u(γ + ηBcr)
+γ ∈ Υ(Br) .
+(2.7)
+By the property of the conditional probability, it is straightforward to see that for
+any u ∈ L1(µ),
+�
+Υ
+u dµ =
+�
+Υ
+��
+Υ(Br)
+uη
+r dµη
+r
+�
+dµ(η) .
+(2.8)
+See, e.g., [DS21a, Prop. 3.44]. For a measurable set Ω ⊂ Υ, define a section Ωη
+r ⊂
+Υ(Br) at η ∈ Υ on Bc
+r by
+Ωη
+r := {γ ∈ Υ(Br) : γ + ηBcr ∈ Ω} .
+(2.9)
+By applying the disintegration formula (2.8) to u = 1Ω, we obtain
+µ(Ω) =
+�
+Υ
+µη
+r(Ωη
+r) dµ(η) .
+(2.10)
+Poisson measure.
+Let (X, τ, ν) be a locally compact Polish space with Radon
+measure ν satisfying ν(X) < ∞. The Poisson measure πν on Υ(X) with intensity ν
+is defined in terms of the symmetric tensor measure ν⊙ as follows:
+πν(·) := e−ν(X)
+∞
+�
+k=1
+ν⊙k�
+· ∩ Υk(X)
+�
+= e−ν(X)
+∞
+�
+k=1
+1
+k!(Pr)#ν⊗k�
+· ∩ Υk(X)
+�
+.
+(2.11)
+L2-transportation distance.
+Let (X, d) be a locally compact complete separable
+metric space.
+For i = 1, 2 let proji : X×2 → X denote the projection to the ith
+coordinate for i = 1, 2. For γ, η ∈ Υ, let Cpl(γ, η) be the set of all couplings of γ
+and η, i.e.,
+Cpl(γ, η) := {q ∈ M (X
+×2): (proj1)♯q = γ , (proj2)♯q = η} .
+Here M (X ×2) denotes the space of all Radon measures on X ×2. The L2-transportation
+extended distance on Υ(X) is
+dΥ(γ, η) :=
+inf
+q∈Cpl(γ,η)
+��
+X×2 d2(x, y) dq(x, y)
+�1/2
+,
+inf ∅ = +∞ .
+(2.12)
+We refer the readers to [DS21a, §4.2, p.52] for details regarding the L2-transportation
+extended distance dΥ.
+It is important to note that dΥ is an extended distance,
+
+10
+K. SUZUKI
+attaining the value +∞ and dΥ is lower semi-continuous with respect to the product
+vague topology τ ×2
+v
+but never τ ×2
+v -continuous.
+We introduce a variant of the L2-transportation extended distance, called L2-
+transportation-type extended distance ¯dΥ defined as
+¯dΥ(γ, η) :=
+
+
+
+dΥ(γ, η)
+if γBcr = ηBcr for some r > 0 ,
++∞
+otherwise .
+(2.13)
+By definition, dΥ ≤ ¯dΥ on Υ, and dΥ = ¯dΥ on Υ(Br) for any r > 0. In particular,
+we have
+Lip(Υ, dΥ) ⊂ Lip(Υ, ¯dΥ) ,
+Lip¯dΥ(u) ≤ LipdΥ(u) ,
+u ∈ Lip(Υ, dΥ) .
+(2.14)
+It can be readily seen that
+¯dΥ(γ, η) < ∞
+⇐⇒
+γBcr = ηBcr , γ(Br) = η(Br)
+for some r > 0 .
+(2.15)
+When we work with the configuration space Υ(Rn) over the n-dimensional Eu-
+clidean space Rn or over any Polish subset in Rn, we always choose the Euclidean
+distance d(x, y) = |x − y| and the L2-transportation distance dΥ and ¯dΥ associated
+with d.
+2.7. sineβ ensemble. Let β > 0 and CβEk be the circular β ensemble on the k-
+particle configuration space, i.e., it is the probability measure Pk,β on the space Υk(S1)
+over the unit circle S1 ⊂ C defined as
+dPk,β :=
+1
+Zk,β
+�
+1≤j<l≤k
+��eiθj − eiθl��β dθ1
+2π · · · dθk
+2π ,
+where the normalisation constant Zk,β is given in terms of Gamma function Γ:
+Zk,β := Γ( 1
+2βk + 1)
+Γ( 1
+2βk + 1)k .
+According to [KS09, Def. 1.6], the circular β ensemble CβE is defined as the limit
+probability measure Pβ whose Laplace transform is determined as
+�
+exp
+�
+−
+�
+x∈γ
+f(x)
+�
+dPβ(γ) = lim
+k→∞
+�
+exp
+�
+−
+k
+�
+i=1
+f(kθi)
+�
+dPk,β(θ1, . . . , θk) ,
+for all f ∈ C0(R). In [VV09] a probability measure µβ on Υ(R) called sine β ensem-
+ble has been constructed by a limit of Gaussian β-ensemble. These two measures Pβ
+and µβ turned out to be identical each other by the work of [Nak14]. Throughout the
+rest of the article, we use the symbol µ = µβ to denote sineβ ensemble (equivalently,
+circular β ensemble) and we do not specify the inverse temperature β as there is no
+particular role played by a special β.
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+11
+Number-rigidity.
+A Borel probability µ on Υ = Υ(Rn) is said to be number
+rigid (in short: (R)) if for any bounded domain E ⊂ Rn, there exists Ω ⊂ Υ so that
+µ(Ω) = 1 and, for any γ, η ∈ Ω
+γEc = ηEc implies γE = ηE .
+(R)
+Namely, the configuration outside E determines the number of particle inside E. The
+number-rigidity has been proven in [Gho15] for the sine2 ensemble and in [NR18],
+[DHLM20] for the sineβ ensemble for general β > 0.
+3. Curvature bound for finite-particle systems
+In this section, we study Dirichlet forms on the configuration space Υ(Br) over
+metric balls Br ⊂ R. We denoted by m and mr the Lebesgue measure on R and
+its restriction on the metric ball Br := [−r, r] respectively, and take the Euclidean
+distance d(x, y) := |x − y| for x, y ∈ Br.
+3.1. Construction of Dirichlet forms on Υk(Br). Let W 1,2
+s (m⊗k
+r ) be the space
+of m⊗k
+r -classes of (1, 2)-Sobolev and symmetric functions on the product space B×k
+r ,
+i.e.,
+W 1,2
+s (m⊗k
+r ) :=
+�
+u ∈ L2
+s(m⊗k
+r ) :
+�
+B×k
+r
+|∇⊗ku|2 dm⊗k
+r
+< ∞
+�
+,
+where ∇⊗k denotes the weak derivative on R×k: ∇⊗ku := (∂1u, . . . , ∂ku). The space
+W 1,2
+s (m⊗k
+r ) consisting of symmetric functions, the projection Pr : B×k
+r
+→ Υk(Br) ∼=
+B×k
+r /Sk naturally acts on W 1,2
+s (m⊗k
+r ) and the resulting quotient space is denoted by
+W 1,2(m⊙k
+r ), which is the (1, 2)-Sobolev space on Υk(Br):
+W 1,2(m⊙k
+r ) :=
+�
+u ∈ L2(m⊙k
+r ) :
+�
+Υk(Br)
+|∇⊙ku|2 dm⊙k
+r
+< ∞
+�
+,
+where ∇⊙k is the quotient operator of the weak gradient operator ∇⊗k through the
+projection Pr and m⊙k
+r
+is the symmetric product measure defined as
+m⊙k
+r
+:= 1
+k!(Pr)#m⊗k
+r
+.
+For 0 < r < R < ∞, k ∈ N0 and η ∈ Υ(Bc
+r), we introduce the following finite
+Borel measure on Υk(Br): for γ = �k
+i=1 δxi
+dµk,η
+r,R(γ) := e−Ψk,η
+r,R(γ) dm⊙k
+r (γ) ,
+(3.1)
+Ψk,η
+r,R(γ) := − log
+� k
+�
+i<j
+|xi − xj|β
+k
+�
+i=1
+�
+y∈ηBcr ,|y|≤R
+���1 − xi
+y
+���
+β
+�
+.
+The corresponding weighted Sobolev norm is denoted by
+EΥ(Br),µk,η
+r,R(u) :=
+�
+Υk(Br)
+|∇⊙ku|2 dµk,η
+r,R ,
+u ∈ Lipb(Υk(Br), dΥ) ,
+(3.2)
+
+12
+K. SUZUKI
+where we note that as Lip(Υk, dΥ) ⊂ W 1,2(m⊙k
+r ) and |∇⊙ku| ≤ LipdΥ(u) due to the
+Rademacher theorem descendent from the one in the product Sobolev space W 1,2(m⊗k
+r )
+through the quotient, the expression |∇⊙ku| and its integral against the probability
+measure µk,η
+r,R make sense for u ∈ Lipb(Υk(Br), dΥ).
+Proposition 3.1. The form (3.2) is well-defined and closable. The closure is a local
+Dirichlet form on L2(µk,η
+r,R) and its domain is denoted by D
+�
+EΥ(Br),µk,η
+r,R).
+Proof. The well-definedness follows from the following inequality:
+�
+Υk(Br)
+|∇⊙ku|2 dµk,η
+r,R ≤
+���e−Ψk,η
+r,R
+���
+L∞(Υk(Br),µk,η
+r,R)
+�
+Υk(Br)
+|∇⊙ku|2 dm⊙k < ∞ .
+(3.3)
+The closability of EΥ(Br),µk,η
+r,R descends from the closability of the corresponding
+Dirichlet form on the product space B×k
+r
+defined on the space of symmetric d×k-
+Lipschitz functions:
+EB×k
+r
+,µk,η
+r,R :=
+�
+B×k
+r
+|∇⊗ku|2e−Ψk,η
+r,R dm⊗k
+r
+,
+where the closability of EB×k
+r
+,µk,η
+r,R is a consequence of the continuity of the den-
+sity e−Ψk,η
+r,R on B×k
+r
+and the standard Hamza-type argument by [MR85, Fuk97], see
+for an accessible reference, e.g., [MR90, pp. 44-45]. The locality of the form is an
+immediate consequence of the locality of the gradient operator ∇⊙k.
+■
+Let µ be the sineβ ensemble. Due to [DHLM20, Thm. 1.1], the following limit
+exists for µ-a.e. η, all x ∈ Br and r > 0:
+lim
+R→∞
+�
+y∈ηBcr ,|y|≤R
+���1 − x
+y
+���
+β
+.
+Recall that µη
+r has been defined in (2.5). By [DHLM20, Thm. 1.1] and the number-
+rigidity (R) of µ, for µ-a.e. η there exists k = k(η) so that
+µη
+r(Υl(Br)) > 0 if and only if l = k(η) ,
+(3.4)
+and for γ = �k
+i=1 δxi,
+dµη
+r = dµk,η
+r
+= e−Ψk,η
+r
+Zη
+r
+dm⊙k
+r
+,
+(3.5)
+Ψk,η
+r (γ) := − log
+� k
+�
+i<j
+|xi − xj|β
+k
+�
+i=1
+lim
+R→∞
+�
+y∈ηBcr ,|y|≤R
+���1 − xi
+y
+���
+β
+�
+,
+where Zη
+r is the normalising constant. The corresponding weighted Sobolev norm is
+defined as
+EΥ(Br),µk,η
+r (u) :=
+�
+Υk(Br)
+|∇⊙ku|2 dµk,η
+r
+,
+u ∈ Lipb(Υk(Br), dΥ) .
+(3.6)
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+13
+Proposition 3.2. Let µ be the sineβ ensemble for β > 0. The form (3.6) is well-
+defined and closable for µ-a.e. η. The closure is a local Dirichlet form on L2(µk,η
+r )
+and its domain is denoted by D(EΥ(Br),µk,η
+r ).
+Proof. As e−Ψk,η
+r,R
+R→∞
+−−−→ e−Ψk,η
+r
+uniformly on Υk(Br) for µ-a.e. η by [DHLM20,
+Lem. 2.3 and Proof of Thm. 2.1 in p. 183], the density e−Ψk,η
+r
+is continuous on
+B⊙k
+r , hence the same proof as Prop. 3.1 applies to conclude the statement.
+■
+3.2. Curvature bound for finite-particle systems. We show that the poten-
+tial Ψk,η
+r,R defined in (3.1) is geodesically convex in (Υ(Br), dΥ).
+Proposition 3.3. Ψk,η
+r,R is geodesically convex in (Υk(Br), dΥ) for any 0 < r < R <
+∞, k ∈ N and η ∈ Υ(Bc
+r),
+Proof. Note that if u1, . . . , uk are convex and α1, . . . , αk ≥ 0, then �k
+i=1 αiui is
+again convex. Note also that for any 0 < r < R, any y ∈ [−R, −r] ∪ [r, R] and any
+i, j ∈ {1, 2, . . . , k} with i ̸= j, the functions − log |xi − xj| and − log |1 − xi
+y | are
+convex in the following areas for any σ ∈ Sk:
+�
+(x1, . . . , xk) ∈ B×k
+r
+: xσ(1) < xσ(2) < · · · < xσ(k)
+�
+.
+The following expression, therefore, concludes that Ψk,η
+r,R is geodesically convex as a
+function on Υk(Br): for any γ = �k
+i=1 δxi
+Ψk,η
+r,R(γ) = −β
+k
+�
+i<j
+log(|xi − xj|) − β
+k
+�
+i=1
+�
+y∈ηBcr ,|y|≤R
+log
+���1 − xi
+y
+��� .
+(3.7)
+The proof is complete.
+■
+Thanks to the geodesical convexity of the potential Ψk,η
+r,R shown in Prop. 3.3,
+the Dirichlet form (EΥ(Br),µk,η
+r,R, D(EΥ(Br),µk,η
+r,R)) satisfies the Riemannian Curvature-
+Dimension condition RCD(0, ∞).
+Proposition 3.4. The space (Υk(Br), dΥ, µk,η
+r,R) satisfies RCD(0, ∞) for every k ∈
+N0, 0 < r < R < ∞ and η ∈ Υ, and it holds that
+�
+EΥ(Br),µk,η
+r,R, D(EΥ(Br),µk,η
+r,R)
+�
+=
+�
+ChdΥ,µk,η
+r,R, W 1,2(Υk(Br), dΥ, µk,η
+r,R)
+�
+.
+Proof. Noting that B×k
+r
+is a convex subset in Rk, the space (B×k
+r , d×k, m⊗k
+r ) is a
+geodesic subspace of Rk and, therefore, satisfies RCD(0, ∞) by the Global-to-Local
+property of RCD(0, ∞), see [AGS14b, Thm. 6.20]. Noting that the k-particle con-
+figuration space (Υk(Br), dΥ, µk,η
+r,R) is the quotient space of (B×k
+r , d×k, m⊗k) with
+respect to the symmetric group Sk and that the property RCD(0, ∞) is preserved
+under the quotient operation with respect to Sk thanks to [GKMS18], we obtain
+that (Υk(Br), dΥ, m⊙k) satisfies RCD(0, ∞) as well. By the geodesical convexity of
+the potential Ψk,η
+r,R shown in Prop. 3.3 and the continuity of the density e−Ψk,η
+r,R, the
+weighted space (Υk(Br), dΥ, µk,η
+r,R) satisfies RCD(0, ∞) by [AGS14b, Prop. 6.21].
+
+14
+K. SUZUKI
+To conclude the statement, it suffices to check the identity
+EΥ(Br),µk,η
+r,R = ChdΥ,µk,η
+r,R .
+By the Rademacher theorem on Υk(Br) descendent from the Rademacher theorem
+on B×k
+r , the slope |DdΥu| coincides with |∇⊙ku| for u ∈ Lip(Υk(Br), dΥ). Thus,
+EΥ(Br),µk,η
+r,R(u) =
+�
+Υk(Br)
+|DdΥu|2 dµk,η
+r,R
+u ∈ Lipb(Υk(Br), dΥ) .
+(3.8)
+Since ChdΥ,µk,η
+r,R is the L2-lower semi-continuous envelope, the functional ChdΥ,µk,η
+r,R is
+the maximal L2-lower semi-continuous functional satisfying
+ChdΥ,µk,η
+r,R(u) ≤
+�
+Υk(Br)
+|DdΥu|2 dµk,η
+r,R .
+As EΥ(Br),µk,η
+r,R is closed by Prop. 3.1, in particular, EΥ(Br),µk,η
+r,R is L2-lower semi-
+continuous. Therefore, combining the maximality of ChdΥ,µk,η
+r,R with (3.8), it holds
+that
+EΥ(Br),µk,η
+r,R ≤ ChdΥ,µk,η
+r,R and W 1,2(Υk(Br), dΥ, µk,η
+r,R) ⊂ D(EΥ(Br),µk,η
+r,R)
+and
+ChdΥ,µk,η
+r,R(u) = EΥ(Br),µk,η
+r,R(u)
+u ∈ Lipb(Υk(Br), dΥ) .
+As Lipb(Υk(Br), dΥ) is dense both in D(EΥ(Br),µk,η
+r,R) and W 1,2(Υk(Br), dΥ, µk,η
+r,R) by
+construction, the proof is completed.
+■
+In view of Prop. 3.4 and the approximation Ψk,η
+r,R to Ψk,η
+r
+as R → ∞, we prove
+that (Υk(Br), dΥ, µk,η
+r ) satisfies RCD(0, ∞) as well.
+Proposition 3.5. Let µ be the sineβ ensemble for β > 0. For any 0 < r < ∞ and
+µ-a.e. η ∈ Υ, the space (Υk(Br), dΥ, µk,η
+r ) satisfies RCD(0, ∞), where k = k(η) as
+in (3.4). Furthermore,
+�
+EΥ(Br),µk,η
+r , D(EΥ(Br),µk,η
+r )
+�
+=
+�
+ChdΥ,µk,η
+r , W 1,2(Υk(Br), dΥ, µk,η
+r )
+�
+.
+Proof. Since the potential Ψk,η
+r,R is geodesically convex for any R and it converges
+pointwise to Ψk,η
+r
+as R → ∞ for µ-a.e. η by [DHLM20, Lem. 2.3 and Proof of Thm.
+2.1 in p. 183], the potential Ψk,η
+r
+is again geodesically convex on (Υk(Br), dΥ). Fur-
+thermore, as the density e−Ψk,η
+r,R converges uniformly to e−Ψk,η
+r
+on Υk(Br) as R → ∞
+for µ-a.e. η by [DHLM20, Lem. 2.3 and Proof of Thm. 2.1 in p. 183], the den-
+sity e−Ψk,η
+r
+is continuous on Υ(Br). Noting the fact that the constant multiplication
+(by the normalisation constant Zη
+r ) does not change the lower Ricci curvature bound
+(see e.g., [Stu06, Prop. 4.13]), the same proof as Prop. 3.4 applies to conclude the
+statement.
+■
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+15
+4. Curvature bound for infinite-particle systems
+In this section, we construct a local Dirichlet form on Υ = Υ(R) associated with
+sineβ ensemble µ and show the BE1(0, ∞) property by the following steps: we first
+construct truncated Dirichlet forms on Υ whose gradient operators are truncated up
+to configurations inside Br. We then identify them with the superposition Dirichlet
+forms lifted from Υ(Br), thanks to which we can show BE1(0, ∞) for the truncated
+forms. We take the monotone limit of the truncated forms to construct a Dirichlet
+form with invariant measure sineβ ensembles µ and BE1(0, ∞) extends to the limit
+form. In the end of this section, we discuss several applications of the BE1(0, ∞)
+property.
+4.1. Superposition of Dirichlet forms from Υ(Br) onto Υ. In this subsection,
+we construct the truncated Dirichlet forms on Υ. We first construct square field
+operators on Υ and Υ(Br) respectively.
+For so doing, we introduce a map Uγ,x
+transferring functions on the configuration space Υ to functions on the base space R.
+For u : Υ → R, define Uγ,x(u) : R → R by
+Uγ,x(u)(y) := u
+�
+1X\{x} ·γ + δy
+�
+− u
+�
+1X\{x} ·γ
+�
+,
+γ ∈ Υ,
+x ∈ γ .
+(4.1)
+In the context of configuration spaces, the operation Uγ,x has been firstly discussed
+in [MR00, Lem. 1.2], see also [DS21a, Lem. 2.16]. We introduce the localisation of
+the operator Uγ,x on Br. Recall that for a measurable function u: Υ → R, r > 0
+and for η ∈ Υ, we set in (2.7)
+uη
+r(γ) := u(γ + ηBcr) for γ ∈ Υ(Br).
+Lemma 4.1. For u : Υ(Br) → R, define Ur
+γ,x(u) : Br → R by
+Ur
+γ,x(u)(y) := u(1X\{x} · γ + δy) − u(1X\{x} · γ)
+γ ∈ Υ(Br), x ∈ γ .
+The operation Ur
+γ,x maps from Lip(Υ(Br), dΥ) to Lip(Br) and Lipschitz constants
+are contracted by Ur
+γ,x for any r > 0:
+Lip(Ur
+γ,x(u)) ≤ LipdΥ(u)
+∀γ ∈ Υ(Br)
+∀x ∈ γ .
+Furthermore, for any u : Υ → R,
+Ur
+γBr ,x(uγ
+r)(y) = Uγ,x(u)(y)
+for every γ ∈ Υ, x ∈ γBr and y ∈ Br .
+Proof. Let u ∈ Lip(Υ(Br), dΥ). Then
+|Ur
+γ,x(u)(y) − Ur
+γ,x(u)(z)| = |u(1X\{x} ·γ + δy) − u(1X\{x} ·γ + δz)|
+≤ LipdΥ(u)dΥ(1X\{x} ·γ + δy, 1X\{x} ·γ + δz)
+= LipdΥ(u)|y − z| ,
+which concludes the first assertion.
+
+16
+K. SUZUKI
+We verify the second assertion. For every x ∈ γBr and y ∈ Br,
+Uγ,x(u)(y) = u(1X\{x} · γ + δy) − u(1X\{x} · γ)
+= u(1X\{x} · γBr + γBcr + δy) − u(1X\{x} · γBr + γBcr)
+= ur,γ(1X\{x} · γBr + δy) − ur,γ(1X\{x} · γBr)
+= Ur
+γBr ,x(uγ
+r)(y) .
+The proof is complete.
+■
+We now define a square field operator on Υ truncated up to particles inside Br.
+Definition 4.2 (Truncated square field on Υ). Let u : Υ → R be a measurable
+function so that Uγ,x(u)|Br ∈ W 1,2(mr) for µ-a.e. γ and every x ∈ γBr. The following
+operator is called the truncated square field ΓΥ
+r ,
+(4.2)
+ΓΥ
+r (u)(γ) :=
+�
+x∈γBr
+|∇Uγ,x(u)|2(x) .
+Thanks to Lem. A.1, Formula (4.2) is well-defined for µ-a.e. γ. Indeed, as Uγ,x(u)|Br ∈
+W 1,2
+loc (mr), the weak gradient ∇Uγ,x(u) is well-defined pointwise on a measurable set
+Σ ⊂ Br with mr(Σc) = 0. By applying Lem. A.1, Formula (4.2) is well-defined on
+the set Ω(r) of µ-full measure.
+Based on the truncated square field ΓΥ
+r , we introduce the truncated form on Υ
+defined on a certain core.
+Definition 4.3 (Core). For r > 0, let Cr be defined as the space of µ-classes of
+measurable functions u so that
+(a) u ∈ L∞(µ);
+(b) uη
+r ∈ Lipb(Υ(Br), dΥ) for µ-a.e. η and r > 0;
+(c) The following integral is finite:
+EΥ,µ
+r
+(u) :=
+�
+Υ
+ΓΥ
+r (u) dµ < ∞ .
+(4.3)
+Note that, thanks to Lem. 4.1, if a measurable function u : Υ → R satisfies (b),
+then Uγ,x(u)|Br ∈ Lip(Br, d) ⊂ W 1,2(mr). Thus, the expression ΓΥ
+r (u) in (4.3) is
+well-posed.
+Definition 4.4 (Square field on Υ(Br)). Fix r > 0 and η ∈ Υ. For a µ-measurable
+function u : Υ(Br) → R satisfying u|Υk(Br) ∈ D(EΥ(Br),µk,η
+r ) for any k ∈ N0, we
+define the following square field operator on Υ(Br):
+ΓΥ(Br)(u) :=
+∞
+�
+k=0
+���∇⊙k�
+u|Υk(Br)
+����
+2
+,
+(4.4)
+and define the following form:
+EΥ(Br),µη
+r(u) :=
+�
+Υ(Br)
+ΓΥ(Br)(u) dµη
+r ,
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+17
+D(EΥ(Br),µη
+r) := {u : Υ(Br) → R, EΥ(Br),µη
+r(u) < ∞} .
+Due to the number-rigidity (R), the Dirichlet form EΥ(Br),µη
+r is equal to EΥ(Br),µk,η
+r
+up to the normalising constant multiplication, therefore, it is a Dirichlet form as
+well. The corresponding semigroup is denoted by {T Υ(Br),µη
+r
+t
+}t≥0.
+Remark 4.5. The number-rigidity (R) is not necessary to conclude that EΥ(Br),µη
+r
+is a Dirichlet form since any countable sum of Dirichlet forms is a Dirichlet form
+(see e.g., [MR90, Exercise 3.9]).
+Before discussing properties of truncated forms, we prepare a lemma, which states
+that the operation (·)η
+r defined in (2.7) maps from Lip(Υ, ¯dΥ) to Lip(Υ(Br), dΥ) and
+contracts Lipschitz constants.
+Lemma 4.6. Let u ∈ Lip(Υ, ¯dΥ). Then, uη
+r ∈ Lip(Υ(Br), dΥ) and
+LipdΥ(uη
+r) ≤ Lip¯dΥ(u) ,
+∀η ∈ Υ ,
+r > 0 .
+(4.5)
+Proof. Let γ, ζ ∈ Υ(Br) and η ∈ Υ. Then,
+|uη(γ) − uη(ζ)| = |u(γ + ηBcr) − u(ζ + ηBcr)| ≤ Lip¯dΥ(u)¯dΥ(γ + ηBcr, ζ + ηBcr)
+= Lip¯dΥ(u)dΥ(γ, ζ) .
+The proof is completed.
+■
+The following proposition relates the two square fields ΓΥ
+r and ΓΥ(Br).
+Proposition 4.7 (Truncated form). The following relations hold on Cr:
+ΓΥ
+r (u)(γ + ηBcr) = ΓΥ(Br)(uη
+r)(γ) ,
+µ-a.e. η, µη
+r-a.e. γ ∈ Υ(Br) ,
+(4.6)
+EΥ,µ
+r
+(u) =
+�
+Υ
+EΥ(Br),µη
+r(uη
+r) dµ(η) ,
+u ∈ Cr .
+Furthermore, the Rademacher-type property holds: Lipb(¯dΥ, µ) ⊂ Cr and
+ΓΥ
+r (u) ≤ Lip¯dΥ(u)2
+∀u ∈ Lipb(¯dΥ, µ) .
+(4.7)
+As a consequence, the form (EΥ,µ
+r
+, Cr) in (4.3) is a densely defined closable Markovian
+form and the closure (EΥ,µ
+r
+, D(EΥ,µ
+r
+)) is a local Dirichlet form on L2(µ). The L2-
+semigroups corresponding to (EΥ,µ
+r
+, D(EΥ,µ
+r
+)) is denoted by {T Υ,µ
+r,t }t≥0.
+Proof. We first prove (4.6). Let u ∈ Cr. Thanks to (b) in Def. 4.3 and Lem. 4.1,
+Uγ,x(u) ∈ Lip(Br, d) ,
+µ-a.e. γ
+∀r > 0 .
+Thus, noting Lip(Br, d) ⊂ W 1,2(mr), the LHS of (4.6) is well-defined. The RHS
+of (4.6) is also well-defined by (b) in Def. 4.3 and the fact that Lipb(Υ(Br), dΥ) ⊂
+D(EΥ(Br),µη
+r) by construction. Thus, for µ-a.e. η, we can take a measurable set Ω =
+Ω(η) ⊂ Υ(Br) with µη
+r(Ω) = 1 so that (4.6) is well-defined for every γ ∈ Ω. As µη
+r is
+absolutely continuous with respect to the Poisson measure πmr, we may assume that
+
+18
+K. SUZUKI
+every γ ∈ Ω does not have multiple points, i.e., γ({x}) ∈ {0, 1} for every x ∈ Br.
+Let γ ∈ Ω ∩ Υk(Br). Then, according to (4.4),
+ΓΥ(Br)(uη
+r)(γ) =
+���∇⊙k�
+uη
+r
+����
+2
+(γ)
+=
+�
+x∈γ
+��∇uη
+r
+�
+1Br\{x} ·γ + δ•
+���2(x)
+=
+�
+x∈γ
+���∇
+�
+uη
+r
+�
+1Br\{x} ·γ + δ•
+�
+− uη
+r
+�
+1Br\{x} ·γ
+�����
+2
+(x)
+=
+�
+x∈γBr
+���∇
+�
+u
+�
+1X\{x} ·(γ + ηBcr) + δ•
+�
+− u
+�
+1X\{x} ·(γ + ηBcr)
+�����
+2
+(x)
+= ΓΥ
+r (u)(γ + ηBcr)
+where the second equality followed from the definition of the symmetric gradient
+operator ∇⊙k, for which we used the fact that γ ∈ Ω does not have multiple points;
+the third equality follows simply as uη
+r
+�
+1Br\{x} ·γ
+�
+does not depend on the variable
+denoted as •, on which the weak gradient ∇ operates; the fifth equality followed
+from the definition of the square field ΓΥ
+r . As this argument holds for arbitrary
+k ∈ N0, (4.6) has been shown. The Markov property and the locality of EΥ,µ
+r
+follow
+immediately from (4.6) since EΥ(Br),µη
+r possesses the corresponding properties by
+construction.
+We now show the Rademacher-type property: Lipb(¯dΥ, µ) ⊂ Cr and
+ΓΥ
+r (u) ≤ Lip¯dΥ(u)2
+∀u ∈ Lipb(¯dΥ, µ)
+∀r > 0 .
+(4.8)
+We first show Lipb(¯dΥ, µ) ⊂ Cr. The verification of (a) in Def. 4.3 is obvious. The
+verification of (b) in Def. 4.3 follows from the Lipschitz contraction (4.5) of the
+operator (·)η
+r. The verification of (c) in Def. 4.3 follows by showing (4.8) as µ is a
+probability measure.
+We now prove (4.8). As the Cheeger energy ChdΥ,µk,η
+r
+coincided with EΥ(Br),µk,η
+r
+by
+Prop. 3.5, the Rademacher-type property for EΥ(Br),µk,η
+r
+follows from that for ChdΥ,µk,η
+r ,
+the latter of which is an immediate consequence by the definition of the Cheeger
+energy. Therefore, we have that
+ΓΥ(Br)(u) ≤ LipdΥ(u)2
+∀u ∈ Lip(Υ(Br), dΥ)
+∀r > 0 .
+(4.9)
+In view of the relation between ΓΥ
+r and ΓΥ(Br) in (4.6) and the Lipschitz contrac-
+tion (4.5) of the operator (·)η
+r, we concluded (4.8).
+Noting that Lipb(dΥ, µ) ⊂ L2(µ) is dense (e.g., [AGS14a, Prop. 4.1]) and the
+fact that Lipb(dΥ, µ) ⊂ Lipb(¯dΥ, µ) ⊂ Cr by (2.14) and (4.8), we obtain that the
+form (EΥ,µ
+r
+, Cr) is densely defined.
+We now show the closability. Noting that EΥ(Br),µη
+r is closable for µ-a.e. η by
+Prop. 3.5, the superposition form ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+)) (defined below in Def. 4.8) is
+closable (indeed it is closed) by [BH91, Prop. V.3.1.1]. As the two forms (EΥ,µ
+r
+, Cr)
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+19
+and ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+)) coincide on Cr and Cr ⊂ D( ¯EΥ,µ
+r
+) by construction, the closability
+of (EΥ,µ
+r
+, Cr) is inherited from the closedness of the superposition form ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+)).
+The proof is complete.
+■
+The superposition of the Dirichlet form EΥ(Br),µη
+r onto Υ is now defined below.
+Definition 4.8 (Superposition Dirichlet form, e.g., [BH91, Prop. V.3.1.1]).
+D( ¯EΥ,µ
+r
+) :=
+�
+u ∈ L2(µ) :
+�
+Υ
+EΥ(Br),µη
+r(uη
+r) dµ(η) < ∞
+�
+,
+(4.10)
+¯EΥ,µ
+r
+(u) :=
+�
+Υ
+EΥ(Br),µη
+r(uη
+r) dµ(η) .
+It is known that ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+)) is a Dirichlet form on L2(µ) [BH91, Prop. V.3.1.1].
+The L2-semigroup and the infinitesimal generator corresponding to ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+))
+are denoted by { ¯T Υ,µ
+r,t }t≥0 and ( ¯AΥ,µ
+r
+, D( ¯AΥ,µ
+r
+)) respectively.
+The semigroup { ¯T Υ,µ
+r,t }t≥0 corresponding to the superposition form ¯EΥ,µ
+r
+can be
+obtained as the superposition of the semigroup {T Υ(Br),µη
+r
+t
+}t≥0 associated with the
+form EΥ(Br),µη
+r. For the following proposition, we refer the reader to [Del21, (iii)
+Prop. 2.13].
+Proposition 4.9 ([Del21, (iii) Prop. 2.13]). The following holds:
+¯T Υ,µ
+r,t u(γ) = T Υ(Br),µγ
+r
+t
+uγ
+r(γBr) ,
+(4.11)
+for µ-a.e. γ ∈ Υ, any t > 0.
+Remark 4.10. The proof of [Del21, (iii) Prop. 2.13] has been given in terms of direct
+integral in a general setting. As the measure µη
+r can be identified to the conditional
+probability µ(· | ·Bcr = ηBcr) by a bi-measure-preserving isomorphism as remarked
+in (2.6), our setting is a particular case of direct integrals discussed in [Del21].
+We now discuss the relation between EΥ,µ
+r
+and ¯EΥ,µ
+r
+. As the former form is con-
+structed as the smallest closed extension of (EΥ,µ
+r
+, Cr), it is clear by definition that
+EΥ,µ
+r
+= ¯EΥ,µ
+r
+on
+Cr ,
+D(EΥ,µ
+r
+) ⊂ D( ¯EΥ,µ
+r
+) .
+The following theorem proves that the opposite inclusion holds as well.
+Theorem 4.11. (EΥ,µ
+r
+, D(EΥ,µ
+r
+)) = ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+)).
+Proof. The inclusion D(EΥ,µ
+r
+) ⊂ D( ¯EΥ,µ
+r
+) with the inequality ¯EΥ,µ
+r
+≤ EΥ,µ
+r
+is straight-
+forward by definition. Noting ¯EΥ,µ
+r
+= EΥ,µ
+r
+on CR and D(EΥ,µ
+r
+) is the closure of Cr,
+it suffices to show that Cr ⊂ D( ¯EΥ,µ
+r
+) is dense. Thanks to Lem. A.4, we only need
+to show that ¯T Υ,µ
+r,t Cr ⊂ Cr.
+As ¯T Υ,µ
+r,t
+is an L∞-contraction semigroup by the sub-Markovian property of the
+semigroup (see, e.g., [MR90, Def. I.4.1]), we obtain ¯T Υ,µ
+r,t Cr ⊂ L∞(µ), which verifies
+(a) in Def. 4.3
+
+20
+K. SUZUKI
+Verification of (b) in Def. 4.3.
+Let u ∈ Cr and we show that ¯T Υ,µ
+r,t u satisfies (b)
+in Def. 4.3. By Prop. 4.9, we can identify the following two operators:
+¯T Υ,µ
+r,t u = T Υ(Br),µ·
+r
+t
+u·
+r(·Br) .
+This implies that
+�
+¯T Υ,µ
+r,t u
+�η
+r(·) = ¯T Υ,µ
+r,t u(· + ηBcr) = T Υ(Br),µη
+r
+t
+uη
+r(·) .
+Take k = k(η) as in (3.4). As the conditional probability µη
+r is supported only on
+Υk(Br), we only need to show
+T Υ(Br),µk,η
+r
+t
+uη
+r ∈ Lipb(Υk(Br), dΥ) .
+(4.12)
+As (Υk(Br), dΥ, µk,γ
+r ) is RCD(0, ∞) for k = k(η) for µ-a.e. η by Prop. 3.5, the corre-
+sponding semigroup satisfies L∞(µk,η
+r )-to-Lipb(Υk(Br), dΥ)-regularisation property
+([AGS14a, Thm. 6.5]), which shows that for µ-a.e. η
+T Υ(Br),µk,η
+r
+t
+v ∈ Lipb(Υk(Br), dΥ)
+∀v ∈ L∞(µk,η
+r ) ,
+and its Lipchitz constant is bounded as
+LipdΥ(T Υ(Br),µk,η
+r
+t
+v) ≤ c(t, K)∥v∥L∞(µk,η
+r
+) ,
+with constant c(t, K) depending only on t and the curvature bound K = 0 (to be
+more precise, c(t, 0) =
+1
+√
+2t). This proves (4.12), which completes the verification
+of (b).
+Verificaiton of (c) in Def. 4.3.
+Let u ∈ Cr. Thanks to the verification of (b), the
+square field ΓΥ
+r ( ¯T Υ,µ
+r,t u) is well-defined, and by (4.6) it holds that for µ-a.e. η
+ΓΥ
+r ( ¯T Υ,µ
+r,t u)(γ + ηBcr) = ΓΥ(Br)�
+( ¯T Υ,µ
+r,t u)η
+r
+�
+(γ)
+µη
+r-a.e. γ ∈ Υ(Br) .
+(4.13)
+In view of the contraction property of the semigroup with respect to the form by
+general theory (see, e.g., [FOT11, p.23, Lem. 1.3.3]), viz.
+EΥ(Br),µη
+r(T Υ(Br),µη
+r
+t
+uη
+r) ≤ EΥ(Br),µη
+r(uη
+r)
+as well as Prop. 4.9 and (4.13), we obtain
+�
+Υ
+ΓΥ
+r ( ¯T Υ,µ
+r,t u) dµ =
+�
+Υ
+EΥ(Br),µη
+r�
+( ¯T Υ,µ
+r,t u)η
+r
+�
+dµ(η)
+=
+�
+Υ
+EΥ(Br),µη
+r(T Υ(Br),µη
+r
+t
+uη
+r) dµ(η)
+≤
+�
+Υ
+EΥ(Br),µη
+r(uη
+r) dµ(η)
+= EΥ,µ
+r
+(u) < ∞ .
+The verification of (c) is completed. Therefore, we confirmed ¯T Υ,µ
+r,t Cr ⊂ Cr, which
+concludes the statement.
+■
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+21
+As a consequence of Thm. 4.11 and Prop. 4.9, we obtain the superposition formula
+for the semigroup {T Υ,µ
+r,t }t≥0 in terms of the semigroup {T Υ(Br),µη
+r
+t
+}t≥0.
+Corollary 4.12 (Coincidence of semigroups). The following three operators coin-
+cide:
+T Υ,µ
+r,t u(γ) = ¯T Υ,µ
+r,t u(γ) = T Υ(Br),µγ
+r
+t
+uγ
+r(γBr) ,
+(4.14)
+for µ-a.e. γ ∈ Υ, any t > 0.
+4.2. Monotone limit form. We now construct a Dirichlet form on Υ with sineβ-
+invariant measure µ as the monotone limit of (EΥ,µ
+r
+, D(EΥ,µ
+r
+) as r → ∞. The follow-
+ing proposition follows immediately from the definitions of the square field ΓΥ
+r and
+the core Cr.
+Proposition 4.13 (Monotonicity). The form (EΥ,µ
+r
+, D(EΥ,µ
+r
+) and the square field
+ΓΥ
+r are monotone increasing as r ↑ ∞, viz.,
+ΓΥ
+r (u) ≤ ΓΥ
+s (u) ,
+EΥ,µ
+r
+(u) ≤ EΥ,µ
+s
+(u) ,
+D(EΥ,µ
+s
+) ⊂ D(EΥ,µ
+r
+)
+r ≤ s .
+Proof. As Cr is a core of the form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)), it suffices to check Cs ⊂ Cr
+and ΓΥ
+r (u) ≤ ΓΥ
+s (u) on Cs. Let u ∈ Cs and we show u ∈ Cr. By a simple reasoning
+similar to the proof of Lem. 4.6, we can see
+uη
+r ∈ Lipb(Υ(Br), dΥ)
+µ-a.e. η
+if uη
+s ∈ Lipb(Υ(Bs), dΥ)
+µ-a.e. η .
+By Def. 4.2, it is straightforward to see ΓΥ
+r (u) ≤ ΓΥ
+s (u). Thus,
+EΥ,µ
+r
+(u) =
+�
+Υ
+ΓΥ
+r (u) dµ ≤
+�
+Υ
+ΓΥ
+s (u) dµ =
+�
+Υ
+ΓΥ
+s (u) dµ = EΥ,µ
+s
+(u) < ∞ .
+Therefore, we conclude u ∈ Cr. The proof is completed.
+■
+We now define a Dirichlet form on Υ whose invariant measure is the sineβ mea-
+sure µ by the monotone limit of (EΥ,µ
+r
+, D(EΥ,µ
+r
+)).
+Definition 4.14 (Monotone limit form). The form (EΥ,µ, D(EΥ,µ)) is defined as the
+monotone limit:
+D(EΥ,µ) := {u ∈ ∩r>0D(EΥ,µ
+r
+) : EΥ,µ(u) = lim
+r→∞ EΥ,µ
+r
+(u) < ∞} ,
+(4.15)
+EΥ,µ(u) := lim
+r→∞ EΥ,µ
+r
+(u) .
+The form (EΥ,µ, D(EΥ,µ)) is a Dirichlet form on L2(µ) as it is the monotone limit
+of Dirichlet forms (e.g., by [MR90, Exercise 3.9]). The square field ΓΥ is defined as
+the monotone limit of ΓΥ
+r as well:
+ΓΥ(u) := lim
+r→∞ ΓΥ
+r (u)
+u ∈ D(EΥ,µ) .
+(4.16)
+The corresponding L2(µ)-semigroup is denoted by {T Υ,µ
+t
+}t≥0.
+
+22
+K. SUZUKI
+We now show that the form (EΥ,µ, D(EΥ,µ)) is a local Dirichlet form on L2(µ) and
+satisfies the Rademacher-type property with respect to the L2-transportation-type
+distance ¯dΥ.
+Proposition 4.15. The form (EΥ,µ, D(EΥ,µ)) is a local Dirichlet form on L2(µ).
+Furthermore, (EΥ,µ, D(EΥ,µ)) satisfies Rademacher-type property:
+Lip(¯dΥ, µ) ⊂ D(EΥ,µ) ,
+ΓΥ(u) ≤ Lip¯dΥ(u)2
+∀u ∈ Lip(¯dΥ, µ) .
+(4.17)
+Proof. The local property of (EΥ,µ, D(EΥ,µ)) follows from the fact that (EΥ,µ, D(EΥ,µ))
+is the monotone limit of the local Dirichlet form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)).
+We show the
+Rademacher-type property. Since ΓΥ is the limit square field of ΓΥ
+r as in (4.16), it
+suffices to show
+Lip(¯dΥ, µ) ⊂ Cr
+and
+ΓΥ
+r (u) ≤ Lip¯dΥ(u)2
+∀u ∈ Lip(¯dΥ, µ)
+∀r > 0 ,
+which has been already proven in Prop. 4.7.
+We verified (4.17).
+The proof is
+complete.
+■
+Proposition 4.16. The semigroup {T Υ,µ
+t
+}t≥0 is the L2(µ)-strong operator limit of
+the semigroups {T Υ,µ
+r,t }t≥0, viz.,
+L2(µ)– lim
+r→∞ T Υ,µ
+r,t u = T Υ,µ
+t
+u
+∀u ∈ L2(µ) ,
+t > 0 .
+Proof. The statement follows from the monotonicity of (EΥ,µ
+r
+, D(EΥ,µ
+r
+) as r ↑ ∞
+proven in Prop. 4.13 and [RS80, S.14, p.373].
+■
+4.3. Bakry–Émery Curvature bound for (EΥ,µ, D(EΥ,µ)). In this subsection,
+we prove the Bakry–Émery curvature bound for the form (EΥ,µ, D(EΥ,µ)).
+Theorem 4.17. Let β > 0 and µ be the sineβ ensemble. The form (EΥ,µ, D(EΥ,µ))
+satisfies the 1-Bakry–Émery curvature dimension condition BE1(0, ∞):
+ΓΥ�
+T Υ,µ
+t
+u
+� 1
+2 ≤ T Υ,µ
+t
+�
+ΓΥ(u)
+1
+2�
+∀u ∈ D(EΥ,µ)
+∀t > 0 .
+(BE1(0, ∞))
+Proof. We first prove BE1(0, ∞) for the form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)). Let u ∈ D(EΥ,µ
+r
+). By
+Prop. 3.5, [Han18, Thm. 1.1], by the expression (3.4) of µη
+r in terms of µk,η
+r
+and
+by the definition (4.4) of ΓΥ(Br), there exists Ξ1
+r ⊂ Υ with µ(Ξ1
+r) = 1 so that for
+every η ∈ Ξ1
+r there exists a measurable set Ω1,η
+r
+⊂ Υ(Br) with µη
+r(Ω1,η
+r ) = 1 satisfying
+that for every γ ∈ Ω1,η
+r , the following 1-Bakry–Émery gradient estimate holds:
+ΓΥ(Br)(T Υ(Br),µη
+r
+t
+uη
+r)
+1
+2(γ) ≤ T Υ(Br),µη
+r
+t
+�
+ΓΥ(Br)(uη
+r)
+� 1
+2(γ) .
+(4.18)
+By Prop. 4.7, there exists Ξ2
+r ⊂ Υ with µ(Ξ2
+r) = 1 so that for every η ∈ Ξ2
+r there
+exists a measurable set Ω2,η
+r
+⊂ Υ(Br) with µη
+r(Ω2,η
+r ) = 1 satisfying that for every
+γ ∈ Ω2,η
+r
+ΓΥ
+r (T Υ,µ
+r,t u)(γ + ηBcr) = ΓΥ(Br)��
+T Υ,µ
+r,t u
+�η
+r
+�
+(γ) ;
+(4.19)
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+23
+ΓΥ
+r (u)(γ + ηBcr) = ΓΥ(Br)(uη
+r)(γ) .
+By Cor. 4.12, there exists Λ3
+r ⊂ Υ with µ(Λ3
+r) = 1 so that for every γ ∈ Λ3
+r
+T Υ,µ
+r,t u(γ) = T Υ(Br),µγ
+r
+t
+uγ
+r(γ) .
+(4.20)
+By the standard disintegration argument, we can write
+Λ3
+r =
+�
+η∈Ξ3r
+pr−1
+r (Ω3,η
+r ) ∩ Υη
+r ,
+where Ω3,η
+r
+= (Λ3
+r)η
+r := {γ ∈ Υ(Br) : γ + ηBcr ∈ Λ3
+r} and Ξ3
+r = prBcr(Λ3
+r), and Υη
+r
+has been defined in (2.4).
+By the disintegration formula (2.10), µ(Ξ3
+r) = 1 and
+µη
+r(Ω3,η
+r ) = 1 for every η ∈ Ξ3
+r.
+Let Ξr := Ξ1
+r ∩ Ξ2
+r ∩ Ξ3
+r and Ωη
+r := Ω1,η
+r
+∩ Ω2,η
+r
+∩ Ω3,η
+r
+for η ∈ Ξr. Set
+Kr :=
+�
+η∈Ξr
+pr−1
+r (Ωη
+r) ∩ Υη
+r .
+By construction, µ(Ξr) = 1 and µη
+r(Ωη
+r) = 1 for every η ∈ Ξr. By (4.18), (4.19) and
+(4.20), the following inequalities hold for every γ ∈ Kr:
+ΓΥ
+r (T Υ,µ
+r,t u)
+1
+2(γ) = ΓΥ
+r (T Υ,µ
+r,t u)
+1
+2(γBr + γBcr)
+(4.21)
+= ΓΥ(Br)((T Υ,µ
+r,t u)γ
+r)
+1
+2(γBr)
+≤ T Υ(Br),µγ
+r
+t
+ΓΥ(Br)(uγ
+r)
+1
+2(γBr)
+= T Υ(Br),µγ
+r
+t
+�
+(ΓΥ
+r (u)γ
+r)
+1
+2�
+(γBr)
+= T Υ,µ
+r,t ΓΥ
+r (u)
+1
+2(γ) .
+Let Θ := {γ ∈ Υ : ΓΥ
+r (T Υ,µ
+r,t u)
+1
+2(γ) ≤ T Υ,µ
+r,t ΓΥ
+r (u)
+1
+2(γ)}. Then Θ is µ-measurable by
+construction, and thanks to (4.21), it holds that Kr ⊂ Θ. By applying Lem. A.2, we
+obtain µ(Θ) = 1, which concludes BE1(0, ∞) for the truncated form (EΥ,µ
+r
+, D(EΥ,µ
+r
+))
+for any r > 0.
+We now prove BE1(0, ∞) of the form (EΥ,µ, D(EΥ,µ)).
+Let u ∈ D(EΥ,µ) ⊂
+∩r>0D(EΥ,µ
+r
+). By the L2(µ)-lower semi-continuity of the square field ΓΥ, the mono-
+tonicity ΓΥ
+r ≤ ΓΥ
+r′ for r ≤ r′ (we will use it in the following displayed formulas in
+the first equality and in the second inequality), the convergence of T Υ,µ
+r′,t
+to T Υ,µ
+t
+as r′ → ∞ in the L2-strong operator sense by Prop. 4.16, and BE1(0, ∞) for the
+truncated form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)) for any r > 0, the following inequalities hold true:
+ΓΥ(T Υ,µ
+t
+u)1/2 = lim
+r→∞ ΓΥ
+r (T Υ,µ
+t
+u)1/2 ≤ lim
+r→∞ lim inf
+r′→∞ ΓΥ
+r (T Υ,µ
+r′,t u)1/2
+≤ lim inf
+r′→∞ ΓΥ
+r′(T Υ,µ
+r′,t u)1/2
+≤ lim inf
+r′→∞ T Υ,µ
+r′,t ΓΥ
+r′(u)1/2
+= T Υ,µ
+t
+ΓΥ(u)1/2 .
+
+24
+K. SUZUKI
+The last equality in the above displayed formulas followed from the following argu-
+ment: by the L2(µ)-contraction property of T Υ,µ
+r′,t , the monotonicity of ΓΥ
+r′ as r′ ↑ ∞,
+and the convergence of the semigroups T Υ,µ
+r′,t
+to T Υ,µ
+t
+as r′ → ∞ in the L2-strong
+operator sense by Prop. 4.16,
+��T Υ,µ
+r′,t ΓΥ
+r′(u)1/2 − T Υ,µ
+t
+ΓΥ(u)1/2��
+L2(µ)
+=
+��T Υ,µ
+r′,t ΓΥ
+r′(u)1/2 − T Υ,µ
+r′,t ΓΥ(u)1/2��
+L2(µ) +
+��T Υ,µ
+r′,t ΓΥ(u)1/2 − T Υ,µ
+t
+ΓΥ(u)1/2��
+L2(µ)
+≤
+��ΓΥ
+r′ (u)1/2 − ΓΥ(u)1/2��
+L2(µ) +
+��T Υ,µ
+r′,t ΓΥ(u)1/2 − T Υ,µ
+t
+ΓΥ(u)1/2��
+L2(µ)
+r′→∞
+−−−→ 0 .
+The proof is completed.
+■
+4.4. Integral Bochner, local Poicaré and local log-Sobolev inequalities.
+As an application of BE(0, ∞) proven in Thm. 4.17, we show several functional
+inequalities. We define the integral Γ2-operator as follows:
+ΓΥ,µ
+2
+(u, ϕ) :=
+�
+Υ
+�1
+2ΓΥ(u)AΥ,µϕ − ΓΥ(u, AΥ,µu)ϕ
+�
+dµ ,
+(4.22)
+D(ΓΥ,µ
+2
+) :=
+�
+(u, ϕ) : D(AΥ,µ)×2 : AΥ,µu ∈ D(EΥ,µ), ϕ, AΥ,µu ∈ L∞(µ)
+�
+,
+where AΥ,µ denotes the L2(µ)-infinitesimal generator associated with (EΥ,µ, D(EΥ,µ)).
+Corollary 4.18. Let µ be the sineβ ensemble with β > 0. The following hold:
+(a) (lntegral Bochner inequality) for every (u, ϕ) ∈ D(ΓΥ,µ
+2
+)
+ΓΥ,µ
+2
+(u, ϕ) ≥ 0 ;
+(b) (local Poincaré inequality) for u ∈ D(EΥ,µ) and t > 0,
+T Υ,µ
+t
+u2 − (T Υ,µ
+t
+u)2 ≤ 2tT Υ,µ
+t
+ΓΥ(u) ,
+T Υ,µ
+t
+u2 − (T Υ,µ
+t
+u)2 ≥ 2tΓΥ(T Υ,µ
+t
+u) ;
+(c) (local logarithmic Sobolev inequality) for non-negative u ∈ D(EΥ,µ)
+and t > 0,
+T Υ,µ
+t
+u log u − T Υ,µ
+t
+u log T Υ,µ
+t
+u ≤ tT Υ,µ
+t
+�ΓΥ(u)
+u
+�
+,
+T Υ,µ
+t
+u log u − T Υ,µ
+t
+u log T Υ,µ
+t
+u ≥ tΓΥ(T Υ,µ
+t
+u)
+T Υ,µ
+t
+u
+.
+Proof. The statement (a) follows from BE(0, ∞) proven in Thm. 4.17 and [AGS15,
+Cor. 2.3]. The statement (b) and (c) are consequences of BE(0, ∞), see e.g., [BGL14,
+Thm.s 4.7.2, 5.5.2.].
+■
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+25
+5. Dimension-free/log Harnack inequalities and Lipschitz
+regularisation
+In this section, we prove functional inequalities involving the Bakry–Émery cur-
+vature bound BE(0, ∞) and the L2-transportation-type extended distance ¯dΥ.
+Theorem 5.1. Let µ be the sineβ ensemble with β > 0. Then the following inequal-
+ities hold:
+(a) (log-Harnack inequality) for every non-negative u ∈ L∞(Υ, µ), t > 0,
+there exists Ω ⊂ Υ so that µ(Ω) = 1 and
+T Υ,µ
+t
+(log u)(γ) ≤ log(T Υ,µ
+t
+u)(η) + ¯dΥ(γ, η)2 ,
+every γ, η ∈ Ω ;
+(b) (dimension-free Harnack inequality) for every non-negative u ∈ L∞(Υ, µ),
+t > 0 and α > 1 there exists Ω ⊂ Υ so that µ(Ω) = 1 and
+(T Υ,µ
+t
+u)α(γ) ≤ T Υ,µ
+t
+uα(η) exp
+�
+α
+2(α − 1)
+¯dΥ(γ, η)2�
+,
+for every γ, η ∈ Ω ;
+(c) (Lipschitz contraction) For u ∈ Lipb(¯dΥ, µ) and t > 0,
+T Υ,µ
+t
+u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ
+t
+u
+and the following estimate holds:
+Lip¯dΥ( ˜T Υ,µ
+t
+u) ≤ Lip¯dΥ(u) ;
+(d) (L∞-to-Lip regularisation) For u ∈ L∞(µ) and any t > 0,
+T Υ,µ
+t
+u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ
+t
+u
+and the following estimate holds:
+Lip¯dΥ( ˜T Υ,µ
+t
+u) ≤
+1
+√
+2t∥u∥L∞(µ) .
+Proof. We prove (a). By the relation between T Υ,µ
+r,t
+and T Υ(Br),µ·
+r
+t
+(·Br) in Prop. 4.12,
+there exists a measurable set Ωr
+sem ⊂ Υ with µ(Ωr
+sem) = 1 so that for every η ∈ Ωr
+sem
+T Υ,µ
+r,t (η) = T Υ(Br),µη
+r
+t
+(ηBr) .
+(5.1)
+Let u ∈ L∞(µ). Thanks to Lem. A.3, there exists Ωr
+∞ ⊂ Υ so that µ(Ωr
+∞) = 1
+and
+uη
+r ∈ L∞(µη
+r),
+∀η ∈ Ωr
+∞,
+∀r ∈ N .
+By Prop. 3.5, there exists a measurable set Ωr
+rcd ⊂ Υ so that µ(Ωr
+rcd) = 1 and
+(Υk, dΥ, µη
+r) is RCD(0, ∞) with k = k(η) as in (3.4) for every η ∈ Ωr
+rcd.
+Let Ωr := Ωr
+sem ∩ Ωr
+∞ ∩ Ωr
+rcd. As the log-Harnack inequality holds in RCD spaces
+(see, [AGS15, Lem. 4.6]), the following holds for every η ∈ Ωr and k = k(η)
+T Υk(Br),µk,η
+r
+t
+(log uη
+r)(γ) ≤ log(T Υk(Br),µk,η
+r
+t
+uη
+r)(ζ) + dΥ(γ, ζ)2 ,
+∀ γ, ζ ∈ Υk(Br) .
+(5.2)
+
+26
+K. SUZUKI
+Noting the convergence of the semigroups {T Υ,µ
+r,t }t≥0 to {T Υ,µ
+t
+}t≥0 in the L2(µ)-
+operator sense by Prop. 4.16, there exist Ωcon ⊂ Υ with µ(Ωcon) = 1 and a (non-
+relabelled) subsequence of {r} so that for every γ ∈ Ωcon
+T Υ,µ
+r,t (log u)(γ)
+r→∞
+−−−→ T Υ,µ
+t
+(log u)(γ) ,
+log(T Υ,µ
+r,t u)(γ)
+r→∞
+−−−→ log(T Υ,µ
+t
+u)(γ) .
+(5.3)
+Let Ω′ = Ωcon ∩r∈N Ωr, which by construction satisfies µ(Ω′) = 1. Our goal is now
+to prove that there exists Ω ⊂ Ω′ with µ(Ω) = 1 so that
+T Υ,µ
+t
+(log u)(γ) ≤ log(T Υ,µ
+t
+u)(η) + ¯dΥ(γ, η)2 ,
+every γ, η ∈ Ω .
+(5.4)
+Thanks to (5.3), Formula (5.4) comes down to the corresponding inequality for the
+semigroup {T Υ,µ
+r,t }t≥0 for any r > 0:
+T Υ,µ
+r,t (log u)(γ) ≤ log(T Υ,µ
+r,t u)(η) + ¯dΥ(γ, η)2 ,
+every γ, η ∈ Ω .
+(5.5)
+We prove (5.5) by contradiction. Suppose that for any Ω ⊂ Ω′ with µ(Ω) = 1,
+there exists γ, η ∈ Ω so that
+T Υ,µ
+r,t (log u)(γ) ≥ log(T Υ,µ
+r,t u)(η) + ¯dΥ(γ, η)2 .
+(5.6)
+We may assume that ¯dΥ(γ, η) < ∞, otherwise, we have nothing to prove. Thus,
+by (2.15), there exists r > 0 so that
+γBcr = ηBcr ,
+γ(Br) = η(Br) .
+(5.7)
+By making use of (5.1), (5.2), (5.7), we obtain
+T Υ,µ
+r,t (log u)(γ) = T Υ,µ
+r,t (log u)(γBr + γBcr)
+(5.8)
+= T Υ(Br),µγ
+r
+t
+(log uγ
+r)(γBr)
+≤ log(T Υ(Br),µγ
+r
+t
+u)(ηBr) + dΥ(γBr, ηBr)2
+= log(T Υ,µ
+r,t u)(η) + ¯dΥ(γ, η)2 ,
+which contradicts (5.6), therefore, the proof of (a) is completed.
+The proof of (b) follows precisely in the same strategy as above by replacing
+T Υ,µ
+t
+(log u), log(T Υ,µ
+t
+u) and ¯dΥ(γ, η)2 by (T Υ,µ
+t
+u)α, T Υ,µ
+t
+uα and
+α
+2(α−1)¯dΥ(γ, η)2 re-
+spectively, and noting that the dimension-free Harnack inequality holds on RCD(K, ∞)
+spaces ([Li15, Thm. 3.1]).
+The proof of (c): Note that uη
+r ∈ Lip(Υ(Br), dΥ) whenever u ∈ Lip(Υ, ¯dΥ) and
+LipdΥ(uη
+r) ≤ Lip¯dΥ(u) by Lem. 4.6. Note also that the sought conclusion of (c) can
+be rephrased as
+˜T Υ,µ
+t
+u(γ) − ˜T Υ,µ
+t
+u(η) ≤ Lip¯dΥ(u)¯dΥ(γ, η)
+∀γ, η ∈ Υ .
+Thus, by the same proof strategy as in (a) replacing T Υ,µ
+t
+(log u)(γ) and log(T Υ,µ
+t
+u)(η)
+with T Υ,µ
+t
+u(γ) and T Υ,µ
+t
+u(η), and noting that the Lipschitz contraction property
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+27
+holds on RCD spaces ([AGS14b, (iv) in Thm. 6.1]), we conclude that there exists
+Ω ⊂ Υ with µ(Ω) = 1 so that
+T Υ,µ
+t
+(γ) − T Υ,µ
+t
+(η) ≤ Lip¯dΥ(u)¯dΥ(γ, η)
+∀γ, η ∈ Ω .
+The conclusion now follows from the McShane extension Theorem [DS21b, Lem. 2.1].
+The proof of (d) is the same as that of (c) but using the L∞-to-Lip property
+([AGS14b, Thm. 6.5]) in RCD(K, ∞) spaces instead of [AGS14b, (iv) in Thm. 6.1]).
+The proof is complete.
+■
+6. Generalisation
+We have been so far working in the case of sineβ ensemble. In this section, we
+seek to generalise the aforementioned statements to general probability measures
+on Υ = Υ(Rn) for n ∈ N. In this section, we denote by m and mr the Lebesgue
+measure on Rn and its restriction on Br(0) respectively, and we take the Euclidean
+distance d(x, y) := |x − y| for x, y ∈ Rn. Let µ be a Borel probability on Υ and
+assume that it is fully supported on Υ with respect to the vague topology τv. Let
+K(µη
+r) ⊂ N0 be defined as
+K(µη
+r) := {k ∈ N0 : µk,η
+r (Υk(Br)) > 0} .
+Assumption 6.1. Let K ∈ R and µ be a fully supported Borel probability with
+respect to the vague topolgoy τv. Assume the following conditions:
+(a) the measure µη
+r is absolutely continuous with respect to the Poisson mea-
+sure πmr, and µk,η
+r
+is equivalent to πmr|Υk(Br) for any k ∈ K(µη
+r), µ-a.e. η and
+any r > 0;
+(b) the density
+dµk,η
+r
+dπmr|Υk(Br)
+is τv-continuous on Υk(Br), and the logarithmic density
+Ψk,η
+r
+= − log
+�
+dµk,η
+r
+dπmr|Υk(Br)
+�
+is K-geodesically convex with respect to dΥ on Υk(Br) for any k ∈ K(µη
+r),
+µ-a.e. η and any r > 0.
+Under (a) in Assumption, the local Dirichlet form (EΥ,µ, D(EΥ,µ)) is constructed
+in the same proof as in the case of sineβ ensemble as we have not use any partic-
+ular property of K = 0. We further show the synthetic curvature bound for the
+form (EΥ,µ, D(EΥ,µ)) and related functional inequalities.
+Theorem 6.2. Suppose that µ satisfies Assumption 6.1. Then the form (EΥ,µ, D(EΥ,µ))
+satisfies
+
+28
+K. SUZUKI
+(a) (Bakry–Émery inequality BE1(K, ∞))
+ΓΥ�
+T Υ,µ
+t
+u
+� 1
+2 ≤ e−KtT Υ,µ
+t
+�
+ΓΥ(u)
+1
+2�
+∀u ∈ D(EΥ,µ) ;
+(b) (lntegral Bochner inequality) for every (u, ϕ) ∈ D(ΓΥ,µ
+2
+)
+ΓΥ,µ
+2
+(u, ϕ) ≥ K
+�
+Υ
+ΓΥ(u)ϕ dµ ;
+(c) (local Poincaré inequality) for u ∈ D(EΥ,µ) and t > 0,
+T Υ,µ
+t
+u2 − (T Υ,µ
+t
+u)2 ≤ 1 − e−2Kt
+K
+T Υ,µ
+t
+ΓΥ(u) ,
+T Υ,µ
+t
+u2 − (T Υ,µ
+t
+u)2 ≥ e−2Kt − 1
+K
+ΓΥ(T Υ,µ
+t
+u) ;
+(d) (local logarithmic Sobolev inequality) for non-negative u ∈ D(EΥ,µ)
+and t > 0,
+T Υ,µ
+t
+u log u − T Υ,µ
+t
+u log T Υ,µ
+t
+u ≤ 1 − e−2Kt
+2K
+T Υ,µ
+t
+�ΓΥ(u)
+u
+�
+,
+T Υ,µ
+t
+u log u − T Υ,µ
+t
+u log T Υ,µ
+t
+u ≥ e−2Kt − 1
+2K
+ΓΥ(T Υ,µ
+t
+u)
+T Υ,µ
+t
+u
+.
+(e) (log Harnack inequality) for every non-negative u ∈ L∞(Υ, µ), t > 0,
+there exists Ω ⊂ Υ so that µ(Ω) = 1 and
+T Υ,µ
+t
+(log u)(γ) ≤ log(T Υ,µ
+t
+u)(η) +
+K
+2(1 − e−2Kt)
+¯dΥ(γ, η)2 ,
+∀γ, η ∈ Ω ;
+(f) (dimension-free Harnack inequality) for every non-negative u ∈ L∞(Υ, µ),
+t > 0 and α > 1 there exists Ω ⊂ Υ so that µ(Ω) = 1 and
+(T Υ,µ
+t
+u)α(γ) ≤ T Υ,µ
+t
+uα(η) exp
+�
+αK
+2(α − 1)(1 − e−2Kt)
+¯dΥ(γ, η)2�
+,
+∀γ, η ∈ Ω ;
+(g) (Lipschitz contraction) For u ∈ Lip(¯dΥ, µ) and t > 0,
+T Υ,µ
+t
+u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ
+t
+u
+and the following estimate holds:
+Lip¯dΥ( ˜T Υ,µ
+t
+u) ≤ e−KtLip¯dΥ(u) ;
+(h) (L∞-to-Lip regularisation) For u ∈ L∞(µ) and t > 0,
+T Υ,µ
+t
+u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ
+t
+u
+and the following estimate holds:
+Lip¯dΥ( ˜T Υ,µ
+t
+u) ≤
+1
+�
+2I2K(t)
+∥u∥L∞(µ)
+∀t > 0 ,
+where IK(t) :=
+� t
+0 eKr dr.
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+29
+Proof. Thanks to Assumption 6.1, the space (Υk(Br), dΥ, µk,η
+r ) satisfies RCD(K, ∞)
+for every k ∈ K(µη
+r) as in the same proof of Prop. 3.4. As we have not used any
+particular properties of K = 0 for the proofs in the case of sineβ, the completely
+same proofs (up to constant multiplication depending only on K) work in the case of
+general K ∈ R and general µ satisfying Assumption 6.1, which therefore concludes
+the statements of Thm. 6.2.
+■
+Appendix A.
+Let m and mr be the Lebesgue measure on Rn and its restriction on Br respectively.
+Set Υ = Υ(Rn).
+Lemma A.1. Let µ be a Borel probability on Υ satisfying that µη
+r is absolutely
+continuous with respect to the Poisson measure πmr for any r > 0 and µ-a.e. η. Let
+Σ ⊂ Br so that mr(Σc) = 0. Let Ω(r) := {γ ∈ Υ : γΣ = γBr}. Then,
+µ
+�
+Ω(r)
+�
+= 1
+∀r > 0 .
+Proof. We fix r > 0 and write simply Ω = Ω(r). By the disintegration formula (2.10),
+µ(Ω) =
+�
+Υ
+µη
+r(Ωη
+r) dµ(η) .
+Thus, it suffices to show µη
+r(Ωη
+r) = 1 for µ-a.e. η. This is equivalent to show
+µη
+r(Ωη
+r) =
+�
+k∈N0
+µk,η
+r (Ωη
+r) = 1 .
+(A.1)
+As µk,η
+r
+is absolutely continuous with respect to πmr|Υk(Br), it suffices to prove
+πmr|Υk(Br)((Ωη
+r)c) = 0 for every k ∈ N0 and η ∈ Υ.
+We show that (recall the definition of symmetric product Σ⊙k in (2.3))
+Σ⊙k ⊂ Ωη
+r ∩ Υk(Br)
+∀η ∈ Υ .
+(A.2)
+Let γ ∈ Σ⊙k. Then by the definition of Ω, it holds that γ + ηBcr ∈ Ω for any η ∈ Υ.
+Thus, by recalling the definition (2.9) of Ωη
+r, we obtain γ ∈ Ωη
+r ∩ Υk(Br). Thus,
+(A.2) holds true.
+By using (A.2), πmr|Υk(Br) = e−mr(Br)m⊙k
+r
+by (2.11) and m⊙k
+r
+�
+(Σ⊙k)c�
+= 0 by
+hypothesis, we conclude that for every η ∈ Υ
+πmr|Υk(Br)((Ωη
+r)c) = e−mr(Br)m⊙k
+r
+��
+Ωη
+r ∩ Υk(Br)
+�c�
+≤ e−mr(Br)m⊙k
+r
+�
+(Σ⊙k)c�
+= 0 .
+The proof is complete.
+■
+We recall that for η ∈ Υ, we set Υη
+r := {γ ∈ Υ : γBcr = ηBcr}.
+Lemma A.2 (disintegration lemma). Assume that there exists a measurable set
+Ξ ⊂ Υ with µ(Ξ) = 1 so that for every η ∈ Ξ, there exists a family of measurable
+
+30
+K. SUZUKI
+sets Ωη ⊂ Υ(Br) so that µη
+r(Ωη) = 1 for every η ∈ Ξ. Let Ω ⊂ Υ be the (not
+necessarily measurable) subset defined by
+Ω :=
+�
+η∈Ξ
+pr−1
+r (Ωη) ∩ Υη
+r .
+Assume further that there exists a measurable set Θ ⊂ Υ so that Ω ⊂ Θ. Then,
+µ(Θ) = 1.
+Caveat.
+As the set Ω is defined as uncountable union of measurable sets, the mea-
+surability of Ω is not necessarily true in general. The disintegration formula (2.10)
+is, therefore, not necessarily applicable directly to Ω, which motivates the aforemen-
+tioned lemma.
+Proof of Lem. A.2. Let Θη
+r = {γ ∈ Υ(Br) : γ + ηBcr ∈ Θ} be a section of Θ at ηBcr
+as in (2.9). Then, Ωη ⊂ Θη
+r by assumption. Thus, µη
+r(Θη
+r) ≥ µη
+r(Ωη) ≥ 1. By the
+disintegration formula in (2.10), we have that
+µ(Θ) =
+�
+Υ
+µη
+r(Θη
+r) dµ(η) ≥ 1 .
+The proof is completed.
+■
+Lemma A.3. Let µ be a Borel probability on Υ. Let Ω ⊂ Υ satisfy µ(Ω) = 1.
+Then, there exists Ω′ ⊂ Ω with µ(Ω′) = 1 and
+µη
+r(Ωη
+r) = 1 ,
+∀η ∈ Ω′ .
+(A.3)
+Proof. By the disintegration formula (2.10),
+1 = µ(Ω) =
+�
+Υ
+µη
+r(Ωη
+r) dµ(η) =
+�
+Ω
+µη
+r(Ωη
+r) dµ(η) ,
+by which the statement is readily concluded.
+■
+Lemma A.4. Let (Q, D(Q)) be a closed form on a complete separable Hilbert
+space H. Let {Tt} and (A, D(A)) be the corresponding semigroup and infinitesi-
+mal generator respectively. Suppose that there exists an algebra C ⊂ D(Q) so that
+C ⊂ H is dense and TtC ⊂ C for any t > 0. Then C is dense in D(Q).
+Proof. It holds that TtD(A) ⊂ D(A) by the general property of semigroups associ-
+ated with closed forms. Thus, combining it with the hypothesis TtC ⊂ C,
+Tt(C ∩ D(A)) ⊂ C ∩ D(A) .
+Thus, by [RS75, Thm. X.49], C ∩ D(A) is dense in the graph norm in the space
+(A, D(A)). Namely, we obtained
+(A, C ∩ D(A)) is essentially self-adjoint .
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+31
+The density C ⊂ D(Q) now follows by the density of C ∩ D(A) in the graph norm,
+by the density of D(A) ⊂ D(Q) due to the general property of closed forms, by the
+density of C ⊂ H and by a simple integration-by-parts argument
+−Q(u, u) = (Au, u)H ≤ ∥Au∥H∥u∥H .
+The proof is complete.
+■
+References
+[AGS14a] Ambrosio, L., Gigli, N., and Savaré, G. Calculus and heat flow in metric
+measure spaces and applications to spaces with Ricci bounds from below.
+Invent. Math., 395:289–391, 2014.
+[AGS14b] Ambrosio, L., Gigli, N., and Savaré, G. Metric measure spaces with
+Riemannian Ricci curvature bounded from below.
+Duke Math. J.,
+163(7):1405–1490, 2014.
+[AGS15] Ambrosio, L., Gigli, N., and Savaré, G.
+Bakry–Émery Curvature-
+Dimension
+Condition
+and
+Riemannian
+Ricci
+Curvature
+Bounds.
+Ann. Probab., 43(1):339–404, 2015.
+[BGL14] Bakry, D., Gentil, I., and Ledoux, M. Analysis and Geometry of Markov
+Diffusion Operators, volume 348 of Grundlehren der mathematischen
+Wissenschaften. Springer, 2014.
+[BH91] Bouleau, N. and Hirsch, F.
+Dirichlet forms and analysis on Wiener
+space. De Gruyter, 1991.
+[Del21] Dello Schiavo, L. Ergodic Decomposition of Dirichlet Forms via Direct
+Integrals and Applications. Potential Anal., 2021.
+[DHLM20] Dereudre, D., Hardy, A., Leblé, T., and Maïda, M. DLR Equations and
+Rigidity for the Sine-Beta Process. Commun. Pure Appl. Math., pages
+172–222, 2020.
+[DS21a] Dello Schiavo, L. and Suzuki, K.
+Configuration spaces over sin-
+gular spaces –I. Dirichlet-Form and Metric Measure Geometry –.
+arXiv:2109.03192v2 (version 2), 2021.
+[DS21b] Dello Schiavo, L. and Suzuki, K. On the Rademacher and Sobolev-to-
+Lipschitz Properties for Strongly Local Dirichlet Spaces. J. Func. Anal.,
+281(11):Online first, 2021.
+[DS22] Dello Schiavo, L. and Suzuki, K. Configuration Spaces over Singular
+Spaces II – Curvature. arXiv:2205.01379, 2022.
+[EH15] Erbar, M. and Huesmann, M. Curvature bounds for configuration spaces.
+Calc. Var., 54:307–430, 2015.
+[FOT11] Fukushima, M., Oshima, Y., and Takeda, M. Dirichlet forms and sym-
+metric Markov processes, volume 19 of De Gruyter Studies in Mathe-
+matics. de Gruyter, extended edition, 2011.
+
+32
+K. SUZUKI
+[Fuk97] Fukushima, M. Distorted Brownian motions and BV functions. Trends
+in Probability and Analysis, N. Kono, N-R. Shieh, eds, pages 143–150,
+1997.
+[Gho15] Ghosh, S. Determinantal processes and completeness of random expo-
+nentials: the critical case. Probab. Theory Relat. Fields, 163(3):643–665,
+2015.
+[GKMS18] Galaz-García, F., Kell, M., Mondino, A., and Sosa, G. On quotients of
+spaces with Ricci curvature bounded below. J. Funct. Anal., 275:1368–
+1446, 2018.
+[Han18] Han, B.X. New characterizations of ricci curvature on rcd metric mea-
+sure spaces.
+Discrete and Continuous Dynamical Systems. Series A,
+38(10):4915–4927, 2018.
+[KS09] Killip, R. and Stoiciu, M. Eigenvalue statistics for cmv matrices: from
+poisson to clock via random matrix ensembles.
+Duke Math. J., 146
+(3)::361–399, 2009.
+[KT10] Katori, M. and Tanemura, H.
+Non-equilibrium dynamics of Dyson’s
+model with an infinite number of particles.
+Comm. Math. Phys.,
+293(2):469–497, 2010.
+[Li15] H. Li. Dimension-Free Harnack Inequalities on RCD(K, ∞) Spaces. J.
+Theoret. Probab., 29:1280–1297, 2015.
+[MR85] Ma, Z.-M. and Röckner, M. Dirichlet forms-closability and change of
+speed measure, Infinite dimensional analysis and stochastic processes.
+Research Notes in Math. S. Albeverio, ed., Pitman, 124:119–144, 1985.
+[MR90] Ma, Z.-M. and Röckner, M.
+Introduction to the Theory of (Non-
+Symmetric) Dirichlet Forms. Springer, 1990.
+[MR00] Ma, Z.-M. and Röckner, M. Construction of Diffusions on Configuration
+Spaces. Osaka J. Math., 37:273–314, 2000.
+[Nak14] Nakano, F. Level statistics for one-dimensional schrödinger operators
+and gaussian beta ensemble. J. Stat. Phys., 156(1):66–93, 2014.
+[NR18] Najnundel, J. and Reda, C. Rigidity of the Sineβ process. Electron.
+Commun. Probab., 23:1–8, 2018.
+[Osa13] Osada, H. Interacting Brownian Motions in Infinite Dimensions with
+Logarithmic Interaction Potentials. Ann. Probab., 41(1):1–49, 2013.
+[RS75] Reed, M. and Simon, B. Methods of Modern Mathematical Physics II –
+Fourier Analysis, Self-Adjointness. Academic Press, New York, London,
+1975.
+[RS80] Reed, M. and Simon, B. Methods of Modern Mathematical Physics I –
+Functional Analysis. Academic Press, New York, London, 1980.
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+33
+[Spo87] Spohn, H. Interacting Brownian Particles: A Study of Dyson’s Model.
+Hydrodynamic Behavior and Interacting Particle Systems, pages 151–
+179, 1987.
+[Stu06] Sturm, K.-T. On the geometry of metric measure spaces. I. Acta Math.,
+196:65–131, 2006.
+[Tsa16] Tsai, L.-C.
+Infinite dimensional stochastic differential equations for
+dyson’s model. Probability Theory and Related Fields, 166:801–850, 2016.
+[VG20] Von Renesse, M.. and Gyünesu, B. Molecules as metric measure spaces
+with kato-bounded ricci curvature.
+Comptes Rendus. Mathématique,
+358:595–602, 2020.
+[VV09] Valkó, B. and Virág, B. Continuum limits of random matrices and the
+brownian carousel. Invent. math., 177(3):463–508, 2009.
+Department of Mathematical Science, Durham University, Science Laboratories,
+South Road, DH1 3LE, United Kingdom
+
diff --git a/0NAyT4oBgHgl3EQfbfc0/content/tmp_files/load_file.txt b/0NAyT4oBgHgl3EQfbfc0/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf,len=1324
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='00262v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='PR]  31 Dec 2022  CURVATURE BOUND OF DYSON BROWNIAN MOTION  KOHEI SUZUKI  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' In this article, we show 1-Bakry–Émery lower Ricci curvature  bound BE1(0, ∞) of a Dirichlet form on the configuration space whose in-  variant measure is sineβ ensemble for any β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  As a particular case of  β = 2, our result proves BE1(0, ∞) for a Dirichlet form related to the unlablled  Dyson Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We prove furthermore several functional inequalities  including the integral Bochner inequality, the local Poincaré and the local log-  Sobolev inequalities as well as the log-Harnack and the dimension-free Harnack  inequalities, the Lipschitz contraction property and the L∞-to-Lipschitz regu-  larisation property of the semigroup with the L2-transportation-type extended  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' At the end of the article, we provide a sufficient condition for the  synthetic lower Ricci curvature bound in the case of general invariant measures  beyond sineβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Introduction  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Notation and Preliminaries  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Curvature bound for finite-particle systems  11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Curvature bound for infinite-particle systems  15  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Dimension-free/log Harnack inequalities and Lipschitz regularisation  25  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Generalisation  27  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  29  References  31  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Introduction  The objective of this article is to reveal the structure of lower curvature bound be-  hind an infinite particle system of diffusions with logarithmic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Such an  interacting particle system is realised as a continuous-time strong Markov process  having continuous paths (called a diffusion process) taking values in the configu-  ration space Υ = Υ(R) over R and having the sineβ (β > 0) ensemble µ as an  Date: 31/12/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Dyson Brownian motion, sine beta ensemble, Ricci curvature bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Department of Mathematical Science, Durham University  E-mail: kohei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='suzuki@durham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='uk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  1  2  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We study a corresponding Dirichlet form (EΥ,µ, D(EΥ,µ)) with  square field ΓΥ (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='15) whose invariant measure is sineβ ensemble µ on the  configuration space Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The case of β = 2 is particularly related to the diffusion  process called (unlabelled) Dyson Brownian motion (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' [Spo87, KT10, Osa13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The  labelled interacting diffusions can be phrased formally as the following infinitely  many stochastic differential equation with logarithmic interaction (see [Tsa16] for a  rigorous construction):  dXk  t = β  2 lim  r→∞  �  i̸=k:|Xk  t −Xi  t|<r  1  Xk  t − Xi  t  dt + dBk  t ,  k ∈ N ,  whereby {Bk  }k∈N are infinitely many independent Brownian motions on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The main result of this article is to show that the aforementioned Dirichlet  form (EΥ,µ, D(EΥ,µ)) satisfies the non-negative lower Ricci curvature bound BE1(0, ∞)  in the sense of Bakry–Émery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  We prove, furthermore, several related functional  inequalities including the integral Bochner inequality with respect to the integral  Γ2-operator (ΓΥ,µ  2  , D(ΓΥ,µ  2  )) (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='22)), the local Poincaré, the local log-Sobolev  inequalities as well as the dimension-free Harnack inequality, the log-Harnack in-  equality, the Lipschitz contraction and the L∞-to-Lipschitz regularisation property  with respect to the L2-transportation-type extended distance ¯dΥ on Υ (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The main results are summarised in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let β > 0 and µ be the sineβ ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The form (EΥ,µ, D(EΥ,µ)) satis-  fies the following:  (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='17) 1-Bakry–Émery estimate BE1(0, ∞): for u ∈ D(EΥ,µ), t > 0,  ΓΥ�  T Υ,µ  t  u  � 1  2 ≤ T Υ,µ  t  �  ΓΥ(u)  1  2�  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='18) Integral Bochner inequality: for every (u, ϕ) ∈ D(ΓΥ,µ  2  )  ΓΥ,µ  2  (u, ϕ) ≥ 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='18) Local Poincaré inequality: for u ∈ D(EΥ,µ), t > 0,  T Υ,µ  t  u2 − (T Υ,µ  t  u)2 ≤ 2tT Υ,µ  t  ΓΥ(u) ,  T Υ,µ  t  u2 − (T Υ,µ  t  u)2 ≥ 2tΓΥ(T Υ,µ  t  u) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='18) Local log-Sobolev inequality: for non-negative u ∈ D(EΥ,µ), t > 0,  T Υ,µ  t  u log u − T Υ,µ  t  u log T Υ,µ  t  u ≤ tT Υ,µ  t  �ΓΥ(u)  u  �  ,  T Υ,µ  t  u log u − T Υ,µ  t  u log T Υ,µ  t  u ≥ tΓΥ(T Υ,µ  t  u)  T Υ,µ  t  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1) Log-Harnack inequality: for every non-negative u ∈ L∞(Υ, µ),  t > 0, there exists Ω ⊂ Υ so that µ(Ω) = 1 and  T Υ,µ  t  (log u)(γ) ≤ log(T Υ,µ  t  u)(η) + ¯dΥ(γ, η)2 ,  ∀γ, η ∈ Ω ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  CURVATURE BOUND OF DYSON BROWNIAN MOTION  3  (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1) Dimension-free Harnack inequality: for every non-negative u ∈  L∞(Υ, µ), t > 0 and α > 1 there exists Ω ⊂ Υ so that µ(Ω) = 1 and  (T Υ,µ  t  u)α(γ) ≤ T Υ,µ  t  uα(η) exp  �  α  2(α − 1)  ¯dΥ(γ, η)2�  ,  ∀γ, η ∈ Ω ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1) Lipschitz contraction: for every u ∈ Lip(¯dΥ, µ) and t > 0  T Υ,µ  t  u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ  t  u  and the following Lipschitz contraction holds:  Lip¯dΥ( ˜T Υ,µ  t  u) ≤ Lip¯dΥ(u) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1) L∞(Υ, µ)-to-Lip(¯dΥ, µ) regularisation by semigroup: For u ∈  L∞(µ) and t > 0,  T Υ,µ  t  u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ  t  u  and the following estimate holds:  Lip¯dΥ( ˜T Υ,µ  t  u) ≤  1  √  2t∥u∥L∞(µ)  ∀u ∈ Lipb(¯dΥ, µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  At the end of this article, Theorem will be extended to general point processes  beyond the sineβ ensemble, see Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Comparison with Literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' To the best knowledge of the author, this is the  first article addressing lower Ricci curvature bound in the setting of interacting and  infinite particle systems of diffusion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' In the case of non-interacting case  where the invariant measure is the Poisson measure, it has been studied in [EH15]  in the case of Riemannian manifolds and in [DS22] in the case of general diffusion  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' In the case of finite particle systems, a variable Ricci curvature bound has  been addressed in [VG20] in the case of Coulomb-type potentials where the curvature  bound depends on the number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Outline of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' After preparing the notation and the preliminaries in  Section 2, we discuss in Section 3 the synthetic lower Ricci curvature bound for  Dirichlet forms (EΥ(Br),µη  r, D(EΥ(Br),µη  r)) on the configuration space Υ(Br) over the  closed metric ball Br with radius r > 0 centred at 0, whose invariant measure is  the projected regular conditional probability µη  r on Υ(Br) conditioned at η on the  compliment Bc  r ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The key point of the proof is that the logarithm of the Radon–  Nikodým density Ψη  r := − log(dµη  r/ dπmr) with respect the Poisson measure πmr on  Υ(Br) with the intensity mr being the Lebesgue measure restricted on Br is geodesi-  cally convex in (Υ(Br), ¯dΥ) due to the following DLR (Dobrushin–Lanford–Ruelle)  equation (⋆) proven in [DHLM20, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1] with the number-rigidity ([Gho15] for  sine2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' [NR18] and [DHLM20] for sineβ): for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η, there exists a unique k =  4  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  k(η) ∈ N0 so that µη  r(Υl(Br)) > 0 if and only if l = k where Υl(Br) := {γ ∈  Υ(Br) : γ(Br) = l}, and for γ = �k  i=1 δxi ∈ Υ(Br)  dµη  r = 1  Zη  r e−Ψk,η  r  dm⊙k  r  ,  (⋆)  Ψk,η  r (γ) := − log  � k  �  i<j  |xi − xj|β  k  �  i=1  lim  R→∞  �  y∈ηBcr ,|y|≤R  ���1 − xi  y  ���  β  �  ,  where m⊙k  r  is the k-symmetric product measure of the Lebesgue measure mr re-  stricted on Br ⊂ R and Zη  r is the normalising constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  In Section 4, we prove BE1(0, ∞) for (EΥ,µ, D(EΥ,µ)) in the following steps: we  first construct the truncated form (EΥ,µ  r  , D(EΥ,µ  r  )) on Υ whose gradient operator is  truncated up to configurations on Br (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We then identify it with the super-  position Dirichlet form ( ¯EΥ,µ  r  , D( ¯EΥ,µ  r  )) lifted from (EΥ(Br),µη  r, D(EΥ(Br),µη  r)) with re-  spect to the conditioning η (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By this identification, we can lift BE(0, ∞)  from the form (EΥ(Br),µη  r, D(EΥ(Br),µη  r)) onto the truncated form (EΥ,µ  r  , D(EΥ,µ  r  )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Showing the monotonicity of the form (EΥ,µ  r  , D(EΥ,µ  r  )) with respect to r and pass-  ing to the limit r → ∞, we prove BE1(0, ∞) for the limit form (EΥ,µ, D(EΥ,µ))  (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' As a consequence of BE1(0, ∞), we obtain local Poincaré and local  log-Sobolev inequalities (Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  In Section 5, we prove the log-Harnack inequality, the dimension-free Harnack  inequality, the Lipschitz contraction and L∞(µ)-to-Lip(¯dΥ) properties (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Our proof strategy is to lift the corresponding functional inequalities from the space  of finitely many configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  In Section 6, we extend Theorem to the case of general point processes beyond  sineβ (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' A large part of the current work has been completed while  the author was at Bielefeld University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' He gratefully acknowledges funding by the  Alexander von Humboldt Stiftung to support his stay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Data Availability Statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Data sharing not applicable to this article as no  datasets were generated or analysed during the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Notation and Preliminaries  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Numbers, Tensors, Function Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We write N := {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' }, N0 =  {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' }, N := N∪{+∞} and N0 := N0 ∪{+∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The uppercase letter N is used  for N ∈ N0, while the lowercase letter n is used for n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We shall adhere to the  following conventions:  the superscript □×N (the subscript □×N) denotes (N-fold) product objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  the superscript □⊗N (the subscript □⊗N) denotes (N-fold) tensor objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  CURVATURE BOUND OF DYSON BROWNIAN MOTION  5  the superscript □⊙N (the subscript □⊙N) denotes (N-fold) symmetric tensor  objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let (X, τ) be a topological space with σ-finite Borel measure ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We use the following  symbols:  (a) Lp(ν) (1 ≤ p ≤ ∞) for the space of ν-equivalence classes of functions u  with |u|p ν-integrable when 1 ≤ p < ∞, and with u ν-essentially bounded  when p = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The Lp(ν)-norm is denoted by ∥u∥p  Lp(ν) := ∥u∥p  p :=  �  X |u|p dν  for 1 ≤ p < ∞, and ∥u∥L∞(ν) := ∥u∥∞ = esssupX u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' In the case of p = 2, the  inner-product is denoted by (u, v)L2(ν) :=  �  X uv dν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (b) Lp  s(ν⊗n) := {u ∈ Lp(ν⊗n) : u is symmetric} where u is said to be symmetric  if and only if u(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' , xk) = u(xσ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' , xσ(k)) for any element σ ∈ S(k) in  the k-symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (c) Cb(X) for the space of τ-continuous bounded functions on X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' if X is locally  compact, C0(X) denotes the space of τ-continuous and compactly supported  functions on X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' C∞  0 (R) for the space of compactly supported smooth func-  tions on R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (d) We write 1A for the indicator function on A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 1A(x) = 1 if x ∈ A, and  1A(x) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We refer the reader to [MR90, BH91] for this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Throughout this paper, a Hilbert space always means a Hilbert space with inner  product (·, ·)H taking value in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Given a bilinear form (Q, D(Q)) on a Hilbert space H, we write  Q(u) := Q(u, u) ,  Qα(u, v) := Q(u, v) + α(u, v)H , α > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let (X, Σ, ν) be a σ-finite measure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' A symmetric Dirichlet form on L2(ν) is  a non-negative definite densely defined closed symmetric bilinear form (Q, D(Q))  on L2(ν) satisfying the Markov property  u0 := 0 ∨ u ∧ 1 ∈ D(Q)  and  Q(u0) ≤ Q(u) ,  u ∈ D(Q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Throughout this article, Dirichlet form always means symmetric Dirichlet form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' If  not otherwise stated, D(Q) is always regarded as a Hilbert space with norm  ∥ · ∥D(Q) := Q1( · )1/2 :=  �  Q( · ) + ∥ · ∥2  L2(ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  In order to distinguish Dirichlet forms defined in different base spaces with different  reference measures, we often use the notation QX,ν to specify the base space X and  the reference measure ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Square field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  A Dirichlet form (Q, D(Q)) admits square field Γ if there exists a  dense subspace H ⊂ D(Q) ∩ L∞(ν) having the following property: for any u ∈ H,  6  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  there exists v ∈ L1(ν) so that  2Q(uh, u) − Q(h, u2) =  �  X  hv dν  ∀h ∈ D(Q) ∩ L∞(ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Such v is denoted by Γ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The square field Γ can be uniquely extended as an  operator on D(Q) × D(Q) → L1(ν) ([BH91, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Semigroups and generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  We refer the reader to [MR90, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' I, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2] for  the following contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let (Q, D(Q)) be a symmetric closed form on a Hilbert  space H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The infinitesimal generator (A, D(A)) corresponding to (Q, D(Q)) is the  unique densely defined closed operator on H satisfying the following integration-by-  parts formula:  −(u, Av)H = Q(u, v)  ∀u ∈ D(Q), v ∈ D(A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The resolvent operator {Rα}α≥0 is the unique bounded linear operator on H satis-  fying  Qα(Rαu, v) = (u, v)H  ∀u ∈ H  v ∈ D(Q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The semigroup {Tt}t≥0 is the unique bounded linear operator on H satisfying  Gαu =  � ∞  0  e−αtTtu dt  u ∈ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let (Q, D(Q)) be a Dirihclet form on L2(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' It is called local ([BH91,  Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2]) if for any F, G ∈ C∞  c (R) and any u ∈ D(Q),  supp[F] ∩ supp[G] = ∅ =⇒ Q(F0 ◦ u, G0 ◦ u) = 0 ,  where F0(x) := F(x) − F(0) and G0(x) := G(x) − G(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let X be any non-empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' A function d: X ×2 → [0, ∞] is  an extended distance if it is symmetric and satisfying the triangle inequality, and it  does not vanish outside the diagonal in X ×2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' d(x, y) = 0 iff x = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' a distance  if it is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let x0 ∈ X and r ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We write Br(x0) := {dx0 ≤ r}, where  dx0 := d(x0, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Lipschitz algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  A function f : X → R is d-Lipschitz if there exists a con-  stant L > 0 so that  ��u(x) − u(y)  �� ≤ L d(x, y) ,  x, y ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1)  The smallest constant L so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1) holds is the (global) Lipschitz constant of u,  denoted by Lipd(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For any non-empty A ⊂ X we write Lip(A, d), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Lipb(A, d)  for the family of all finite, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' bounded, d-Lipschitz functions on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For simplic-  ity of notation, further let Lip(d) := Lip(X, d), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Lipb(d) := Lipb(X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Set also  Lip1(d) := {u ∈ Lip(d) : Lipd(u) ≤ 1} and Lip1  b(d) := Lip1(d) ∩ Lipb(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For a given  measure ν, we set  Lip(d, ν) := {u ∈ Lip(d) : u is ν-measurable} ,  CURVATURE BOUND OF DYSON BROWNIAN MOTION  7  as well as Lipb(d, ν) and Lip1  b(d, ν) denoting the corresponding subspaces of ν-  measurable functions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Geodesical convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  A metric space (X, d) is called a geodesic space if for any  x0, x1 ∈ X there exists a constant speed geodesic ω : [0, 1] → X connecting x0 and x1:  ω0 = x0 ,  ω1 = x1 ,  d(ωt, ωs) = |t − s|d(ω0, ω1)  ∀t, s ∈ [0, 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  For a function U : X → R ∪ {+∞}, define D(U) := {x ∈ X : U(x) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We say  that U is K-geodesically convex for K ∈ R if for any x0, x1 ∈ D(U) there exists a  constant speed geodesic ω : [0, 1] → X with ω0 = x0 and ω1 = x1 and  U(ωt) ≤ (1 − t)U(ω0) + tU(ω1) − K  2 t(1 − t)d2(ω0, ω1)  ∀t ∈ [0, 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  When K = 0, we say that U is geodesically convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Cheeger energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' A complete separable geodesic metric space (X, d) equipped  with fully supported Radon measure ν with finite total mass ν(X) < ∞ is called a  metric measure space in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let (X, d, ν) be a metric measure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For  u ∈ Lip(d), the slope |Ddu|(x) is defined as  |Ddu|(x) :=  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  lim sup  y→x  |u(x) − u(y)|  d(x, y)  if x is not isolated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  0  otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The slope is universally measurable, see [AGS14a, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The Cheeger energy  Chd,ν : L2(ν) → R ∪ {+∞} is defined as the L2(ν)-lower semi-continuous envelope  of  �  X |Ddu|2 dν:  Chd,ν(u) := inf  �  lim inf  n→∞  �  X  |Ddun|2 dν : un ∈ Lip(d) ∩ L2(ν)  L2  −→ u  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The domain is denoted by W 1,2(X, d, ν) := {u ∈ L2(ν) : Chd,ν(u) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The  Cheeger energy Chd,ν can be expressed by the following integration, see [AGS14a,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5] : there exists a measurable function |∇u|∗ ∈ L2(ν) so that |∇u|∗ ≤ |Ddu|  ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' for every u ∈ Lip(d) and  Chd,ν(u) =  �  X  |∇u|2  ∗ dν  ∀u ∈ W 1,2(X, d, ν) ,  where |∇u|∗ is called minimal relaxed slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Riemannian Curvature-dimension condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let (X, d, ν) be a metric  measure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The following definition is an equivalent characterisation of RCD(K, ∞)  by [AGS15, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We say that (X, d, ν) satisfies the Riemannian Curvature-  Dimension Condition RCD(K, ∞) for K ∈ R if  (i) Chd,ν is quadratic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', Chd,ν(u + v) + Chd,ν(u − v) = 2Chd,ν(u) + 2Chd,ν(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (ii) Sobolev-to-Lipschitz property holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', every u ∈ W 1,2(X, d, ν) with |∇u|∗ ≤  1 has a d-Lipschitz ν-representative ˜u satisfying Lip(˜u) ≤ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  8  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  (iii) Chd,ν satisfies BE2(K, ∞), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', |∇Ttu|2  ∗ ≤ e−2KtTt|∇u|2  ∗ for every u ∈ W 1,2(X, d, ν)  and t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  In this case, the Cheeger energy Chd,ν is a local Dirichlet form ([AGS14b, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We  note that, while [AGS15, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='18] is stated in terms of the minimal weak upper  gradient denoted by |∇ · |w, it is identical to the minimal relaxed slope |∇ · |∗ due to  [AGS14a, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' A configuration on a locally compact Polish space X  is any N0-valued Radon measure γ on X, which can be expressed by γ = �N  ii δxi  for N ∈ ¯N, where δx denotes the Dirac measure at x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', δx(A) = 1 if and only if  x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The configuration space Υ = Υ(X) is the space of all configurations over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The space Υ is equipped with the vague topology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', the topology generated by  the duality of the space C0(X) of continuous functions with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We  write the restriction γA := γ ⇂A for a Polish subspace A ⊂ X and the corresponding  restriction map is denoted by  prA : Υ −→ Υ(A): γ �−→ γA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2)  The N-particle configuration space is denoted by  ΥN := {γ ∈ Υ : γ(X) = N} ,  N ∈ N0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let Sk be the k-symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' It can be readily seen that the k-particle config-  uration space Υk is isomorphic to the quotient space X×k/Sk:  Υk ∼= X⊙k := X×k/Sk ,  k ∈ N0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3)  The associated projection map from X×k to the quotient space X×k/Sk is denoted  by Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For η ∈ Υ and r > 0, we set  Υη  r := {γ ∈ Υ : γBcr = ηBcr} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4)  Conditional probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let µ be a Borel probability measure on Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let  µ(· | prBcr(·) = ηBcr)  denote the regular conditional probability of µ conditioned at η ∈ Υ with respect  to the σ-field generated by the projection map γ ∈ Υ �→ prr(γ) = γBr ∈ Υ(Br) (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', [DS21a, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='32] for the precise definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let µη  r be the probability measure  on Υ(Br) defined as  µη  r := (prr)#µ(· | prBcr(·) = ηBcr) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5)  and its restriction on Υk(Br) is denoted by µk,η  r  := µη  r|Υk(Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Note: The conditional probability µ(· | prBcr(·) = ηBcr) is a probability measure on  the whole space Υ whose support is Υη  r = {γ ∈ Υ : γBcr = ηBcr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We may project the  CURVATURE BOUND OF DYSON BROWNIAN MOTION  9  conditional probability to the probability measure µη  r on Υ(Br) as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5) without  loss of information in the sense that  prr : Υη  r → Υ(Br) is a bi-measure-preserving bijection .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6)  Namely, the projection map prr is bijective with the inverse map pr−1  r  defined  as pr−1  r (γ) := γ + η, and both prr and pr−1  r  are measure-preserving between the  two measures µ(· | prBcr(·) = ηBcr) and µη  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  For a measurable function u: Υ → R, r > 0 and for η ∈ Υ, we set  uη  r(γ) := u(γ + ηBcr)  γ ∈ Υ(Br) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7)  By the property of the conditional probability, it is straightforward to see that for  any u ∈ L1(µ),  �  Υ  u dµ =  �  Υ  ��  Υ(Br)  uη  r dµη  r  �  dµ(η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='8)  See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', [DS21a, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For a measurable set Ω ⊂ Υ, define a section Ωη  r ⊂  Υ(Br) at η ∈ Υ on Bc  r by  Ωη  r := {γ ∈ Υ(Br) : γ + ηBcr ∈ Ω} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='9)  By applying the disintegration formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='8) to u = 1Ω, we obtain  µ(Ω) =  �  Υ  µη  r(Ωη  r) dµ(η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='10)  Poisson measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let (X, τ, ν) be a locally compact Polish space with Radon  measure ν satisfying ν(X) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The Poisson measure πν on Υ(X) with intensity ν  is defined in terms of the symmetric tensor measure ν⊙ as follows:  πν(·) := e−ν(X)  ∞  �  k=1  ν⊙k�  ∩ Υk(X)  �  = e−ν(X)  ∞  �  k=1  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' (Pr)#ν⊗k�  ∩ Υk(X)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='11)  L2-transportation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let (X, d) be a locally compact complete separable  metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  For i = 1, 2 let proji : X×2 → X denote the projection to the ith  coordinate for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For γ, η ∈ Υ, let Cpl(γ, η) be the set of all couplings of γ  and η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=',  Cpl(γ, η) := {q ∈ M (X  ×2): (proj1)♯q = γ , (proj2)♯q = η} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Here M (X ×2) denotes the space of all Radon measures on X ×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The L2-transportation  extended distance on Υ(X) is  dΥ(γ, η) :=  inf  q∈Cpl(γ,η)  ��  X×2 d2(x, y) dq(x, y)  �1/2  ,  inf ∅ = +∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='12)  We refer the readers to [DS21a, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='52] for details regarding the L2-transportation  extended distance dΥ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  It is important to note that dΥ is an extended distance,  10  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  attaining the value +∞ and dΥ is lower semi-continuous with respect to the product  vague topology τ ×2  v  but never τ ×2  v -continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  We introduce a variant of the L2-transportation extended distance, called L2-  transportation-type extended distance ¯dΥ defined as  ¯dΥ(γ, η) :=  \uf8f1  \uf8f2  \uf8f3  dΥ(γ, η)  if γBcr = ηBcr for some r > 0 ,  +∞  otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='13)  By definition, dΥ ≤ ¯dΥ on Υ, and dΥ = ¯dΥ on Υ(Br) for any r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' In particular,  we have  Lip(Υ, dΥ) ⊂ Lip(Υ, ¯dΥ) ,  Lip¯dΥ(u) ≤ LipdΥ(u) ,  u ∈ Lip(Υ, dΥ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='14)  It can be readily seen that  ¯dΥ(γ, η) < ∞  ⇐⇒  γBcr = ηBcr , γ(Br) = η(Br)  for some r > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='15)  When we work with the configuration space Υ(Rn) over the n-dimensional Eu-  clidean space Rn or over any Polish subset in Rn, we always choose the Euclidean  distance d(x, y) = |x − y| and the L2-transportation distance dΥ and ¯dΥ associated  with d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' sineβ ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let β > 0 and CβEk be the circular β ensemble on the k-  particle configuration space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', it is the probability measure Pk,β on the space Υk(S1)  over the unit circle S1 ⊂ C defined as  dPk,β :=  1  Zk,β  �  1≤j<l≤k  ��eiθj − eiθl��β dθ1  2π · · · dθk  2π ,  where the normalisation constant Zk,β is given in terms of Gamma function Γ:  Zk,β := Γ( 1  2βk + 1)  Γ( 1  2βk + 1)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  According to [KS09, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6], the circular β ensemble CβE is defined as the limit  probability measure Pβ whose Laplace transform is determined as  �  exp  �  −  �  x∈γ  f(x)  �  dPβ(γ) = lim  k→∞  �  exp  �  −  k  �  i=1  f(kθi)  �  dPk,β(θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' , θk) ,  for all f ∈ C0(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' In [VV09] a probability measure µβ on Υ(R) called sine β ensem-  ble has been constructed by a limit of Gaussian β-ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' These two measures Pβ  and µβ turned out to be identical each other by the work of [Nak14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Throughout the  rest of the article, we use the symbol µ = µβ to denote sineβ ensemble (equivalently,  circular β ensemble) and we do not specify the inverse temperature β as there is no  particular role played by a special β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  CURVATURE BOUND OF DYSON BROWNIAN MOTION  11  Number-rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  A Borel probability µ on Υ = Υ(Rn) is said to be number  rigid (in short: (R)) if for any bounded domain E ⊂ Rn, there exists Ω ⊂ Υ so that  µ(Ω) = 1 and, for any γ, η ∈ Ω  γEc = ηEc implies γE = ηE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (R)  Namely, the configuration outside E determines the number of particle inside E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The  number-rigidity has been proven in [Gho15] for the sine2 ensemble and in [NR18],  [DHLM20] for the sineβ ensemble for general β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Curvature bound for finite-particle systems  In this section, we study Dirichlet forms on the configuration space Υ(Br) over  metric balls Br ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We denoted by m and mr the Lebesgue measure on R and  its restriction on the metric ball Br := [−r, r] respectively, and take the Euclidean  distance d(x, y) := |x − y| for x, y ∈ Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Construction of Dirichlet forms on Υk(Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let W 1,2  s (m⊗k  r ) be the space  of m⊗k  r -classes of (1, 2)-Sobolev and symmetric functions on the product space B×k  r ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=',  W 1,2  s (m⊗k  r ) :=  �  u ∈ L2  s(m⊗k  r ) :  �  B×k  r  |∇⊗ku|2 dm⊗k  r  < ∞  �  ,  where ∇⊗k denotes the weak derivative on R×k: ∇⊗ku := (∂1u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' , ∂ku).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The space  W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2  s (m⊗k  r ) consisting of symmetric functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' the projection Pr : B×k  r  → Υk(Br) ∼=  B×k  r /Sk naturally acts on W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2  s (m⊗k  r ) and the resulting quotient space is denoted by  W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2(m⊙k  r ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' which is the (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2)-Sobolev space on Υk(Br):  W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2(m⊙k  r ) :=  �  u ∈ L2(m⊙k  r ) :  �  Υk(Br)  |∇⊙ku|2 dm⊙k  r  < ∞  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  where ∇⊙k is the quotient operator of the weak gradient operator ∇⊗k through the  projection Pr and m⊙k  r  is the symmetric product measure defined as  m⊙k  r  := 1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' (Pr)#m⊗k  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  For 0 < r < R < ∞, k ∈ N0 and η ∈ Υ(Bc  r), we introduce the following finite  Borel measure on Υk(Br): for γ = �k  i=1 δxi  dµk,η  r,R(γ) := e−Ψk,η  r,R(γ) dm⊙k  r (γ) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1)  Ψk,η  r,R(γ) := − log  � k  �  i<j  |xi − xj|β  k  �  i=1  �  y∈ηBcr ,|y|≤R  ���1 − xi  y  ���  β  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The corresponding weighted Sobolev norm is denoted by  EΥ(Br),µk,η  r,R(u) :=  �  Υk(Br)  |∇⊙ku|2 dµk,η  r,R ,  u ∈ Lipb(Υk(Br), dΥ) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2)  12  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  where we note that as Lip(Υk, dΥ) ⊂ W 1,2(m⊙k  r ) and |∇⊙ku| ≤ LipdΥ(u) due to the  Rademacher theorem descendent from the one in the product Sobolev space W 1,2(m⊗k  r )  through the quotient, the expression |∇⊙ku| and its integral against the probability  measure µk,η  r,R make sense for u ∈ Lipb(Υk(Br), dΥ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2) is well-defined and closable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The closure is a local  Dirichlet form on L2(µk,η  r,R) and its domain is denoted by D  �  EΥ(Br),µk,η  r,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The well-definedness follows from the following inequality:  �  Υk(Br)  |∇⊙ku|2 dµk,η  r,R ≤  ���e−Ψk,η  r,R  ���  L∞(Υk(Br),µk,η  r,R)  �  Υk(Br)  |∇⊙ku|2 dm⊙k < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3)  The closability of EΥ(Br),µk,η  r,R descends from the closability of the corresponding  Dirichlet form on the product space B×k  r  defined on the space of symmetric d×k-  Lipschitz functions:  EB×k  r  ,µk,η  r,R :=  �  B×k  r  |∇⊗ku|2e−Ψk,η  r,R dm⊗k  r  ,  where the closability of EB×k  r  ,µk,η  r,R is a consequence of the continuity of the den-  sity e−Ψk,η  r,R on B×k  r  and the standard Hamza-type argument by [MR85, Fuk97], see  for an accessible reference, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', [MR90, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 44-45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The locality of the form is an  immediate consequence of the locality of the gradient operator ∇⊙k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  Let µ be the sineβ ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Due to [DHLM20, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1], the following limit  exists for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η, all x ∈ Br and r > 0:  lim  R→∞  �  y∈ηBcr ,|y|≤R  ���1 − x  y  ���  β  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Recall that µη  r has been defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By [DHLM20, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1] and the number-  rigidity (R) of µ, for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η there exists k = k(η) so that  µη  r(Υl(Br)) > 0 if and only if l = k(η) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4)  and for γ = �k  i=1 δxi,  dµη  r = dµk,η  r  = e−Ψk,η  r  Zη  r  dm⊙k  r  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5)  Ψk,η  r (γ) := − log  � k  �  i<j  |xi − xj|β  k  �  i=1  lim  R→∞  �  y∈ηBcr ,|y|≤R  ���1 − xi  y  ���  β  �  ,  where Zη  r is the normalising constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The corresponding weighted Sobolev norm is  defined as  EΥ(Br),µk,η  r (u) :=  �  Υk(Br)  |∇⊙ku|2 dµk,η  r  ,  u ∈ Lipb(Υk(Br), dΥ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6)  CURVATURE BOUND OF DYSON BROWNIAN MOTION  13  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let µ be the sineβ ensemble for β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6) is well-  defined and closable for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The closure is a local Dirichlet form on L2(µk,η  r )  and its domain is denoted by D(EΥ(Br),µk,η  r ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' As e−Ψk,η  r,R  R→∞  −−−→ e−Ψk,η  r  uniformly on Υk(Br) for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η by [DHLM20,  Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3 and Proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1 in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 183], the density e−Ψk,η  r  is continuous on  B⊙k  r , hence the same proof as Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1 applies to conclude the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Curvature bound for finite-particle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We show that the poten-  tial Ψk,η  r,R defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1) is geodesically convex in (Υ(Br), dΥ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Ψk,η  r,R is geodesically convex in (Υk(Br), dΥ) for any 0 < r < R <  ∞, k ∈ N and η ∈ Υ(Bc  r),  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Note that if u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' , uk are convex and α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' , αk ≥ 0, then �k  i=1 αiui is  again convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Note also that for any 0 < r < R, any y ∈ [−R, −r] ∪ [r, R] and any  i, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' , k} with i ̸= j, the functions − log |xi − xj| and − log |1 − xi  y | are  convex in the following areas for any σ ∈ Sk:  �  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' , xk) ∈ B×k  r  : xσ(1) < xσ(2) < · · · < xσ(k)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The following expression, therefore, concludes that Ψk,η  r,R is geodesically convex as a  function on Υk(Br): for any γ = �k  i=1 δxi  Ψk,η  r,R(γ) = −β  k  �  i<j  log(|xi − xj|) − β  k  �  i=1  �  y∈ηBcr ,|y|≤R  log  ���1 − xi  y  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7)  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  Thanks to the geodesical convexity of the potential Ψk,η  r,R shown in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3,  the Dirichlet form (EΥ(Br),µk,η  r,R, D(EΥ(Br),µk,η  r,R)) satisfies the Riemannian Curvature-  Dimension condition RCD(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The space (Υk(Br), dΥ, µk,η  r,R) satisfies RCD(0, ∞) for every k ∈  N0, 0 < r < R < ∞ and η ∈ Υ, and it holds that  �  EΥ(Br),µk,η  r,R, D(EΥ(Br),µk,η  r,R)  �  =  �  ChdΥ,µk,η  r,R, W 1,2(Υk(Br), dΥ, µk,η  r,R)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Noting that B×k  r  is a convex subset in Rk, the space (B×k  r , d×k, m⊗k  r ) is a  geodesic subspace of Rk and, therefore, satisfies RCD(0, ∞) by the Global-to-Local  property of RCD(0, ∞), see [AGS14b, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Noting that the k-particle con-  figuration space (Υk(Br), dΥ, µk,η  r,R) is the quotient space of (B×k  r , d×k, m⊗k) with  respect to the symmetric group Sk and that the property RCD(0, ∞) is preserved  under the quotient operation with respect to Sk thanks to [GKMS18], we obtain  that (Υk(Br), dΥ, m⊙k) satisfies RCD(0, ∞) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By the geodesical convexity of  the potential Ψk,η  r,R shown in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3 and the continuity of the density e−Ψk,η  r,R, the  weighted space (Υk(Br), dΥ, µk,η  r,R) satisfies RCD(0, ∞) by [AGS14b, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  14  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  To conclude the statement, it suffices to check the identity  EΥ(Br),µk,η  r,R = ChdΥ,µk,η  r,R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  By the Rademacher theorem on Υk(Br) descendent from the Rademacher theorem  on B×k  r , the slope |DdΥu| coincides with |∇⊙ku| for u ∈ Lip(Υk(Br), dΥ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thus,  EΥ(Br),µk,η  r,R(u) =  �  Υk(Br)  |DdΥu|2 dµk,η  r,R  u ∈ Lipb(Υk(Br), dΥ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='8)  Since ChdΥ,µk,η  r,R is the L2-lower semi-continuous envelope, the functional ChdΥ,µk,η  r,R is  the maximal L2-lower semi-continuous functional satisfying  ChdΥ,µk,η  r,R(u) ≤  �  Υk(Br)  |DdΥu|2 dµk,η  r,R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  As EΥ(Br),µk,η  r,R is closed by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1, in particular, EΥ(Br),µk,η  r,R is L2-lower semi-  continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Therefore, combining the maximality of ChdΥ,µk,η  r,R with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='8), it holds  that  EΥ(Br),µk,η  r,R ≤ ChdΥ,µk,η  r,R and W 1,2(Υk(Br), dΥ, µk,η  r,R) ⊂ D(EΥ(Br),µk,η  r,R)  and  ChdΥ,µk,η  r,R(u) = EΥ(Br),µk,η  r,R(u)  u ∈ Lipb(Υk(Br), dΥ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  As Lipb(Υk(Br), dΥ) is dense both in D(EΥ(Br),µk,η  r,R) and W 1,2(Υk(Br), dΥ, µk,η  r,R) by  construction, the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  In view of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4 and the approximation Ψk,η  r,R to Ψk,η  r  as R → ∞, we prove  that (Υk(Br), dΥ, µk,η  r ) satisfies RCD(0, ∞) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let µ be the sineβ ensemble for β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For any 0 < r < ∞ and  µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η ∈ Υ, the space (Υk(Br), dΥ, µk,η  r ) satisfies RCD(0, ∞), where k = k(η) as  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Furthermore,  �  EΥ(Br),µk,η  r , D(EΥ(Br),µk,η  r )  �  =  �  ChdΥ,µk,η  r , W 1,2(Υk(Br), dΥ, µk,η  r )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Since the potential Ψk,η  r,R is geodesically convex for any R and it converges  pointwise to Ψk,η  r  as R → ∞ for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η by [DHLM20, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3 and Proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1 in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 183], the potential Ψk,η  r  is again geodesically convex on (Υk(Br), dΥ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Fur-  thermore, as the density e−Ψk,η  r,R converges uniformly to e−Ψk,η  r  on Υk(Br) as R → ∞  for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η by [DHLM20, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3 and Proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1 in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 183], the den-  sity e−Ψk,η  r  is continuous on Υ(Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Noting the fact that the constant multiplication  (by the normalisation constant Zη  r ) does not change the lower Ricci curvature bound  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', [Stu06, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='13]), the same proof as Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4 applies to conclude the  statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  CURVATURE BOUND OF DYSON BROWNIAN MOTION  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Curvature bound for infinite-particle systems  In this section, we construct a local Dirichlet form on Υ = Υ(R) associated with  sineβ ensemble µ and show the BE1(0, ∞) property by the following steps: we first  construct truncated Dirichlet forms on Υ whose gradient operators are truncated up  to configurations inside Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We then identify them with the superposition Dirichlet  forms lifted from Υ(Br), thanks to which we can show BE1(0, ∞) for the truncated  forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We take the monotone limit of the truncated forms to construct a Dirichlet  form with invariant measure sineβ ensembles µ and BE1(0, ∞) extends to the limit  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' In the end of this section, we discuss several applications of the BE1(0, ∞)  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Superposition of Dirichlet forms from Υ(Br) onto Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' In this subsection,  we construct the truncated Dirichlet forms on Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We first construct square field  operators on Υ and Υ(Br) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  For so doing, we introduce a map Uγ,x  transferring functions on the configuration space Υ to functions on the base space R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  For u : Υ → R, define Uγ,x(u) : R → R by  Uγ,x(u)(y) := u  �  1X\\{x} ·γ + δy  �  − u  �  1X\\{x} ·γ  �  ,  γ ∈ Υ,  x ∈ γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1)  In the context of configuration spaces, the operation Uγ,x has been firstly discussed  in [MR00, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2], see also [DS21a, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We introduce the localisation of  the operator Uγ,x on Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Recall that for a measurable function u: Υ → R, r > 0  and for η ∈ Υ, we set in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7)  uη  r(γ) := u(γ + ηBcr) for γ ∈ Υ(Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For u : Υ(Br) → R, define Ur  γ,x(u) : Br → R by  Ur  γ,x(u)(y) := u(1X\\{x} · γ + δy) − u(1X\\{x} · γ)  γ ∈ Υ(Br), x ∈ γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The operation Ur  γ,x maps from Lip(Υ(Br), dΥ) to Lip(Br) and Lipschitz constants  are contracted by Ur  γ,x for any r > 0:  Lip(Ur  γ,x(u)) ≤ LipdΥ(u)  ∀γ ∈ Υ(Br)  ∀x ∈ γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Furthermore, for any u : Υ → R,  Ur  γBr ,x(uγ  r)(y) = Uγ,x(u)(y)  for every γ ∈ Υ, x ∈ γBr and y ∈ Br .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let u ∈ Lip(Υ(Br), dΥ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Then  |Ur  γ,x(u)(y) − Ur  γ,x(u)(z)| = |u(1X\\{x} ·γ + δy) − u(1X\\{x} ·γ + δz)|  ≤ LipdΥ(u)dΥ(1X\\{x} ·γ + δy, 1X\\{x} ·γ + δz)  = LipdΥ(u)|y − z| ,  which concludes the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  16  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  We verify the second assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For every x ∈ γBr and y ∈ Br,  Uγ,x(u)(y) = u(1X\\{x} · γ + δy) − u(1X\\{x} · γ)  = u(1X\\{x} · γBr + γBcr + δy) − u(1X\\{x} · γBr + γBcr)  = ur,γ(1X\\{x} · γBr + δy) − ur,γ(1X\\{x} · γBr)  = Ur  γBr ,x(uγ  r)(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  We now define a square field operator on Υ truncated up to particles inside Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2 (Truncated square field on Υ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let u : Υ → R be a measurable  function so that Uγ,x(u)|Br ∈ W 1,2(mr) for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' γ and every x ∈ γBr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The following  operator is called the truncated square field ΓΥ  r ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2)  ΓΥ  r (u)(γ) :=  �  x∈γBr  |∇Uγ,x(u)|2(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Thanks to Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1, Formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2) is well-defined for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Indeed, as Uγ,x(u)|Br ∈  W 1,2  loc (mr), the weak gradient ∇Uγ,x(u) is well-defined pointwise on a measurable set  Σ ⊂ Br with mr(Σc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By applying Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1, Formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2) is well-defined on  the set Ω(r) of µ-full measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Based on the truncated square field ΓΥ  r , we introduce the truncated form on Υ  defined on a certain core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3 (Core).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For r > 0, let Cr be defined as the space of µ-classes of  measurable functions u so that  (a) u ∈ L∞(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (b) uη  r ∈ Lipb(Υ(Br), dΥ) for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η and r > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (c) The following integral is finite:  EΥ,µ  r  (u) :=  �  Υ  ΓΥ  r (u) dµ < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3)  Note that, thanks to Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1, if a measurable function u : Υ → R satisfies (b),  then Uγ,x(u)|Br ∈ Lip(Br, d) ⊂ W 1,2(mr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thus, the expression ΓΥ  r (u) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3) is  well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4 (Square field on Υ(Br)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Fix r > 0 and η ∈ Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For a µ-measurable  function u : Υ(Br) → R satisfying u|Υk(Br) ∈ D(EΥ(Br),µk,η  r ) for any k ∈ N0, we  define the following square field operator on Υ(Br):  ΓΥ(Br)(u) :=  ∞  �  k=0  ���∇⊙k�  u|Υk(Br)  ����  2  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4)  and define the following form:  EΥ(Br),µη  r(u) :=  �  Υ(Br)  ΓΥ(Br)(u) dµη  r ,  CURVATURE BOUND OF DYSON BROWNIAN MOTION  17  D(EΥ(Br),µη  r) := {u : Υ(Br) → R, EΥ(Br),µη  r(u) < ∞} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Due to the number-rigidity (R), the Dirichlet form EΥ(Br),µη  r is equal to EΥ(Br),µk,η  r  up to the normalising constant multiplication, therefore, it is a Dirichlet form as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The corresponding semigroup is denoted by {T Υ(Br),µη  r  t  }t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The number-rigidity (R) is not necessary to conclude that EΥ(Br),µη  r  is a Dirichlet form since any countable sum of Dirichlet forms is a Dirichlet form  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', [MR90, Exercise 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Before discussing properties of truncated forms, we prepare a lemma, which states  that the operation (·)η  r defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7) maps from Lip(Υ, ¯dΥ) to Lip(Υ(Br), dΥ) and  contracts Lipschitz constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let u ∈ Lip(Υ, ¯dΥ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Then, uη  r ∈ Lip(Υ(Br), dΥ) and  LipdΥ(uη  r) ≤ Lip¯dΥ(u) ,  ∀η ∈ Υ ,  r > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let γ, ζ ∈ Υ(Br) and η ∈ Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Then,  |uη(γ) − uη(ζ)| = |u(γ + ηBcr) − u(ζ + ηBcr)| ≤ Lip¯dΥ(u)¯dΥ(γ + ηBcr, ζ + ηBcr)  = Lip¯dΥ(u)dΥ(γ, ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  The following proposition relates the two square fields ΓΥ  r and ΓΥ(Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7 (Truncated form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The following relations hold on Cr:  ΓΥ  r (u)(γ + ηBcr) = ΓΥ(Br)(uη  r)(γ) ,  µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η, µη  r-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' γ ∈ Υ(Br) ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6)  EΥ,µ  r  (u) =  �  Υ  EΥ(Br),µη  r(uη  r) dµ(η) ,  u ∈ Cr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Furthermore, the Rademacher-type property holds: Lipb(¯dΥ, µ) ⊂ Cr and  ΓΥ  r (u) ≤ Lip¯dΥ(u)2  ∀u ∈ Lipb(¯dΥ, µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7)  As a consequence, the form (EΥ,µ  r  , Cr) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3) is a densely defined closable Markovian  form and the closure (EΥ,µ  r  , D(EΥ,µ  r  )) is a local Dirichlet form on L2(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The L2-  semigroups corresponding to (EΥ,µ  r  , D(EΥ,µ  r  )) is denoted by {T Υ,µ  r,t }t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We first prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let u ∈ Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thanks to (b) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3 and Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1,  Uγ,x(u) ∈ Lip(Br, d) ,  µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' γ  ∀r > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Thus, noting Lip(Br, d) ⊂ W 1,2(mr), the LHS of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The RHS  of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6) is also well-defined by (b) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3 and the fact that Lipb(Υ(Br), dΥ) ⊂  D(EΥ(Br),µη  r) by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thus, for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η, we can take a measurable set Ω =  Ω(η) ⊂ Υ(Br) with µη  r(Ω) = 1 so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6) is well-defined for every γ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' As µη  r is  absolutely continuous with respect to the Poisson measure πmr, we may assume that  18  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  every γ ∈ Ω does not have multiple points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', γ({x}) ∈ {0, 1} for every x ∈ Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let γ ∈ Ω ∩ Υk(Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Then, according to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4),  ΓΥ(Br)(uη  r)(γ) =  ���∇⊙k�  uη  r  ����  2  (γ)  =  �  x∈γ  ��∇uη  r  �  1Br\\{x} ·γ + δ•  ���2(x)  =  �  x∈γ  ���∇  �  uη  r  �  1Br\\{x} ·γ + δ•  �  − uη  r  �  1Br\\{x} ·γ  �����  2  (x)  =  �  x∈γBr  ���∇  �  u  �  1X\\{x} ·(γ + ηBcr) + δ•  �  − u  �  1X\\{x} ·(γ + ηBcr)  �����  2  (x)  = ΓΥ  r (u)(γ + ηBcr)  where the second equality followed from the definition of the symmetric gradient  operator ∇⊙k, for which we used the fact that γ ∈ Ω does not have multiple points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  the third equality follows simply as uη  r  �  1Br\\{x} ·γ  �  does not depend on the variable  denoted as •, on which the weak gradient ∇ operates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' the fifth equality followed  from the definition of the square field ΓΥ  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' As this argument holds for arbitrary  k ∈ N0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6) has been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The Markov property and the locality of EΥ,µ  r  follow  immediately from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6) since EΥ(Br),µη  r possesses the corresponding properties by  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  We now show the Rademacher-type property: Lipb(¯dΥ, µ) ⊂ Cr and  ΓΥ  r (u) ≤ Lip¯dΥ(u)2  ∀u ∈ Lipb(¯dΥ, µ)  ∀r > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='8)  We first show Lipb(¯dΥ, µ) ⊂ Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The verification of (a) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3 is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The  verification of (b) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3 follows from the Lipschitz contraction (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5) of the  operator (·)η  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The verification of (c) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3 follows by showing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='8) as µ is a  probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  We now prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' As the Cheeger energy ChdΥ,µk,η  r  coincided with EΥ(Br),µk,η  r  by  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5, the Rademacher-type property for EΥ(Br),µk,η  r  follows from that for ChdΥ,µk,η  r ,  the latter of which is an immediate consequence by the definition of the Cheeger  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Therefore, we have that  ΓΥ(Br)(u) ≤ LipdΥ(u)2  ∀u ∈ Lip(Υ(Br), dΥ)  ∀r > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='9)  In view of the relation between ΓΥ  r and ΓΥ(Br) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6) and the Lipschitz contrac-  tion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5) of the operator (·)η  r, we concluded (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Noting that Lipb(dΥ, µ) ⊂ L2(µ) is dense (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', [AGS14a, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1]) and the  fact that Lipb(dΥ, µ) ⊂ Lipb(¯dΥ, µ) ⊂ Cr by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='8), we obtain that the  form (EΥ,µ  r  , Cr) is densely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  We now show the closability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Noting that EΥ(Br),µη  r is closable for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η by  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5, the superposition form ( ¯EΥ,µ  r  , D( ¯EΥ,µ  r  )) (defined below in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='8) is  closable (indeed it is closed) by [BH91, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' As the two forms (EΥ,µ  r  , Cr)  CURVATURE BOUND OF DYSON BROWNIAN MOTION  19  and ( ¯EΥ,µ  r  , D( ¯EΥ,µ  r  )) coincide on Cr and Cr ⊂ D( ¯EΥ,µ  r  ) by construction, the closability  of (EΥ,µ  r  , Cr) is inherited from the closedness of the superposition form ( ¯EΥ,µ  r  , D( ¯EΥ,µ  r  )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  The superposition of the Dirichlet form EΥ(Br),µη  r onto Υ is now defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='8 (Superposition Dirichlet form, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', [BH91, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  D( ¯EΥ,µ  r  ) :=  �  u ∈ L2(µ) :  �  Υ  EΥ(Br),µη  r(uη  r) dµ(η) < ∞  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='10)  ¯EΥ,µ  r  (u) :=  �  Υ  EΥ(Br),µη  r(uη  r) dµ(η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  It is known that ( ¯EΥ,µ  r  , D( ¯EΥ,µ  r  )) is a Dirichlet form on L2(µ) [BH91, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The L2-semigroup and the infinitesimal generator corresponding to ( ¯EΥ,µ  r  , D( ¯EΥ,µ  r  ))  are denoted by { ¯T Υ,µ  r,t }t≥0 and ( ¯AΥ,µ  r  , D( ¯AΥ,µ  r  )) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The semigroup { ¯T Υ,µ  r,t }t≥0 corresponding to the superposition form ¯EΥ,µ  r  can be  obtained as the superposition of the semigroup {T Υ(Br),µη  r  t  }t≥0 associated with the  form EΥ(Br),µη  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' For the following proposition, we refer the reader to [Del21, (iii)  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='9 ([Del21, (iii) Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The following holds:  ¯T Υ,µ  r,t u(γ) = T Υ(Br),µγ  r  t  uγ  r(γBr) ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='11)  for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' γ ∈ Υ, any t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The proof of [Del21, (iii) Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='13] has been given in terms of direct  integral in a general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' As the measure µη  r can be identified to the conditional  probability µ(· | ·Bcr = ηBcr) by a bi-measure-preserving isomorphism as remarked  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6), our setting is a particular case of direct integrals discussed in [Del21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  We now discuss the relation between EΥ,µ  r  and ¯EΥ,µ  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' As the former form is con-  structed as the smallest closed extension of (EΥ,µ  r  , Cr), it is clear by definition that  EΥ,µ  r  = ¯EΥ,µ  r  on  Cr ,  D(EΥ,µ  r  ) ⊂ D( ¯EΥ,µ  r  ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The following theorem proves that the opposite inclusion holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' (EΥ,µ  r  , D(EΥ,µ  r  )) = ( ¯EΥ,µ  r  , D( ¯EΥ,µ  r  )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The inclusion D(EΥ,µ  r  ) ⊂ D( ¯EΥ,µ  r  ) with the inequality ¯EΥ,µ  r  ≤ EΥ,µ  r  is straight-  forward by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Noting ¯EΥ,µ  r  = EΥ,µ  r  on CR and D(EΥ,µ  r  ) is the closure of Cr,  it suffices to show that Cr ⊂ D( ¯EΥ,µ  r  ) is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thanks to Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4, we only need  to show that ¯T Υ,µ  r,t Cr ⊂ Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  As ¯T Υ,µ  r,t  is an L∞-contraction semigroup by the sub-Markovian property of the  semigroup (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', [MR90, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1]), we obtain ¯T Υ,µ  r,t Cr ⊂ L∞(µ), which verifies  (a) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3  20  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  Verification of (b) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let u ∈ Cr and we show that ¯T Υ,µ  r,t u satisfies (b)  in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='9, we can identify the following two operators:  ¯T Υ,µ  r,t u = T Υ(Br),µ·  r  t  u·  r(·Br) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  This implies that  �  ¯T Υ,µ  r,t u  �η  r(·) = ¯T Υ,µ  r,t u(· + ηBcr) = T Υ(Br),µη  r  t  uη  r(·) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Take k = k(η) as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' As the conditional probability µη  r is supported only on  Υk(Br), we only need to show  T Υ(Br),µk,η  r  t  uη  r ∈ Lipb(Υk(Br), dΥ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='12)  As (Υk(Br), dΥ, µk,γ  r ) is RCD(0, ∞) for k = k(η) for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5, the corre-  sponding semigroup satisfies L∞(µk,η  r )-to-Lipb(Υk(Br), dΥ)-regularisation property  ([AGS14a, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5]), which shows that for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η  T Υ(Br),µk,η  r  t  v ∈ Lipb(Υk(Br), dΥ)  ∀v ∈ L∞(µk,η  r ) ,  and its Lipchitz constant is bounded as  LipdΥ(T Υ(Br),µk,η  r  t  v) ≤ c(t, K)∥v∥L∞(µk,η  r  ) ,  with constant c(t, K) depending only on t and the curvature bound K = 0 (to be  more precise, c(t, 0) =  1  √  2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' This proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='12), which completes the verification  of (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Verificaiton of (c) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let u ∈ Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thanks to the verification of (b), the  square field ΓΥ  r ( ¯T Υ,µ  r,t u) is well-defined, and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6) it holds that for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η  ΓΥ  r ( ¯T Υ,µ  r,t u)(γ + ηBcr) = ΓΥ(Br)�  ( ¯T Υ,µ  r,t u)η  r  �  (γ)  µη  r-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' γ ∈ Υ(Br) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='13)  In view of the contraction property of the semigroup with respect to the form by  general theory (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', [FOT11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='23, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3]), viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  EΥ(Br),µη  r(T Υ(Br),µη  r  t  uη  r) ≤ EΥ(Br),µη  r(uη  r)  as well as Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='9 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='13), we obtain  �  Υ  ΓΥ  r ( ¯T Υ,µ  r,t u) dµ =  �  Υ  EΥ(Br),µη  r�  ( ¯T Υ,µ  r,t u)η  r  �  dµ(η)  =  �  Υ  EΥ(Br),µη  r(T Υ(Br),µη  r  t  uη  r) dµ(η)  ≤  �  Υ  EΥ(Br),µη  r(uη  r) dµ(η)  = EΥ,µ  r  (u) < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The verification of (c) is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Therefore, we confirmed ¯T Υ,µ  r,t Cr ⊂ Cr, which  concludes the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  CURVATURE BOUND OF DYSON BROWNIAN MOTION  21  As a consequence of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='11 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='9, we obtain the superposition formula  for the semigroup {T Υ,µ  r,t }t≥0 in terms of the semigroup {T Υ(Br),µη  r  t  }t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='12 (Coincidence of semigroups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The following three operators coin-  cide:  T Υ,µ  r,t u(γ) = ¯T Υ,µ  r,t u(γ) = T Υ(Br),µγ  r  t  uγ  r(γBr) ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='14)  for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' γ ∈ Υ, any t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Monotone limit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We now construct a Dirichlet form on Υ with sineβ-  invariant measure µ as the monotone limit of (EΥ,µ  r  , D(EΥ,µ  r  ) as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The follow-  ing proposition follows immediately from the definitions of the square field ΓΥ  r and  the core Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='13 (Monotonicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The form (EΥ,µ  r  , D(EΥ,µ  r  ) and the square field  ΓΥ  r are monotone increasing as r ↑ ∞, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=',  ΓΥ  r (u) ≤ ΓΥ  s (u) ,  EΥ,µ  r  (u) ≤ EΥ,µ  s  (u) ,  D(EΥ,µ  s  ) ⊂ D(EΥ,µ  r  )  r ≤ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' As Cr is a core of the form (EΥ,µ  r  , D(EΥ,µ  r  )), it suffices to check Cs ⊂ Cr  and ΓΥ  r (u) ≤ ΓΥ  s (u) on Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let u ∈ Cs and we show u ∈ Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By a simple reasoning  similar to the proof of Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6, we can see  uη  r ∈ Lipb(Υ(Br), dΥ)  µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η  if uη  s ∈ Lipb(Υ(Bs), dΥ)  µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  By Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2, it is straightforward to see ΓΥ  r (u) ≤ ΓΥ  s (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thus,  EΥ,µ  r  (u) =  �  Υ  ΓΥ  r (u) dµ ≤  �  Υ  ΓΥ  s (u) dµ =  �  Υ  ΓΥ  s (u) dµ = EΥ,µ  s  (u) < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Therefore, we conclude u ∈ Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  We now define a Dirichlet form on Υ whose invariant measure is the sineβ mea-  sure µ by the monotone limit of (EΥ,µ  r  , D(EΥ,µ  r  )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='14 (Monotone limit form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The form (EΥ,µ, D(EΥ,µ)) is defined as the  monotone limit:  D(EΥ,µ) := {u ∈ ∩r>0D(EΥ,µ  r  ) : EΥ,µ(u) = lim  r→∞ EΥ,µ  r  (u) < ∞} ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='15)  EΥ,µ(u) := lim  r→∞ EΥ,µ  r  (u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The form (EΥ,µ, D(EΥ,µ)) is a Dirichlet form on L2(µ) as it is the monotone limit  of Dirichlet forms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', by [MR90, Exercise 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The square field ΓΥ is defined as  the monotone limit of ΓΥ  r as well:  ΓΥ(u) := lim  r→∞ ΓΥ  r (u)  u ∈ D(EΥ,µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='16)  The corresponding L2(µ)-semigroup is denoted by {T Υ,µ  t  }t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  22  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  We now show that the form (EΥ,µ, D(EΥ,µ)) is a local Dirichlet form on L2(µ) and  satisfies the Rademacher-type property with respect to the L2-transportation-type  distance ¯dΥ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The form (EΥ,µ, D(EΥ,µ)) is a local Dirichlet form on L2(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Furthermore, (EΥ,µ, D(EΥ,µ)) satisfies Rademacher-type property:  Lip(¯dΥ, µ) ⊂ D(EΥ,µ) ,  ΓΥ(u) ≤ Lip¯dΥ(u)2  ∀u ∈ Lip(¯dΥ, µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='17)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The local property of (EΥ,µ, D(EΥ,µ)) follows from the fact that (EΥ,µ, D(EΥ,µ))  is the monotone limit of the local Dirichlet form (EΥ,µ  r  , D(EΥ,µ  r  )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  We show the  Rademacher-type property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Since ΓΥ is the limit square field of ΓΥ  r as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='16), it  suffices to show  Lip(¯dΥ, µ) ⊂ Cr  and  ΓΥ  r (u) ≤ Lip¯dΥ(u)2  ∀u ∈ Lip(¯dΥ, µ)  ∀r > 0 ,  which has been already proven in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  We verified (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The proof is  complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The semigroup {T Υ,µ  t  }t≥0 is the L2(µ)-strong operator limit of  the semigroups {T Υ,µ  r,t }t≥0, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=',  L2(µ)– lim  r→∞ T Υ,µ  r,t u = T Υ,µ  t  u  ∀u ∈ L2(µ) ,  t > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The statement follows from the monotonicity of (EΥ,µ  r  , D(EΥ,µ  r  ) as r ↑ ∞  proven in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='13 and [RS80, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='373].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Bakry–Émery Curvature bound for (EΥ,µ, D(EΥ,µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' In this subsection,  we prove the Bakry–Émery curvature bound for the form (EΥ,µ, D(EΥ,µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let β > 0 and µ be the sineβ ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The form (EΥ,µ, D(EΥ,µ))  satisfies the 1-Bakry–Émery curvature dimension condition BE1(0, ∞):  ΓΥ�  T Υ,µ  t  u  � 1  2 ≤ T Υ,µ  t  �  ΓΥ(u)  1  2�  ∀u ∈ D(EΥ,µ)  ∀t > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (BE1(0, ∞))  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We first prove BE1(0, ∞) for the form (EΥ,µ  r  , D(EΥ,µ  r  )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let u ∈ D(EΥ,µ  r  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5, [Han18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1], by the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4) of µη  r in terms of µk,η  r  and  by the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4) of ΓΥ(Br), there exists Ξ1  r ⊂ Υ with µ(Ξ1  r) = 1 so that for  every η ∈ Ξ1  r there exists a measurable set Ω1,η  r  ⊂ Υ(Br) with µη  r(Ω1,η  r ) = 1 satisfying  that for every γ ∈ Ω1,η  r , the following 1-Bakry–Émery gradient estimate holds:  ΓΥ(Br)(T Υ(Br),µη  r  t  uη  r)  1  2(γ) ≤ T Υ(Br),µη  r  t  �  ΓΥ(Br)(uη  r)  � 1  2(γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='18)  By Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7, there exists Ξ2  r ⊂ Υ with µ(Ξ2  r) = 1 so that for every η ∈ Ξ2  r there  exists a measurable set Ω2,η  r  ⊂ Υ(Br) with µη  r(Ω2,η  r ) = 1 satisfying that for every  γ ∈ Ω2,η  r  ΓΥ  r (T Υ,µ  r,t u)(γ + ηBcr) = ΓΥ(Br)��  T Υ,µ  r,t u  �η  r  �  (γ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='19)  CURVATURE BOUND OF DYSON BROWNIAN MOTION  23  ΓΥ  r (u)(γ + ηBcr) = ΓΥ(Br)(uη  r)(γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  By Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='12, there exists Λ3  r ⊂ Υ with µ(Λ3  r) = 1 so that for every γ ∈ Λ3  r  T Υ,µ  r,t u(γ) = T Υ(Br),µγ  r  t  uγ  r(γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='20)  By the standard disintegration argument, we can write  Λ3  r =  �  η∈Ξ3r  pr−1  r (Ω3,η  r ) ∩ Υη  r ,  where Ω3,η  r  = (Λ3  r)η  r := {γ ∈ Υ(Br) : γ + ηBcr ∈ Λ3  r} and Ξ3  r = prBcr(Λ3  r), and Υη  r  has been defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  By the disintegration formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='10), µ(Ξ3  r) = 1 and  µη  r(Ω3,η  r ) = 1 for every η ∈ Ξ3  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let Ξr := Ξ1  r ∩ Ξ2  r ∩ Ξ3  r and Ωη  r := Ω1,η  r  ∩ Ω2,η  r  ∩ Ω3,η  r  for η ∈ Ξr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Set  Kr :=  �  η∈Ξr  pr−1  r (Ωη  r) ∩ Υη  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  By construction, µ(Ξr) = 1 and µη  r(Ωη  r) = 1 for every η ∈ Ξr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='18), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='19) and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='20), the following inequalities hold for every γ ∈ Kr:  ΓΥ  r (T Υ,µ  r,t u)  1  2(γ) = ΓΥ  r (T Υ,µ  r,t u)  1  2(γBr + γBcr)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='21)  = ΓΥ(Br)((T Υ,µ  r,t u)γ  r)  1  2(γBr)  ≤ T Υ(Br),µγ  r  t  ΓΥ(Br)(uγ  r)  1  2(γBr)  = T Υ(Br),µγ  r  t  �  (ΓΥ  r (u)γ  r)  1  2�  (γBr)  = T Υ,µ  r,t ΓΥ  r (u)  1  2(γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let Θ := {γ ∈ Υ : ΓΥ  r (T Υ,µ  r,t u)  1  2(γ) ≤ T Υ,µ  r,t ΓΥ  r (u)  1  2(γ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Then Θ is µ-measurable by  construction, and thanks to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='21), it holds that Kr ⊂ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By applying Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2, we  obtain µ(Θ) = 1, which concludes BE1(0, ∞) for the truncated form (EΥ,µ  r  , D(EΥ,µ  r  ))  for any r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  We now prove BE1(0, ∞) of the form (EΥ,µ, D(EΥ,µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let u ∈ D(EΥ,µ) ⊂  ∩r>0D(EΥ,µ  r  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By the L2(µ)-lower semi-continuity of the square field ΓΥ, the mono-  tonicity ΓΥ  r ≤ ΓΥ  r′ for r ≤ r′ (we will use it in the following displayed formulas in  the first equality and in the second inequality), the convergence of T Υ,µ  r′,t  to T Υ,µ  t  as r′ → ∞ in the L2-strong operator sense by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='16, and BE1(0, ∞) for the  truncated form (EΥ,µ  r  , D(EΥ,µ  r  )) for any r > 0, the following inequalities hold true:  ΓΥ(T Υ,µ  t  u)1/2 = lim  r→∞ ΓΥ  r (T Υ,µ  t  u)1/2 ≤ lim  r→∞ lim inf  r′→∞ ΓΥ  r (T Υ,µ  r′,t u)1/2  ≤ lim inf  r′→∞ ΓΥ  r′(T Υ,µ  r′,t u)1/2  ≤ lim inf  r′→∞ T Υ,µ  r′,t ΓΥ  r′(u)1/2  = T Υ,µ  t  ΓΥ(u)1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  24  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  The last equality in the above displayed formulas followed from the following argu-  ment: by the L2(µ)-contraction property of T Υ,µ  r′,t , the monotonicity of ΓΥ  r′ as r′ ↑ ∞,  and the convergence of the semigroups T Υ,µ  r′,t  to T Υ,µ  t  as r′ → ∞ in the L2-strong  operator sense by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='16,  ��T Υ,µ  r′,t ΓΥ  r′(u)1/2 − T Υ,µ  t  ΓΥ(u)1/2��  L2(µ)  =  ��T Υ,µ  r′,t ΓΥ  r′(u)1/2 − T Υ,µ  r′,t ΓΥ(u)1/2��  L2(µ) +  ��T Υ,µ  r′,t ΓΥ(u)1/2 − T Υ,µ  t  ΓΥ(u)1/2��  L2(µ)  ≤  ��ΓΥ  r′ (u)1/2 − ΓΥ(u)1/2��  L2(µ) +  ��T Υ,µ  r′,t ΓΥ(u)1/2 − T Υ,µ  t  ΓΥ(u)1/2��  L2(µ)  r′→∞  −−−→ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Integral Bochner, local Poicaré and local log-Sobolev inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  As an application of BE(0, ∞) proven in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='17, we show several functional  inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We define the integral Γ2-operator as follows:  ΓΥ,µ  2  (u, ϕ) :=  �  Υ  �1  2ΓΥ(u)AΥ,µϕ − ΓΥ(u, AΥ,µu)ϕ  �  dµ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='22)  D(ΓΥ,µ  2  ) :=  �  (u, ϕ) : D(AΥ,µ)×2 : AΥ,µu ∈ D(EΥ,µ), ϕ, AΥ,µu ∈ L∞(µ)  �  ,  where AΥ,µ denotes the L2(µ)-infinitesimal generator associated with (EΥ,µ, D(EΥ,µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let µ be the sineβ ensemble with β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The following hold:  (a) (lntegral Bochner inequality) for every (u, ϕ) ∈ D(ΓΥ,µ  2  )  ΓΥ,µ  2  (u, ϕ) ≥ 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (b) (local Poincaré inequality) for u ∈ D(EΥ,µ) and t > 0,  T Υ,µ  t  u2 − (T Υ,µ  t  u)2 ≤ 2tT Υ,µ  t  ΓΥ(u) ,  T Υ,µ  t  u2 − (T Υ,µ  t  u)2 ≥ 2tΓΥ(T Υ,µ  t  u) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (c) (local logarithmic Sobolev inequality) for non-negative u ∈ D(EΥ,µ)  and t > 0,  T Υ,µ  t  u log u − T Υ,µ  t  u log T Υ,µ  t  u ≤ tT Υ,µ  t  �ΓΥ(u)  u  �  ,  T Υ,µ  t  u log u − T Υ,µ  t  u log T Υ,µ  t  u ≥ tΓΥ(T Υ,µ  t  u)  T Υ,µ  t  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The statement (a) follows from BE(0, ∞) proven in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='17 and [AGS15,  Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The statement (b) and (c) are consequences of BE(0, ∞), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', [BGL14,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  CURVATURE BOUND OF DYSON BROWNIAN MOTION  25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Dimension-free/log Harnack inequalities and Lipschitz  regularisation  In this section, we prove functional inequalities involving the Bakry–Émery cur-  vature bound BE(0, ∞) and the L2-transportation-type extended distance ¯dΥ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let µ be the sineβ ensemble with β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Then the following inequal-  ities hold:  (a) (log-Harnack inequality) for every non-negative u ∈ L∞(Υ, µ), t > 0,  there exists Ω ⊂ Υ so that µ(Ω) = 1 and  T Υ,µ  t  (log u)(γ) ≤ log(T Υ,µ  t  u)(η) + ¯dΥ(γ, η)2 ,  every γ, η ∈ Ω ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (b) (dimension-free Harnack inequality) for every non-negative u ∈ L∞(Υ, µ),  t > 0 and α > 1 there exists Ω ⊂ Υ so that µ(Ω) = 1 and  (T Υ,µ  t  u)α(γ) ≤ T Υ,µ  t  uα(η) exp  �  α  2(α − 1)  ¯dΥ(γ, η)2�  ,  for every γ, η ∈ Ω ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (c) (Lipschitz contraction) For u ∈ Lipb(¯dΥ, µ) and t > 0,  T Υ,µ  t  u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ  t  u  and the following estimate holds:  Lip¯dΥ( ˜T Υ,µ  t  u) ≤ Lip¯dΥ(u) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (d) (L∞-to-Lip regularisation) For u ∈ L∞(µ) and any t > 0,  T Υ,µ  t  u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ  t  u  and the following estimate holds:  Lip¯dΥ( ˜T Υ,µ  t  u) ≤  1  √  2t∥u∥L∞(µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We prove (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By the relation between T Υ,µ  r,t  and T Υ(Br),µ·  r  t  (·Br) in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='12,  there exists a measurable set Ωr  sem ⊂ Υ with µ(Ωr  sem) = 1 so that for every η ∈ Ωr  sem  T Υ,µ  r,t (η) = T Υ(Br),µη  r  t  (ηBr) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1)  Let u ∈ L∞(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thanks to Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3, there exists Ωr  ∞ ⊂ Υ so that µ(Ωr  ∞) = 1  and  uη  r ∈ L∞(µη  r),  ∀η ∈ Ωr  ∞,  ∀r ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  By Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5, there exists a measurable set Ωr  rcd ⊂ Υ so that µ(Ωr  rcd) = 1 and  (Υk, dΥ, µη  r) is RCD(0, ∞) with k = k(η) as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4) for every η ∈ Ωr  rcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let Ωr := Ωr  sem ∩ Ωr  ∞ ∩ Ωr  rcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' As the log-Harnack inequality holds in RCD spaces  (see, [AGS15, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6]), the following holds for every η ∈ Ωr and k = k(η)  T Υk(Br),µk,η  r  t  (log uη  r)(γ) ≤ log(T Υk(Br),µk,η  r  t  uη  r)(ζ) + dΥ(γ, ζ)2 ,  ∀ γ, ζ ∈ Υk(Br) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2)  26  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  Noting the convergence of the semigroups {T Υ,µ  r,t }t≥0 to {T Υ,µ  t  }t≥0 in the L2(µ)-  operator sense by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='16, there exist Ωcon ⊂ Υ with µ(Ωcon) = 1 and a (non-  relabelled) subsequence of {r} so that for every γ ∈ Ωcon  T Υ,µ  r,t (log u)(γ)  r→∞  −−−→ T Υ,µ  t  (log u)(γ) ,  log(T Υ,µ  r,t u)(γ)  r→∞  −−−→ log(T Υ,µ  t  u)(γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3)  Let Ω′ = Ωcon ∩r∈N Ωr, which by construction satisfies µ(Ω′) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Our goal is now  to prove that there exists Ω ⊂ Ω′ with µ(Ω) = 1 so that  T Υ,µ  t  (log u)(γ) ≤ log(T Υ,µ  t  u)(η) + ¯dΥ(γ, η)2 ,  every γ, η ∈ Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4)  Thanks to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3), Formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4) comes down to the corresponding inequality for the  semigroup {T Υ,µ  r,t }t≥0 for any r > 0:  T Υ,µ  r,t (log u)(γ) ≤ log(T Υ,µ  r,t u)(η) + ¯dΥ(γ, η)2 ,  every γ, η ∈ Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5)  We prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5) by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Suppose that for any Ω ⊂ Ω′ with µ(Ω) = 1,  there exists γ, η ∈ Ω so that  T Υ,µ  r,t (log u)(γ) ≥ log(T Υ,µ  r,t u)(η) + ¯dΥ(γ, η)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6)  We may assume that ¯dΥ(γ, η) < ∞, otherwise, we have nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thus,  by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='15), there exists r > 0 so that  γBcr = ηBcr ,  γ(Br) = η(Br) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7)  By making use of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='7), we obtain  T Υ,µ  r,t (log u)(γ) = T Υ,µ  r,t (log u)(γBr + γBcr)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='8)  = T Υ(Br),µγ  r  t  (log uγ  r)(γBr)  ≤ log(T Υ(Br),µγ  r  t  u)(ηBr) + dΥ(γBr, ηBr)2  = log(T Υ,µ  r,t u)(η) + ¯dΥ(γ, η)2 ,  which contradicts (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6), therefore, the proof of (a) is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The proof of (b) follows precisely in the same strategy as above by replacing  T Υ,µ  t  (log u), log(T Υ,µ  t  u) and ¯dΥ(γ, η)2 by (T Υ,µ  t  u)α, T Υ,µ  t  uα and  α  2(α−1)¯dΥ(γ, η)2 re-  spectively, and noting that the dimension-free Harnack inequality holds on RCD(K, ∞)  spaces ([Li15, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The proof of (c): Note that uη  r ∈ Lip(Υ(Br), dΥ) whenever u ∈ Lip(Υ, ¯dΥ) and  LipdΥ(uη  r) ≤ Lip¯dΥ(u) by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Note also that the sought conclusion of (c) can  be rephrased as  ˜T Υ,µ  t  u(γ) − ˜T Υ,µ  t  u(η) ≤ Lip¯dΥ(u)¯dΥ(γ, η)  ∀γ, η ∈ Υ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Thus, by the same proof strategy as in (a) replacing T Υ,µ  t  (log u)(γ) and log(T Υ,µ  t  u)(η)  with T Υ,µ  t  u(γ) and T Υ,µ  t  u(η), and noting that the Lipschitz contraction property  CURVATURE BOUND OF DYSON BROWNIAN MOTION  27  holds on RCD spaces ([AGS14b, (iv) in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1]), we conclude that there exists  Ω ⊂ Υ with µ(Ω) = 1 so that  T Υ,µ  t  (γ) − T Υ,µ  t  (η) ≤ Lip¯dΥ(u)¯dΥ(γ, η)  ∀γ, η ∈ Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The conclusion now follows from the McShane extension Theorem [DS21b, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The proof of (d) is the same as that of (c) but using the L∞-to-Lip property  ([AGS14b, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='5]) in RCD(K, ∞) spaces instead of [AGS14b, (iv) in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Generalisation  We have been so far working in the case of sineβ ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' In this section, we  seek to generalise the aforementioned statements to general probability measures  on Υ = Υ(Rn) for n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' In this section, we denote by m and mr the Lebesgue  measure on Rn and its restriction on Br(0) respectively, and we take the Euclidean  distance d(x, y) := |x − y| for x, y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let µ be a Borel probability on Υ and  assume that it is fully supported on Υ with respect to the vague topology τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let  K(µη  r) ⊂ N0 be defined as  K(µη  r) := {k ∈ N0 : µk,η  r (Υk(Br)) > 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let K ∈ R and µ be a fully supported Borel probability with  respect to the vague topolgoy τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Assume the following conditions:  (a) the measure µη  r is absolutely continuous with respect to the Poisson mea-  sure πmr, and µk,η  r  is equivalent to πmr|Υk(Br) for any k ∈ K(µη  r), µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η and  any r > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (b) the density  dµk,η  r  dπmr|Υk(Br)  is τv-continuous on Υk(Br), and the logarithmic density  Ψk,η  r  = − log  �  dµk,η  r  dπmr|Υk(Br)  �  is K-geodesically convex with respect to dΥ on Υk(Br) for any k ∈ K(µη  r),  µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η and any r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Under (a) in Assumption, the local Dirichlet form (EΥ,µ, D(EΥ,µ)) is constructed  in the same proof as in the case of sineβ ensemble as we have not use any partic-  ular property of K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We further show the synthetic curvature bound for the  form (EΥ,µ, D(EΥ,µ)) and related functional inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Suppose that µ satisfies Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Then the form (EΥ,µ, D(EΥ,µ))  satisfies  28  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  (a) (Bakry–Émery inequality BE1(K, ∞))  ΓΥ�  T Υ,µ  t  u  � 1  2 ≤ e−KtT Υ,µ  t  �  ΓΥ(u)  1  2�  ∀u ∈ D(EΥ,µ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (b) (lntegral Bochner inequality) for every (u, ϕ) ∈ D(ΓΥ,µ  2  )  ΓΥ,µ  2  (u, ϕ) ≥ K  �  Υ  ΓΥ(u)ϕ dµ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (c) (local Poincaré inequality) for u ∈ D(EΥ,µ) and t > 0,  T Υ,µ  t  u2 − (T Υ,µ  t  u)2 ≤ 1 − e−2Kt  K  T Υ,µ  t  ΓΥ(u) ,  T Υ,µ  t  u2 − (T Υ,µ  t  u)2 ≥ e−2Kt − 1  K  ΓΥ(T Υ,µ  t  u) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (d) (local logarithmic Sobolev inequality) for non-negative u ∈ D(EΥ,µ)  and t > 0,  T Υ,µ  t  u log u − T Υ,µ  t  u log T Υ,µ  t  u ≤ 1 − e−2Kt  2K  T Υ,µ  t  �ΓΥ(u)  u  �  ,  T Υ,µ  t  u log u − T Υ,µ  t  u log T Υ,µ  t  u ≥ e−2Kt − 1  2K  ΓΥ(T Υ,µ  t  u)  T Υ,µ  t  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (e) (log Harnack inequality) for every non-negative u ∈ L∞(Υ, µ), t > 0,  there exists Ω ⊂ Υ so that µ(Ω) = 1 and  T Υ,µ  t  (log u)(γ) ≤ log(T Υ,µ  t  u)(η) +  K  2(1 − e−2Kt)  ¯dΥ(γ, η)2 ,  ∀γ, η ∈ Ω ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (f) (dimension-free Harnack inequality) for every non-negative u ∈ L∞(Υ, µ),  t > 0 and α > 1 there exists Ω ⊂ Υ so that µ(Ω) = 1 and  (T Υ,µ  t  u)α(γ) ≤ T Υ,µ  t  uα(η) exp  �  αK  2(α − 1)(1 − e−2Kt)  ¯dΥ(γ, η)2�  ,  ∀γ, η ∈ Ω ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (g) (Lipschitz contraction) For u ∈ Lip(¯dΥ, µ) and t > 0,  T Υ,µ  t  u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ  t  u  and the following estimate holds:  Lip¯dΥ( ˜T Υ,µ  t  u) ≤ e−KtLip¯dΥ(u) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (h) (L∞-to-Lip regularisation) For u ∈ L∞(µ) and t > 0,  T Υ,µ  t  u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ  t  u  and the following estimate holds:  Lip¯dΥ( ˜T Υ,µ  t  u) ≤  1  �  2I2K(t)  ∥u∥L∞(µ)  ∀t > 0 ,  where IK(t) :=  � t  0 eKr dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  CURVATURE BOUND OF DYSON BROWNIAN MOTION  29  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thanks to Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1, the space (Υk(Br), dΥ, µk,η  r ) satisfies RCD(K, ∞)  for every k ∈ K(µη  r) as in the same proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' As we have not used any  particular properties of K = 0 for the proofs in the case of sineβ, the completely  same proofs (up to constant multiplication depending only on K) work in the case of  general K ∈ R and general µ satisfying Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1, which therefore concludes  the statements of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Let m and mr be the Lebesgue measure on Rn and its restriction on Br respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Set Υ = Υ(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let µ be a Borel probability on Υ satisfying that µη  r is absolutely  continuous with respect to the Poisson measure πmr for any r > 0 and µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let  Σ ⊂ Br so that mr(Σc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let Ω(r) := {γ ∈ Υ : γΣ = γBr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Then,  µ  �  Ω(r)  �  = 1  ∀r > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' We fix r > 0 and write simply Ω = Ω(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By the disintegration formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='10),  µ(Ω) =  �  Υ  µη  r(Ωη  r) dµ(η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Thus, it suffices to show µη  r(Ωη  r) = 1 for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' This is equivalent to show  µη  r(Ωη  r) =  �  k∈N0  µk,η  r (Ωη  r) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='1)  As µk,η  r  is absolutely continuous with respect to πmr|Υk(Br), it suffices to prove  πmr|Υk(Br)((Ωη  r)c) = 0 for every k ∈ N0 and η ∈ Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  We show that (recall the definition of symmetric product Σ⊙k in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3))  Σ⊙k ⊂ Ωη  r ∩ Υk(Br)  ∀η ∈ Υ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2)  Let γ ∈ Σ⊙k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Then by the definition of Ω, it holds that γ + ηBcr ∈ Ω for any η ∈ Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Thus, by recalling the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='9) of Ωη  r, we obtain γ ∈ Ωη  r ∩ Υk(Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thus,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  By using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2), πmr|Υk(Br) = e−mr(Br)m⊙k  r  by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='11) and m⊙k  r  �  (Σ⊙k)c�  = 0 by  hypothesis, we conclude that for every η ∈ Υ  πmr|Υk(Br)((Ωη  r)c) = e−mr(Br)m⊙k  r  ��  Ωη  r ∩ Υk(Br)  �c�  ≤ e−mr(Br)m⊙k  r  �  (Σ⊙k)c�  = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  We recall that for η ∈ Υ, we set Υη  r := {γ ∈ Υ : γBcr = ηBcr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2 (disintegration lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Assume that there exists a measurable set  Ξ ⊂ Υ with µ(Ξ) = 1 so that for every η ∈ Ξ, there exists a family of measurable  30  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  sets Ωη ⊂ Υ(Br) so that µη  r(Ωη) = 1 for every η ∈ Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let Ω ⊂ Υ be the (not  necessarily measurable) subset defined by  Ω :=  �  η∈Ξ  pr−1  r (Ωη) ∩ Υη  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Assume further that there exists a measurable set Θ ⊂ Υ so that Ω ⊂ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Then,  µ(Θ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Caveat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  As the set Ω is defined as uncountable union of measurable sets, the mea-  surability of Ω is not necessarily true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' The disintegration formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='10)  is, therefore, not necessarily applicable directly to Ω, which motivates the aforemen-  tioned lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof of Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let Θη  r = {γ ∈ Υ(Br) : γ + ηBcr ∈ Θ} be a section of Θ at ηBcr  as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Then, Ωη ⊂ Θη  r by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thus, µη  r(Θη  r) ≥ µη  r(Ωη) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By the  disintegration formula in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='10), we have that  µ(Θ) =  �  Υ  µη  r(Θη  r) dµ(η) ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let µ be a Borel probability on Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let Ω ⊂ Υ satisfy µ(Ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Then, there exists Ω′ ⊂ Ω with µ(Ω′) = 1 and  µη  r(Ωη  r) = 1 ,  ∀η ∈ Ω′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' By the disintegration formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='10),  1 = µ(Ω) =  �  Υ  µη  r(Ωη  r) dµ(η) =  �  Ω  µη  r(Ωη  r) dµ(η) ,  by which the statement is readily concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let (Q, D(Q)) be a closed form on a complete separable Hilbert  space H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Let {Tt} and (A, D(A)) be the corresponding semigroup and infinitesi-  mal generator respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Suppose that there exists an algebra C ⊂ D(Q) so that  C ⊂ H is dense and TtC ⊂ C for any t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Then C is dense in D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' It holds that TtD(A) ⊂ D(A) by the general property of semigroups associ-  ated with closed forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Thus, combining it with the hypothesis TtC ⊂ C,  Tt(C ∩ D(A)) ⊂ C ∩ D(A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Thus, by [RS75, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='49], C ∩ D(A) is dense in the graph norm in the space  (A, D(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Namely, we obtained  (A, C ∩ D(A)) is essentially self-adjoint .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  CURVATURE BOUND OF DYSON BROWNIAN MOTION  31  The density C ⊂ D(Q) now follows by the density of C ∩ D(A) in the graph norm,  by the density of D(A) ⊂ D(Q) due to the general property of closed forms, by the  density of C ⊂ H and by a simple integration-by-parts argument  −Q(u, u) = (Au, u)H ≤ ∥Au∥H∥u∥H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  ■  References  [AGS14a] Ambrosio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', Gigli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', and Savaré, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Calculus and heat flow in metric  measure spaces and applications to spaces with Ricci bounds from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 395:289–391, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [AGS14b] Ambrosio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', Gigli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', and Savaré, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Metric measure spaces with  Riemannian Ricci curvature bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=',  163(7):1405–1490, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [AGS15] Ambrosio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', Gigli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', and Savaré, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Bakry–Émery Curvature-  Dimension  Condition  and  Riemannian  Ricci  Curvature  Bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 43(1):339–404, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [BGL14] Bakry, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', Gentil, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', and Ledoux, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Analysis and Geometry of Markov  Diffusion Operators, volume 348 of Grundlehren der mathematischen  Wissenschaften.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Springer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [BH91] Bouleau, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Hirsch, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Dirichlet forms and analysis on Wiener  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' De Gruyter, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [Del21] Dello Schiavo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Ergodic Decomposition of Dirichlet Forms via Direct  Integrals and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Potential Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [DHLM20] Dereudre, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', Hardy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', Leblé, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', and Maïda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' DLR Equations and  Rigidity for the Sine-Beta Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', pages  172–222, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [DS21a] Dello Schiavo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Suzuki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Configuration spaces over sin-  gular spaces –I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Dirichlet-Form and Metric Measure Geometry –.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='03192v2 (version 2), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [DS21b] Dello Schiavo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Suzuki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' On the Rademacher and Sobolev-to-  Lipschitz Properties for Strongly Local Dirichlet Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Func.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=',  281(11):Online first, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [DS22] Dello Schiavo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Suzuki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Configuration Spaces over Singular  Spaces II – Curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='01379, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [EH15] Erbar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Huesmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Curvature bounds for configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 54:307–430, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [FOT11] Fukushima, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', Oshima, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', and Takeda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Dirichlet forms and sym-  metric Markov processes, volume 19 of De Gruyter Studies in Mathe-  matics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' de Gruyter, extended edition, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  32  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' SUZUKI  [Fuk97] Fukushima, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Distorted Brownian motions and BV functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Trends  in Probability and Analysis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Kono, N-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Shieh, eds, pages 143–150,  1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [Gho15] Ghosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Determinantal processes and completeness of random expo-  nentials: the critical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Theory Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Fields, 163(3):643–665,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [GKMS18] Galaz-García, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', Kell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', Mondino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', and Sosa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' On quotients of  spaces with Ricci curvature bounded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 275:1368–  1446, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [Han18] Han, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' New characterizations of ricci curvature on rcd metric mea-  sure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Discrete and Continuous Dynamical Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Series A,  38(10):4915–4927, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [KS09] Killip, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Stoiciu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Eigenvalue statistics for cmv matrices: from  poisson to clock via random matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 146  (3)::361–399, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [KT10] Katori, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Tanemura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Non-equilibrium dynamics of Dyson’s  model with an infinite number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=',  293(2):469–497, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [Li15] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Dimension-Free Harnack Inequalities on RCD(K, ∞) Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Theoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 29:1280–1297, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [MR85] Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Röckner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Dirichlet forms-closability and change of  speed measure, Infinite dimensional analysis and stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Research Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Albeverio, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', Pitman, 124:119–144, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [MR90] Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Röckner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Introduction to the Theory of (Non-  Symmetric) Dirichlet Forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Springer, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [MR00] Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Röckner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Construction of Diffusions on Configuration  Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Osaka J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 37:273–314, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [Nak14] Nakano, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Level statistics for one-dimensional schrödinger operators  and gaussian beta ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 156(1):66–93, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [NR18] Najnundel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Reda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Rigidity of the Sineβ process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 23:1–8, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [Osa13] Osada, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Interacting Brownian Motions in Infinite Dimensions with  Logarithmic Interaction Potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 41(1):1–49, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [RS75] Reed, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Simon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Methods of Modern Mathematical Physics II –  Fourier Analysis, Self-Adjointness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Academic Press, New York, London,  1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [RS80] Reed, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Simon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Methods of Modern Mathematical Physics I –  Functional Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Academic Press, New York, London, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  CURVATURE BOUND OF DYSON BROWNIAN MOTION  33  [Spo87] Spohn, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Interacting Brownian Particles: A Study of Dyson’s Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Hydrodynamic Behavior and Interacting Particle Systems, pages 151–  179, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [Stu06] Sturm, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' On the geometry of metric measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=',  196:65–131, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [Tsa16] Tsai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Infinite dimensional stochastic differential equations for  dyson’s model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Probability Theory and Related Fields, 166:801–850, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [VG20] Von Renesse, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='. and Gyünesu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Molecules as metric measure spaces  with kato-bounded ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Comptes Rendus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Mathématique,  358:595–602, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  [VV09] Valkó, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' and Virág, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Continuum limits of random matrices and the  brownian carousel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=' math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content=', 177(3):463–508, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
+page_content='  Department of Mathematical Science, Durham University, Science Laboratories,  South Road, DH1 3LE, United Kingdom' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfbfc0/content/2301.00262v1.pdf'}
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+Hybrid Quantum-Classical Autoencoders for
+End-to-End Radio Communication
+Zsolt Tabi∗, Bence Bak´o∗†, D´aniel T. R. Nagy∗‡, P´eter Vaderna‡, Zs´ofia Kallus‡,
+P´eter H´aga‡, Zolt´an Zimbor´as∗†
+∗E¨otv¨os Lor´and University, Budapest, Hungary
+Email: zsolttabi@ik.elte.hu
+†Wigner Research Centre for Physics, Budapest, Hungary
+Email: bako.bence@wigner.hu, zimboras.zoltan@wigner.hu
+‡Ericsson Research, Budapest, Hungary
+Email: {daniel.a.nagy, peter.vaderna, zsofia.kallus, peter.haga}@ericsson.com
+Abstract—Quantum neural networks are emerging as poten-
+tial candidates to leverage noisy quantum processing units for
+applications. Here we introduce hybrid quantum-classical au-
+toencoders for end-to-end radio communication. In the physical
+layer of classical wireless systems, we study the performance
+of simulated architectures for standard encoded radio signals
+over a noisy channel. We implement a hybrid model, where
+a quantum decoder in the receiver works with a classical
+encoder in the transmitter part. Besides learning a latent space
+representation of the input symbols with good robustness against
+signal degradation, a generalized data re-uploading scheme for
+the qubit-based circuits allows to meet inference-time constraints
+of the application.
+Index Terms—variational quantum algorithms, quantum ma-
+chine learning, quantum autoencoder, radio communication
+I. INTRODUCTION
+One of the most popular Quantum Machine Learning
+(QML) methods are Quantum Neural Networks (QNNs) [1],
+[2]. These are special variational quantum circuits, designed as
+the quantum analogues of classical neural networks. QNNs can
+be optimized with gradient-based or gradient-free optimization
+algorithms forming hybrid quantum-classical training loops
+[3], [4]. Various QNN architectures have been proposed such
+as quantum convolutional neural networks [5], generative mod-
+els [6], long short-term memories [7] and autoencoders [8]–
+[10]. Beside the high activity in algorithmic research within
+QML, their novel benchmarking and requirement setting ap-
+plications are also motivating a wide variety of works [11].
+Although QML is still in a phase of basic research with
+many open questions, its early implementations in wireless
+communication systems spark both scientific curiosity and
+commercial interest [12]. However, high-performance, near
+real-time applications might impose a new set of requirements
+on these solutions.
+Wireless communication has undergone tremendous evolu-
+tion during the last decades. The increasing adoption of AI
+and ML methods is opening up new development possibilities
+in various parts of the radio stack. In the design of the sixth
+generation (6G) wireless networks, AI and ML technologies
+are considered to be tightly integrated into the system and
+smart algorithms can be applied to all aspects of network
+operations and procedures [13]. Considering the improvement
+of quantum computers it is envisioned that quantum algorithms
+and especially QML will play a significant role in future
+networks [12].
+The structure of this paper is as follows. In Sec. II, we
+give a high-level overview of wireless communication systems
+together with an autoencoder solution used in the radio phys-
+ical layer. Our novel hybrid classical-quantum autoencoder
+prototype is presented in Sec. III. We discuss our result in
+Sec. IV. Finally, we conclude with an outlook in Sec. V.
+II. AUTOENCODER ARCHITECTURE IN END-TO-END
+COMMUNICATION
+I
+Q
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+(a)
+I
+Q
+(b)
+Fig. 1: 16-QAM constellation diagrams. (a) Transmitted
+symbols; (b) Received noisy signal.
+s
+Transmitter
+Channel
+Receiver
+ˆs
+x
+y
+Fig. 2: High-level representation of a communication sys-
+tem. A message is transmitted through a noisy communication
+channel to be recovered by the receiver.
+The components of communication networks are organized
+in a layered architecture where each layer is responsible for
+different communication aspects [14]. The physical layer is the
+arXiv:2301.02609v1  [quant-ph]  6 Jan 2023
+
+Fig. 3: The proposed hybrid quantum-classical autoencoder embedded into the end-to-end communication architecture.
+The transmitter maps each message s to a symbol, then sends it through the channel. The channel is represented by an
+additive noise acting on the signal x. The receiver, realized by a quantum decoder, consists of a multi-layer QNN adapting
+data re-uploading. It decodes the noisy signal and gives an estimate of the original message.
+lowest layer. It provides the means of transmitting a stream of
+raw bits over a data channel connecting the network elements.
+The physical layer on the transmitter side converts data in
+the form of bits to electromagnetic waves to be transmitted
+wirelessly, while the receiver converts the electromagnetic
+waves received by an antenna to binary data. The main
+challenge in wireless data transmission is to overcome the
+channel impairments so that the messages can be recovered
+with small error rate.
+The role of modulation is to convert digital data into radio
+waves. It can be achieved in different ways, the informa-
+tion can be encoded by varying (shifting) either amplitude,
+frequency or phase of the electromagnetic wave. The more
+states the modulation has, the more bits are transferred in one
+symbol, resulting in higher data rate. However, with higher
+order modulation the signal is more sensitive to channel errors,
+so the applied modulation usually depends on the channel
+quality. Fig. 1 shows a constellation diagram of the symbol
+representation of 16-QAM modulation, where 4-bit strings
+can be represented as complex numbers in a scheme resistant
+to general noise patterns while achieving high data rate with
+minimal channel uses.
+Based on [15], we model a simple communications system,
+shown in Fig. 2, consisting of a transmitter that modulates
+message s ∈ M = {0, . . . , M − 1} into a signal x and
+sends it over a noisy channel to the receiver that tries to
+decode the received signal y resulting in the received message
+ˆs. The transmitted signal x suffers degradation due to the
+noise present on the channel. In case of transmission over a
+complex channel with n discrete channel uses, the transmitter
+can be represented as the transformation f : M �→ R2n,
+mapping the message s to x ∈ R2n signal with certain
+constraints imposed by the transmitting hardware (e.g., energy
+constraint or average power constraint). The channel can be
+modeled as a conditional probability density function p(y|x)
+that produces the output signal y ∈ R2n given the input signal,
+usually via some noise model (e.g., additive white Gaussian
+noise (AWGN)). The receiver is represented as the mapping
+g : R2n �→ M that recovers some estimate ˆs of the original
+message from the received signal. In this work, we focus on
+the case of single channel use (n = 1), however, this model
+can be easily adopted to cases of n > 1. Also, the number of
+transmit-receive pairs can be increased to get a Multiple-Input
+and Multiple-Output (MIMO) system [15], [16].
+Both of f and g transformations can be created in various
+ways. In case of simple noise models applied in the channel
+the transformations can be designed as explicit mathematical
+formulas. In the case of complex noise scenarios that are
+difficult to describe with mathematical models, a possible way
+is to train deep neural networks, especially autoencoders, to
+solve the encoding and decoding tasks [15].
+An autoencoder is a special type of deep neural network,
+with the aim to compress or denoise data [17]–[19]. Autoen-
+coders consist of an encoder function f : RD �→ RL, and
+a decoder function g : RL �→ RD. The encoder transforms
+its input χ into a latent space representation f(χ) ∈ RL,
+whereas the decoder tries to reconstruct it: ˆχ = g(f(χ)).
+Usually we have L < D, i.e., the encoder produces a compact
+representation of the data. f and g are typically deep neural
+networks trained jointly to minimize a loss function of the
+form L(χ, g(f(χ)).
+In telecommunication, opposite to the general compressing
+and denoising interpretation, autoencoders can be effectively
+used in the presented communication system to learn how to
+represent input messages as signals [15]. This model differs
+from the “typical” autoencoder concept in the sense that it does
+not try to remove noise from the input, instead it learns how
+to represent the input in a way that is robust against a given
+noisy channel acting in the latent space of the autoencoder.
+As a result of the training process, the latent space (or hidden
+layer) of the autoencoder contains the learned constellation of
+symbols (or codebook). The learned constellation is optimized
+for best mapping of the input messages to signals that can be
+accurately decoded with the largest success probability for the
+
+Quantum Decoder
+Encoder
+cod
+Noise model p(y|x)
+argmax
+Message vector
+Qubit encoding
+Qubit encoding
+Normalization
+ayer
+0.1
+abit readou
+La
+·
+0.8
+3
+0.05
+Transmitter
+Receiver
+hannelEncoding
+First Layer
+· · ·
+· · ·
+· · ·
+· · ·
+|0⟩
+Rx(y1)
+R(α1, β1, γ1)
+|0⟩
+Rx(y2)
+R(α2, β2, γ2)
+|0⟩
+R(α3, β3, γ3)
+|0⟩
+R(α4, β4, γ4)
+(a)
+First Layer with Encoding
+· · ·
+· · ·
+· · ·
+· · ·
+|0⟩
+Rx(y1)
+R(α1, β1, γ1)
+|0⟩
+Rx(y2)
+R(α2, β2, γ2)
+|0⟩
+R(α3, β3, γ3)
+|0⟩
+R(α4, β4, γ4)
+(b)
+First Layer with Double Encoding
+· · ·
+· · ·
+· · ·
+· · ·
+|0⟩
+Rx(y1)
+R(α1, β1, γ1)
+|0⟩
+Rx(y2)
+R(α2, β2, γ2)
+|0⟩
+Rx(y1)
+R(α3, β3, γ3)
+|0⟩
+Rx(y2)
+R(α4, β4, γ4)
+(c)
+First Layer with Weighted Double Encoding
+· · ·
+· · ·
+· · ·
+· · ·
+|0⟩
+Rx(w1 · y1)
+R(α1, β1, γ1)
+|0⟩
+Rx(w2 · y2)
+R(α2, β2, γ2)
+|0⟩
+Rx(w3 · y1)
+R(α3, β3, γ3)
+|0⟩
+Rx(w4 · y2)
+R(α4, β4, γ4)
+(d)
+Fig. 4: Quantum decoder implementations with encoding schemes. Ansatz circuits with (a) simple data encoding; (b) simple
+data re-uploading; (c) double data re-uploading; (d) weighted double data re-uploading.
+specific channel model. Whereas the encoder learns how to
+produce optimal symbols, the receiver learns how to decode
+these symbols after they have been corrupted by the channel,
+i.e., how to recover x after sampling from p(y|x).
+III. HYBRID QUANTUM AUTOENCODER FOR RADIO
+PHYSICAL LAYER
+A. Hybrid quantum autoencoder overview
+Quantum autoencoder architectures have previously been
+proposed to compress as well as denoise quantum data [8],
+[10]. Hybrid quantum-classical autoencoders enable many
+variations for quantum or classical encoding/decoding or the
+use of classical data. In this work, a hybrid quantum-classical
+autoencoder is applied for processing classical information.
+Building on the physical layer autoencoder presented in
+Sec. II, we propose a hybrid quantum-classical autoencoder
+with classical encoder on the transmitter side and a quantum
+decoder on the receiver side – trained in an end-to-end
+solution. The encoder projects the original message to a lower
+dimensional representation, robust to the channel degradation
+effect. Once the signal is passed to the quantum decoder, the
+compressed information is mapped to a higher dimensional
+Hilbert space of the qubits by a QNN that has been previously
+shown to be efficient for classification tasks [20], [21].
+In our model, the classical encoder consists of an embedding
+followed by a normalization. A simple linear embedding is
+used to produce the constellation, satisfying the average power
+constraint by normalization. The decoder is realized by a
+general strongly connected quantum neural network which
+we refer to as a quantum decoder. By simulating increasing
+levels of noise in the channel, we can present a performance
+evaluation of the various neural network architectures.
+B. Quantum decoder architectures
+A general QNN architecture has three main components
+as shown in Fig. 3: qubit encoding for embedding the input
+data, the parameterized QNN layers, and the qubit readout
+given as a probability distribution over the possible con-
+stellation symbols obtained from suitable measurements with
+high enough number of shots. To encode the output of the
+channel, we choose angle embedding with parameterized Rx
+rotations [22]. With this embedding, there are multiple ways
+to encode two-dimensional feature vectors into four qubits. As
+for the variational ansatz, we use strongly entangling layers
+introduced in quantum classifiers as they are known to be
+expressive reaching ‘wide corners of the Hilbert space’ [23].
+The measurements are performed in the computational basis
+and the obtained probability distribution over the 16 basis
+states is the output of the decoder.
+The simplest single-layer realization of such a QNN struc-
+ture is presented in Fig. 4a. To improve this ansatz, we can
+apply the data re-uploading trick recently introduced in [24].
+This technique, as shown in Fig. 4b, repeats the input encoding
+block before each layer of the QNN circuit. The intuition be-
+hind the effectiveness of this method is that by re-introducing
+the input before each layer, one can mimic the computational
+structure of typical classical deep neural networks, where the
+copying of the classical information is readily available, which
+
+would be, without this trick, prohibited by the no-cloning
+theorem in quantum machine learning. The expressivity of a
+model can be further increased by applying the encoding on
+different subsystems in parallel [25]. With this in mind, we
+further enhance the ansatz by encoding the first feature into
+both qubit no. 1 and no. 3 and the second input feature into
+both qubit no. 2 and no. 4. This double data re-uploading
+ansatz is presented in Fig. 4c. As a final improvement, we
+considered the role of the number of trainable parameters. As
+the expressive power of the ansatz is highly dependent on the
+number of trainable parameters, one should try to include as
+many parameters as possible. One way to increase the number
+of parameters while keeping the circuit as shallow as possible
+– to respect the limited hardware capabilities and the inference
+time constraints of the application – is to introduce trainable
+weights in the data re-uploading blocks, as shown in Fig. 4d.
+This modification keeps the depth constant.
+C. Training and fine-tuning
+For our hybrid autoencoder to achieve low estimation errors,
+the training of the end-to-end system requires to be further
+improved via hyper-parameter tuning.
+First, the training of the hybrid model is done on batches
+uniformly sampled from the set of messages {0, . . . , 15}.
+These are sent as two dimensional encoded symbols through
+the AWGN channel with SNR of 15 dB and i.i.d. noise.
+The accuracy of the model is measured by evaluating
+the Symbol Error Rate (SER), a key performance indica-
+tor commonly used in radio communication. The network
+weight updates are calculated with the sparse categorical cross-
+entropy of the distribution generated by the decoder and the
+ground truth symbols. This loss function is used to calculate
+gradients in a mini-batch gradient descent with batch size of 64
+and Adam optimizer [26]. We simulate the hybrid autoencoder
+using PennyLane [27], a quantum machine learning framework
+with its TensorFlow [28] backend. Second, we evaluate the
+reached model accuracy at various hyper-parameter settings.
+The search is conducted by KerasTuner [29] after partitioning
+the space as the simulator compute times are prohibitive of a
+full grid search.
+We start by first evaluating the learning rate parameter set
+η ∈ {0.1, 0.01, 0.001} using the simple ansatz presented on
+Fig. 4a with L = 8 layers with 1000-shot measurements and
+1000 training steps. Based on these results, the only viable
+value of η = 0.1 is set for the rest of this study.
+We continue with evaluating modifications to the basic
+ansatz but keeping the number of layers L = 8 and 1000
+training steps fixed, to minimize the overall computation time.
+The results are shown in Fig. 5. For the basic circuit, the SER
+fluctuates around its initial value without showing convergence
+to a desirable level. A significant accuracy improvement of
+roughly 40% is achieved by implementing single data re-
+uploading (Fig. 4b with ansatz of 1×DR). Introducing the
+double data re-uploading layer (2×DR with ansatz of Fig. 4c)
+leads to another 15% improvement. Finally, we can even fur-
+ther increase the performance by another 20% when using the
+0
+200
+400
+600
+800
+1000
+Number of training steps
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Symbol Error Rate
+basic
+1x DR
+2x DR
+2x wDR
+Fig. 5: Learning curves of circuit architectures. The number
+of data re-uploading (none, single, double) and the weighted
+data encoding have high impact on the convergence properties
+of the quantum autoencoder.
+weighted double data re-uploading technique (2×wDR with
+ansatz Fig. 4d). Based on these results, the weighted double
+data re-uploading ansatz is chosen for further experiments.
+As a last step, we optimize the number of layers. The hybrid
+autoencoder using the best performing ansatz is trained with
+8 to 24 layers. Increasing the number of layers clearly shows
+the improvement in SER as well as in convergence time as
+seen in Fig. 6.
+IV. PERFORMANCE EVALUATION
+A. Validation
+Comparing our hybrid architecture to the classical method is
+crucial to validate the solution. Based on the learning curves
+presented, the shallowest network reaching accuracy similar
+to the classical solution contains L = 16 layers. Further
+increasing the number of layers leads to small improvements
+in accuracy but it is suboptimal in terms of circuit depth.
+Although the hybrid quantum autoencoder models are
+trained at SNR of 15 dB we further validate the results at
+different values. The evaluation is shown in Fig. 7. We see
+that the trained networks generalize well on previously unseen
+SNR values, and reach performance similar to the classical
+baseline.
+In Fig. 8, the constellation diagrams produced by autoen-
+coders having different numbers of layers are shown. If the
+trained autoencoder has good performance, it is expected that
+the symbols are uniformly distributed in the diagram, similarly
+to Fig 1. We see that increasing the number of layers leads to
+a more balanced distribution of symbols in the Q − I space,
+which implies that the symbols can be well separated in case
+of noisy channels.
+B. Time characteristics
+In radio telecommunication, the latency of the data trans-
+mission is also an important performance metric. In some use
+
+0
+200
+400
+600
+800
+1000
+Number of training steps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Symbol Error Rate
+L=8
+L=12
+L=16
+L=24
+classical
+Fig. 6: Learning curves of classical and hybrid autoen-
+coders for a set of layer numbers. We find that that
+the minimal number of layers necessary to achieve results
+comparable to the classical baseline is 16. Throughout these
+tests, we used ansatz according to Fig. 4d.
+cases it is even critical that the end-to-end delay falls below a
+certain threshold. In 5G networks, it is possible to achieve
+ms level latency. Hence, in addition to the accuracy it is
+inevitable to investigate the time characteristics of the autoen-
+coder model. After transpiling [30] the circuit ansatz to IBM
+QPU backend ibmq_belem and ibmq_santiago [31] and
+constructing the pulse-level scheduling, we can calculate the
+theoretical execution times on both QPUs. The transpiled
+circuits are deeper than the original ansatz, because we need
+SWAP gates due to limited qubit connectivity and the basis
+gate-set of the device can differ from the one used in Fig 4.
+In Table I, we present the circuit depth and the approximate
+per shot execution times of quantum decoders depending on
+the number of layers. The time values in the table suggest
+the following feasibility considerations for running QNN in a
+real-time system. The number of shots highly determines the
+reliability of the result of the inference. When the quantum
+decoder is executed with 1000 shots (a level already acceptable
+in current systems for this problem size), the inference time
+is the order of magnitude of 100ms which is higher than the
+accepted level in real-time radio systems. However, this can
+be reduced to the accepted level of below 10ms because the
+probability distribution is expected to be highly peaked for
+well-trained autoencoders.
+V. CONCLUSION AND OUTLOOK
+We presented a novel hybrid implementation of a quantum-
+classical autoencoder for end-to-end radio communication.
+The decoder was implemented as a variational quantum circuit.
+We showed that the use of advanced double re-uploading
+encoding schemes allows for the inference-time constraints of
+the application to be met without losing accuracy required
+from the autoencoder.
+By implementing a combination of parallel encodings and
+weighted data re-uploading, we showed how these schemes
+5
+0
+5
+10
+15
+20
+Signal-to-noise Ratio [dB]
+10
+2
+10
+1
+10
+0
+Symbol Error Rate
+L=8
+L=12
+L=16
+L=24
+classical
+Fig. 7: Validation of inference accuracy of the trained
+classical and hybrid autoencoders. With increasing SNR
+values, hybrid models generalize to validation data on par with
+the classical.
+I
+Q
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+(a)
+I
+Q
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+(b)
+Fig. 8: Constellations (latent space representations) learned
+by the hybrid autoencoder trained with SNR=15. (a) L = 8
+layers (b) L = 24 layers.
+TABLE I: Estimated execution times of the quantum
+decoder. The circuit was run on the ibmq_belem and
+ibmq_santiago depending on the number of layers, cal-
+culated with Qiskit’s transpiler.
+ibmq_belem
+ibmq_santiago
+# layers
+depth
+time [µs/shot]
+depth
+time [µs/shot]
+8
+125
+54.3
+145
+30.4
+12
+187
+78.4
+221
+43.6
+16
+260
+111.8
+297
+56.9
+20
+311
+124.2
+373
+70.12
+24
+379
+149.8
+449
+83.4
+can improve not just the QNN expressivity but also the
+performance of the whole autoencoder model. We expect these
+quantum-enhanced models to outperform classical ones in
+more complex channel noise scenarios, a direction for future
+study.
+
+ACKNOWLEDGMENT
+Zsolt Tabi and Zimbor´as Zolt´an would like to thank the
+support of the Hungarian National Research, Development and
+Innovation Office (NKFIH) through the Quantum Information
+National Laboratory of Hungary and through the Grants No.
+FK 135220, K124351 and TKP2021-NVA-29.
+REFERENCES
+[1] M. Schuld, I. Sinayskiy, and F. Petruccione, “The quest for a quantum
+neural network,” Quantum Information Processing, vol. 13, no. 11, pp.
+2567–2586, Nov 2014. https://doi.org/10.1007/s11128-014-0809-8
+[2] N.
+Killoran,
+T.
+R.
+Bromley,
+J.
+M.
+Arrazola,
+M.
+Schuld,
+N. Quesada, and S. Lloyd, “Continuous-variable quantum neural
+networks,”
+Phys. Rev.
+Research,
+vol.
+1,
+p.
+033063, Oct
+2019.
+https://link.aps.org/doi/10.1103/PhysRevResearch.1.033063
+[3] D. Wierichs, J. Izaac, C. Wang, and C. Y.-Y. Lin, “General parameter-
+shift rules for quantum gradients,” Quantum, vol. 6, p. 677, Mar. 2022.
+https://doi.org/10.22331/q-2022-03-30-677
+[4] V. Bergholm, J. Izaac, M. Schuld, C. Gogolin, M. Sohaib Alam,
+S. Ahmed et al., “Pennylane: Automatic differentiation of hybrid
+quantum-classical computations,” arXiv e-prints, p. arXiv:1811.04968,
+Nov. 2018.
+[5] S. Wei, Y. Chen, Z. Zhou, and G. Long, “A quantum convolutional
+neural network on nisq devices,” AAPPS Bulletin, vol. 32, no. 1, p. 2,
+Jan 2022. https://doi.org/10.1007/s43673-021-00030-3
+[6] S.
+Lloyd
+and
+C.
+Weedbrook,
+“Quantum
+generative
+adversarial
+learning,”
+Phys.
+Rev.
+Lett.,
+vol.
+121,
+p.
+040502,
+Jul
+2018.
+https://link.aps.org/doi/10.1103/PhysRevLett.121.040502
+[7] S. Y.-C. Chen, S. Yoo, and Y.-L. L. Fang, “Quantum long short-term
+memory,” in APS March Meeting Abstracts, ser. APS Meeting Abstracts,
+vol. 2021, Jan. 2021, p. V32.009.
+[8] C.-J. Huang, H. Ma, Q. Yin, J.-F. Tang, D. Dong, C. Chen et al.,
+“Realization of a quantum autoencoder for lossless compression
+of quantum data,” Phys. Rev. A, vol. 102, p. 032412, Sep 2020.
+https://link.aps.org/doi/10.1103/PhysRevA.102.032412
+[9] J.
+Romero,
+J.
+P.
+Olson,
+and
+A.
+Aspuru-Guzik,
+“Quantum
+autoencoders for efficient compression of quantum data,” Quantum
+Science and Technology, vol. 2, no. 4, p. 045001, aug 2017.
+https://doi.org/10.1088/2058-9565/aa8072
+[10] D. Bondarenko and P. Feldmann, “Quantum autoencoders to denoise
+quantum data,” Phys. Rev. Lett., vol. 124, p. 130502, Mar 2020.
+https://link.aps.org/doi/10.1103/PhysRevLett.124.130502
+[11] S. Chakrabarti, R. Krishnakumar, G. Mazzola, N. Stamatopoulos,
+S. Woerner, and W. J. Zeng, “A Threshold for Quantum Advantage
+in
+Derivative
+Pricing,”
+Quantum,
+vol.
+5,
+p.
+463,
+Jun.
+2021.
+https://doi.org/10.22331/q-2021-06-01-463
+[12] S. J. Nawaz, S. K. Sharma, S. Wyne, M. N. Patwary, and M. Asaduz-
+zaman, “Quantum machine learning for 6G communication networks:
+State-of-the-art and vision for the future,” IEEE Access, vol. 7, pp.
+46 317–46 350, 2019.
+[13] “6G
+–
+connecting
+a
+cyber-physical
+world
+-
+a
+re-
+search
+outlook
+toward
+2030,”
+Ericsson,
+Tech.
+Rep.,
+February 2022. https://www.ericsson.com/4927de/assets/local/reports-
+papers/white-papers/6g–connecting-a-cyber-physical-world.pdf
+[14] A. S. Tanenbaum and D. Wetherall, Computer networks, 5th Edition.
+Pearson, 2011. https://www.worldcat.org/oclc/698581231
+[15] T. O’Shea and J. Hoydis, “An introduction to deep learning for the
+physical layer,” IEEE Transactions on Cognitive Communications and
+Networking, vol. 3, no. 4, pp. 563–575, 2017.
+[16] D. Garc´ıa, J. O. Lacruz, D. Badini, D. De Donno, and J. Widmer,
+“Model-free machine learning of wireless siso/mimo communications,”
+Computer
+Communications,
+vol.
+181,
+pp.
+192–202,
+2022.
+https://www.sciencedirect.com/science/article/pii/S0140366421003704
+[17] G. E. Hinton and R. R. Salakhutdinov, “Reducing the dimensionality of
+data with neural networks,” Science, vol. 313, no. 5786, pp. 504–507,
+Jul. 2006. https://doi.org/10.1126/science.1127647
+[18] I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning. MIT Press,
+2016, http://www.deeplearningbook.org.
+[19] H. Neji, J. Nogueras-Iso, J. Lacasta, M. Ben Halima, and A. M. Alimi,
+“Adversarial autoencoders for denoising digitized historical documents:
+The use case of incunabula,” in 2019 International Conference on
+Document Analysis and Recognition Workshops (ICDARW), vol. 6, 2019,
+pp. 31–34.
+[20] E. Farhi and H. Neven, “Classification with quantum neural networks
+on near term processors,” arXiv: Quantum Physics, 2018.
+[21] A. Mari, T. R. Bromley, J. Izaac, M. Schuld, and N. Killoran, “Transfer
+learning in hybrid classical-quantum neural networks,” Quantum, vol. 4,
+p. 340, Oct. 2020.
+[22] M. Schuld and F. Petruccione, Machine Learning with Quantum
+Computers,
+ser.
+Quantum
+Science
+and
+Technology.
+Springer
+International
+Publishing,
+2021.
+https://books.google.hu/books?id=-
+N5IEAAAQBAJ
+[23] M. Schuld, A. Bocharov, K. M. Svore, and N. Wiebe, “Circuit-centric
+quantum classifiers,” Phys. Rev. A, vol. 101, p. 032308, Mar 2020.
+https://link.aps.org/doi/10.1103/PhysRevA.101.032308
+[24] A. P’erez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre,
+“Data re-uploading for a universal quantum classifier,” Quantum, vol. 4,
+p. 226, 2020.
+[25] M.
+Schuld,
+R.
+Sweke,
+and
+J.
+J.
+Meyer,
+“Effect
+of
+data
+encoding on the expressive power of variational quantum-machine-
+learning models,” Phys. Rev. A, vol. 103, p. 032430, Mar 2021.
+https://link.aps.org/doi/10.1103/PhysRevA.103.032430
+[26] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,”
+arXiv preprint arXiv:1412.6980, 2014.
+[27] V. Bergholm, J. A. Izaac, M. Schuld, C. Gogolin, and N. Killoran,
+“Pennylane: Automatic differentiation of hybrid quantum-classical com-
+putations,” ArXiv, vol. abs/1811.04968, 2018.
+[28] M.
+Abadi,
+A.
+Agarwal,
+P.
+Barham,
+E.
+Brevdo,
+Z.
+Chen,
+C.
+Citro
+et
+al.,
+“TensorFlow:
+Large-scale
+machine
+learning
+on
+heterogeneous systems,” 2015, software available from tensorflow.org.
+https://www.tensorflow.org/
+[29] T. O’Malley, E. Bursztein, J. Long, F. Chollet, H. Jin, L. Invernizzi et al.,
+“Kerastuner,” https://github.com/keras-team/keras-tuner, 2019.
+[30] M. S. ANIS, Abby-Mitchell, H. Abraham, AduOffei, R. Agarwal,
+G. Agliardi et al., “Qiskit: An open-source framework for quantum
+computing,” 2021.
+[31] “IBM quantum,” 2021. https://quantum-computing.ibm.com/
+
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+page_content='Hybrid Quantum-Classical Autoencoders for  End-to-End Radio Communication  Zsolt Tabi∗, Bence Bak´o∗†, D´aniel T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Nagy∗‡, P´eter Vaderna‡, Zs´ofia Kallus‡,  P´eter H´aga‡, Zolt´an Zimbor´as∗†  ∗E¨otv¨os Lor´and University, Budapest, Hungary  Email: zsolttabi@ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='elte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='hu  †Wigner Research Centre for Physics, Budapest, Hungary  Email: bako.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='bence@wigner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='hu, zimboras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='zoltan@wigner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='hu  ‡Ericsson Research, Budapest, Hungary  Email: {daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='nagy, peter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='vaderna, zsofia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='kallus, peter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='haga}@ericsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='com  Abstract—Quantum neural networks are emerging as poten-  tial candidates to leverage noisy quantum processing units for  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Here we introduce hybrid quantum-classical au-  toencoders for end-to-end radio communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' In the physical  layer of classical wireless systems, we study the performance  of simulated architectures for standard encoded radio signals  over a noisy channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' We implement a hybrid model, where  a quantum decoder in the receiver works with a classical  encoder in the transmitter part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Besides learning a latent space  representation of the input symbols with good robustness against  signal degradation, a generalized data re-uploading scheme for  the qubit-based circuits allows to meet inference-time constraints  of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Index Terms—variational quantum algorithms, quantum ma-  chine learning, quantum autoencoder, radio communication  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' INTRODUCTION  One of the most popular Quantum Machine Learning  (QML) methods are Quantum Neural Networks (QNNs) [1],  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' These are special variational quantum circuits, designed as  the quantum analogues of classical neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' QNNs can  be optimized with gradient-based or gradient-free optimization  algorithms forming hybrid quantum-classical training loops  [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Various QNN architectures have been proposed such  as quantum convolutional neural networks [5], generative mod-  els [6], long short-term memories [7] and autoencoders [8]–  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Beside the high activity in algorithmic research within  QML, their novel benchmarking and requirement setting ap-  plications are also motivating a wide variety of works [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Although QML is still in a phase of basic research with  many open questions, its early implementations in wireless  communication systems spark both scientific curiosity and  commercial interest [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' However, high-performance, near  real-time applications might impose a new set of requirements  on these solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Wireless communication has undergone tremendous evolu-  tion during the last decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The increasing adoption of AI  and ML methods is opening up new development possibilities  in various parts of the radio stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' In the design of the sixth  generation (6G) wireless networks, AI and ML technologies  are considered to be tightly integrated into the system and  smart algorithms can be applied to all aspects of network  operations and procedures [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Considering the improvement  of quantum computers it is envisioned that quantum algorithms  and especially QML will play a significant role in future  networks [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  The structure of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' II, we  give a high-level overview of wireless communication systems  together with an autoencoder solution used in the radio phys-  ical layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Our novel hybrid classical-quantum autoencoder  prototype is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' We discuss our result in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Finally, we conclude with an outlook in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' AUTOENCODER ARCHITECTURE IN END-TO-END  COMMUNICATION  I  Q  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  (a)  I  Q  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 1: 16-QAM constellation diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' (a) Transmitted  symbols;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' (b) Received noisy signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  s  Transmitter  Channel  Receiver  ˆs  x  y  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 2: High-level representation of a communication sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' A message is transmitted through a noisy communication  channel to be recovered by the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  The components of communication networks are organized  in a layered architecture where each layer is responsible for  different communication aspects [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The physical layer is the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='02609v1  [quant-ph]  6 Jan 2023  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 3: The proposed hybrid quantum-classical autoencoder embedded into the end-to-end communication architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  The transmitter maps each message s to a symbol, then sends it through the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The channel is represented by an  additive noise acting on the signal x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The receiver, realized by a quantum decoder, consists of a multi-layer QNN adapting  data re-uploading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' It decodes the noisy signal and gives an estimate of the original message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  lowest layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' It provides the means of transmitting a stream of  raw bits over a data channel connecting the network elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  The physical layer on the transmitter side converts data in  the form of bits to electromagnetic waves to be transmitted  wirelessly, while the receiver converts the electromagnetic  waves received by an antenna to binary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The main  challenge in wireless data transmission is to overcome the  channel impairments so that the messages can be recovered  with small error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  The role of modulation is to convert digital data into radio  waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' It can be achieved in different ways, the informa-  tion can be encoded by varying (shifting) either amplitude,  frequency or phase of the electromagnetic wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The more  states the modulation has, the more bits are transferred in one  symbol, resulting in higher data rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' However, with higher  order modulation the signal is more sensitive to channel errors,  so the applied modulation usually depends on the channel  quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 1 shows a constellation diagram of the symbol  representation of 16-QAM modulation, where 4-bit strings  can be represented as complex numbers in a scheme resistant  to general noise patterns while achieving high data rate with  minimal channel uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Based on [15], we model a simple communications system,  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 2, consisting of a transmitter that modulates  message s ∈ M = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' , M − 1} into a signal x and  sends it over a noisy channel to the receiver that tries to  decode the received signal y resulting in the received message  ˆs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The transmitted signal x suffers degradation due to the  noise present on the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' In case of transmission over a  complex channel with n discrete channel uses, the transmitter  can be represented as the transformation f : M �→ R2n,  mapping the message s to x ∈ R2n signal with certain  constraints imposed by the transmitting hardware (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=', energy  constraint or average power constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The channel can be  modeled as a conditional probability density function p(y|x)  that produces the output signal y ∈ R2n given the input signal,  usually via some noise model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=', additive white Gaussian  noise (AWGN)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The receiver is represented as the mapping  g : R2n �→ M that recovers some estimate ˆs of the original  message from the received signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' In this work, we focus on  the case of single channel use (n = 1), however, this model  can be easily adopted to cases of n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Also, the number of  transmit-receive pairs can be increased to get a Multiple-Input  and Multiple-Output (MIMO) system [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Both of f and g transformations can be created in various  ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' In case of simple noise models applied in the channel  the transformations can be designed as explicit mathematical  formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' In the case of complex noise scenarios that are  difficult to describe with mathematical models, a possible way  is to train deep neural networks, especially autoencoders, to  solve the encoding and decoding tasks [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  An autoencoder is a special type of deep neural network,  with the aim to compress or denoise data [17]–[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Autoen-  coders consist of an encoder function f : RD �→ RL, and  a decoder function g : RL �→ RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The encoder transforms  its input χ into a latent space representation f(χ) ∈ RL,  whereas the decoder tries to reconstruct it: ˆχ = g(f(χ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Usually we have L < D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=', the encoder produces a compact  representation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' f and g are typically deep neural  networks trained jointly to minimize a loss function of the  form L(χ, g(f(χ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  In telecommunication, opposite to the general compressing  and denoising interpretation, autoencoders can be effectively  used in the presented communication system to learn how to  represent input messages as signals [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' This model differs  from the “typical” autoencoder concept in the sense that it does  not try to remove noise from the input, instead it learns how  to represent the input in a way that is robust against a given  noisy channel acting in the latent space of the autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  As a result of the training process, the latent space (or hidden  layer) of the autoencoder contains the learned constellation of  symbols (or codebook).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The learned constellation is optimized  for best mapping of the input messages to signals that can be  accurately decoded with the largest success probability for the  Quantum Decoder  Encoder  cod  Noise model p(y|x)  argmax  Message vector  Qubit encoding  Qubit encoding  Normalization  ayer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1  abit readou  La 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='8  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='05  Transmitter  Receiver  hannelEncoding  First Layer  · ·  · ·  · ·  · ·  |0⟩  Rx(y1)  R(α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ1)  |0⟩  Rx(y2)  R(α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ2)  |0⟩  R(α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ3)  |0⟩  R(α4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ4)  (a)  First Layer with Encoding  · ·  · ·  · ·  · ·  |0⟩  Rx(y1)  R(α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ1)  |0⟩  Rx(y2)  R(α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ2)  |0⟩  R(α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ3)  |0⟩  R(α4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ4)  (b)  First Layer with Double Encoding  · ·  · ·  · ·  · ·  |0⟩  Rx(y1)  R(α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ1)  |0⟩  Rx(y2)  R(α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ2)  |0⟩  Rx(y1)  R(α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ3)  |0⟩  Rx(y2)  R(α4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ4)  (c)  First Layer with Weighted Double Encoding  · ·  · ·  · ·  · ·  |0⟩  Rx(w1 · y1)  R(α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ1)  |0⟩  Rx(w2 · y2)  R(α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ2)  |0⟩  Rx(w3 · y1)  R(α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ3)  |0⟩  Rx(w4 · y2)  R(α4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' β4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' γ4)  (d)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4: Quantum decoder implementations with encoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Ansatz circuits with (a) simple data encoding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' (b) simple  data re-uploading;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' (c) double data re-uploading;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' (d) weighted double data re-uploading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  specific channel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Whereas the encoder learns how to  produce optimal symbols, the receiver learns how to decode  these symbols after they have been corrupted by the channel,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=', how to recover x after sampling from p(y|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' HYBRID QUANTUM AUTOENCODER FOR RADIO  PHYSICAL LAYER  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Hybrid quantum autoencoder overview  Quantum autoencoder architectures have previously been  proposed to compress as well as denoise quantum data [8],  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Hybrid quantum-classical autoencoders enable many  variations for quantum or classical encoding/decoding or the  use of classical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' In this work, a hybrid quantum-classical  autoencoder is applied for processing classical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Building on the physical layer autoencoder presented in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' II, we propose a hybrid quantum-classical autoencoder  with classical encoder on the transmitter side and a quantum  decoder on the receiver side – trained in an end-to-end  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The encoder projects the original message to a lower  dimensional representation, robust to the channel degradation  effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Once the signal is passed to the quantum decoder, the  compressed information is mapped to a higher dimensional  Hilbert space of the qubits by a QNN that has been previously  shown to be efficient for classification tasks [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  In our model, the classical encoder consists of an embedding  followed by a normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' A simple linear embedding is  used to produce the constellation, satisfying the average power  constraint by normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The decoder is realized by a  general strongly connected quantum neural network which  we refer to as a quantum decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' By simulating increasing  levels of noise in the channel, we can present a performance  evaluation of the various neural network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Quantum decoder architectures  A general QNN architecture has three main components  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 3: qubit encoding for embedding the input  data, the parameterized QNN layers, and the qubit readout  given as a probability distribution over the possible con-  stellation symbols obtained from suitable measurements with  high enough number of shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' To encode the output of the  channel, we choose angle embedding with parameterized Rx  rotations [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' With this embedding, there are multiple ways  to encode two-dimensional feature vectors into four qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' As  for the variational ansatz, we use strongly entangling layers  introduced in quantum classifiers as they are known to be  expressive reaching ‘wide corners of the Hilbert space’ [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  The measurements are performed in the computational basis  and the obtained probability distribution over the 16 basis  states is the output of the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  The simplest single-layer realization of such a QNN struc-  ture is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' To improve this ansatz, we can  apply the data re-uploading trick recently introduced in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  This technique, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4b, repeats the input encoding  block before each layer of the QNN circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The intuition be-  hind the effectiveness of this method is that by re-introducing  the input before each layer, one can mimic the computational  structure of typical classical deep neural networks, where the  copying of the classical information is readily available, which  would be, without this trick, prohibited by the no-cloning  theorem in quantum machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The expressivity of a  model can be further increased by applying the encoding on  different subsystems in parallel [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' With this in mind, we  further enhance the ansatz by encoding the first feature into  both qubit no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 1 and no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 3 and the second input feature into  both qubit no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 2 and no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' This double data re-uploading  ansatz is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' As a final improvement, we  considered the role of the number of trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' As  the expressive power of the ansatz is highly dependent on the  number of trainable parameters, one should try to include as  many parameters as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' One way to increase the number  of parameters while keeping the circuit as shallow as possible  – to respect the limited hardware capabilities and the inference  time constraints of the application – is to introduce trainable  weights in the data re-uploading blocks, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  This modification keeps the depth constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Training and fine-tuning  For our hybrid autoencoder to achieve low estimation errors,  the training of the end-to-end system requires to be further  improved via hyper-parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  First, the training of the hybrid model is done on batches  uniformly sampled from the set of messages {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' , 15}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  These are sent as two dimensional encoded symbols through  the AWGN channel with SNR of 15 dB and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  The accuracy of the model is measured by evaluating  the Symbol Error Rate (SER), a key performance indica-  tor commonly used in radio communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The network  weight updates are calculated with the sparse categorical cross-  entropy of the distribution generated by the decoder and the  ground truth symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' This loss function is used to calculate  gradients in a mini-batch gradient descent with batch size of 64  and Adam optimizer [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' We simulate the hybrid autoencoder  using PennyLane [27], a quantum machine learning framework  with its TensorFlow [28] backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Second, we evaluate the  reached model accuracy at various hyper-parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  The search is conducted by KerasTuner [29] after partitioning  the space as the simulator compute times are prohibitive of a  full grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  We start by first evaluating the learning rate parameter set  η ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='001} using the simple ansatz presented on  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4a with L = 8 layers with 1000-shot measurements and  1000 training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Based on these results, the only viable  value of η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1 is set for the rest of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  We continue with evaluating modifications to the basic  ansatz but keeping the number of layers L = 8 and 1000  training steps fixed, to minimize the overall computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' For the basic circuit, the SER  fluctuates around its initial value without showing convergence  to a desirable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' A significant accuracy improvement of  roughly 40% is achieved by implementing single data re-  uploading (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4b with ansatz of 1×DR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Introducing the  double data re-uploading layer (2×DR with ansatz of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4c)  leads to another 15% improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Finally, we can even fur-  ther increase the performance by another 20% when using the  0  200  400  600  800  1000  Number of training steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='8  Symbol Error Rate  basic  1x DR  2x DR  2x wDR  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 5: Learning curves of circuit architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The number  of data re-uploading (none, single, double) and the weighted  data encoding have high impact on the convergence properties  of the quantum autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  weighted double data re-uploading technique (2×wDR with  ansatz Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Based on these results, the weighted double  data re-uploading ansatz is chosen for further experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  As a last step, we optimize the number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The hybrid  autoencoder using the best performing ansatz is trained with  8 to 24 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Increasing the number of layers clearly shows  the improvement in SER as well as in convergence time as  seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' PERFORMANCE EVALUATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Validation  Comparing our hybrid architecture to the classical method is  crucial to validate the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Based on the learning curves  presented, the shallowest network reaching accuracy similar  to the classical solution contains L = 16 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Further  increasing the number of layers leads to small improvements  in accuracy but it is suboptimal in terms of circuit depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Although the hybrid quantum autoencoder models are  trained at SNR of 15 dB we further validate the results at  different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The evaluation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' We see  that the trained networks generalize well on previously unseen  SNR values, and reach performance similar to the classical  baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 8, the constellation diagrams produced by autoen-  coders having different numbers of layers are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' If the  trained autoencoder has good performance, it is expected that  the symbols are uniformly distributed in the diagram, similarly  to Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' We see that increasing the number of layers leads to  a more balanced distribution of symbols in the Q − I space,  which implies that the symbols can be well separated in case  of noisy channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Time characteristics  In radio telecommunication, the latency of the data trans-  mission is also an important performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' In some use  0  200  400  600  800  1000  Number of training steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='7  Symbol Error Rate  L=8  L=12  L=16  L=24  classical  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 6: Learning curves of classical and hybrid autoen-  coders for a set of layer numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' We find that that  the minimal number of layers necessary to achieve results  comparable to the classical baseline is 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Throughout these  tests, we used ansatz according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  cases it is even critical that the end-to-end delay falls below a  certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' In 5G networks, it is possible to achieve  ms level latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Hence, in addition to the accuracy it is  inevitable to investigate the time characteristics of the autoen-  coder model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' After transpiling [30] the circuit ansatz to IBM  QPU backend ibmq_belem and ibmq_santiago [31] and  constructing the pulse-level scheduling, we can calculate the  theoretical execution times on both QPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The transpiled  circuits are deeper than the original ansatz, because we need  SWAP gates due to limited qubit connectivity and the basis  gate-set of the device can differ from the one used in Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  In Table I, we present the circuit depth and the approximate  per shot execution times of quantum decoders depending on  the number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The time values in the table suggest  the following feasibility considerations for running QNN in a  real-time system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The number of shots highly determines the  reliability of the result of the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' When the quantum  decoder is executed with 1000 shots (a level already acceptable  in current systems for this problem size), the inference time  is the order of magnitude of 100ms which is higher than the  accepted level in real-time radio systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' However, this can  be reduced to the accepted level of below 10ms because the  probability distribution is expected to be highly peaked for  well-trained autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' CONCLUSION AND OUTLOOK  We presented a novel hybrid implementation of a quantum-  classical autoencoder for end-to-end radio communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  The decoder was implemented as a variational quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  We showed that the use of advanced double re-uploading  encoding schemes allows for the inference-time constraints of  the application to be met without losing accuracy required  from the autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  By implementing a combination of parallel encodings and  weighted data re-uploading, we showed how these schemes  5  0  5  10  15  20  Signal-to-noise Ratio [dB]  10  2  10  1  10  0  Symbol Error Rate  L=8  L=12  L=16  L=24  classical  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 7: Validation of inference accuracy of the trained  classical and hybrid autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' With increasing SNR  values, hybrid models generalize to validation data on par with  the classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  I  Q  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  (a)  I  Q  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 8: Constellations (latent space representations) learned  by the hybrid autoencoder trained with SNR=15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' (a) L = 8  layers (b) L = 24 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  TABLE I: Estimated execution times of the quantum  decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' The circuit was run on the ibmq_belem and  ibmq_santiago depending on the number of layers, cal-  culated with Qiskit’s transpiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  ibmq_belem  ibmq_santiago  # layers  depth  time [µs/shot]  depth  time [µs/shot]  8  125  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='3  145  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='4  12  187  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='4  221  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='6  16  260  111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='8  297  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='9  20  311  124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='2  373  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='12  24  379  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='8  449  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='4  can improve not just the QNN expressivity but also the  performance of the whole autoencoder model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' We expect these  quantum-enhanced models to outperform classical ones in  more complex channel noise scenarios, a direction for future  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  ACKNOWLEDGMENT  Zsolt Tabi and Zimbor´as Zolt´an would like to thank the  support of the Hungarian National Research, Development and  Innovation Office (NKFIH) through the Quantum Information  National Laboratory of Hungary and through the Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  FK 135220, K124351 and TKP2021-NVA-29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  REFERENCES  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Schuld, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Sinayskiy, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Petruccione, “The quest for a quantum  neural network,” Quantum Information Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  2567–2586, Nov 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1007/s11128-014-0809-8  [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Killoran,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Bromley,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Arrazola,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Schuld,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Quesada, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Lloyd, “Continuous-variable quantum neural  networks,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Research,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  1,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  033063, Oct  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1103/PhysRevResearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='033063  [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Wierichs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Izaac, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Wang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Lin, “General parameter-  shift rules for quantum gradients,” Quantum, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 677, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='22331/q-2022-03-30-677  [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Bergholm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Izaac, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Schuld, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Gogolin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Sohaib Alam,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=', “Pennylane: Automatic differentiation of hybrid  quantum-classical computations,” arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='04968,  Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Wei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Zhou, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Long, “A quantum convolutional  neural network on nisq devices,” AAPPS Bulletin, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 2,  Jan 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1007/s43673-021-00030-3  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Lloyd  and  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Weedbrook,  “Quantum  generative  adversarial  learning,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=',  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  121,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  040502,  Jul  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='040502  [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Yoo, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Fang, “Quantum long short-term  memory,” in APS March Meeting Abstracts, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' APS Meeting Abstracts,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 2021, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 2021, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' V32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Ma, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Yin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Tang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Dong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=',  “Realization of a quantum autoencoder for lossless compression  of quantum data,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 102, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 032412, Sep 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='032412  [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Romero,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Olson,  and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Aspuru-Guzik,  “Quantum  autoencoders for efficient compression of quantum data,” Quantum  Science and Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 045001, aug 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1088/2058-9565/aa8072  [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Bondarenko and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Feldmann, “Quantum autoencoders to denoise  quantum data,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 124, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 130502, Mar 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='130502  [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Chakrabarti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Krishnakumar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Mazzola, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Stamatopoulos,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Woerner, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Zeng, “A Threshold for Quantum Advantage  in  Derivative  Pricing,”  Quantum,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  5,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  463,  Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='22331/q-2021-06-01-463  [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Nawaz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Sharma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Wyne, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Patwary, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Asaduz-  zaman, “Quantum machine learning for 6G communication networks:  State-of-the-art and vision for the future,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  46 317–46 350, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [13] “6G  –  connecting  a  cyber-physical  world a  re-  search  outlook  toward  2030,”  Ericsson,  Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=',  February 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='ericsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='com/4927de/assets/local/reports-  papers/white-papers/6g–connecting-a-cyber-physical-world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='pdf  [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Tanenbaum and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Wetherall, Computer networks, 5th Edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Pearson, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='worldcat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/oclc/698581231  [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' O’Shea and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Hoydis, “An introduction to deep learning for the  physical layer,” IEEE Transactions on Cognitive Communications and  Networking, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 563–575, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Garc´ıa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Lacruz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Badini, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' De Donno, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Widmer,  “Model-free machine learning of wireless siso/mimo communications,”  Computer  Communications,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  181,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  192–202,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='com/science/article/pii/S0140366421003704  [17] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Hinton and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Salakhutdinov, “Reducing the dimensionality of  data with neural networks,” Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 313, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 5786, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 504–507,  Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1127647  [18] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Goodfellow, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Bengio, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Courville, Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' MIT Press,  2016, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='deeplearningbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [19] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Neji, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Nogueras-Iso, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Lacasta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Ben Halima, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Alimi,  “Adversarial autoencoders for denoising digitized historical documents:  The use case of incunabula,” in 2019 International Conference on  Document Analysis and Recognition Workshops (ICDARW), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 6, 2019,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 31–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [20] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Farhi and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Neven, “Classification with quantum neural networks  on near term processors,” arXiv: Quantum Physics, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Mari, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Bromley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Izaac, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Schuld, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Killoran, “Transfer  learning in hybrid classical-quantum neural networks,” Quantum, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 340, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Schuld and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Petruccione, Machine Learning with Quantum  Computers,  ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Quantum  Science  and  Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Springer  International  Publishing,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  https://books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='hu/books?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='id=-  N5IEAAAQBAJ  [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Schuld, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Bocharov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Svore, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Wiebe, “Circuit-centric  quantum classifiers,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 101, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 032308, Mar 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='032308  [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' P’erez-Salinas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Cervera-Lierta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Gil-Fuster, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Latorre,  “Data re-uploading for a universal quantum classifier,” Quantum, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 4,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 226, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Schuld,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Sweke,  and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Meyer,  “Effect  of  data  encoding on the expressive power of variational quantum-machine-  learning models,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 103, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' 032430, Mar 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='032430  [26] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Kingma and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Ba, “Adam: A method for stochastic optimization,”  arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='6980, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [27] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Bergholm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Izaac, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Schuld, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Gogolin, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Killoran,  “Pennylane: Automatic differentiation of hybrid quantum-classical com-  putations,” ArXiv, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' abs/1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='04968, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Abadi,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Agarwal,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Barham,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Brevdo,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Chen,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  Citro  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=',  “TensorFlow:  Large-scale  machine  learning  on  heterogeneous systems,” 2015, software available from tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='org/  [29] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' O’Malley, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Bursztein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Long, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Chollet, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Jin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Invernizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=',  “Kerastuner,” https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='com/keras-team/keras-tuner, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' ANIS, Abby-Mitchell, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Abraham, AduOffei, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Agarwal,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' Agliardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=', “Qiskit: An open-source framework for quantum  computing,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='  [31] “IBM quantum,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content=' https://quantum-computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
+page_content='com/' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfuwF7/content/2301.02609v1.pdf'}
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+arXiv:2301.02618v1  [math.RT]  6 Jan 2023
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Abstract. We calculate the dg algebra of global functions on commuting stacks of complex reductive
+groups using tools from Betti Geometric Langlands. In particular, we prove that the ring of invariant
+functions on the commuting scheme is reduced. Our main technical results include: a semi-orthogonal
+decomposition of the cocenter of the affine Hecke category; and the calculation of endomorphisms of a
+Whittaker sheaf in a diagram organizing parabolic induction of character sheaves.
+Contents
+1.
+Introduction
+2
+1.1.
+Application to commuting stacks
+2
+1.2.
+Background: universal affine Hecke category and its cocenter
+3
+1.3.
+Main automorphic results
+5
+1.4.
+Parabolic character sheaves and semi-orthogonal decomposition of the cocenter
+7
+1.5.
+Further results
+8
+1.6.
+Conventions
+9
+1.7.
+Acknowledgements
+9
+2.
+Universal affine Hecke category
+10
+2.1.
+Hecke categories
+10
+2.2.
+Whittaker objects
+12
+2.3.
+Langlands duality
+14
+2.4.
+Coxeter presentation
+15
+2.5.
+Hochschild homology and cocenters
+15
+2.6.
+Spectral realization
+18
+2.7.
+Automorphic realization
+21
+2.8.
+Descended trace of Whittaker object
+25
+3.
+Horocycle descent
+29
+3.1.
+Preliminaries
+29
+3.2.
+Descent for smooth stacks
+30
+3.3.
+Singular and ind-version
+36
+4.
+Harder-Narasimhan subcategories
+37
+4.1.
+Combinatorial pieces
+37
+4.2.
+The space B
+42
+4.3.
+Geometric pieces and sheaves on them
+47
+4.4.
+Semi-orthogonal decomposition of the cocenter
+53
+5.
+Endomorphisms of Whittaker functional
+59
+5.1.
+Combinatorial descriptions of character sheaves
+60
+5.2.
+Fourier-Sato transform and Whittaker sheaf
+64
+5.3.
+Calculation of endormorphisms of the Whittaker sheaf
+67
+5.4.
+Additional applications
+68
+6.
+Functions on the commuting stack
+72
+6.1.
+Almost commuting pairs of semisimple elements
+72
+6.2.
+Chevalley restriction theorem for the commuting stack
+73
+Date: January 9, 2023.
+1
+
+2
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+6.3.
+From groups to Lie algebras
+75
+Appendix A.
+Calculating some colimits
+78
+A.1.
+Recollements and stratifications
+78
+A.2.
+Interaction of colimits and recollement
+80
+A.3.
+Posets and cosheaves
+81
+A.4.
+Expansion of cosheaves
+84
+A.5.
+A categorical contraction principle
+86
+Appendix B.
+Functoriality of sheaves with Lagrangian singular support
+89
+Appendix C.
+Universal version of Tao-Travkin Theorem
+91
+References
+92
+1. Introduction
+This paper is part of a broader study of the cocenter of the universal affine Hecke category. Its main
+results include (i) a semi-orthogonal decomposition of the cocenter, as found in Theorem 1.4.4, whose
+distinguished case spelled out in Theorem 1.3.4 proves a conjecture of [LN21]; and (ii) the calculation of
+endomorphisms of a distinguished Whittaker object in the cocenter, as found in Theorem 1.3.6, which
+builds on work of [Lia]. In a sequel, we will combine the results of this paper with those of [NYa] to
+identify the cocenter of the universal affine Hecke category with the genus one automorphic category in
+Betti Geometric Langlands, proving the Betti Geometric Langlands Conjecture in genus one. But already
+a concrete output of the results proved here is a calculation of the dg algebra of global functions on the
+commuting stack of a reductive group. We will first explain this application, then turn to more details of
+the main results of the paper.
+1.1. Application to commuting stacks.
+1.1.1. Classical commuting stacks. Let us start with the underived statements.
+Let G∨ be a connected reductive group over the complex numbers C.
+Let C2
+G∨ be the commuting
+scheme of G∨, the closed subscheme of G∨ × G∨ consisting of (g1, g2) ∈ G∨ × G∨ satisfying the equation
+g1g2g−1
+1 g−1
+2
+= 1. The group G∨ acts on C2
+G∨ by simultaneous conjugation.
+Following [BFM], C2
+G∨ decomposes into open and closed subschemes indexed by π1(G∨,der), the funda-
+mental group of the derived group G∨,der of G∨. For each c ∈ π1(G∨,der), one defines a torus T ∨
+c , as the
+abelianization of a Levi subgroup L∨
+c of G∨, and a map of stacks
+ιc : (T ∨
+c × T ∨
+c )/Wc → C2
+G∨/G∨
+where Wc = NG∨(L∨
+c )/L∨
+c is the relative Weyl group of L∨
+c . When c = 1 ∈ π1(G∨,der) is the identity,
+T ∨
+c = T ∨ is a maximal torus of G∨, and Wc is the usual Weyl group W = W(G∨, T ∨). For more details
+of this construction, we refer to Section 6.1.
+1.1.2. Theorem (See Theorem 6.2.1). Let G∨ be a connected reductive group over C. Assume Ansatz 1.2.5
+holds.
+Then the maps ιc for c ∈ π1(G∨,der) induce an isomorphism on rings of invariant functions
+O(C2
+G∨)G∨ ≃
+�
+c∈π1(G∨,der)
+O(T ∨
+c × T ∨
+c )Wc.
+In particular, O(C2
+G∨)G∨ is reduced.
+1.1.3. Remark. The Ansatz 1.2.5 that we assume in Theorem 1.1.2 is a universal monodromic version of
+Bezrukavnikov’s equivalence [Bez16] in the Betti setting, which we expect to be proved by similar methods
+to [Bez16].
+From Theorem 1.1.2, we also deduce the following description of invariant functions on the Lie algebra
+commuting scheme and Lie algebra-Lie group commuting scheme.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+3
+1.1.4. Theorem (See Theorem 6.3.1 and Theorem 6.3.3). Let G∨ be a connected reductive group over C
+with maximal torus T ∨ ⊂ G∨, and respective Lie algebras t∨ ⊂ g∨. Assume Ansatz 1.2.5 holds.
+Let C2
+g∨ be the Lie algebra commuting scheme of pairs (X1, X2) ∈ g∨ × g∨ satisfying adX1(X2) = 0, and
+Cg∨,G∨ the Lie algebra-Lie group commuting scheme of pairs (X, g) ∈ g∨ × G∨ satisfying Adg(X) = X.
+Then restrictions to t∨ × t∨ and t∨ × T ∨ respectively give isomorphisms on rings of invariant functions
+O(C2
+g∨)G∨
+∼
+→ O(t∨ × t∨)W ,
+O(Cg∨,G∨)G∨
+∼
+→ O(t∨ × T ∨)W .
+In fact, it is possible to adapt our methods and prove Theorem 1.1.4 directly. With this approach, we
+do not need to assume Ansatz 1.2.5, but can simply appeal to Bezrukavnikov’s equivalence [Bez16]. Since
+we do not take this approach in this paper, we include Ansatz 1.2.5 in the statement of Theorem 1.1.4.
+1.1.5. Derived commuting stack. We deduce Theorem 1.1.2 from a derived statement which we present
+next.
+The derived commuting stack Z2
+G∨ is the derived moduli of pairs g1, g2 ∈ G∨ with g1g2g−1
+1 g−1
+2
+= 1 up
+to conjugation. More formally, it is the adjoint quotient of the derived fiber product
+Z2
+G∨ ≃ ((G∨ × G∨) ×R
+G∨ {1})/G∨
+with respect to the commutator map G∨ × G∨ → G∨, (g1, g2) �→ g1g2g−1
+1 g−1
+2
+and unit element 1 ∈ G∨.
+The underlying classical stack of Z2
+G∨ is C2
+G∨/G∨. More geometrically, it is the moduli
+Z2
+G∨ ≃ LocG∨(T 2)
+of G∨-local systems on the two-torus T 2, where g1, g2 are the monodromies around the two factors.
+We describe the entire dg algebra of derived global functions on Z2
+G∨.
+1.1.6. Theorem (Corollary 6.2.5). Let G∨ be a connected reductive group over C. Assume Ansatz 1.2.5
+holds.
+Then there is an equivalence of dg algebras
+O(Z2
+G∨) ≃
+�
+c∈π1(G∨,der)
+O(Z2
+T ∨
+c )Wc ≃
+�
+c∈π1(G∨,der)
+O(T ∨
+c × T ∨
+c × t∨
+c [−1])Wc.
+We explain some notation in the theorem. Here t∨
+c = LieT ∨
+c . The notation t∨
+c [−1] denotes the affine
+derived scheme with coordinate ring O(t∨
+c [−1]) equal to the exterior algebra Sym∗((t∨
+c )∗[1]) generated in
+degree −1.
+In particular, when G∨,der is simply-connected, we have a simple equality
+O(Z2
+G∨) ≃ O(T ∨ × T ∨ × t∨[−1])W .
+In this case, the above isomorphism was conjectured in [BRY22, Conjecture 1].
+1.2. Background: universal affine Hecke category and its cocenter. Although Theorem 1.1.6 only
+involves G∨, our proof focuses on the Langlands dual group G and in particular its loop group.
+Throughout the remainder of the introduction, we will make the simplifying assumption that G is
+almost simple and simply-connected. Most results we state here are proved for general reductive G either
+in the literature or in this paper.
+Below we summarize known results about the affine Hecke category (or variants thereof) that will be
+used in this paper.
+1.2.1. Universal affine Hecke category. Let B ⊂ G be a Borel subgroup, U ⊂ B its unipotent radical, and
+H = B/U the universal Cartan. Let G = G((t)) be the loop group, I ⊂ G the standard Iwahori subgroup
+corresponding to B, Iu ⊂ I its pro-unipotent radical, and note H ≃ I/Iu.
+Fix a maximal torus T ⊂ B ⊂ G. Let W = NG(T )/T be the Weyl group of G, X∗(T ) = Hom(Gm, T )
+the coweight lattice of T , and �
+W = NG(T )/T [[t]] ≃ X∗(H) ⋊ W the extended affine Weyl group of G. Let
+W a ⊂ �
+W be the affine Weyl group generated by affine simple reflections, and set Ω = �
+W/W a ∼= NG(I)/I.
+
+4
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+The universal finite Hecke category of G (resp. universal affine Hecke category of G) is the convolution
+monoidal dg derived category of H-bimonodromic complexes of sheaves of C-modules
+HG = Shbimon(U\G/U)
+(resp. HG = colimw∈�
+W Shbimon(Iu\G≤w/Iu) )
+where the latter is with respect to the natural ind-scheme structure G = colimw∈�
+W G≤w.
+1.2.2. Cocenter. We will regard HG and HG as algebra objects in C-linear stable presentable ∞-categories
+StL
+C (where morphisms are left adjoints) and make constructions therein. It is also sometimes useful to re-
+gard HG and HG as algebra objects in the C-linear stable presentable bimodule ∞-category BimodHH(StL
+C)
+where the finite Hecke category HH = Shmon(H) for the universal Cartan naturally acts by left and right
+convolution. We will formulate our results in the absolute setting, but most hold in this relative setting
+as well; in fact, certain proofs are made easier by shifting to the relative setting.
+Recall the cocenter of a monoidal category A is the Hochschild homology category
+hh(A) = A ⊗A⊗Aop A
+The trace map is the natural projection
+tr : A
+� A ⊗A⊗Aop A = hh(A)
+induced by the unit of A in the first factor of the tensor.
+The closed embedding G ⊂ G induces a natural fully faithful monoidal functor HG → HG, and in turn
+a natural map on cocenters we will denote by
+(1.2.1)
+a : hh(HG)
+� hh(HG).
+As a special case of a more general coequalization result in Section 3, we show for the universal finite
+Hecke category HG, there is a natural identification of its cocenter (see Theorem 2.7.2)
+(1.2.2)
+hh(HG) ≃ ShN (G/G)
+where ShN (G/G) is the dg derived category of complexes of sheaves of C-modules with nilpotent singular
+support on the adjoint-quotient G/G. One can think of ShN (G/G) as a version of character sheaves on
+G.
+A guiding goal is to arrive at a similar geometric description of the cocenter hh(HG) of the universal
+affine Hecke category HG. As a starting point, we will use a universal monodromic version of a result of
+Tao-Travkin [TT]. In Appendix C, we provide the necessary extensions to apply their arguments to the
+universal monodromic case.
+Let I be the set of simple roots of G and Ia be the set of affine simple roots of G. For each proper
+subset J ⫋ Ia, let LJ ⊂ G be the corresponding Levihoric, and HLJ ⊂ HG the universal finite Hecke
+category of LJ. For J ⊂ J′ ⫋ Ia, we have a natural diagram of monoidal inclusions HLJ ⊂ HLJ′ .
+1.2.3. Theorem (Tao-Travkin [TT], see Appendix C). Assume G is simply-connected. Then the natural
+maps induce a monoidal equivalence
+colim⊗
+J⫋Ia HLJ
+∼
+� HG
+where the colimit is of monoidal categories.
+1.2.4. Spectral realization. To deduce spectral consequences of our automorphic results, we will use a
+universal version of a result of Bezrukavnikov [Bez16]. Since it is not yet in the literature, we state it here
+as an Ansatz; we will provide a proof in a sequel.
+Given a Borel subgroup B∨ ⊂ G∨, the universal Steinberg stack is the fiber product of adjoint quotients
+StG∨ = B∨/B∨ ×G∨/G∨ B∨/B∨
+(Here the derived and naive fiber product agree.) More geometrically, it is the moduli
+StG∨ ≃ LocG∨,B∨(S1 × [0, 1], S1 × {0, 1})
+of G∨-local systems on the cylinder S1 × [0, 1] with B∨-reductions along the boundary S1 × {0, 1}
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+5
+The universal spectral affine Hecke category is the monoidal convolution category IndCoh(StG∨) of
+ind-coherent sheaves on the universal Steinberg stack.
+1.2.5. Ansatz. There is a monoidal equivalence of universal affine Hecke categories
+Φ : IndCoh(StG∨)
+∼
+� HG
+with the following properties:
+(1) Φ identifies the structure sheaf OStG∨ with the universal affine Whittaker object WhG as coalgebra
+objects. (See Section 2.6.6 for the coalgebra structure on OStG∨ , and Section 2.2.7 for the definition
+of WhG and its coalgebra structure.)
+(2) Φ is naturally an equivalence of algebras in bimodules over QC(H∨). (Here we regard the HH-
+bimodule HG as a QC(H∨)-bimodule using the canonical monoidal equivalence HH ≃ QC(H∨).)
+1.2.6. Remark. Ansatz 1.2.5 (2) is not used in this paper, we only record it for conceptual completeness.
+With this in hand, we can invoke the spectral calculations of [BNP17] to conclude:
+1.2.7. Theorem. Assuming Ansatz (1.2.5), there is a canonical equivalence
+(1.2.3)
+IndCohN (Z2
+G∨) ≃ hh(HG).
+1.3. Main automorphic results. Now we formulate our main results regarding the cocenter hh(HG)
+that will lead to a proof of Theorem 1.1.6.
+1.3.1. Whittaker objects. By the Whittaker functional on ShN (G/G), we mean the functor
+WG/G : ShN (G/G)
+� ModC
+WG(F) = ϕ0χ∗r!(F)
+given by the right-adjoint transport χ∗r! across the Whittaker correspondence
+G/G
+U/U
+r
+�
+χ
+� A1
+followed by vanishing cycles ϕ0 for the coordinate function on A1.
+Here χ is induced from a generic
+character U → U/[U, U] ≃ ⊕i∈IA1
+i → A1
+By the Whittaker object WhG/G ∈ ShN (G/G), we mean the object corepresenting WG/G in the sense
+of a natural equivalence WG/G(F) ≃ Hom(WhG/G, F) for F ∈ ShN (G/G).
+Recall the natural maps a : ShN (G/G) ≃ hh(HG) → hh(HG) in (1.2.1) and (1.2.2). For the purpose of
+introduction, we define the cocenter Whittaker object WhG/G as
+(1.3.1)
+WhG/G := a(WhG/G) ∈ hh(HG).
+The actual definition of WhG/G in the main text uses descended trace, which makes the above identity a
+nontrivial theorem (see Theorem 2.8.5).
+1.3.2. Theorem (See Corollary 2.8.7). Assume Ansatz (1.2.5). Then under the equivalence (1.2.3), the
+structure sheaf OZ2
+G∨ corresponds to the cocenter Whittaker object WhG/G.
+Using this theorem, to calculate the dg algebra O(Z2
+G∨), which is the derived endomorphism ring of
+OZ2
+G∨ , it suffices to calculate the derived endomorphism ring of WhG/G.
+1.3.3. Fully faithful embedding from colimit of character sheaves to affine cocenter. To calculate the derived
+endomorphism ring of WhG/G), we will identify a full subcategory of hh(HG) containing WhG/G, in which
+the endomorphism ring is easier to compute.
+Recall the notion of the cocenter of a monoidal category A generalizes to the Hochschild homology
+category of any A-bimodule M. One defines
+hh(A, M) = A ⊗A⊗Aop M
+with trace map the natural projection
+tr : M
+� A ⊗A⊗Aop M = hh(A, M)
+
+6
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+induced by the unit of A in the first factor of the tensor. So the cocenter hh(A) is the Hochschild homology
+category of the regular A-bimodule category A.
+To begin, from Theorem 1.2.3, for the bimodule category HG, we obtain an equivalence after passing
+to Hochschild homology categories
+colimJ⫋Ia hh(HLJ, HG)
+∼
+� hh(HG, HG) = hh(HG).
+We can restrict the domain of this equivalence to obtain a functor relating cocenters
+colimJ⫋Ia hh(HLJ) = colimJ⫋Ia hh(HLJ, HLJ)
+� hh(HG, HG) = hh(HG).
+Our first main result is the following theorem conjectured in [LN21, see Claim 1.12].
+1.3.4. Theorem. Recall G is assumed to be almost simple and simply-connected for simplicity. The natural
+map is a fully faithful embedding
+colimJ⫋Ia hh(HLJ) = colimJ⫋Ia hh(HLJ, HLJ)� �
+� hh(HG, HG) = hh(HG).
+In fact, Theorem 1.3.4 is the “semistable” part of a more general result that describes a semi-orthogonal
+decomposition of hh(HG) indexed by “Harder-Narasimhan” strata. The generalization is not needed for
+the specific applications of this paper, but is a key input to the work in progress [LNY]. We will give a
+precise statement of the semi-orthogonal decomposition in Section 1.4.
+Recall WhG/G is the image of WhG/G ∈ ShN (G/G) ≃ hh(HG) under the map a defined in (1.2.1). Note
+G = LI, for I ⊂ Ia the finite simple roots. Let us write WhG,I for the image of WhG/G under the natural
+map
+ShN (G/G) ≃ hh(HG) = hh(HLI)
+� colimJ⫋Ia hh(HLJ) ≃ colimJ⫋Ia ShN (LJ/LJ).
+By Theorem 1.3.4, to prove Theorem 1.1.6, remains to calculate the derived endomorphism ring of
+WhG,I as an object in colimJ⫋Ia ShN (LJ/LJ).
+1.3.5. Calculation of endomorphisms of WhG,I. To state the result of the calculation of End(WhG,I), we
+need to recall some constructions from generalized Springer theory.
+Let Z(G) be the center of G, and Irr(Z(G)) be the group of irreducible complex characters of Z(G).
+To each χ ∈ Irr(Z(G)/Z0(G)), under the generalized Springer correspondence, the local system on the
+regular nilpotent orbit of G with central character χ appears in the parabolic induction of a cuspidal local
+system on a Levi subgroup Lχ ⊂ G, unique up to conjugation. Set Tχ to be the connected center of Lχ
+with Lie algebra tχ, and let T ∨
+χ denote the dual torus with Lie algebra t∨
+χ. Set Wχ = NG(Lχ)/Lχ to be
+the relative Weyl group which acts on Tχ and tχ, hence also on T ∨
+χ and t∨
+χ.
+1.3.6. Theorem (See Theorem 5.3.1). There is a canonical equivalence of dg algebras
+End(WhG,I) ≃ ⊕χ∈Irr(Z(G))O(T ∨
+χ × T ∨
+χ × t∨
+χ[−1])Wχ.
+Note that Irr(Z(G)) can be canonically identified with π1(G∨), therefore the index set of the above
+decomposition can be replaced with π1(G∨), which is what appeared in Theorem 1.1.6.
+The proof of the theorem uses generalized Springer theory and the decomposition of character sheaves
+of [Lia].
+Theorems (1.3.4) and 1.3.6 together imply an unconditional description of the derived endomorphism
+ring of the cocenter Whittaker object WhG/G ∈ hh(HG).
+1.3.7. Theorem (See Corollary 5.3.2). There is an equivalence of dg algebras
+End(WhG/G) ≃ ⊕χ∈Irr(Z(G))O(T ∨
+χ × T ∨
+χ × t∨
+χ[−1])Wχ.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+7
+1.4. Parabolic character sheaves and semi-orthogonal decomposition of the cocenter. We give
+more details that lead to the proof of Theorem 1.3.4. The results we are about to present here are not used
+in full strength in the proof of Theorem 1.3.4, but they will be used in future work to identify the affine
+cocenter with the genus one Betti geometric Langlands automorphic category, and in doing so, establish
+Betti geometric Langlands in genus one.
+For simplicity, we will continue to assume here G is almost simple and simply-connected.
+1.4.1. Parabolic character sheaves and Hochschild homology. In Section 3, we give a geometric realization
+of the Hochschild homology hh(HLJ, HG) in terms of Lusztig’s theory of parabolic (or rather parahoric)
+character sheaves.
+For J ⫋ Ia, let PJ ⊂ G be the standard parahoric subgroup with Levi quotient LJ, and Pu
+J ⊂ PJ its
+unipotent radical. Let HG,J be the full subcategory
+HG,J = ShN
+�Pu
+J \G/Pu
+J
+Ad(LJ)
+�
+where ShN (−) means sheaves whose singular support is nilpotent under the moment map for the left (or
+right) LJ-action when pulled back to Pu
+J \G/Pu
+J. Then HG,J can be viewed as a Betti version of Lusztig’s
+parabolic character sheaves for loop groups.
+1.4.2. Theorem (see Theorem 2.7.10). For J ⫋ Ia, there is a canonical equivalence
+hh(HLJ, HG) ≃ HG,J.
+Moreover, for J ⊂ J′ ⫋ Ia, the natural functor hh(HLJ, HG) → hh(HL′
+J, HG) gets transported under
+the above equivalence to the functor HG,J → HG,J′ given by a natural horocycle correspondence.
+Under the identifications in the theorem, the diagram of full subcategories
+J ⫋ Ia �→ hh(HLJ, HLJ) = ShN (LJ/LJ)
+is identified with the full subcategory of HG,J of sheaves supported on Pu
+J \PJ/Pu
+J
+LJ
+, which is essentially the
+adjoint quotient LJ/LJ. The transition functors for J ⊂ J′ are given by parabolic induction.
+Theorem 1.4.2 is a consequence of a very general result we prove in Section 3, which says that for
+very general HL-bimodules M coming from geometry (where L is a reductive group), the trace map
+M → hh(HL, M) can be geometrically realized as the pull-push functor along a horocycle correspondence.
+1.4.3. Semi-orthogonal decomposition of the cocenter. To state the semi-orthogonal decomposition of hh(HG),
+we need the notion of Newton points. First, there is a Newton point map ν : W a → X∗(T )+
+Q from the
+affine Weyl group W a to rational dominant coweights: for any w ∈ W a, and sufficiently divisible n, we
+have wn ∈ X∗(T ), and set ν(w) ∈ X∗(T )+
+Q to be the rational dominant coweight so that nν(w) and wn
+are in the same W-orbit. The Newton point map ν is invariant under conjugation by W a; we denote by
+NP ⊂ X∗(T )+
+Q its image. Note NP has a natural positive coroot partial ordering but we will work with a
+coarser linear order.
+Next, to each Newton point ν ∈ NP, we associate a finite simplicial complex1 B♥
+ν . For example, when
+ν = 0, B♥
+0 recovers the fundamental alcove of A. For each facet σ of B♥
+ν , we attach a category of (possibly
+twisted) character sheaves ShN (Yν,σ). Together they form a cosheaf of categories on the poset opposite
+to the set of facets of B♥
+ν , where the transition functors are given by (twisted) parabolic induction. For
+example, over B♥
+0 , we recover the cosheaf of categories J �→ ShN (LJ/LJ) for J ⫋ Ia, with transition
+maps given by the usual parabolic induction.
+1.4.4. Theorem. The cocenter category hh(HG) has a semi-orthogonal decomposition indexed by non-
+negative integers n ≥ 0 with the n-th associated graded category of the form
+hh(HG)n =
+�
+ν∈NP,⟨2ρ,ν⟩=n
+hh(HG)ν
+1Strictly speaking, B♥
+ν may only be a simplicial complex after a barycentric subdivision: as naturally constructed, the
+intersection of two simplices in B♥
+ν may be a union of more than one simplex.
+
+8
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+and
+hh(HG)ν ≃ colimσ⊂B♥
+ν ShN (Yν,σ)
+In particular, the first filtered piece hh(HG)0 is a full subcategory of hh(HG), and the theorem identifies
+it with the colimit colimJ⫋Ia ShN (LJ/LJ). This leads to the fully faithfulness asserted in Theorem 1.3.4.
+Theorem 1.4.4 is a categorification of a result of Xuhua He [He18] where, among other things, he gave
+a decomposition of the cocenter of the affine Hecke algebra indexed by Newton points.
+1.4.5. Idea of proof of Theorem 1.4.4. The essential idea of the proof of Theorem 1.4.4 is to perform a
+categorical version of Morse theory on a cosheaf on a certain topological space Bν (or rather the poset of
+its facets) that encodes the combinatorics of conjugacy classes in the affine Weyl group W a.
+We sketch the construction of Bν. Let A be the apartment associated to T in the building of G. We
+regard A as a labelled simplicial complex, with each facet labelled by its type J ⊊ Ia. We view the
+labelling as a simplicial map A → A/W a ≃ ∆ where ∆ is the fundamental alcove whose facets are in
+bijection with J ⫋ Ia.
+Given a conjugacy class [w] ⊂ W a of an element w ∈ W a, with centralizer Cw ⊂ W a, we may identify
+[w] ≃ W a/Cw with the open alcoves (open facets, or equivalently, J-facets with J = ∅) in the quotient
+space X[w] = A/Cw which depends only on the conjugacy class [w]. In general, the J-facets of Xw, for any
+J ⫋ Ia, index the image of O[w] under the natural projection to the adjoint quotient W a → W a/Ad(WJ)
+For each ν ∈ NP, we glue together the X[w], for conjugacy classes of [w] with Newton point ν, into a
+simplicial complex 2 Bν. The gluing procedure relies on the combinatorics called pieces for the affine Weyl
+group (see Section 4.1.1). The space B♥
+ν mentioned in Section 1.4.3 is a subspace of Bν which we call the
+essential part of Bν.
+There is a natural function fν : Bν → R obtained by gluing a certain quadratic function on X[w]
+introduced by He and Nie [HN14] in their work on minimal length elements in conjugacy classes. The
+critical locus Crit(fν) ⊂ Bν is contained in the subspace B♥
+ν ⊂ Bν.
+For example, when ν = 0, B0 is obtained by gluing X[w] for all conjugacy classes [w] of finite order.
+The critical locus of f0, which coincides with B♥
+0 in this case, is exactly the image of X[1] → B0, which
+can be identified with the fundamental alcove ∆.
+For each J, the category of parahoric character sheaves HG,J has a semi-orthogonal decomposition given
+by the stratification of Pu
+J \G/Pu
+J
+LJ
+by geometric pieces, which were introduced by Lusztig [Lusa]. The strata
+in HG,J are indexed by J-facets of B = �
+ν∈NP Bν. When we try to compute the colimit colimJ HG,J, the
+complication is that the transition functors do not respect the semi-orthogonal decompositions. However,
+by performing a categorical version of Morse theory on Bν, we are able to show that only the pieces of
+HG,J indexed by the essential part B♥
+ν ⊂ Bν contribute to the cocenter.
+There are two underlying reasons we are able to implement categorical Morse theory (and specifically,
+a contraction principle) in our situation. One is that the strata categories of parahoric character sheaves
+attached to facets of Bν have a certain local constancy property. This is a consequence of a geometric
+result proved by He [He]. Another is that the embedding B♥
+ν ֒→ Bν is a homotopy equivalence, which uses
+the gradient flow of the function fν. In Appendix A, we collect general methods of calculating colimits
+indexed by posets, and prove a general contraction principle for cosheaves of categories (see Theorem
+A.5.1).
+1.5. Further results. In Section 5.4, we use similar techniques to calculate the derived endomorphism
+rings of two other natural objects in hh(HG) in terms of spectral data. One of the calculations can be
+interpreted as a derived spherical Hecke algebra, and the other one is the endomorphism ring of the
+universal Eisenstein series in the genus one automorphic category.
+The methods of this paper lead not only to a calculation of the endomorphisms of objects in the cocenter
+but to a full description of the cocenter of the universal affine Hecke category. The primary additional
+inputs are:
+2Same comment as above: Bν may only be a simplicial complex after a barycentric subdivision: as naturally constructed,
+the intersection of two simplices in Bν may be a union of more than one simplex.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+9
+(1) The automorphic gluing construction under nodal degenerations of curves [NYa].
+(2) The description of nilpotent sheaves on degree zero semistable G-bundles on a genus one curve [LN21].
+Combined with the methods of this paper, we are able to prove the following to appear in a sequel [LNY].
+For X a smooth projective curve, its Betti automorphic category ShN (BunG(X)) is the dg derived
+category of complexes of sheaves of C-modules on the moduli of G-bundles on X.
+1.5.1. Conjecture. For E a smooth projective genus one curve, there is a natural equivalence from the
+cocenter of the universal affine Hecke category to the Betti automorphic category
+hh(HG)
+∼
+� ShN (BunG(E))
+We can invoke the spectral calculations of [BNP17] to deduce the Betti geometric Langlands conjecture
+in genus one. Recall for X be a smooth projective curve, there is a natural spectral action on its Betti au-
+tomorphic category ShN (BunG(X)) by the tensor category QCoh(LocG∨(E)) of quasi-coherent complexes
+on the moduli of G∨-local systems on X.
+1.5.2. Corollary (of Conjecture 1.5.1). For E a smooth projective genus one curve, the Betti geometric
+Langlands conjecture holds: there is an equivalence of QCoh(LocG∨(E))-module categories
+IndCohN (LocG∨(E))
+∼
+� ShN (BunG(E))
+1.6. Conventions. In the rest of the paper, we will use the following notations.
+1.6.1.
+We fix a field k of characteristic 0 as the coefficient field for our sheaves and categories.
+1.6.2.
+We denote by StL (resp. StR) the ∞-category of stable presentable categories with morphisms left
+adjoints (resp. right adjoints).
+We denote by StL
+k (resp. StR
+k ) the ∞-category of stable presentable k-linear categories with morphisms
+left adjoints (resp. right adjoints). In the many body of the paper, we will be working primarily with StL
+k .
+By a monoidal category, we will typically mean an algebra object in StL
+k .
+1.6.3.
+For a complex algebraic stack X, we denote by Sh(X) the k-linear dg derived category of complexes
+of sheaves in k-vector spaces on X under the analytic topology. We will refer to objects in Sh(X) as sheaves.
+If X is smooth and Λ ⊂ T ∗X is a conical closed subset, we denote by ShΛ(X) the full subcategory of
+Sh(X) consisting of sheaves with singular support in Λ. In particular, using 0 to denote the zero section,
+Sh0(X) is the full subcategory of sheaves with locally constant cohomology sheaves.
+1.6.4.
+Let G be a connected reductive group over C. When needed, we will choose a maximal torus T
+and a pair of opposite Borel subgroups B and B− containing T . Let U and U − be the unipotent radicals
+of B and B−. The quotient H = B/U (the universal Cartan) is canonically independent of the choice of
+B. Let r = dim H be the rank of G. Let W = W(G, T ) be the Weyl group of G. Let g, t, b, u, · · · denote
+the Lie algebras of G, T, B, U, · · ·.
+We will fix an Ad(G)-invariant non-degenerate symmetric bilinear form on g to identify g and g∗.
+1.6.5.
+For an affine algebraic group L over C, let BL = [(Spec C)/L] be its classifying space, regarded as
+an Artin stack.
+1.7. Acknowledgements. We thank David Ben-Zvi and Quoc P. Ho for inspiring discussions, and Tsao-
+Hsien Chen, Peter Haine and James Tao for generous technical help. We thank Xuhua He especially for
+providing a key geometric ingredient needed in this paper in the form of [He].
+PL was partially supported by the National Natural Science Foundation of China (Grant No. 12101348).
+DN was partially supported by NSF grant DMS-2101466. ZY was partially supported by the Simons
+Investigatorship and the Packard Fellowship.
+
+10
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+2. Universal affine Hecke category
+In this section, we provide more details about the universal affine Hecke category as introduced in
+Section 1.2.1.
+But here and in the rest of the paper, unless otherwise stated, we follow the setup of
+Section 1.6.4, and work with G a general connected complex reductive group.
+2.1. Hecke categories.
+2.1.1. Finite Hecke categories. Consider the quotient stack U\G/U, and its convolution diagram
+U\G/U × U\G/U
+p1
+�♥♥♥♥♥♥♥♥♥♥♥♥
+p2
+�❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+U\G ×U G/U
+δ
+�
+π
+� U\G/U
+U\G/U
+U\G/U
+Note δ is smooth (a base change of the diagonal BU → BU × BU).
+The map π is given by the
+multiplication on G. It is a base change of the projection BU → BG with fibers isomorphic to G/U. It
+behaves like a proper map for H-monodromic sheaves on the source. More precisely, π has a factorization
+π : U\G ×U G/U
+h
+� U\G ×B G/U
+π′
+� U\G/U
+where h is an H-torsor (a base change of BU → BB with fibers isomorphic to H = B/U), and π′ is proper
+(a base change of the projection BB → BG with fibers isomorphic to G/B). In general, for any H-torsor
+p : E → X, and a sheaf F ∈ Sh(E) that is H-monodromic, so locally constant along the fibers of p, we
+have a canonical isomorphism
+(2.1.1)
+p!F ≃ p∗F[−r]
+where as usual r = dim H is the rank of G. Indeed, we write H(C) = H>0Hc where Hc is the compact
+real form of H and H>0 the neutral component of the split real form of H. Then p factors as
+p : E
+p0 � E/H>0
+pc
+� X.
+Now pc is proper and H>0 is contractible so F descends to F ∈ Sh(E/H>0). We have p!F = pc!p0!F =
+pc∗p0!(p∗
+0F) ≃ pc∗(F ⊗ p0!k), and p∗F ≃ pc∗F. The relative fundamental class of p0 gives a canonical
+isomorphism p0!k ≃ k[−r] ∈ Sh(E/H>0), hence a natural isomorphism p!F ≃ p∗F[−r].
+Returning to the convolution diagram, for any sheaf F ∈ Sh(U\G ×U G/U) that is H-monodromic for
+the action t · (g1, g2) = (g1t−1, tg2) (for t ∈ H, g1, g2 ∈ G), so locally constant along the fibers of h, we
+have canonical isomorphisms
+π!F ≃ π′
+!h!F ≃ π′
+!h∗F[−r] ≃ π′
+∗h∗F[−r] ≃ π∗F[−r].
+Thus for such sheaves, the pushforward π! satisfies all the base change identities of a proper map.
+Consider the closed embedding of the unit coset
+U\B/U
+u
+� U\G/U.
+Let qH : U\B/U → H be the natural projection, which is a U-gerbe. Let exp : h → H be the universal
+cover, and introduce the universal local system
+Luniv = q∗
+H exp! Dh ∈ Sh0(U\B/U)
+where Dh ≃ kh[r] ∈ Sh0(h) is the Verdier dualizing sheaf. Note that Luniv is concentrated in degree −r
+with stalks isomorphic to the group algebra k[X∗(H)].
+2.1.2. Definition. The universal finite Hecke category of G is the monoidal category of H-bimonodromic
+sheaves on U\G/U (under the left and right translations of H)
+HG = Shbimon(U\G/U)
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+11
+equipped with convolution
+⋆ : HG ⊗ HG
+� HG
+F1 ⋆ F2 = π!δ∗(p∗
+1F1 ⊠ p∗
+2F2)
+and unit object e = u!Luniv.
+It is easy to see that H-bimonodromic sheaves on U\G/U are exactly U × U-equivariant sheaves
+with nilpotent singular support when pulled back to G. Therefore we also denote Shbimon(U\G/U) by
+ShN (U\G/U).
+2.1.3. Example. For the torus H, we have HH = Sh0(H) the dg derived category of locally constant
+sheaves on H. Convolution is simply the pushforward L1 ⋆L2 = m!(L1 ⊠L2) along the multiplication map
+m : H × H → H, and the unit object is the universal local system e = Luniv = exp! Dh.
+2.1.4. Remark. We can also naturally regard HG as a monoidal category in HH-bimodule categories.
+2.1.5. Loop group and parahorics. Let G = G((t)) be the loop group of G, I ⊂ G the Iwahori subgroup
+given by the preimage of B under the reduction mod t map G[[t]] → G. Let Ia be the set of simple (affine)
+roots of G with respect to I, and I ⊂ Ia the subset of simple roots of G with respect to B.
+Let �
+W = X∗(H) ⋊ W be the extended affine Weyl group of G and W a ⊂ �
+W the affine Weyl group
+generated by affine simple reflections. We think of �
+W as acting on the standard apartment A = X∗(T )R.
+For a simple reflection s ∈ W a, let αs ∈ Ia denote the corresponding affine simple root.
+A subset J ⊂ Ia is of finite type if the subgroup WJ generated by simple reflections s for αs ∈ J is
+finite. We use the notation J ⊂ft Ia to mean that J is a finite type subset of Ia. When G is almost
+simple, J ⊂ft Ia simply means that J ⫋ Ia.
+Given J ⊂ft Ia, let PJ ⊂ G be the standard parahoric subgroup containing I of type J, i.e., if Pu
+J ⊂ PJ
+denotes its pro-unipotent radical, and LJ = PJ/Pu
+J its Levi quotient, then J are the simple roots of LJ.
+Note that BJ = I/(I ∩ Pu
+J ) is a Borel subgroup of LJ, with unipotent radical UJ = Iu/(Iu ∩ Pu
+J ).
+When J = ∅, we have I = P∅ and H = L∅, and we write Iu = Pu
+∅. When J = I, we have PI = G[[t]],
+LI = G, B = BI and U = UI.
+We identify �
+W with NG(T )/T [[t]]. For w ∈ �
+W and any lift ˙w ∈ NG(T ), the subspace I ˙wI ⊂ G is
+independent of the choices of T and ˙w, and we denote it by Gw.
+Let ≤ denote the Bruhat order on
+W a extended to �
+W by declaring w1 and w2 are incomparable if they are in different cosets of W a. Let
+G≤w = ∪w′≤wG(w′). It is well-known that Gw/I ⊂ G/I is isomorphic to an affine space of dimension ℓ(w),
+and its closure is G≤w/I.
+2.1.6. Affine Hecke categories. As in the finite-dimensional case, consider the quotient stack Iu\G/Iu,
+and its convolution diagram
+Iu\G/Iu × Iu\G/Iu
+p1
+�♠♠♠♠♠♠♠♠♠♠♠♠♠
+p2
+�❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+Iu\G ×Iu G/Iu
+δ
+�
+π
+� Iu\G/Iu
+Iu\G/Iu
+Iu\G/Iu
+Note δ is pro-smooth (a base change of the diagonal BIu → BIu ×BIu), and π has the similar “almost”
+ind-proper property as in the finite-dimensional case for H-monodromic sheaves. More precisely, consider
+the factorization
+π : Iu\G ×Iu G/Iu
+h
+� Iu\G ×B G/Iu
+π′
+� Iu\G/Iu
+where h is an H-torsor (a base change of BIu → BI with fibers isomorphic to H ≃ I/Iu), and π′ is ind-
+proper (a base change of the projection BI → BG with fibers isomorphic to the affine flag variety G/I). For
+any sheaf F ∈ Sh(Iu\G×IuG/Iu) that is H-monodromic with respect to the action t·(g1, g2) = (g1t−1, tg2),
+so locally constant along the fibers of h, we have a canonical isomorphism by (2.1.1)
+h!F ≃ h∗F[−r].
+
+12
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+On the other hand, if in addition F is supported on Iu\G≤w1 ×Iu G≤w2/Iu for some w1, w2 ∈ �
+W, then
+since π′ is proper when restricted to Iu\G≤w1 ×I G≤w2/Iu, we have canonical isomorphisms
+π!F ≃ π′
+!h!F ≃ π′
+!h∗F[−r] ≃ π′
+∗h∗F[−r] ≃ π∗F[−r]
+Thus for H-monodromic sheaves F suppored on some Iu\G≤w1 ×Iu G≤w2/Iu, the pushforward π! satisfies
+all the base change identities of a proper map.
+Consider the closed embedding of the unit coset
+Iu\I/Iu
+u
+� Iu\G/Iu.
+Let qH : Iu\I/Iu → H be the natural projection, which is a Iu-gerbe. Introduce the universal local
+system
+Luniv = q∗
+H exp! Dh ∈ Sh0(Iu\I/Iu)
+where Dh ≃ kh[r] ∈ Sh0(h) is the Verdier dualizing sheaf.
+2.1.7. Definition. The universal affine Hecke category of G is the colimit of H-bimonodromic sheaves on
+Iu\G≤w/Iu for w ∈ �
+W
+HG = colimw∈�
+W Shbimon(Iu\G≤w/Iu)
+with respect to the full embeddings Shbimon(Iu\G≤w1/Iu) ֒→ Shbimon(Iu\G≤w2/Iu) whenever w1 ≤ w2.
+It is equipped with a monoidal structure given by convolution
+⋆ : HG ⊗ HG
+� HG
+F1 ⋆ F2 = π!δ∗(p∗
+1F1 ⊠ p∗
+2F2)
+and unit object e = u!Luniv.
+2.1.8. Example. For the torus H, we have HH = Sh0(H × X∗(H)) the dg derived category of locally
+constant sheaves. Convolution is simply the pushforward L1 ⋆ L2 = m!(L1 ⊠ L2) along the multiplication
+and addition map m : (H × X∗(H)) × (H × X∗(H)) → H × X∗(H), and the unit object is the universal
+local system e = Luniv = exp! Dh supported on H × {0}.
+2.1.9. Remark. As with HG, we can also naturally regard HG as a monoidal category in HH-bimodule
+categories.
+2.2. Whittaker objects. Fix a maximal torus T ⊂ B ⊂ G, and let B− ⊂ G be the opposite Borel
+subgroup, and U − ⊂ B− its unipotent radical.
+2.2.1. Finite case. Consider the diagram
+A1
+U −
+χ
+�
+r− � U\G/U
+where r− is induced by the inclusion U − ⊂ G, and χ : U → U/[U, U] → A1 is a non-degenerate character
+(i.e., nontrivial on each simple root group).
+Consider the natural factorization of r−
+r− : U −
+i−
+� G/U
+q
+� U\G/U
+Note i− is a closed embedding transverse to the B-orbits in G/U, and q is smooth (a base change of
+pt → BU). Thus for any F ∈ Sh(U\G/U) that is left H-monodromic, so locally constant along the left
+H-orbits in U\G/U, we have canonical isomorphisms
+r!
+−F ≃ i!
+−q!F ≃ i∗
+−q∗F[2 dim r−] ≃ r∗
+−F[2 dim r−]
+Let ϕχ,1 : Sh(U −) → k-mod denote the vanishing cycles at the identity 1 ∈ U − with respect to the
+non-degenerate character χ : U − → A1.
+2.2.2. Definition.
+(1) The Whittaker functor is the composition
+WG : HG
+� k-mod
+WG(F) = ϕχ,1r!
+−F[− dim r−].
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+13
+(2) The universal finite Whittaker sheaf WhG ∈ HG is the object corepresenting the Whittaker functor
+WG(F) ≃ HomHG(WhG, F),
+for all F ∈ HG.
+2.2.3. Remark. The shift in the definition of WG is chosen to make WG exact for the perverse t-structure
+on HG. In particular, for the unit object e ∈ HG, we have WG(e) ∼= k[X∗(H)] is concentrated in degree 0.
+2.2.4. Coalgebra structure on Whittaker sheaf. Here we define the natural coalgebra structure on WhG.
+First, introduce the analogous Whittaker functor on U\G ×U G/U:
+WU\G×U G/U : Sh(U\G ×U G/U)
+� k-mod
+WU\G×U G/U(F) = ϕχ+χ,1r!
+−F[− dim r−].
+where
+A1
+U − × U −
+χ+χ
+�
+r− � U\G ×U G/U
+Now observe the Whittaker functor WG is naturally lax monoidal: for any F1, F2 ∈ HG, we have a
+natural map:
+WG(F1) ⊗ WG(F2) ≃ WG×G(F1 ⊠ F2) ≃ WU\G×U G/U(δ∗(F1 ⊠ F2))
+−→ WU\G×U G/U(π!π!δ∗(F1 ⊠ F2)) ≃ WG(F1 ⋆ F2)
+In other words, we have a natural transformation of functors
+WG ⊠ WG → WG ◦ m : HG ⊗ HG → k-mod
+This induces a map between co-representing objects:
+(2.2.1)
+α : mℓ(WhG) −→ WhG ⊗ WhG
+By adjunction, we obtain a map:
+β : WhG −→ WhG ⋆ WhG
+which defines a comultiplication on WhG.
+The counit and coherences can be constructed analogously. In fact, the full additive ∞-subcategory
+of HG generated by the monoidal unit e and Whittaker object WhG is closed under convolution and
+in fact a classical category (i.e. equivalent as an ∞-category to a discrete category, since its Homs are
+concentrated in degree 0). The coalgebra structure on WhG comes from a coalgebra structure in this
+classical subcategory, and thus its coherences are all strictly determined.
+2.2.5. Microlocal description for Whittaker sheaf. We will not need the following microlocal interpretation
+but mention it for conceptual clarity. We will use an analogue for character sheaves discussed in Section 5.2
+below.
+Consider the cotangent bundle T ∗(G/U), and note its fiber at the identity coset U/U ∈ G/U is naturally
+isomorphic to (g/u)∗. Thus the differential dχ : u− → A1 gives a covector
+ξ : g/u
+� g/b ≃ u−
+dχ � A1.
+Let Ξ : HG → k-mod denote the ∗-pullback along q : G/U → U\G/U, followed by the microstalk at
+ξ ∈ T ∗
+U/U(G/U) (normalized so that the microstalks of perverse sheaves are concentrated in degree 0).
+The following is a standard calculation. For a general version in Betti Geometric Langlands, see [NT].
+2.2.6. Lemma. The Whittaker functor is naturally isomorphic to the microstalk functor
+WG ≃ Ξ : HG
+� k-mod.
+
+14
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+2.2.7. Affine case. Consider the natural closed embedding
+i : U\G/U
+� Iu\G/Iu
+induced by the inclusion of constant loops G ⊂ G.
+2.2.8. Definition.
+(1) The affine Whittaker functor is the composition
+WG : HG
+� k-mod
+WG(F) = WG(i!F).
+(2) The universal affine Whittaker sheaf WhG ∈ HG is the object corepresenting the affine Whittaker
+functor
+WG(F) ≃ HomHG(WhG, F),
+for all F ∈ HG.
+2.2.9. Lemma. The universal affine Whittaker sheaf is the pushforward of the finite Whittaker finite sheaf
+WhG ≃ i!WhG
+In particular, for the unit object e ∈ HG, we have WG(e) ∼= k[X∗(H)].
+Proof. By adjunction,
+WG(F) = WG(i!F) ≃ HomHG(WhG, i!F) ≃ HomHG(i!WhG, F)
+□
+Note that i! is monoidal, and therefore WhG has the induced coalgebra structure from WhG.
+2.3. Langlands duality. Let G∨ be the dual group of G, viewed as a split group over k. Let B∨ ⊂ G∨ be
+the distinguished Borel subgroup, U ∨ ⊂ B∨ its unipotent radical, and H∨ = B∨/U ∨ its universal Cartan.
+We may identify the Weyl group of G∨ with W.
+Recall the canonical identification B∨/B∨ ≃ �
+G∨/G∨ where �
+G∨/G∨ is the Grothendieck-Springer stack
+of pairs (g, E) of an element g ∈ G∨, and a Borel subgroup E ⊂ G∨, such that g ∈ E, all up to conjugation.
+Recall under the above identification, the map of adjoint quotients B∨/B∨ → G∨/G∨ corresponds to the
+Grothendieck-Springer map �
+G∨/G∨ → G∨/G∨ that forgets the Borel subgroup E ⊂ G∨.
+The projection B∨ → H∨ factors through B∨/B∨ → H∨, which corresponds to the projection
+�
+G∨/G∨ → H∨ that takes a pair (g, E) to the class [g] ∈ E/Eu ≃ H∨.
+2.3.1. Definition.
+(1) The universal Steinberg stack is the derived fiber product of adjoint quotients
+StG∨ = B∨/B∨ ×R
+G∨/G∨ B∨/B∨
+(2) The universal spectral affine Hecke category is the monoidal convolution category
+IndCoh(StG∨) = IndCoh(B∨/B∨ ×R
+G∨/G∨ B∨/B∨)
+2.3.2. Remark. The smallness of the Grothendieck alteration �
+G∨ → G∨ implies that the underived fiber
+product B∨/B∨ ×G∨/G∨ B∨/B∨ has the expected dimension zero. Since G∨/G∨ and B∨/B∨ are both
+smooth, we see that the derived structure on StG∨ is in fact trivial.
+2.3.3. Remark. Note we can also naturally regard IndCoh(StG∨) as a monoidal category in QC(H∨)-
+bimodule categories.
+To deduce spectral consequences of results on HG, we will use a universal version of a result of
+Bezrukavnikov [Bez16]. Since it is not yet in the literature, we state it here as an Ansatz; we will provide
+a proof in a sequel.
+2.3.4. Ansatz. There is a monoidal equivalence of universal affine Hecke categories
+(2.3.1)
+Φ : IndCoh(StG∨)
+∼
+� HG
+with the following properties:
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+15
+(1) Φ identifies the structure sheaf OStG∨ with the universal affine Whittaker object WhG as coalgebra
+objects.
+(2) Φ is naturally an equivalence of algebras in bimodules over QC(H∨). (Here we regard the HH-
+bimodule HG as a QC(H∨)-bimodule using the canonical monoidal equivalence HH ≃ Sh0(H) ≃
+QC(H∨).)
+2.3.5. Remark. Ansatz 2.3.4 (2) is not used in this paper, we only record it for conceptual completeness.
+2.4. Coxeter presentation. To make calculations in the cocenter of HG, we will use a universal mon-
+odromic version of a result of Tao-Travkin [TT]. In Appendix C, we provide the necessary extensions to
+apply their arguments to the universal monodromic case.
+Let G◦ be the neutral component of the loop group G, and let HG◦ ⊂ HG be the full subcategory of
+objects supported on G◦. Then HG◦ is a monoidal subcategory of HG.
+For each finite type subset J ⊂ft Ia, the finite universal Hecke category HLJ embeds into HG◦ as
+a monoidal full subcategory of sheaves supported on Iu\PJ/Iu, which is a pro-unipotent gerbe over
+UJ\LJ/UJ. These embeddings are compatible with inclusions J ⊂ J′ ⊂ft Ia in an obvious sense, and
+induce a monoidal functor
+(2.4.1)
+colim⊗
+J⊂ftIa HLJ
+� HG◦
+Here we regard all the Hecke categories as objects in stable presentable k-linear categories StL
+k with
+morphisms left adjoints, and we write colim⊗ to emphasize that the colimit is of monoidal categories in
+HH-bimodues, i.e. of algebra objects in HH-bimodues in StL
+k .
+2.4.1. Theorem (Tao-Travkin [TT], see Appendix C). The functor (2.4.1) is a monoidal equivalence of
+HH-bimodule categories.
+In [TT], the authors worked with the bi-I-equivariant version of the affine Hecke category and assumed
+G was simply-connected. We sketch the necessary modifications to their argument in Appendix C.
+2.4.2. Remark. Formulating a generalization of Theorem 2.4.1 for the whole category HG when G◦ ⫋ G is
+more combinatorially complicated. We will not need it since Theorem 2.4.1 will suffice for our application
+to the cocenter of any reductive G. See Sect. 2.7.5.
+2.5. Hochschild homology and cocenters. We work in the setting of stable presentable k-linear cate-
+gories StL
+k with morphisms left adjoints. All higher algebra constructions will be following [Lur12].
+2.5.1. Definition. Let A be an algebra object in StL
+k , and Aop the opposite algebra.
+(1) Let M be an A-bimodule, i.e. an A ⊗ Aop-module.
+The Hochschild homology of A with values in M is the tensor
+hh(A, M) = A ⊗A⊗Aop M
+The trace map is the natural projection
+tr : M
+� A ⊗A⊗Aop M = hh(A, M)
+induced by the unit of A in the first factor of the tensor.
+(2) The cocenter of A is the Hocschild homology of A with values in the regular bimodule
+hh(A) = hh(A, A) = A ⊗A⊗Aop A
+The trace map is the natural projection
+tr : A
+� A ⊗A⊗Aop A = hh(A, A) = hh(A)
+induced by the unit of A in the first factor of the tensor.
+
+16
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Recall one typically calculates hh(A, M) as the geometric realization of the Hochschild complex (sim-
+plicial object)
+B•(A, M) = [M
+M ⊗ A
+��
+M ⊗ A ⊗ A · · · ]
+���
+This results from tensoring the regular A-bimodule with the bar resolution
+M
+[M ⊗ A
+�
+M ⊗ A ⊗ A
+��
+M ⊗ A ⊗ A ⊗ A · · · ]
+���
+Given a monoidal subcategory A′ ⊂ A, there is a minor variation on the Hochschild complex that
+equally well calculates hh(A, M). Note we can regard A as an algebra in A′-bimodules, and likewise,
+regard M as an A-bimodule in A′-bimodules. Then to calculate hh(A, M), we can form the relative
+Hochschild complex
+B•(A, M)A′ = [hh(A′, M)
+hh(A′, M ⊗A′ A)
+��
+hh(A′, M ⊗A′ A ⊗A′ A) · · · ]
+���
+This results from tensoring the regular A-bimodule with the relative bar resolution inside of A′-bimodules
+M
+[M ⊗A′ A
+�
+M ⊗A′ A ⊗A′ A
+��
+M ⊗A′ A ⊗A′ A ⊗A′ A · · · ]
+���
+In particular, we can take A′ = ⟨1A⟩ ⊂ A to be the full subcategory generated by the unit.
+2.5.2. Descended trace. The following general construction will provide a useful way to characterize certain
+objects of the cocenter of a monoidal category. Let ∆ denote the simplex category of non-empty finite
+ordered sets [n] = {0, . . . , n}, n ≥ 0.
+Suppose A is a monoidal category with product denoted by ⋆.
+Given an algebra object a ∈ A, let L = LModa(A) and R = RModa(A) denote the category of left and
+right a-modules in A. Let B = Bimoda(A) denote the category of a-bimodules in A. Note B is naturally
+monoidal with product given by
+m ⋆a n = colim∆op[m ⋆ n
+m ⋆ a ⋆ n
+��
+m ⋆ a ⋆ a ⋆ n · · · ]
+���
+Similarly, L is a B ⊗ A-bimodule, and R is a A ⊗ B-bimodule. In fact, L and R are in duality with unit
+and counit maps denoted by
+(2.5.1)
+u : B
+� L ⊗A R
+c : R ⊗B L
+� A
+Here, u is the inverse of the natural functor L ⊗A R → B (sending x ⊗ y to x ⋆ y), which is an equivalence
+by [BFN10, Prop. 4.1].
+Recall the dual bimodules L and R, with their unit and counit maps, provide a map on cocenters
+(2.5.2)
+h : hh(B) = B ⊗B⊗Bop B
+u′
+� (L ⊗A R) ⊗B⊗Bop B ≃ A ⊗A⊗Aop (R ⊗B L)
+c′
+� A ⊗A⊗Aop A = hh(A)
+where u′ is induced by u, and c′ by c.
+To avoid confusion, let us denote the usual trace maps by
+trA : A
+� hh(A)
+trB : B
+� hh(B)
+Given m ∈ B, we can take its B-trace trB(m) ∈ hh(B) then its image h(trB(m)) ∈ hh(A).
+2.5.3. Definition. Given m ∈ B, we call
+tr(m) := h(trB(m)) ∈ hh(A)
+the descended trace of m.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+17
+2.5.4. Functoriality of descended trace. Suppose f : A → A′ is a monoidal functor of monoidal categories.
+Then there is an evident commutative diagram
+(2.5.3)
+A
+tr
+�
+f
+� A′
+tr
+�
+hh(A)
+hh(f) � hh(A′)
+Suppose in addition a ∈ A is an algebra object with image algebra object a′ = f(a) ∈ A′. Then
+applying f to bimodules provides a natural monoidal functor F : Bimoda(A) → Bimoda′(A′). We have
+an evident commutative diagram
+(2.5.4)
+Bimoda(A)
+tr
+�
+F
+� Bimoda′(A′)
+tr
+�
+hh(Bimoda(A))
+h
+�
+hh(F )� hh(Bimoda′(A′))
+h
+�
+hh(A)
+hh(f)
+� hh(A′)
+where each h denotes the map (2.5.2) from the cocenter of bimodules to the cocenter induced by the left
+and right module categories.
+We immediately conclude:
+2.5.5. Lemma. Suppose f : A → A′ is a monoidal functor of monoidal categories.
+Suppose a ∈ A is an algebra object with image algebra object a′ = f(a) ∈ A′.
+Under the induced functor hh(f) : hh(A) → hh(A′), we have an identification of descended traces
+hh(f)(tr(m)) ≃ tr(F(m))
+m ∈ Bimoda(A).
+In particular, for the regular bimodule m = a with image F(m) = f(a) = a′, an identification of descended
+traces
+hh(f)(tr(a)) ≃ tr(a′)
+2.5.6. Formula for descended trace. We provide here a formula for the descended trace.
+Let ∆+ denote the augmented simplex category of (possibly empty) finite ordered sets [n] = {0, . . . , n},
+n ≥ −1. Given m ∈ B = Bimoda(A), consider its bar augmented simplicial object m• : ∆op
++ → B given
+by the assignments: mn = m ⋆ a⋆n, with each face map a contraction via the multiplication of a, and each
+degeneracy map an insertion of the unit of a. We can picture the diagram of face maps in the usual way
+m• = [m
+m ⋆ a
+�
+m ⋆ a ⋆ a
+��
+m ⋆ a ⋆ a ⋆ a · · · ]
+���
+and the augmentation descends to an equivalence on the geometric realization
+m
+colim∆op[m ⋆ a
+∼
+�
+m ⋆ a ⋆ a
+��
+m ⋆ a ⋆ a ⋆ a · · · ]
+���
+Now let us take the descended trace of the bar augmented simplicial object to obtain a resolution
+tr(m)
+colim∆op[tr(m ⋆ a)
+∼
+�
+tr(m ⋆ a ⋆ a)
+��
+tr(m ⋆ a ⋆ a ⋆ a) · · · ]
+���
+using that we work in the setting of continuous functors.
+Unwinding the definitions, for any a-bimodule of the form ℓ ⋆ r where ℓ ∈ L = LModa(A) and r ∈
+R = RModa(A), we have a natural isomorphism tr(ℓ ⋆ r) ≃ trA(r ⋆a ℓ). Thus the above resolution gives a
+concrete formula for the descended trace
+(2.5.5)
+tr(m)
+colim∆op[trA(m)
+∼
+�
+trA(m ⋆ a)
+��
+trA(m ⋆ a ⋆ a) · · · ]
+���
+
+18
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+where the face maps are given by contractions via the multiplication of a (and degeneracy maps given by
+an insertion of the unit of a). In particular, the new face map trA(m ⋆ a⋆(n+1)) → trA(m ⋆ a⋆n) is induced
+by the left multiplication of the last factor on the first available thanks to the cyclic symmetry of the trace.
+2.5.7. Descended trace for coalgebras. One can repeat the construction of the descended trace starting
+with a coalgebra c ∈ A rather than an algebra a ∈ A. We only prefer the algebra formulation due to our
+lack of familiarity with “bi-comodules”. But in any case, we will only work with the coalgebra c itself. In
+this case, we can take the following concrete formula as definition of the descended trace
+tr(c) := lim∆[trA(c)
+�� trA(c ⋆ c)
+��� trA(c ⋆ c ⋆ c) · · · ]
+where the coface maps are given by expansions via the comultiplication of c (and degeneracy maps given
+by an insertion of the counit of c). In particular, the new coface map trA(c⋆n) → trA(c⋆n+1) is induced
+by the left comultiplication of the last factor on the first available thanks to the cyclic symmetry of the
+trace.
+In our applications, where some additional hypotheses hold, we can relate the above definition for
+coalgebras with the prior theory for algebras, and in particular take advantage of the prior recorded
+functoriality.
+Assume that A is compactly generated, and the monoidal product preserves the category of compact
+objects Ac ⊂ A, so that A ≃ IndAc as monoidal categories. Then the categories L, R, B, hh(B), hh(A) are
+all compactly generated, and the functor tr preserves compact objects. Suppose our coalgebra is compact
+c ∈ Ac, and denote by a ∈ Aop
+c
+the same object regarded as an algebra in the dual monoidal category
+A∨ ≃ IndAop
+c . Suppose in addition a ∈ Bimoda(A∨) is compact (for example, it is a summand of a ⋆ a).
+Then chasing definitions, under the canonical equivalence hh(A∨)c ≃ hh(Aop
+c ) ≃ hh(Ac)op ≃ hh(A)∨
+c , the
+colimit calculating the algebra descended trace tr(a) ∈ hh(Aop
+c ) is equivalent to the limit calculating the
+coalgebra descended trace tr(c) ∈ hh(Ac).
+2.6. Spectral realization. In this subsection, we consider the spectral realization IndCoh(StG∨) of the
+universal affine Hecke category. In particular, we compute the descended trace of the structure sheaf
+OStG∨ as a coalgebra object.
+The arguments are quite general, so we will work with an abstract setup.
+We stress that in this
+subsection, all fiber products of stacks are derived fiber products.
+Let X → Y be a proper morphism between smooth stacks. Then IndCoh(X ×Y X) is naturally a
+monoidal category via convolution: F ⋆ G = m∗δ∗
+23(F ⊠ G) where δ23 is the diagonal map and m the
+forgetful map in the correspondence
+X ×Y X × X ×Y X
+X ×Y X ×Y X
+δ23
+�
+m
+� X ×Y X
+2.6.1. Example. In our application, we will take the induction map X = B∨/B∨ → G∨/G∨ = Y so that
+X ×Y X = StG∨.
+Alternatively, we could define the !-convolution: F⋆!G = m∗δ!
+23(F⊠G). We will denote by IndCoh(X×Y
+X)! the resulting monoidal category with the !-convolution. Note that δ23 is quasi-smooth, so δ!
+23 and δ∗
+23,
+and hence ⋆ and ⋆! as well, only differ by an invertible twist.
+Serre duality gives an equivalence of compact objects Coh(X ×Y X)op ≃ Coh(X ×Y X) and hence an
+equivalence of their ind-completions IndCoh(X ×Y X)∨ ≃ IndCoh(X ×Y X).
+Note the dual IndCoh(X×Y X)∨ inherits a monoidal structure from IndCoh(X×Y X). The identification
+IndCoh(X ×Y X)∨ ≃ IndCoh(X ×Y X) naturally lifts to an equivalence of monoidal categories
+(2.6.1)
+DSerre
+X×Y X : IndCoh(X ×Y X)∨
+∼
+� IndCoh(X ×Y X)!
+Now denote by LY = Hom(S1, Y ) the derived loop space of Y , and ΛX/Y = π∗δ!(T ∗−1
+X×Y X) ⊂ T ∗(LY )
+the Lagrangian defined via the correspondence
+(2.6.2)
+X ×Y X
+(X ×Y X) ×X×X X ≃ LY ×Y X
+δ
+�
+π
+� LY
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+19
+2.6.2. Example. Continuing in the setup of Example 2.6.1, we find the commuting stack
+LY = L(G∨/G∨) = Z2
+G∨ = (G∨/G∨ × G∨/G∨) ×G∨/G∨ ({1}/G∨) ≃ ((G∨ × G∨) ×G∨ {1})/G∨
+the derived fiber product of the commutator map c : G∨ × G∨ → G∨, and the inclusion of the identity
+1 ∈ G∨, all up to conjugation. We also find the Lagrangian of nilpotent codirections ΛX/Y = N as
+calculated in [BNP17].
+Thanks to [BNP17], we have the following:
+2.6.3. Theorem.
+(1) The functor π∗δ∗ lands in IndCohΛX/Y (LY ) and fits into a commutative diagram
+with the trace map tr inducing a horizontal equivalence
+IndCoh(X ×Y X)
+tr
+�
+π∗δ∗
+�❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+hh(IndCoh(X ×Y X))
+∼
+� IndCohΛX/Y (LY )
+(2) Similarly, the functor π∗δ! lands in IndCohΛX/Y (LY ) and fits into a commutative diagram with
+the trace map tr inducing a horizontal equivalence
+IndCoh(X ×Y X)!
+tr
+�
+π∗δ!
+�❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+hh(IndCoh(X ×Y X)!)
+∼
+� IndCohΛX/Y (LY )
+2.6.4. Remark. It follows that the equivalence on trace categories induced by (2.6.1) is also given by Serre
+duality, i.e, the following diagram naturally commutes
+(2.6.3)
+hh(IndCoh(X ×Y X))∨
+hh(DSerre
+X×Y X) �
+∼
+�
+hh(IndCoh(X ×Y X)!)
+∼
+�
+IndCohΛX/Y (LY )∨
+DSerre
+LY
+� IndCohΛX/Y (LY )
+2.6.5. Example. In the above situation, we compute tr(∆∗OX), where ∆ : X → X ×Y X is the diagonal
+map. We have a commutative diagram with a derived Cartesian square on the left
+X
+∆
+�
+LX
+pX
+�
+ϕ
+�
+Lf
+�❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+X ×Y X
+LY ×Y X
+δ
+�
+π
+� LY
+By base change we have
+tr(∆∗OX) = π∗δ∗∆∗OX ≃ π∗ϕ∗p∗
+XOX = (Lf)∗OLX.
+2.6.6. Decended trace of structure/dualizing sheaf. We continue with the above general setup.
+The structure sheaf OX×Y X is naturally a coalgebra object in IndCoh(X ×Y X) under convolution: its
+comultiplication is given by the unit of adjunction
+(2.6.4)
+OX×Y X
+� m∗m∗OX×Y X ≃ OX×Y X ⋆ OX×Y X.
+Similarly, the dualizing sheaf ωX×Y X is naturally an algebra object in in IndCoh(X ×Y X)! under
+!-convolution: its multiplication is given by the counit of adjunction
+ωX×Y X ⋆! ωX×Y X ≃ m∗m!ωX×Y X
+� ωX×Y X
+
+20
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+In fact, the coalgebra OX×Y X, viewed as an algebra in IndCoh(X ×Y X)∨, and the algebra ωX×Y X are
+identified under the monoidal equivalence (2.6.1).
+Now we have the following calculations of the descended traces of OX×Y X and ωX×Y X:
+2.6.7. Proposition. Assume further that X → Y is surjective.
+(1) Under the equivalence of Theorem 2.6.3(1)
+hh(IndCoh(X ×Y X))
+∼
+� IndCohΛX/Y (LY )
+there is a canonical identification of the descended trace of the coalgebra object OX×Y X with the
+structure sheaf
+tr(OX×Y X) ≃ OLY .
+(2) Similarly, under the equivalence of Theorem 2.6.3(2)
+hh(IndCoh(X ×Y X)!)
+∼
+� IndCohΛX/Y (LY )
+there is a canonical identification of the descended trace of the algebra object ωX×Y X with the
+dualizing sheaf
+tr(ωX×Y X) ≃ ωLY .
+Proof. We prove (2); then (1) follows from the Serre duality equivalences (2.6.1), (2.6.3).
+For the natural map π : LY ×Y X → LY , we have the adjunction
+π∗ : IndCoh(LY ×Y X) ⇄ IndCoh(LY ) : π!
+Moreover, π! is conservative (because π is proper and surjective) and preserves colimits (because π is
+quasi-smooth). Therefore by Berr-Beck, the canonical resolution associated to the comonad π∗π! is an
+equivalence:
+ωLY
+colim∆op[π∗π!ωLY
+∼
+�
+π∗π!π∗π!ωLY
+��
+π∗π!π∗π!π∗π!ωLY · · · ]
+���
+By standard identities including base-change, the canonical resolution can be identified with the simplicial
+object
+tr(ωX×Y X)
+tr(ωX×Y X ⋆! ωX×Y X)
+��
+tr(ωX×Y X ⋆! ωX×Y X ⋆! ωX×Y X) · · ·
+���
+computing the descended trace of ωY ×XY .
+□
+2.6.8. Remark. Here is a more direct proof of (2) of the theorem that in fact motivates the definition of
+the descended trace.
+Set A = IndCoh(X ×Y X), B = BimodOX×Y X(IndCoh(X ×Y X)). By !-descent, we have a monoidal
+equivalence B ≃ QCoh(Y ) where the monoidal structure on QCoh(Y ) is given by the tensor product.
+Moreover, under this equivalence the regular bimodule OX×Y X ∈ B corresponds to the structure sheaf
+OY ∈ QCoh(Y ).
+Thanks to [BFN10], we have a commutative diagram
+QCoh(Y )
+tr
+�
+p∗
+�P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+hh(QCoh(Y ))
+∼
+� QCoh(LY )
+where p : LY → Y is the natural base-point projection. Hence we have an equivalence
+QCoh(LY ) ≃ hh(QCoh(Y )) ≃ hh(B)
+Under this equivalence, the natural map h : hh(B) → hh(A) is identified with the inclusion i : QCoh(LY ) ֒→
+IndCohΛX/Y (LY ), where we view QCoh(LY ) ⊂ IndCoh(LY ) as ind-coherent sheaves with singular support
+in the zero-section.
+Thus we conclude
+tr(OX×Y X) = h(trB(OX×Y X)) ≃ i(tr(OY )) ≃ p∗OY ≃ OLY .
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+21
+2.6.9. Application to Steinberg stack. Now we specialize our prior setup to X → Y the induction map
+B∨/B∨ → G∨/G∨ so that X ×Y X = StG∨. By Example 2.6.2, we have LY = Z2
+G∨ and ΛX/Y = N the
+nilpotent codirections in T ∗−1Z2
+G∨. We immediately conclude the following:
+2.6.10. Corollary. Under the equivalence
+(2.6.5)
+hh(IndCoh(StG∨))
+∼
+� IndCohN (Z2
+G∨)
+there is a canonical identification of the descended trace of the coalgebra object OStG∨ with the structure
+sheaf
+tr(OStG∨ ) ≃ OZ2
+G∨ .
+2.7. Automorphic realization. In this subsection, we give a presentation of the cocenter of the universal
+Hecke category HG using “partial cocenters”. The partial cocenters are then interpretated geometrically
+using parabolic character sheaves introduced by Lusztig.
+2.7.1. Character sheaves as cocenter. We first review the interpretation of character sheaves on G as the
+cocenter of HG. Various versions of this statement appear in the literature, see [BNa], [BFO12] and [Lusb]
+We will state here a universal monodromic version.
+Let ShN (G/G) be the full subcategory of G-equivariant sheaves on G with nilpotent singular support.
+This is a universally monodromic version of character sheaves. Consider the horocycle correspondence
+(2.7.1)
+U\G/U
+G
+U
+δG
+�
+πG � G
+G
+Then we have the horocycle functor
+(2.7.2)
+γ := πG!δ∗
+G : HG = Shbimon(U\G/U)
+� ShN (G/G).
+The following result is a special case of Theorem 3.2.3 which we will prove in Section 3.
+2.7.2. Theorem. There is a canonical equivalence
+hh(HG)
+∼
+� ShN (G/G)
+such that the composition HG
+trG
+−−→ hh(HG) ≃ ShN (G/G) is identified with γ.
+2.7.3. Hochschild homology under HG◦. Recall G◦ is the neutral component of the loop group G and
+HG◦ ⊂ HG is the full subcategory of objects supported on G◦.
+For any finite type J ⊂ft Ia, we have the (universal) finite Hecke category HLJ of the Levihoric
+LJ, which lies inside HG◦.
+Any HG◦-bimodule can be viewed as a HLJ-bimodule and we can form
+the Hochschild homology hh(HLJ, M).
+For J ⊂ J′ both of finite type, we have a natural functor
+hh(HLJ, M) → hh(HLJ′, M).
+2.7.4. Corollary (of Theorem 2.4.1). For any HG◦-bimodule M, the natural maps induce an equivalence
+(2.7.3)
+colimJ⊂ftIa hh(HLJ, M)
+∼
+� hh(HG◦, M)
+Moreover, for each J ⊂ft Ia, the equivalence naturally extends to a commutative diagram
+M
+trJ
+�❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥
+�
+tr
+�❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+hh(HLJ, M)
+� colimJ′⊂ftIa hh(HLJ′ , M)
+∼
+� hh(HG◦, M)
+where the diagonal arrows are traces, and the left horizontal arrow is the natural map.
+
+22
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Proof. By Theorem 2.4.1, we have HG◦ ≃ colimJ⊂ftIa HLJ where the colimit is taken within monoidal
+categories. For a right HG◦-module M1 and a left HG◦-module M2, we then have
+M1 ⊗HG◦ M2 ≃ colimJ⊂ftIa M1 ⊗HLJ M2.
+Applying this to the right HG◦-module HG◦ and the left HG◦-module M, we get an equivalence of HG◦-
+bimodules
+M ≃ HG◦ ⊗HG◦ M ≃ colimJ⊂ftIa HG◦ ⊗HLJ M.
+Taking hh(HG◦, −) on both sides, we get
+(2.7.4)
+hh(HG◦, M) ≃ colimJ⊂ftIa hh(HG◦, HG◦ ⊗HLJ M).
+Finally, observe that hh(HG◦, HG◦ ⊗HLJ M) ≃ HG◦ ⊗HLJ ⊗Hop
+G◦ M ≃ hh(HLJ, M).
+Therefore (2.7.4)
+implies (2.7.3). The rest of the corollary is clear.
+□
+2.7.5. Hochschild homology under HG. Let G be a connected reductive group.
+Now we would like to
+calculate hh(HG, M) for a HG-bimodule M in terms of Hochschild homology under various finite Hecke
+categories.
+First we recall some constructions of Varshavsky. Let D◦ be the poset of finite type subsets J ⊂ft Ia
+under inclusion. Consider the abelian group Ω = NG(I)/I with its action on Ia and induced action on
+D◦. Let D be the category with objects J ⊂ft Ia, morphisms J → J′ given by ω ∈ Ω with ω(J) ⊂ J′, and
+compositions induced by multiplication in Ω. In other words, D is the groupoid D◦/Ω. Note the natural
+(faithful but not full) functor i : D◦ → D which is an equivalence if and only if G is simply-connected.
+For each ω ∈ Ω we define a monoidal auto-equivalence of HG as follows. Choose a lifting ˙ω of ω in
+NG(I). Using ˙ω as the base point, we identify I ˙ωI/Iu with H (via ˙ωh ↔ h). Let C ˙ω ∈ HG be the
+extension by zero of Luniv supported on I ˙ωI/Iu ≃ H. Note that C ˙ω−1 is the inverse of C ˙ω under the
+monoidal structure of HG. We get a monoidal auto-equivalence
+c ˙ω : HG
+C ˙ω⋆(−)⋆C ˙ω−1 � HG.
+We claim that c ˙ω is canonically independent of the choice of the lifting ˙ω. Indeed, for a different lifting
+¨ω = ˙ωh for some h ∈ H, we have a canonical isomorphism
+C ˙ω ≃ C¨ω ⊗R Luniv,h.
+Here R = k[X∗(H)] and Luniv,h is the stalk of Luniv at h ∈ H, an invertible R-module. On the other
+hand,
+C ˙ω−1 ≃ C¨ω−1 ⊗R Luniv,h−1.
+Since Luniv,h and Luniv,h−1 are inverse to each other as invertible R-modules, the operations c ˙ω = C ˙ω ⋆
+(−) ⋆ C ˙ω−1 and c¨ω = C¨ω ⋆ (−) ⋆ C¨ω−1 are canonically identified. Therefore we get a canonical monoidal
+auto-equivalence
+(2.7.5)
+cω : HG → HG.
+The canonicity of cω implies that they together give an action of Ω on HG as a monoidal category.
+The same construction shows that: for any HG-bimodule M, there is a canonical action of Ω on M
+such that ω ∈ Ω acts by C ˙ω ⋆ (−) ⋆ C ˙ω−1, for any lifting ˙ω of ω in NG(I).
+If ω ∈ HomD(J, J′), cω sends HLJ to HLJ′ . Therefore, the diagram of Hecke categories J �→ HLJ, for
+J ⊂ft Ia, naturally extends along i to a functor from D to monoidal categories. Using these functors, for
+any HG-bimodule M, restriction to HLJ for J ⊂ft Ia, naturally extends to a functor from D to bimodules.
+2.7.6. Proposition. Let G be a reductive group. For any HG-bimodule M, the natural maps induce an
+equivalence
+colimD hh(HLJ, M)
+∼
+� hh(HG, M)
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+23
+Moreover, for each J ⊂ft Ia, the equivalence naturally extends to a commutative diagram
+M
+tr
+�❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦
+trJ
+�
+tr
+�❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+hh(HLJ, M)
+� colimD hh(HLJ, M)
+∼
+� hh(HG, M)
+where the diagonal arrows are traces, and the left horizontal arrow is the natural map.
+Proof. For any HG-bimodule M, there is a canonical map
+cM : colimD hh(HLJ, M)
+� hh(HG, M).
+We need to show that this is an equivalence. It suffices to set M = HG ⊗ HG (where HG acts on the first
+factor of HG ⊗ HG on the right and the second on the left) and prove the natural map
+(2.7.6)
+c : colimD hh(HLJ, HG ⊗ HG) ≃ colimD HG ⊗HLJ HG
+� HG ≃ hh(HG, HG ⊗ HG)
+is an equivalence of HG-bimodules (where the HG-action on both sides is induced by its action on the left
+of the first factor of HG ⊗ HG and on the right of the second). Note c is induced by HG-bimodule maps,
+so is naturally an HG-bimodule map. Thus it suffices to check c is an equivalence.
+Consider the following commutative diagram
+colimD◦ HG◦ ⊗HLJ HG
+i
+�
+∼ � colimD◦ hh(HLJ, HG◦ ⊗ HG)
+∼
+� hh(HG◦, HG◦ ⊗ HG)
+∼
+�
+colimD HG ⊗HLJ HG
+∼ � colimD hh(HLJ, HG ⊗ HG)
+c
+� hh(HG, HG ⊗ HG)
+Here the top middle arrow is an equivalence by Corollary 2.7.4, the right vertical map is the evident
+induction equivalence (both are equivalent to HG), and i is the natural map induced by i : D◦ → D and
+the inclusion HG◦ ֒→ HG. Thus to show c is an equivalence, it suffices to show i is an equivalence.
+Let Φ : D◦ → Cat∞ be the functor given by J �→ HG◦ ⊗HLJ HG.
+The forgetful functor i! :
+Fun(D, Cat∞) → Fun(D◦, Cat∞) admits a left adjoint (left Kan extension along i) i! : Fun(D◦, Cat∞) →
+Fun(D, Cat∞), so that colimD◦ Φ = colimD i!Φ. Using this we can rewrite i as
+(2.7.7)
+colimD(i!Φ)(J) → colimD HG ⊗HLJ HG
+induced by the termwise functor φJ : (i!Φ)(J) → HG ⊗HLJ HG for J ∈ D.
+We show that φJ is an
+equivalence for each J ⊂ft Ia. Indeed,
+(i!Φ)(J) =
+�
+ω∈Ω
+Φ(ω(J)) =
+�
+ω∈Ω
+HG◦ ⊗HLω(J) HG.
+We have embeddings
+iω : HG◦ ⊗HLω(J) HG → HG ⊗HLJ HG
+given by x ⊗ y �→ (x ⋆ C ˙ω) ⊗ (C ˙ω−1 ⋆ y) (which is again canonically independent of the lifting ˙ω). The
+functor φJ is the direct sum of iω
+⊕iω :
+�
+ω∈Ω
+HG◦ ⊗HLω(J) HG → HG ⊗HLJ HG,
+which is easily seen to be an equivalence. Since φJ is an equivalence for all J ∈ D, (2.7.7) is an equivalence.
+□
+Recall that Ω acts on any HG-bimodule M by conjugation, and it acts on HG◦ by monoidal auto-
+equivalences, compatible with the bimodule structure on M. These actions induce an action of Ω on the
+Hochshild homology hh(HG◦, M).
+
+24
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+2.7.7. Corollary. For any G-bimodule M, there is a canonical equivalence between the Ω-coinvariants on
+hh(HG◦, M) and hh(HG, M):
+(2.7.8)
+hh(HG◦, M)Ω
+∼
+→ hh(HG, M).
+In particular,
+hh(HG◦, HG)Ω ≃ hh(HG).
+Proof. Let C : D◦ → Cat∞ be the functor given by CJ = hh(HLJ, M). Then this functor has a canonical
+Ω-equivariant structure, hence colimJ∈D◦ CJ carries an action of Ω. There is a natural map
+(2.7.9)
+(colimJ∈D◦ CJ)Ω → colimJ∈D CJ.
+By Corollary 2.7.4 and Proposition 2.7.6, the two sides above are equivalent to the two sides of (2.7.8).
+Thus it suffices to show that (2.7.9) is an equivalence. Note that D = D◦/Ω. Consider the projection
+π : D → BΩ, which is a coCartesian fibration. The left Kan extension π!C is colimJ∈D◦ CJ as a category
+with Ω-action. By [Lur09, Proposition 4.3.3.10], we have
+colimD C ≃ colimBΩ π!C ≃ (colimD◦ C)Ω.
+□
+2.7.8. Geometry of trace. For J ⊂ft Ia, define the ind-stack
+YJ := Pu
+J \G/Pu
+J
+LJ
+,
+where
+·
+LJ denotes the quotient by the conjugation action of LJ. Consider the horocycle correspondence
+(2.7.10)
+Iu\G/Iu
+Pu
+J \G/Pu
+J
+UJ⊂ft
+δJ
+�
+πJ
+� Pu
+J \G/Pu
+J
+LJ
+= YJ
+To simplify the notation, we will set
+HG,J = ShN (YJ) := colimw∈{WJ\�
+W/WJ } ShN
+�Pu
+J \G≤w/Pu
+J
+LJ
+�
+.
+Here the colimit is taken over longest representatives in the WJ-double cosets of �
+W (so that G≤w is a
+union of PJ-double cosets). The notation ShN (−) means, viewed as sheaves on Pu
+J \G/Pu
+J , the singular
+support has nilpotent image under the moment map for the LJ × LJ-action by left and right translations.
+Note that when J = ∅, HG,∅ imposes an Ad(H)-equivariance structure on sheaves on Iu\G/Iu.
+2.7.9. Remark. We will see in Section 4.3 that HG,J is closely related to the notion of parabolic character
+sheaves for the loop group G defined by Lusztig in [Lusa].
+Consider the functor
+πJ!δ∗
+J : HG = Shbimon(Iu\G/Iu)
+� Sh(YJ).
+It is easy to check that the image of πJ!δ∗
+J lands in the full subcategory ShN (YJ) = HG,J (it suffices to
+check on each PJ-double coset of G). Hence we get a functor
+(2.7.11)
+πJ!δ∗
+J : HG
+� HG,J.
+The following result gives a geometric interpretation of the partial cocenters hh(HLJ, HG). It is a special
+case of Theorem 3.3.2, which we will state and prove in Section 3.
+2.7.10. Theorem. For J ⊂ft Ia, the functor πJ!δ∗
+J fits into a commutative diagram with the trace map tr
+inducing a horizontal equivalence
+HG
+tr
+�
+πJ!δ∗
+J
+�▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+hh(HLJ, HG)
+∼
+� HG,J
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+25
+Substituting Theorem 2.7.10 into Corollary 2.7.4 and Proposition 2.7.6, we immediately obtain:
+2.7.11. Corollary. We have equivalences
+colimJ∈D◦ HG,J ≃
+hh(HG◦, HG),
+colimJ∈D HG,J ≃
+hh(HG).
+Moreover, for each J ⊂ft Ia, the equivalence above naturally extends to a commutative diagram
+HG
+πJ!δ∗
+J
+�qqqqqqqqqqq
+�
+tr
+�
+HG,J
+� colimD HG,J′
+∼
+� hh(HG)
+where the left horizontal arrow is the natural map.
+2.7.12. Connected components. For ω ∈ Ω, let Hω
+G be the full subcategory of HG consisting of sheaves
+supported on the ω-component of G. Similarly, define Hω
+G,J. Note the action of Ω on HG preserves each
+Hω
+G. Therefore we have a decomposition by support
+hh(HG) =
+�
+ω∈Ω
+hh(HG)ω
+where
+hh(HG)ω ≃ colimJ∈D Hω
+G,J.
+2.8. Descended trace of Whittaker object. Recall that WhG is naturally a coalgebra in HG, therefore
+it makes sense to take its descended trace tr(WhG) ∈ hh(HG). Let
+WhG/G := tr(WhG) ∈ hh(HG).
+The goal of this subsection is to calculate the WhG/G in terms of character sheaves on G.
+2.8.1. Reduction from G to G. Recall the monoidal functor i! : HG → HG. It induces a functor by passing
+to cocenters
+(2.8.1)
+a : ShN (G/G)
+∼
+� hh(HG)
+hh(i!)� hh(HG)
+where the first equivalence is given by Theorem 2.7.2.
+Recall I ⊂ Ia are the simple roots of G. The corresponding maximal parahoric PI ⊂ G is the arc group
+G0 = G[[t]] with pro-unipotent radical Pu
+I ⊂ PI the arcs based at the identity Gu
+0 = ker(G[[t]] → G). By
+writing hh(i!) as the composition
+hh(HG) → hh(HG, HG) → hh(HG),
+and using Corollary 2.7.11 (applied to J = I), the functor a factors as the composition
+a : ShN (G/G)� � iG/G � HG,I
+� hh(HG)
+where iG/G is the full embedding given by the direct image along the natural map
+G
+G → Gu
+0 \G/Gu
+0
+G
+= YI.
+2.8.2. Lemma. We have the following relation between the descended traces of WhG and WhG:
+tr(WhG) ≃ a(tr(WhG)) ∈ hh(HG).
+Proof. By Lemma 2.2.9, the universal affine Whittaker sheaf is the extension by zero of the finite Whittaker
+finite sheaf WG ≃ i!WhG. The statement then follows from Lemma 2.5.5.
+□
+
+26
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+2.8.3. Whittaker character sheaf. Consider the diagram
+A1
+U −/U −
+χ
+�
+r− � G/G
+where r− is induced by the inclusion U − ⊂ G, and χ is induced by the same-named non-degenerate
+character χ : U → U/[U, U] → Ga = A1 used in Section 2.2.1.
+Let ϕχ,1 : Sh(U −/U −) → k-mod denote the vanishing cycles at the identity 1 ∈ U − with respect to the
+function χ : U −/U − → A1.
+2.8.4. Definition.
+(1) The Whittaker functor on character sheaves is the composition
+WG/G : ShN (G/G)
+� k-mod
+WG/G(F) = ϕχ,1r!
+−F.
+(2) The Whittaker character sheaf WhG/G ∈ ShN (G/G) is the object corepresenting the Whittaker
+functor
+WG/G(F) ≃ HomShN (G/G)(WhG/G, F),
+for all F ∈ ShN (G/G).
+Now we arrive at the main result of this section. A generalization in the context of nodal degenerations
+of curves will appear in [NYb].
+2.8.5. Theorem.
+(1) For the trace map
+trG = γ : HG = ShN (U\G/U)
+� ShN (G/G) ≃ hh(HG)
+there is a canonical identification of the descended trace of the universal finite Whittaker sheaf
+tr(WhG) ∈ hh(HG) with the Whittaker character sheaf WhG/G ∈ ShN (G/G).
+(2) We have a canonical identification of the descended trace of WhG:
+WhG/G = tr(WhG) ≃ a(WhG/G) ∈ hh(HG),
+where a is defined in (2.8.1).
+Proof. (2) follows immediately from (1) by Lemma 2.8.2.
+To prove (1), it will be convenient to view HG as an algebra in HH = Sh0(H)-bimodules. Here Sh0
+means sheaves with singular support within the zero-section, or equivalently local systems. Note HH ⊂ HG
+is the full monoidal subcategory generated by the monoidal unit HH = ⟨1HG⟩ ⊂ HG.
+Recall hh(HG) is canonically independent of whether we work absolutely or in HH-bimodules. To be
+more precise, set H(n)
+G
+= HG ⊗ · · · ⊗ HG (n copies of HG) and H(n)H
+G
+= HG ⊗HH · · · ⊗HH HG (n copies of
+HG). So in particular HG = H(1)
+G = H(1)H
+G
+. Set also H(n)
+G,H = hh(HH, H(n)H
+G
+), so in particular
+HG,H = H(1)
+G,H = ShN
+�U\G/U
+H
+�
+Here as usual ShN means sheaves with nilpotent singular support, or equivalently monodromic sheaves
+with respect to the remaining H-action.
+Then the natural map from the absolute to HH-relative Hochschild complexes induces an equivalence
+on colimits
+(2.8.2)
+hh(HG)
+≃
+�
+[HG
+γ
+�
+q!=q(1)
+!
+�
+H(2)
+G
+q(2)
+!
+�
+��
+H(3)
+G · · · ]
+q(3)
+!
+�
+���
+hh(HG)
+[HG,H
+γH
+�
+H(2)
+G,H
+��
+H(3)
+G,H · · · ]
+���
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+27
+Here the maps q(n)
+!
+are the !-pushforwards along the natural quotient maps. Moreover, the augmentations
+are given by γ = πG!δ∗
+G and γH = πG,H!δ∗
+G,H as appear in the diagram
+G
+G
+G
+U
+�
+πG
+�
+δG � U\G/U
+q
+�
+G
+B
+πG,H
+�❁❁❁❁❁❁❁❁
+δG,H � U\G/U
+H
+with Cartesian square. Note by base-change, γ = πG!δ∗
+G indeed admits the natural factorization
+γ : HG
+q!
+� HG,H
+γH=πG,H!δ∗
+G,H� ShN ( G
+G)
+For 1 ≤ i ≤ n, let mn,i : H(n)
+G,H → H(n−1)
+G,H
+denote the face map of the lower simplicial diagram (2.8.2)
+given by convolving the cyclically-ordered ith and (i + 1)st terms. (By convention, we have m1,1 = γH,
+and H(0)
+G,H = ShN(G/G).)
+Note by standard base-change identities, the natural base-change map is an isomorphism mℓ
+n,imn,i ≃
+mn+1,i+1mℓ
+n+1,i, for all 1 ≤ i ≤ n. Let γ(n)
+H
+: H(n)
+G,H → hh(HG) denote the canonical map given by γH
+applied to any cyclically-ordered total convolution.
+Set Wh(n)
+G
+= WhG ⊗ · · · ⊗ WhG (n copies of WhG) and Wh(n)
+G,H = q(n)
+!
+Wh(n)
+G . In particular, we have
+WhG = Wh(1)
+G and write WhG,H = Wh(1)
+G,H. By convention, set Wh(0)
+G,H = WhG/G ∈ H(0)
+G,H.
+The map α of (2.2.1) gives
+αH : mℓWhG,H → Wh(2)
+G,H,
+which yields for any n ≥ 1, and 1 ≤ i ≤ n a map:
+αH
+n,i : mℓ
+n+1,iWh(n)
+G,H → Wh(n+1)
+G,H
+Similarly, we have αH
+0 : γℓ
+HWhG/G → WhG,H (in fact, it is constructed in the next lemma).
+By adjunction, we obtain
+βH
+n,i : Wh(n)
+G,H → mn+1,iWh(n+1)
+G,H
+Note that, by construction, this is exactly the natural map induced by the coalgebra structure of WhG
+(after applying q(n)
+!
+). Similarly, by adjunction, we have βH
+0 : WhG/G → γHWhG,H.
+Now by Proposition 2.8.6 below, αH
+0 and all αH
+n,i are isomorphisms. Thus unwinding the identities, we
+conclude the canonical resolution of WhG/G given by the monad T = γHγℓ
+H is precisely the resolution
+calculating the descended trace of WhG ∈ HG regarded as a coalgebra:
+WhG/G
+∼
+→ [γHγℓ
+HWhG/G
+�� (γHγℓ
+H)2WhG/G
+��� (γHγℓ
+H)3WhG/G · · · ]
+∼
+→ [γHγℓ
+HWhG/G
+�� γ(2)
+H γℓ
+H
+(2)WhG/G
+��� γ(3)
+H γℓ
+H
+(3)WhG/G · · · ]
+≃ [γHWhG,H
+�� γ(2)
+H Wh(2)
+G,H
+��� γ(3)
+H Wh(3)
+G,H · · · ]
+≃ [γWhG
+�� γ(WhG ⋆ WhG)
+��� γ(WhG ⋆ WhG ⋆ WhG) · · · ]
+□
+2.8.6. Proposition. The maps αH
+0 : γℓ
+HWhG/G → WhG,H and αH
+n,i : mℓ
+n+1,iWh(n)
+G,H → Wh(n+1)
+G,H , for all
+n ≥ 1 and 1 ≤ i ≤ n, are isomorphisms.
+
+28
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Proof. By standard identities, it suffices to prove this for αH
+0 and αH
+1,1. We will give an argument for αH
+0 ;
+a similar but simpler argument works for αH
+1,1.
+Set WG,H = Hom(WhG,H, −), WG/G = Hom(WhG/G, −). We will prove there is a canonical isomor-
+phism of functors
+WG,H(−)[−2ν] ≃ WG/G(γH(−)) : HG,H
+� k-mod
+where ν = dim N. In particular, we will then obtain γℓ
+H(WhG/G) ≃ WhG,H by adjunction.
+Consider the commutative diagram whose right hand square is Cartesian
+(2.8.3)
+U −\(G/U × G/U)/H
+s
+�
+(U − × B)/U −
+�δ
+�
+r
+�
+�π
+� U −/U −
+r
+�
+G\(G/U × G/U)/H
+(G × B)/G
+δ
+�
+π
+� G/G
+Here B = G/B is the flag variety of G, and G acts on G×B by the adjoint action on G and the translation
+action on B. Thus (G × B)/G is isomorphic to the quotient of G by the adjoint action of B. Similarly,
+U − acts on U − × B by the adjoint action on G and the translation action on B. From this perspective,
+π(g, g1) = g, �π(u, g1) = u, and δ(g, g1) = (gg1, g1), �δ(u, g1) = (ug1, g1).
+Set �f = χ ◦ ˜π : (U − × B)/U − → A1. By base change we have
+WG/G(τ(F)) ≃ φ0f∗r!π∗δ∗(F) ≃ φ0 �f∗r!δ∗(F).
+Since δ is smooth of relative dimension ν = dim N, we have δ∗(F) ≃ δ!F[−2ν]. Therefore
+(2.8.4)
+WG/G(τ(F)) ≃ φ0 �f∗r!δ!(F)[−2ν] ≃ φ0 �f∗�δ!s!F[−2ν].
+Now consider the leftward map �δ : (U − × B)/U − → U −\(G/U × G/U)/H at the top of (2.8.3).
+Stratify B by U −-orbits B = ⊔w∈W Bw where B1 = U −B/B denotes the open U −-orbit, and in general
+Bw = U − ˙w ≃ U −/U −
+w where U −
+w = U − ∩ wB where wB = ˙wB ˙w−1.
+Stratify (U −×B)/U − by the pullback of the U −-orbits on B. So we have the strata (U −×U −/U −
+w )/U −,
+in particular the open stratum (U − × U −)/U − ≃ U −. Stratify U −\(G/U × G/U)/H by the pullback of
+the U − × U −-orbits on B × B. So we have the strata U −\(U −/U −
+w × U −/U −
+w′)/H.
+The map �δ restricted to the w-stratum takes the form
+�δw : (U − × Bw)/U −
+� U −\(Bw ×BH Bw).
+We claim that for w ̸= 1, for any Fw ∈ Sh(U −\(Bw ×BH Bw)), we have
+(2.8.5)
+φ0 �f∗�δ!
+wFw ≃ 0.
+To prove the claim, note that w ̸= 1 implies U −
+w contains U−αi ≃ A1 for some simple root αi. We are
+studying the correspondence
+U −
+w \U −/U −
+w
+U −/U −
+w
+�δw
+�
+�π
+� U −/U −
+f
+� A1
+where the second and third quotients are by the adjoint action. Let U −
+0 /U − ⊂ U −/U − denote the pre-
+image of 0 ∈ A1. Then the action provides an isomorphism U − ≃ U−αi × U −
+0 with �f = f ◦ �π given simply
+by the projection to U−αi ≃ A1. Now U−αi ⊂ U −
+w implies that �π∗�δ!
+wFw is constant along U−αi ≃ A1, and
+hence φ0 �f∗π1∗�δ!
+wFw ≃ 0.
+By the claim, we have
+φ0 �f∗�δ!s!F ≃ φ0 �f∗�δ!
+1F1
+where F1 is the restriction of s!F to the open stratum U −\(B1 ×BH B1). Observe that the composition
+s ◦ �δ1 is nothing more than the composition
+U −
+r− � U\G/U
+q
+� (U\G/U)/H
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+29
+used to define WG,H.
+We conclude that
+φ0 �f∗�δ!s!F ≃ φ0�a∗�δ!
+1F1 ≃ WG,H(F).
+Combined with (2.8.4) we get a canonical isomorphism
+WG,H(F)[−2ν] ≃ WG/G(τ(F)).
+□
+Finally, we combine the spectral and automorphic realizations of the descended Whittaker object.
+Assuming Ansatz (1.2.5), taking cocenters of both sides of Φ we get an equivalence
+hh(Φ) : hh(IndCoh(StG∨)) ≃ hh(HG).
+Combined with (2.6.5), we get an equivalence
+(2.8.6)
+IndCohN (Z2
+G∨) ≃ hh(HG).
+Combining Corollary 2.6.10 and Theorem 2.8.5, we get:
+2.8.7. Corollary. Assume Ansatz (1.2.5).
+Under the equivalence (2.8.6), the structure sheaf OZ2
+G∨ ∈
+IndCohN (Z2
+G∨) corresponds to WhG/G ∈ hh(HG).
+3. Horocycle descent
+This section contains a proof of Theorem 2.7.10, or more precisely its generalization to Theorem 3.3.2.
+3.1. Preliminaries.
+3.1.1. Horocycle diagrams. Let G be a connected reductive group, B ⊂ G a Borel subgroup, N = [B, B]
+its unipotent radical, and H = B/N the universal Cartan.
+Consider the basic horocycle diagram
+(3.1.1)
+BG
+BB
+ǫ
+�
+δ
+� BB ×BH BB
+as a diagram over B(G × G) ≃ BG × BG. Note ǫ is smooth and proper, with fibers isomorphic to G/B,
+and δ is smooth, with fibers isomorphic to N.
+Let Z be an ind-stack with a G × G-action. We often turn the second G-action as a right action on Z.
+For subgroups G1 ⊂ G and G2 ⊂ G, we write G1\Z/G2 instead of Z/(G1 × G2).
+We can base-change diagram (3.1.1) along Z/(G × G) → B(G × G) and get a Z-horocycle diagram
+(3.1.2)
+Z/∆G
+Z/∆B
+δ
+�
+ǫ
+�
+(N\Z/N)/∆H
+where we write ∆G, ∆B to emphasize the diagonal action.
+3.1.2. Horocycle adjunctions. Let ν = dim N, and define functors
+hc! = δ!ǫ∗[−2ν] : Sh(Z/∆G) → Sh((N\Z/N)/∆H)
+ch = ǫ∗δ∗ = ǫ!δ∗ : Sh((N\Z/N)/∆H) → Sh(Z/∆G)
+hc∗ = δ∗ǫ! : Sh(Z/∆G) → Sh((N\Z/N)/∆H)
+Since ǫ is proper and δ is smooth of relative dimension ν, we have adjunctions (hc!, ch) and (ch, hc∗).
+Assume Z is smooth and let Λ ⊂ T ∗Z be a closed conic G × G-invariant subset. For any subgroup
+G′ ⊂ G × G, consider the full subcategory ShΛ(Z/G′) ⊂ Sh(Z/G′) of G′-equivariant complexes F on Z
+with singular support satisfying ss(F) ⊂ Λ.
+The following statement is a special case of [MV88, Lemma 1.2].
+3.1.3. Lemma. The functors hc! and hc∗ send ShΛ(Z/∆G) to ShΛ((N\Z/N)/∆H), and the functor ch
+sends ShΛ((N\Z/N)/∆H) to ShΛ(Z/∆G). In particular, if we let hcΛ,! : ShΛ(Z/∆G) → ShΛ((N\Z/N)/∆H)
+be the restriction of hc!, and similarly define chΛ and hcΛ,∗, then there are adjunctions (hcΛ,!, chΛ) and
+(chΛ, hcΛ,∗).
+
+30
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+3.1.4. Unit diagrams. Consider the basic unit diagram
+(3.1.3)
+B\G/B ≃ BB ×BG BB
+BB
+d
+�
+p
+� BH
+as a diagram over B(H × H) ≃ BH × BH. Here d is the relative diagonal and p is the natural projection.
+Note d is a closed embedding and p is smooth, with fibers isomorphic to BN.
+Let Z be an ind-stack with a H × H-action. We can base-change diagram (3.1.3) along H\Z/H →
+B(H × H) and get a Z-unit diagram
+(3.1.4)
+Z′ := Z ×H×H N\G/N
+Z ×H×H N\B/N ≃ Z/∆B
+p
+�
+d
+�
+Z/∆H
+where we write ∆H, ∆B to emphasize the diagonal action. In forming the middle term Z/∆B, the action
+of ∆B factors through ∆H.
+3.1.5. Unit adjunctions. Note p∗ : Sh(Z/∆H) → Sh(Z/∆B) is an equivalence since the ∆B-action on Z
+factors through ∆H and the kernel is the unipotent ∆N. Recall ν = dim N, so −ν = dim BN. Define
+functors
+uℓ = p!d∗[2ν] : Sh(Z′) → Sh(Z/∆H)
+u = d∗p∗ = d!p∗ : Sh(Z/∆H) → Sh(Z′)
+ur = p∗d! : Sh(Z′) → Sh(Z/∆H)
+Since d is proper and p is smooth of relative dimension −ν, we have adjunctions (uℓ, u) and (u, ur).
+Assume Z is smooth and let Λ ⊂ T ∗Z be a closed conic H × H-invariant subset. Consider the full
+subcategory ShΛ(Z/∆H) ⊂ Sh(Z/∆H) of ∆H-equivariant complexes F on Z with singular support
+satisfying ss(F) ⊂ Λ. Consider as well the full subcategory ShΛ(Z′) ⊂ Sh(Z′) of H × H-equivariant
+complexes F on Z × N\G/N with singular support satisfying ss(F) ⊂ Λ × N ′, where N ′ ⊂ T ∗(N\G/N)
+denotes the N × N-reduction of G × N ∗ ⊂ G × g∗ ≃ T ∗G.
+3.1.6. Lemma. The functors uℓ and ur send ShΛ(Z′) to ShΛ(Z/∆H), and the functor u sends ShΛ(Z/∆H)
+to ShΛ(Z′). In particular, if we let uΛ,ℓ : ShΛ(Z′) → ShΛ(Z/∆H) be the restriction of uℓ, and similarly
+define uΛ and uΛ,r, then there are adjunctions (uΛ,ℓ, uΛ) and (uΛ, uΛ,r).
+Proof. First, we show u respects the singular support conditions. Given F ∈ ShΛ(Z/∆H), viewed as a
+∆H-equivariant complex on Z, note p∗F ≃ F ⊠ F0 where F0 denotes the constant sheaf on N\B/N. Let
+d0 : BB → N\G/B be the closed embedding so that d = id×d0. Then ss(d!(F⊠F0)) = ss(F)×ss(d0!F0) ⊂
+Λ × N ′ since d0!F0 is H × H-bimonodromic hence has singular support in N ′.
+Finally, we show uℓ, ur respect the singular support conditions. Given F ∈ ShΛ(Z′), view F as an
+H ×H-equivariant complex on Z ×N\G/N. Using the estimate of singular support for pullbacks d∗F and
+d!F as in [KS90, Corollary 6.4.4, Remark 6.2.8], we see that ss(d∗F) and ss(d!F) are both contained in Λ
+when viewed as sheaves on Z. Therefore the same is true for p!d∗F and p∗d!F since p is an N-gerbe.
+□
+3.2. Descent for smooth stacks. Let Z be a smooth stack with a left G × G-action. We will write the
+first G-action as a left action and turn the second G-action as a right action.
+Let Λ ⊂ T ∗Z be a closed G × G-invariant subset such that under the moment map µ : T ∗Z → g∗ × g∗,
+we have µ(Λ) ⊂ N ∗ × N ∗ where N ∗ ⊂ g∗ denotes the nilcone in the dual to the Lie algebra.
+3.2.1. Example. In many situations of interest, one has Z = Z− × Z+, Λ = Λ− × Λ+, where Z± are
+smooth stacks with G-action, and Λ± ⊂ T ∗Z± is a closed G-invariant subset such that under the moment
+map µ± : T ∗Z± → g∗, we have µ±(Λ±) ⊂ N ∗. In this case, we view the action of G on Z− as a right
+action and the one on Z+ as a left action. Then the Z-horocycle diagram (3.1.2) and Z-unit diagram
+(3.1.4) take the form
+(3.2.1)
+Z− ×G Z+
+Z− ×B Z+
+δ
+�
+ǫ
+�
+Z−/N ×H N\Z+
+Z′ = Z− ×H N\G/N ×H Z+
+Z− ×H N\B/N ×H Z+ ≃ Z− ×B Z+
+p
+�
+d
+�
+Z− ×H Z+
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+31
+3.2.2. Example. A basic example is Z = G, with its natural G × G-action, and Λ = G × N ∗ ⊂ G × g∗.
+The category ShΛ(N\Z/N) as a HG-bimodule.
+The assumptions on Λ imply that any object of
+ShΛ(N\Z/N) is H × H-monodromic, therefore we can also regard ShΛ(N\Z/N) as a HG-bimodule in
+HH = Sh0(H)-bimodules since Sh0(H) = Mod(End(e)) ⊂ HG is the full monoidal subcategory generated
+by the monoidal unit e ∈ HG. Note that
+hh(HH, ShΛ(N\Z/N))
+∼
+� ShΛ((N\Z/N)/∆H).
+Here is the main technical theorem of this section. A generalization to “nilpotent categorical bimodules”
+will appear in [NYb]; in particular the assumption here that Λ ⊂ T ∗Z is Lagrangian is not necessary but
+allows the proof to call on the generalities of Proposition B.0.1.
+3.2.3. Theorem. Let Z be a smooth stack with a G × G-action. Let Λ ⊂ T ∗Z be a closed G × G-invariant
+conic Lagrangian such that under the moment map µ : T ∗Z → g∗ × g∗, we have µ(Λ) ⊂ N ∗ × N ∗.
+Then there is a canonical equivalence
+hh(HG, ShΛ(N\Z/N))
+∼
+� ShΛ(Z/∆G)
+such that the functor ch defined in Section 3.1.2 factors as the composition
+ch : ShΛ((N\Z/N)/∆H) ≃ hh(HH, ShΛ(N\Z/N))
+� hh(HG, ShΛ(N\Z/N)) ≃ ShΛ(Z/∆G).
+3.2.4. Example. In the setting of Example 3.2.1, the theorem gives an equivalence
+ShΛ+(Z−/N) ⊗HG ShΛ−(N\Z+)
+∼
+� ShΛ(Z− ×G Z+).
+3.2.5. Example. In the setting of Example 3.2.2, the theorem gives the equivalence
+hh(HG)
+∼
+� ShN (G/G)
+that we stated in Theorem 2.7.2.
+3.2.6. Remark. In [NYb] we will prove an abstract version of Theorem 3.2.3 where ShΛ(Z) is replaced with
+a Sh(G)-bimodule category satisfying a certain nilpotence condition reflecting that Λ maps to N ∗ × N ∗
+under the moment map. In particular, one can remove the assumption that Λ is a Lagrangian in Theorem
+3.2.3.
+To prove Theorem 3.2.3, we will identify each term of the (relative) Hochschild complex computing
+hh(HG, ShΛ(N\Z/N)) as sheaves on a certain space, and augment the Hochschild complex by adding
+the term ShΛ(Z/∆G). Finally we will apply Lurie’s criterion [Lur12, Corollary 4.7.6.3] to verify that the
+augmented Hochschild complex is a colimit diagram.
+3.2.7. Nonlinear diagrams. To start, we present some general patterns for constructing the (relative)
+Hochschild complex for spaces, following [BNb]. In the subsequent Section 3.2.9, we specialize to the case
+we will use in the proof of Theorem 3.2.3.
+Let K be an algebraic group. Let CorrK be the category of stacks with K-action with morphisms given
+by K-equivariant correspondences.
+Let CorrK×K be the monoidal category of stacks with K × K-action with morphisms given by K × K-
+equivariant correspondences. The monoidal structure on CorrK×K is given by U ⋆ V := U ×K V , where
+the quotient of K uses the second action of K on U and the first action of K on V . Note K with its
+regular K × K-action is the monoidal unit in CorrK×K.
+Note that CorrK is naturally a module category for CorrK×K: for U ∈ CorrK×K and V ∈ CorrK, we
+define the action of U on V to be U ⋆ V := U ×K V ∈ CorrK (quotient using the second action of K on
+U and the action of K on V ; the first action of K on U induces a K-action on U ⋆ V ).
+
+32
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Consider a diagram of stacks with Cartesian square
+(3.2.2)
+X0
+p0
+�
+b′
+� X
+f
+�
+p
+�
+Y
+pt
+b
+� BK
+In other words, we are given a stack X0 with K-action and a K-invariant map X0 → Y .
+Given (3.2.2), observe that A = X0 ×Y X0 is naturally an algebra object in CorrK×K with product
+given by the correspondence
+(3.2.3)
+A ⋆ A = (X0 ×Y X0) ×K (X0 ×Y X0)
+X0 ×Y X ×Y X0
+δ
+�
+πf
+� X0 ×Y X0 = A.
+Here δ is induced by the natural map X = X0/K → X0 ×K X0 of the middle factors (which in turn is
+induced by the diagonal map X0 → X0 × X0), and πf is induced by the projection of the middle factor
+f : X → Y . The unit of A is given by the correspondence
+(3.2.4)
+K = pt ×BK pt
+pt ×BK X ×BK pt ≃ X0 ×X X0
+ǫp
+�
+δf
+� X0 ×Y X0 = A.
+Here ǫp is induced by p : X → BK, and δf is induced by f : X → Y . Note swapping the factors of A gives
+an equivalence with its monoidal opposite.
+Given any map of stacks q : W → Y , observe that W0 := X0 ×Y W is naturally an A-module object in
+CorrK with action given by the correspondence
+(3.2.5)
+A ⋆ W0 = (X0 ×Y X0) ×K (X0 ×Y W)
+X0 ×Y X ×Y W
+δ
+�
+πf
+� X0 ×Y W = W0
+where δ and πf are defined in the same as in (3.2.3).
+Similarly, given any map of stacks q1 × q2 : W → Y × Y , W0 := X0 ×Y W ×Y X0 is naturally an
+A-bimodule object in CorrK×K.
+The following is elementary to check; we leave further details to the reader. We refer to Section 2.5 for
+the terminology of relative Hochschild complex, which we borrow here for the monoidal category CorrK×K.
+3.2.8. Lemma. Given any map of stacks q1 × q2 : W → Y × Y , we have an augmented simplicial object
+B(A, W)• in CorrK×K such that:
+(1) The underlying simplicial object of B(A, W)• is the relative Hochschild complex for the A-bimodule
+W0 in CorrK×K (relative to the unit object K ∈ CorrK×K which is also an algebra object, and the
+unit map K → A given in (3.2.4) is a map of algebras):
+· · ·
+��� W0 ×K×K A
+��
+�� W0/∆K
+�
+(2) The augmentation map B(A, W)0 → B(A, W)−1 is given by the correspondence
+W0/∆K = (X0 ×Y W ×Y X0)/∆K
+W ×Y ×Y X
+δ
+�
+πf
+� W ×Y ×Y Y
+where δ is induced by the natural map X = X0/K → X0 ×K X0, and πf is induced by f : X → Y .
+3.2.9. Our case of interest. We specialize the preceding constructions when the initial diagram (3.2.2)
+takes the form
+BN
+p0
+�
+b′
+� BB
+f
+�
+p
+�
+BG
+pt
+b
+� BH
+where the maps are the embeddings U ⊂ B ⊂ G and projection B ։ H. So here K is simply the universal
+Cartan H.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+33
+Inside of CorrH×H, we have the algebra A = BN ×BG BN ≃ N\G/N with the multiplication dia-
+gram (3.2.3) given by usual convolution
+A ⋆ A = (N\G/N) ×H (N\G/N)
+N\G ×B G/N
+δ
+�
+πf
+� N\G/N = A
+and the unit diagram (3.2.4) taking the form
+(3.2.6)
+H
+N\B/N
+ǫp
+�
+δf
+� N\G/N = A.
+Given a stack Z with G-action, applying the general discussion about A-modules to the induced map
+q : W = G\Z → BG, then we have W0 = N\Z ∈ CorrH is naturally an A-module with action (3.2.5)
+given by usual convolution
+A ⋆ W0 = (N\G/N) ×H (N\Z)
+N\G ×B Z
+δ
+�
+πf
+� N\Z = W0.
+Similarly, given a stack Z with G × G-action, applying the general discussion about A-bimodules to the
+map q1 × q2 : W = G\Z/G → BG × BG, then W0 = N\Z/N ∈ CorrH×H is naturally an A-bimodule
+object.
+Now Lemma 3.2.8 takes the following form.
+3.2.10. Lemma. Given a stack Z with G×G-action, we have an augmented simplicial object B(A, Z)• (in
+the notation of Lemma 3.2.8 it should be called B(A, W)•, where W = G\Z/G) in CorrH×H such that:
+(1) The underlying simplicial object of B(A, Z)• is the relative Hochschild complex of the A-bimodule
+N\Z/N (relative to the algebra map H → A in CorrH×H given by the unit diagram (3.2.6)):
+· · ·
+��� (N\Z/N) ×H×H (N\G/N)
+��
+�� (N\Z/N)/∆H.
+�
+(2) The augmentation map B(A, Z)0 → B(A, Z)−1 is given by the horocycle correspondence
+(N\Z/N)/∆H
+Z/∆B
+δ
+�
+πf
+� Z/∆G.
+3.2.11. Augmented Hochschild complex. For the next step towards the proof of Theorem 3.2.3, we shall
+take the categories of sheaves termwise for the augmented simplicial object B(A, Z)• in CorrH×H provided
+by Lemma 3.2.10, and impose singular support conditions to obtain the augmented Hochschild complex
+for the HG-bimodule ShΛ(N\Z/N).
+Consider the monoidal category HH = Sh0(H). For any X ∈ CorrH×H, Sh(X) is a bimodule for HH.
+Thanks to [GR19], passing to categories of sheaves gives a monoidal functor
+CorrH×H → BimodHH(St)
+where the target is the 2-category of stable presentable ∞-categories that are HH-bimodules (and the
+monoidal structure is tensor product over the middle copy of HH). For a morphism from X to Y in
+CorrH×H, i.e., a H × H-equivariant correspondence
+X
+C
+p
+�
+q
+� Y
+the functor Sh(X) → Sh(Y ) is given by q!p∗. Passing to the categories of all sheaves termwise for B(A, Z)•.
+We obtain an augmented simplicial object Sh(B(A, Z))• in BimodHH(St).
+Now we impose singular support conditions. Let Λ−1 ⊂ T ∗(Z/∆G) be the ∆G-reduction of Λ. For
+n ≥ 0, B(A, Z)n can be written as the quotient
+B(A, Z)n ≃ (Z × Gn)/(B ×H B)n+1
+where for n ≥ 1, the ith factor (0 ≤ i ≤ n) of B ×H B acts on Z × Gn by
+(b, b′) · (z, g1, · · · , gn) =
+
+
+
+
+
+(zb, b′−1g1, · · · , gn)
+i = 0,
+(z, · · · , gib, b′−1gi+1, · · · , gn)
+1 ≤ i ≤ n − 1
+(b′−1z, g1, · · · , gn−1, gnb)
+i = n.
+
+34
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+For n = 0 the action of B ×H B acts on Z is by (b, b′) · z = b′−1zb.
+For n ≥ 0, set Λn ⊂ T ∗B(A, Z)n ≃ T ∗((Z × Gn)/(B ×H B)n+1) be the reduction of Λ × (G × N ∗)n.
+3.2.12. Lemma. The augmented simplicial category Sh(B(A, Z))• restricts to an augmented simplicial
+object C• in BimodHH(StL
+k ) with terms Cn = ShΛn(B(A, Z)n) for n ≥ −1. Moreover:
+(1) Let M = ShΛ(N\Z/N). Then the underlying simplicial object of of C• is equivalent to the relative
+Hochschild complex B(HG, M)•:
+· · ·
+��� M ⊗HH⊗Hop
+H HG
+��
+�� M ⊗HH⊗Hop
+H HH.
+�
+(2) The augmentation map C0 → C−1 is given by the transform
+ch : M ⊗HH⊗Hop
+H HH ≃ ShΛ0((N\Z/N)/∆H)
+� ShΛ−1(Z/∆G)
+associated to the horocycle transform construction (3.1.2) applied to Z.
+Proof. The description of Cn in terms of M and HG follows from the categorical K¨unneth formula for
+sheaves with prescribed singular support conditions on twisted products X1 ×H X2, see [NYa, Lemma
+A.4.3]. It remains to check that the prescribed singular supports are respected by the given functors in
+Sh(B(A, Z))•.
+For n ≥ 0, and the injection ϕ : [n − 1] → [n] whose image misses i, the corresponding face map of
+B(A, Z)• is given by the functor ch resulting from the horocycle transform construction (3.1.2) applied
+to Zi
+n = (Z × Gn)/Bi
+n. Here Bi
+n ⊂ (B ×H B)n+1 is the subgroup where we replace the ith factor by the
+trivial group and keep the other factors unchanged. The G × G-action on Zi
+n is the natural action along
+where the ith factor of (B ×H B)n+1 originally acted. By Lemma 3.1.3, the associated horocycle functors,
+in particular the face map ch, respect the prescribed singular support.
+Similarly, for n ≥ 0, and the surjection ϕ : [n + 1] → [n] that identifies i and i + 1, the corresponding
+degeneracy map of B(A, Z)• respects the singular support as follow. Observe the degeneracy map is given
+by the unit functor u (see Section 3.1.5) resulting from the unit transform construction (3.1.4) applied to
+Zn,i = (Z × Gn)/Bn,i. Here Bn,i ⊂ (B ×H B)n+1 is the subgroup where we replace the ith factor by the
+group N × N and keep the other factors unchanged. The H × H-action on Zn,i is the natural action along
+where the ith factor of (B ×H B)n+1 originally acted. By Lemma 3.1.6, the associated unit functors, in
+particular the degeneracy map u, respect the prescribed singular support.
+□
+3.2.13. Finish of the proof of Theorem 3.2.3. To prove Theorem 3.2.3, it remains to prove that the the
+augmented simplicial object C• exhibits C−1 = ShΛ(Z/∆G) as the colimit of the underlying simplicial
+object of C•, i.e., the Hochschild complex B(HG, M)•. Equivalently, letting C• be the augmented cosim-
+plicial object obtained from C• by passing to right adjoints (note these are available by Lemmas 3.1.3 and
+3.1.6), it suffices to show that C• exhibits C−1 as the limit of the underlying cosimplicial object {Cn}n≥0.
+For this it suffices to check that C• satisfies the following strong form of the criteria of [Lur12, Corollary
+4.7.6.3]:
+(1) The augmentation map d0
+−1 = hc∗ : C−1 → C0 is: (a) conservative and (b) continuous, i.e.
+preserves colimits;
+(2) The following commutative squares are left adjointable for any order-preserving map α : [m] → [n]
+(where m, n ≥ −1)
+Cm
+α
+�
+d0
+m � Cm+1
+α′
+�
+Cn
+d0
+n
+� Cn+1
+Here d0
+m is the inclusion [m] → [m + 1] that whose image misses 0; d0
+n is defined similarly; and
+α′ : [m + 1] → [n + 1] is the map defined by α′(0) = 0 and α′(i + 1) = α(i) + 1 for i ∈ [m].
+Left adjointability of the above square means the face maps d0
+m, d0
+n admit respective left adjoints
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+35
+(d0
+m)ℓ, (d0
+n)ℓ (which in our case are already given by the construction of d0
+m, d0
+n as right adjoints),
+and the associated base change map is an equivalence
+(d0
+n)ℓ ◦ α′
+∼
+� α ◦ (d0
+m)ℓ.
+(1a) First, we check d0
+−1 = hc∗ is conservative. It suffices to show that ch ◦ hc∗ contains the identity
+functor as a direct summand. This is essentially [MV88, Theorem 3.6].
+We use notation from the diagram (3.1.2). By definition ch ◦ hc∗ = ǫ!δ∗δ∗ǫ!. The fiber square along δ
+can be identified with
+Z/∆B
+(Z × N)/∆B
+p1
+�
+a1
+�
+Z/∆B.
+Here the ∆B-action on N is by conjugation. The map a1 is the action map of N on Z via N ×{1} → G×G,
+and p1 is the projection. Since δ is smooth of relative dimension ν = dim N, δ∗δ∗ ∼= p1∗a∗
+1 ∼= p1∗a!
+1[−2ν].
+Hence
+ch ◦ hc∗ ∼= ǫ∗p1∗a!
+1ǫ![−2ν] = p′
+1∗a′!
+1[−2ν]
+where p′
+1 = ǫ ◦ p1, a′
+1 = ǫ ◦ a1. We have a commutative diagram
+(Z × N)/∆B
+π
+�
+p′
+1
+�◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+a′
+1
+�qqqqqqqqqqq
+Z/∆G
+(Z × G)/∆G
+π1
+�
+α1
+�
+Z/∆B
+Here the ∆G-action (resp. ∆B-action) on G (resp. N) is by conjugation, π1 is projection, and α1 is the
+action map of G on Z via G × {1} → G × G. The map π is the base change of the Springer resolution
+N
+Ad(B) →
+G
+Ad(G). Hence for F ∈ ShΛ(Z/∆G), we have
+ch(hc∗(F)) ∼= p′
+1∗a′!
+1F[−2ν] ∼= π1∗π∗π!α!
+1F[−2ν] ∼= π1∗Hom(π!k, α!
+1F)[−2ν].
+The Springer sheaf contains the skyscraper sheaf at 1 ∈ G as a direct summand, hence π!k contains
+i∗k[−2ν] as a direct summand, where i : Z/∆G ֒→ (Z × G)/∆G corresponds to the inclusion of 1 into G.
+Therefore ch(hc∗(F)) contains as a direct summand
+π1∗Hom(i∗k[−2ν], α!F)[−2ν] ∼= π1∗i∗i!α!F ∼= F.
+We have shown that ch ◦ hc∗ contains the identity functor as a direct summand.
+(1b) follows from Proposition B.0.1.
+(2) We will check the required left adjointability for the augmentation map α = d0
+−1 : [−1] → [0]; the
+verification for other maps is similar.
+Consider the relevant categories and functors
+ShΛ(Z/∆G)
+hc∗
+� ShΛ((N\Z/N)/∆H)
+ch
+�
+hc∗0 � ShΛ(N\Z/N) ⊗HH⊗Hop
+H HG.
+ch1
+�
+Here ch1 is the HG-action on ShΛ((N\Z/N)/∆H) induced by the right G-action on Z, and hc∗0 is right
+adjoint to the HG-action on ShΛ((N\Z/N)/∆H) induced by the left G-action on Z.
+We seek to show the natural adjunction transformation
+(3.2.7)
+τ : ch1 ◦ hc∗0
+� hc∗ ◦ ch
+is an equivalence. Indeed, by proper base change, hc∗ ◦ ch = δ∗ǫ!ǫ!δ∗ can be identified with the functor
+c∗p![−2ν] constructed from the diagram
+(3.2.8)
+(N\Z/N)/∆H
+∆B\(Z × G/B)
+c
+�
+p
+�
+(N\Z/N)/∆H
+where in the middle term, b ∈ ∆B acts by b(z, gB) = (bzb−1, bgB), and the maps p and c are defined by
+p(z, g) = z and c(z, g) = gzg−1.
+
+36
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+On the other hand, ch1 ◦ hc∗0 can be identified with the functor m1∗m!
+0[−2ν] constructed from the
+diagram
+(3.2.9)
+(N\Z/N)/∆H
+Z×BG
+∆B
+m1 �
+m0
+�
+(N\Z/N)/∆H
+where in the middle term, b ∈ ∆B acts by b(z, g) = (bz, gb−1), and the maps m0 and m1 are defined by
+m0(z, g) = gz and m1(z, g) = zg.
+We have an isomorphism between the two diagrams (3.2.8) and (3.2.9) that is the identity on (N\Z/N)/∆H
+and on the middle terms it takes the form (z, gB) �→ (g−1z, g) ∈
+Z×BG
+∆B . This isomorphism identifies
+c∗p![−2ν] with m1∗m!
+0[−2ν]. One checks that τ is the composition
+ch1 ◦ hc∗0 ≃ m1∗m!
+0[−2ν] ≃ c∗p![−2ν] ≃ hc∗ ◦ ch.
+Therefore τ is an equivalence. This concludes the proof of Theorem 3.2.3.
+3.3. Singular and ind-version. We will generalize Theorem 3.2.3 to the situation of stratified ind-stacks
+(Theorem 3.3.2).
+3.3.1. Setup. Let Z be an ind-stack equipped with a partition into smooth locally closed substacks Z =
+⊔α∈P Z◦
+α indexed by a poset P. For α ∈ P, assume {β ∈ P|β < α} is finite and Zα := ∪β≤αZ◦
+β is closed
+in Z. Let i◦
+α : Z◦
+α → Z be the inclusion.
+Assume Z has a G × G-action preserving each Zα ⊂ Z. Let Λα ⊂ T ∗Z◦
+α be a closed G × G-invariant
+conic Lagrangian such that under the moment map µ : T ∗Z◦
+α → g∗ × g∗, we have µ(Λα) ⊂ N ∗ × N ∗.
+Assume for β < α ∈ P, the composition (i◦
+β)∗(i◦
+α)∗ : Sh(Z◦
+α) → Sh(Z◦
+β) takes ShΛα(Z◦
+α) to ShΛβ(Z◦
+β).
+Define ShΛ(Z) to be the full subcategory of objects F such that (i◦
+α)∗F ∈ ShΛα(Z◦
+α), for all α ∈ P.
+For any Q ⊂ P that is locally down-closed, i.e., Q = Q1 \ Q2 where Q1 and Q2 are down-closed,
+let ZQ = ∪α∈QZ◦
+α be the corresponding locally closed substack of Z, we can define the full subcategory
+ShΛ(ZQ) ⊂ Sh(ZQ) in the same way by requiring objects to have image under (i◦
+α)∗ lying in ShΛα(Z◦
+α).
+Now for a locally down-closed Q and a downclosed subset Q′ ⊂ Q with complement Q′′ = Q \ Q′, the
+assumptions guarantee that the usual functors in the recollement diagram of Sh(ZQ), Sh(ZQ′) and Sh(ZQ′′)
+restrict to a recollement diagram
+ShΛ(ZQ′′)
+j!
+�
+j∗
+�
+ShΛ(ZQ)
+j!=j∗
+�
+i∗
+�
+i!
+�
+ShΛ(ZQ′)
+i∗=i!
+�
+where i : ZQ′ ֒→ ZQ and j : ZQ′′ ֒→ ZQ are the closed and open inclusions. From this we see that ShΛ(Z)
+admits a stratification indexed by P in the sense of Section A.1.2, with strata categories ShΛα(Z◦
+α) for
+α ∈ P.
+For any subgroup G′ ⊂ G×G, let ShΛ(Z/G′) ⊂ Sh(Z/G′) denote the full subcategory of G′-equivariant
+complexes F on Z whose underlying object lies in ShΛ(Z). The assumptions on Λ imply that any object
+of ShΛ(N\Z/N) is H ×H-monodromic. The G×G actions on Z equip ShΛ(N\Z/N) with a HG-bimodule
+structure.
+3.3.2. Theorem. With the above setup, there is a canonical equivalence of stable ∞-categories
+hh(HG, ShΛ(N\Z/N))
+∼
+� ShΛ(Z/∆G)
+such that the functor ch defined in Section 3.1.2 factors as the composition
+ch : ShΛ((N\Z/N)/∆H) ≃ hh(HH, ShΛ(N\Z/N))
+� hh(HG, ShΛ(N\Z/N)) ≃ ShΛ(Z/∆G).
+Proof. We explain that the steps in the proof of Theorem 3.2.3 can be made to work for the ind-stack
+Z with slight modifications. The augmented simplicial diagram in Lemma 3.2.10 makes sense for the
+ind-stack Z. Now for each term B(A, Z)n = (Z × Gn)/(B ×H B)n+1 (where n ≥ 0), we assign the full
+subcategory category Cn ⊂ Sh(B(A, Z)n) consisting of objects F whose pullback to Z◦
+α × Gn has singular
+support contained in Λα × (G × N ∗)n, for all i ∈ P. Let C−1 = ShΛ(Z/∆G) as is already defined.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+37
+With this definition of C• ∈ BimodHH(StL
+k ), we claim that the statement of Lemma 3.2.12 holds. We
+need to check the maps in the augmented simplicial object Sh(B(A, Z)•) preserve the subcategories C•.
+For α ∈ P, applying Lemma 3.2.12 to Z◦
+α together with the Lagrangian Λα, we get an augmented
+simplicial object Cα,• in BimodHH(StL
+k ) whose terms are ShΛα,n(B(A, Z◦
+α)n). Let i◦
+α,n : B(A, Z◦
+α)n ֒→
+B(A, Z)n be the locally closed embedding induced by i◦
+α.
+We claim that the functors i◦
+α,n! : Cα,n →
+Sh(B(A, Z)n) induce a functor of augmented simplicial objects
+i◦
+α,•! : Cα,• → Sh(B(A, Z)•).
+Indeed, for an injection ϕ : [n − 1] → [n], the corresponding face maps chα,ϕ : Cα,n → Cα,n−1 and
+chϕ : Sh(B(A, Z)n) → Sh(B(A, Z)n−1) are given by special cases of the ch functor defined in Section 3.1.2.
+The isomorphism
+chϕ ◦ i◦
+α,n! ≃ i◦
+α,n−1! ◦ chα,ϕ
+follows from proper base change. For a surjection ϕ : [n + 1] → [n], the corresponding degeneracy maps
+uα,ϕ : Cα,n → Cα,n+1 and uϕ : Sh(B(A, Z)n) → Sh(B(A, Z)n+1) are given by special cases of the unit
+functor u defined in Section 3.1.5. The isomorphism
+uϕ ◦ i◦
+α,n! ≃ i◦
+α,n+1! ◦ uα,ϕ
+again follows from proper base change.
+Now observe that for n ≥ −1, Cn ⊂ Sh(B(A, Z)n) is generated by the images of i◦
+α,n! (for α ∈ P) under
+colimits. We have checked above that the face and degeneracy maps for Sh(B(A, Z)•) preserve the images
+of i◦
+α,•! for fixed α ∈ P. Since the face and degeneracy maps are continuous (they are left adjoints), they
+preserve C•, therefore we get an augmented simplicial object C•, and the analog of Lemma 3.2.12 holds.
+The last step is to check that the augmented cosimplicial object C•, obtained from C• by passing to
+right adjoints, satisfies Lurie’s criterion [Lur12, Corollary 4.7.6.3] for a limit diagram. The same argument
+as in Section 3.2.13 works for the current situation without change.
+□
+4. Harder-Narasimhan subcategories
+This main result of this section is Theorem 4.4.7 giving a recollement structure on the cocenter hh(HG)
+of the universal affine Hecke category HG. The properties of this recollement mirror those of the recollement
+structure on the Betti Langlands automorphic category ShN (BunG(E)) for a genus one curve E induced
+by the Harder-Narasimhan stratification of BunG(E). Most basically, the filtrations of both recollements
+are indexed by Newton points NP ⊂ X∗(T )+
+Q (see Section 4.1.10). In a sequel, we will prove that hh(HG)
+is equivalent to ShN (BunG(E)) so that the recollements match. In this paper, we will focus on the specific
+consequence stated in Theorem 4.4.2 that the associated graded for the minimum index 0 ∈ NP embeds
+fully faithfully in hh(HG).
+4.1. Combinatorial pieces. Our starting point for the analysis of hh(HG) is the colimit description of
+Corollary 2.7.11:
+colimD HG,J
+∼
+� hh(HG)
+We will begin with the group theory of B´edard and Lusztig [Lusa] underlying the natural decomposition
+of each HG,J into pieces.
+4.1.1. Combinatorial pieces. Let W a be the affine Weyl group of G and �
+W = X∗(T ) ⋊ W be the extended
+affine Weyl group. For J ⊂ft Ia, let WJ ⊂ W a be the subgroup generated by J; it is the Weyl group of
+the Levi LJ of the parahoric subgroup PJ. Let J�
+W (resp. �
+W J) be the set of minimal length elements in
+the cosets WJ\�
+W (resp. �
+W/WJ). Let J �
+W J′ be the set of minimal length elements in the double cosets
+WJ\�
+W/WJ′.
+For a group Γ and a subgroup Γ′ ⊂ Γ, let Γ
+Γ′ denote the set of Γ′-conjugacy classes in Γ. More generally,
+if δ is an automorphism of Γ, let
+Γ
+Adδ(Γ′) denote the set of orbits of Γ′ acting on Γ by twisted conjugation
+γ′ · γ = γ′γδ(γ′−1). If γ ∈ Γ, we use
+Γ
+Adγ(Γ′) to mean the
+Γ
+AdAd(γ)(Γ′).
+
+38
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Let us first review a combinatorial procedure of B´edard, as found in [Lusa]. Let J ⊂ft Ia. Let SJ be
+the set of sequences (Jn, J′
+n, un)n≥0 such that
+(1) J0 = J and Jn = Jn−1 ∩ Ad(u0 · · · un−1)Jn−1 for n ≥ 1.
+(2) J′
+0 = J and J′
+n = Jn−1 ∩ Ad(u0 · · · un−1)−1Jn−1 for n ≥ 1.
+(3) un ∈ J′
+n(WJn−1)Jn for n ≥ 0.
+We call elements in SJ combinatorial J-pieces. Note that Jn and J′
+n stabilize for n large hence un = 1 for
+n large. As explained in [Lusa, Prop. 2.5], the map (Jn, J′
+n, un)n≥0 �→ u0u1 · · · um (for m ≫ 0) defines a
+bijection
+(4.1.1)
+SJ
+∼
+→ J �
+W.
+For u ∈ J�
+W, we denote the corresponding element in SJ by
+u
+J ∈ SJ.
+4.1.2. The map σJ. For any J ⊂ft Ia, we define a map
+σJ :
+�
+W
+WJ
+→ SJ
+as follows. For c ∈
+�
+W
+WJ , we set c0 = c, J0 = J′
+0 = J and u0 ∈ J �
+W J to be the WJ-double coset containing the
+WJ-conjugacy class c. We will inductively construct a sequence (cn, Jn, J′
+n, un)n≥0 satisfying the following
+conditions:
+(1) (Jn, J′
+n, un)n≥0 ∈ SJ.
+(2) For n ≥ 1, cn ∈
+WJn−1
+Adu0···un−1 (WJ′n) is characterized by
+(4.1.2)
+cn = {x ∈ WJn−1|u0 · · · un−1x ∈ c}.
+(3) For n ≥ 1, un ∈ J′
+n(WJn−1)Jn is the minimal length element in the (WJ′n, WJn)-double coset of cn.
+These conditions clearly characterize (cn, Jn, J′
+n, un)n≥0 uniquely, given the initial terms (c0 = 0, J0 =
+J, J′
+0 = J, u0) as above. To confirm this inductive procedure is well-defined, we need to show that, given
+(ci, Ji, J′
+i, ui)0≤i≤n−1 satisfying the above conditions (so that Jn and J′
+n can already be defined using Jn−1
+and u0 · · · un−1 as dictated by the requirement that (Jn, J′
+n, un)n≥0 ∈ SJ), the set cn defined using (4.1.2)
+is non-empty and consists of a single (u0 · · · un−1)-twisted conjugacy class under WJ′n.
+First, to see cn is non-empty: from the inductive hypothesis, we know that for x ∈ WJn−2 (we understand
+WJ−1 to be �
+W), u0 · · · un−2x ∈ c if and only if x ∈ cn−1. From the construction of un−1, we see any x ∈ cn−1
+can be written as x = aun−1b for some a ∈ WJ′
+n−1 and b ∈ WJn−1 = Ad(u0 · · · un−2)WJ′
+n−1. Take any such
+x = aun−1b and
+u0 · · · un−2x = u0 · · · un−2aun−1b = a′u0 · · · un−1b
+where a′ = Ad(u0 · · · un−2)a ∈ WJn−1. The above element then lies in the same WJn−1-conjugacy class as
+u0 · · · un−1ba′. This implies u0 · · · un−1WJn−1 ∩ c ̸= ∅.
+Next, to see cn is a single (u0 · · · un−1)-twisted conjugacy class under WJ′
+n: suppose y, y′ ∈ WJn−1 are
+such that u0 · · · un−1y, u0 · · · un−1y′ ∈ c. By the inductive hypothesis, un−1y, un−1y′ ∈ cn−1, hence there
+exists z ∈ WJ′
+n−1 such that un−1y = zun−1y′Ad(u0 · · · un−2)z−1. Let z′ = u−1
+n−1zun−1. Then
+(4.1.3)
+y = u−1
+n−1zun−1y′Ad(u0 · · · un−2)z−1 = z′y′Ad(u0 · · · un−1)z′−1.
+Now z′ = yAd(u0 · · · un−2)zy′−1. Note z′ ∈ Ad(u−1
+n−1)WJ′
+n−1 and yAd(u0 · · · un−2)zy′−1 ∈ WJn−1. There-
+fore z′ ∈ Ad(u−1
+n−1)WJ′
+n−1 ∩ WJn−1. Since u−1
+n−1 ∈ Jn−1(WJn−2)J′
+n−1, we have Ad(u−1
+n−1)WJ′
+n−1 ∩ WJn−1 =
+WAd(un−1)−1J′
+n−1∩Jn−1.
+Note that Ad(un−1)−1J′
+n−1 = Ad(u0 · · · un−1)−1Jn−1 hence Ad(u−1
+n−1)J′
+n−1 ∩
+Jn−1 = Ad(u0 · · · un−1)−1Jn−1 ∩ Jn−1 = J′
+n.
+We conclude that z′ ∈ WJ′
+n.
+The equation (4.1.3) im-
+plies y and y′ lie in the same (u0 · · · un−1)-twisted conjugacy class of WJ′n in WJn−1.
+This completes the definition of the map σJ.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+39
+Composing σJ with the bijection (4.1.1) we get a map
+pJ :
+�
+W
+WJ
+σJ
+−−→ SJ
+∼
+→ J�
+W.
+4.1.3. Lemma. Let (Jn, J′
+n, un)n≥0 ∈ SJ and u = u0u1 · · · un ∈ J �
+W (for large n) be its image under the
+bijection (4.1.1). Let K be the stable value of Jn.
+(1) (Compare [He07, Lemma 1.4]) K is the largest subset of J that is stable under Ad(u), and K is
+also the stable value of J′
+n.
+We denote K by I(J, u).
+(2) The preimage of u
+J under σJ consists of WJ-conjugacy classes of uy where y ∈ WK. Two elements
+uy and uy′ (y, y′ ∈ WK) are in the same WJ-conjugacy class if and only if y, y′ lie in the same
+u-twisted conjugacy class of WK, i.e., y �→ uy gives a canonical bijection
+uWK
+WK
+∼=
+WK
+Adu(WK)
+∼
+→ σ−1
+J ( u
+J ).
+Proof. (1) For n large so that Jn−1 = K, we have Jn = Jn−1 ∩ Ad(u)Jn−1 and Jn−1 = Jn, hence Jn is
+stable under Ad(u). Also J′
+n = Jn−1 ∩ Ad(u)−1Jn−1 = Jn−1 = K.
+Conversely, if K1 ⊂ J is stable under Ad(u), we show inductively that K1 ⊂ Jn for all n ≥ 0, hence
+K1 ⊂ K = I(J, u). For n = 0, this is clear. Assume K1 ⊂ Jn−1 for some n ≥ 1. Let s ∈ K and let αs
+be the corresponding simple root. For K′ ⊂ft Ia, let ΦK′ be the sub root system of the affine roots of
+G spanned by K′. Since K1 is stable under Ad(u), αi = uαi′ for some αi ∈ K1. Write u = u0 · · · un−1x
+where x ∈ WJn−1, then αi = u0 · · · un−1β where β = xαi′ ∈ ΦJn−1. Since u0 · · · un−1 ∈ �
+W Jn−1, u0 · · · un−1
+sends Φ+
+Jn−1 to positive roots, therefore β is also a simple root in ΦJn−1 for otherwise u0 · · · un−1β cannot
+be a simple root.
+The equality αi = u0 · · · un−1β then implies αi ∈ Ad(u0 · · · un−1)Jn−1.
+Therefore
+αi ∈ Jn−1 ∩ Ad(u0 · · · un−1)Jn−1 = Jn for any αi ∈ K1, hence K1 ⊂ Jn.
+(2) is immediate from the construction of σJ.
+□
+4.1.4. Relevant affine subspaces. Let A be the standard apartment with the action of �
+W. By a relevant
+affine subspace of A, we will mean the intersection of a set of affine root hyperplanes. Let E be the set
+of relevant affine subspaces of A, and let E be the set of W a-orbits on E. Each subset K ⊂ft Ia gives
+a relevant affine subspace A(K) = {x ∈ A|α(x) = 0, ∀α ∈ K}. Every relevant affine subspace is W a-
+conjugate to one of the form A(K). This induces a surjection {K ⊂ft Ia} ։ E. This map may not be a
+bijection in general: for K, K′ ⊂ft Ia, A(K) is in the W a-orbit of A(K′) if and only if there exists w ∈ W a
+such that wKw−1 = K′.
+For E ∈ E, we denote its image in E by [E]. The set E is partially ordered such that [E] ≥ [E′] if and
+only if E ⊃ wE′ for some w ∈ W a.
+For J ⊂ft Ia, let EJ be the subset of E consisting of relevant affine subspaces that contain A(J). Clearly
+WJ acts on EJ. For K ⊂ J, we have A(K) ∈ EJ.
+4.1.5. Definition. Let J ⊂ft Ia and u
+J ∈ SJ. Let K = I(J, u) ⊂ J as specified in Lemma 4.1.3.
+(1) We define the J-type of u
+J to be the element τJ( u
+J ) ∈ WJ\EJ given by the image of A(K), i.e., the
+relevant affine space A(K) up to WJ-action.
+(2) We define the coarse type of u
+J to be the element τ( u
+J ) ∈ E = W a\E given by the image of A(K),
+i.e., the relevant affine space A(K) up to W a-action.
+Taking the coarse type of a J-piece defines a map
+τ : S :=
+�
+J⊂ftIa
+SJ → E.
+Next, we give a way to compute the J-type starting from any WJ-conjugacy class of �
+W. For w ∈ �
+W and
+J ⊂ft Ia, the set Ew
+J := {E ∈ EJ|w(E) = E} is non-empty (since A ∈ Ew
+J ) and closed under intersection,
+hence has a unique minimal element which we denote by EJ,w. The next lemma explains the relation
+between J-type and EJ,w.
+
+40
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+4.1.6. Lemma. Let J ⊂ft Ia, u ∈ J �
+W.
+(1) Let K = I(J, u), then EJ,uy = A(K) for any y ∈ WK.
+(2) Suppose w ∈ �
+W is such that σJ(w) = u
+J ∈ SJ. Then the J-type of u
+J is the WJ-orbit of EJ,w.
+(3) Suppose w, w′ ∈ �
+W. Then σJ(w) = σJ(w′) if and only if w′ is WJ-conjugate to an element w′′
+with EJ,w = EJ,w′′ and w|EJ,w = w′′|EJ,w′′ .
+Proof. Let K = I(J, u). Then the J-type of u
+J is the WJ-orbit of A(K). By Lemma 4.1.3, w = xuyx−1
+where y ∈ WK and x ∈ WJ. We have EJ,w = xEJ,uy. Therefore it suffices to consider the case w = uy.
+Hence (2) follows from (1).
+We prove (1). Since Ad(u)K = K, u stabilizes A(K), hence uy stabilizes A(K). By the minimality of
+E := EJ,uy, we have E ⊂ A(K). We claim that E = A(K). Let F be the fundamental alcove. Consider
+the relevant affine subspace Span(E ∩ F) spanned by E ∩ F. We have Span(E ∩ F) = A(K′) for some
+K′ ⊂ft Ia. Since A(J) ⊂ E ⊂ A(K), we have K ⊂ K′ ⊂ J. Note that A(K′) = Span(E ∩ CJ), where
+CJ ⊂ A is the dominant WJ-chamber centered along A(J). On the other hand, since u ∈ J �
+W, we have
+uF ∈ CJ, and therefore Span(uyE ∩ uF) = Span(E ∩ uF) ⊂ Span(E ∩ CJ) = A(K′). The action of
+u sends Span(E ∩ F) = Span(yE ∩ F) (note that y ∈ WK acts by identity on A(K)) isomorphically
+to Span(uyE ∩ uF) = Span(E ∩ uF), hence we must have Span(E ∩ uF) = A(K′), and in particular
+u stabilizes A(K′).
+Now u stabilizes the affine roots ΦK′ spanned by K′ ⊂ J, and u ∈ J �
+W implies
+that u−1 sends positive roots Φ+
+J ⊂ ΦJ to positive affine roots, therefore u preserves Φ+
+K′, and hence
+preserves K′. By the maximality of K as a u-stable subset of J, we must have K′ = K. Therefore,
+E ⊃ Span(E ∩ F) = A(K′) = A(K), forcing E = A(K).
+We prove (3). The statement is invariant under WJ-conjugation of w and w′ separately. Therefore
+we may assume w = u. Let K = I(J, u). Suppose w′ is WJ-conjugate to w′′ with A(K) = EJ,w′′ and
+w|A(K) = w′′|A(K). Then w′′ = uy for some y ∈ WK, hence σJ(w′) = σJ(w′′) = u
+J . Conversely, suppose
+σJ(w′) = u
+J , then by Lemma 4.1.3, w′ is WJ-conjugate to uy for some y ∈ WK, hence EJ,uy = A(K) by
+(1) and uy|A(K) = u|A(K) since y ∈ WK acts by identity on A(K).
+□
+4.1.7. Proposition. Let J ⊂ J′ ⊂ft Ia. Let πJ′
+J :
+�
+W
+WJ →
+�
+W
+WJ′ be the projection. Then there is a unique
+map δJ′
+J : SJ → SJ′ making the following diagram commutative
+�
+W
+WJ
+σJ
+�
+πJ′
+J
+�
+�
+W
+WJ′
+σJ′
+�
+SJ
+δJ′
+J
+� SJ′
+In particular, for J ⊂ J′ ⊂ J′′ ⊂ft Ia, δJ′′
+J′ ◦ δJ′
+J = δJ′′
+J .
+Proof. Since σJ is surjective, δJ′
+J is unique if it exists.
+For the existence of δJ′
+J , we need to show the following. Let w ∈ �
+W and σJ(w) =
+u
+J . We need to
+show that σJ′(w) depends only on u and not on w. By Lemma 4.1.3, up to WJ-conjugacy we may assume
+w = uy for some y ∈ WK, where K = I(J, u). We need to show that σJ′(u) = σJ′(uy) for all y ∈ WK.
+For this, we use the criterion in Lemma 4.1.6(3). By Lemma 4.1.6(1), EJ,u = EJ,uy = A(K). Therefore
+EJ′,u, EJ′,uy ⊂ A(K). Since y acts by identity on A(K), we have EJ′,u = EJ′,uy, and u|EJ′,u = uy|EJ′,uy.
+Therefore σJ′(u) = σJ′(uy) by Lemma 4.1.6(3).
+□
+4.1.8. Lemma. Let J ⊂ J′ ⊂ft Ia, u ∈ J�
+W and u′ ∈ J′�
+W such that δJ′
+J ( u
+J ) = u′
+J′ . Then
+(1) ℓ(u) ≥ ℓ(u′).
+(2) A representative of the J-type of
+u
+J (as a relevant subspace containing A(J), up to WJ-action)
+contains a representative of the J′-type of u′
+J′ . In particular, τ( u
+J ) ≥ τ( u′
+J′ ) under the partial order
+on E defined in Section 4.1.4.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+41
+Proof. (1) By [HN14, Theorem 2.5], u′ has minimal length among σ−1
+J′ ( u′
+J′ ). Since u ∈ σ−1
+J′ ( u′
+J′ ) by con-
+struction, we see that ℓ(u) ≥ ℓ(u′).
+(2) By Lemma 4.1.6, the J-type of u
+J is represented by EJ,u, and the J′-type of u′
+J′ is represented by
+EJ′,u. Clearly EJ′,u ⊂ EJ,u.
+□
+4.1.9. Definition. Let J ⊂ J′ ⊂ft Ia. Let u
+J ∈ SJ and δJ′
+J ( u
+J ) = u′
+J′ ∈ SJ′.
+(1) We say that u
+J is quasi-J′-reduced if ℓ(u) = ℓ(u′).
+(2) We say that u
+J is J′-reduced if ℓ(u) = ℓ(u′), and τ( u
+J ) = τ( u′
+J′ ).
+4.1.10. Newton point. Recall the Newton point of w ∈ �
+W is a point ν(w) ∈ X∗(T )+
+Q (rational dominant
+cone) characterized by the following property: for sufficiently divisible n, wn ∈ X∗(T ) is in the same
+W-orbit as the translation element given by nν(w). The Newton point is constant on each �
+W-conjugacy
+class. Therefore we have a map
+ν :
+�
+W
+�
+W
+→ X∗(T )+
+Q.
+Let NP ⊂ X∗(T )+
+Q be the image of this map.
+We also have the natural projection
+κ : �
+W → �
+W/W a =: Ω
+that factors through the conjugacy classes of �
+W because Ω is abelian. Combining ν and κ we get the
+enhanced Newton map
+�ν :
+�
+W
+�
+W
+→ NP × Ω.
+Let �
+NP ⊂ NP × Ω be the image of �ν.
+4.1.11. Lemma. For J ⊂ft Ia, there is a unique map �νJ : SJ → �
+NP such that the following diagram is
+commutative
+(4.1.4)
+�
+W
+WJ
+�
+σJ
+� SJ
+�νJ
+�
+�
+W
+�
+W
+�ν
+� �
+NP
+where the left vertical map is the natural projection. Moreover, for J ⊂ J′ ⊂ft Ia, we have νJ′ ◦ δJ′
+J = �νJ :
+SJ → �
+NP.
+Proof. Since σJ is surjective, the uniqueness of �νJ is clear: it sends
+u
+J to �ν(u),where u ∈ J �
+W.
+To
+show the diagram (4.1.4) is commutative, by Lemma 4.1.3, it suffices to show that ν(uy) = ν(u) and
+κ(uy) = κ(u) for all y ∈ WK, where K = I(J, u).
+Since WK ⊂ W a, we have κ(uy) = κ(u) ∈ Ω.
+Now we show ν(uy) = ν(u). For any n ≥ 1 we have (uy)n = Ad(u)y · Ad(u2)y · · · Ad(un)y · un. Each
+Ad(ui)y ∈ WK since K is stable under Ad(u). Let m be the order of Ad(u) on K. Then if n is divisible by
+m|WK|, Ad(u)y ·Ad(u2)y · · · Ad(un)y = (Ad(u)y ·Ad(u2)y · · · Ad(um)y)n/m ∈ (WK)n/m = {1}. Therefore
+(uy)n = un for n sufficiently divisible. This implies ν(uy) = ν(u).
+□
+4.1.12. Straight elements. Following Krammer [Kra09], one says w ∈ �
+W is straight if ℓ(wn) = nℓ(w) for
+all n ≥ 1. A conjugacy class c ∈
+�
+W
+Wa is called straight if it contains a straight element.
+4.1.13. Lemma. Let w ∈ �
+W with Newton point ν ∈ NP. Then ℓ(w) ≥ ⟨2ρ, ν⟩, and the equality holds if
+and only if w is a straight element.
+
+42
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Proof. For any translation element tλ ∈ �
+W corresponding to a dominant λ ∈ X∗(T ), we have ℓ(tλ) =
+⟨2ρ, λ+⟩ for the unique dominant element λ+ in the W-orbit of λ. Therefore, if w ∈ �
+W has Newton
+point ν ∈ NP, wn is conjugate to tnν for sufficiently divisible n, hence ℓ(wn) = ⟨2ρ, nν⟩.
+Therefore
+ℓ(w) ≥ 1
+nℓ(wn) = ⟨2ρ, ν⟩, and equality holds if and only if w is a straight element.
+□
+By [HN14, Theorem 3.3], there is a unique straight W a-conjugacy class for each enhanced Newton point
+�ν ∈ �
+NP.
+4.2. The space B. We will introduce a topological space B obtained by gluing copies of quotients of the
+apartment in the building of G and passing to quotients, using the combinatorial pieces for the affine Weyl
+group. The space B will serve as an organizational tool of subquotient categories of HG,J that we study
+in the next section.
+When G is almost simple, B will be a ∆-complex (a weaker notion than a simplicial complex), which is
+a union of simplices where the intersection of two simplices is not necessarily a common face. In general,
+B will be a union of poly-simplices (product of simplices).
+4.2.1. D◦-sets. Recall from Section 2.7.5 that D◦ denotes the poset of finite type subsets J ⊂ft Ia under
+inclusion. A D◦-set X is a functor
+X : D◦ → Sets.
+In other words, it is an assignment J �→ XJ ∈ Sets for each J ⊂ft Ia, and a map XJ → XJ′ whenever
+J ⊂ J′, compatible with three-term inclusions.
+For a D◦-set X, Tot(X) := �
+J⊂ftIa XJ is a poset whose relations are of the form xJ ≤ yJ′, if J ⊂
+J′ ⊂ft Ia and xJ ∈ XJ maps to yJ′ ∈ XJ′ under the map XJ → XJ′.
+For a D◦-set X, we shall define a topological space |X| as follows. For J ∈ D◦, we let |∆J| ⊂ A be the
+standard facet in the (reduced) apartment for T indexed by J (so that for J = ∅, |∆J| is the fundamental
+alcove in A). Then |∆J| is isomorphic to a product of simplices with codimension #J in A. Let |X| be the
+space obtained as quotient of ⊔J⊂ftIaXJ × |∆J| by the relation (xJ, t) ∼ (yJ′, t) where J ⊂ J′, xJ �→ yJ′
+under XJ → XJ′ and t ∈ |∆J| ⊂ |∆J′|. The image of {xJ} × |∆J| in |X| (for xJ ∈ XJ) is called a J-facet
+of |X|; they are parametrized by XJ.
+When G is almost simple of rank r, each J-facet of |X| is a simplex of dimension r − #J, and |X| is a
+∆-complex. In general, |X| is a union of poly-simplices.
+The set Tot(X) is the set of all facets of |X|. The partial order on Tot(X) is the opposite of the closure
+order of faces.
+4.2.2. Example.
+(1) Let δ be the D◦-set given by the constant functor valued in the singleton set.
+The geometric realization |δ| can be identified with the fundamental alcove in A.
+(2) The standard apartment A of G gives rise to a D◦-set Fac(A): Fac(A)J is the set of J-facets in
+A, and the transition maps Fac(A)J → Fac(A)J′ sends a J-facet F to the unique J′-facet in its
+closure. Then we have a canonical homeomorphism |Fac(A)| ∼= A respecting the poly-simplicial
+structures.
+(3) The same construction of (2) applies to any closed subset E ⊂ A that is a union of facets. It gives
+a D◦-subset Fac(E) ⊂ Fac(A), and |Fac(E)| is identified with E as a subspace of |Fac(A)| ∼= A.
+4.2.3. Remark. For a D◦-set X, the poset Tot(X) also has a geometric realization |Tot(X)| (see Sec-
+tion A.3.1).
+Then |Tot(X)| is always a simplicial complex, and it is a subdivision of |X|.
+When G
+is almost simple, |Tot(X)| is the barycentric subdivision of |X|, i.e., |Tot(X)| ∼= sd(|X|) as simplicial
+complexes.
+4.2.4. Construction of B. We define a D◦-set S whose value at J ⊂ft Ia is SJ, and for J ⊂ J′ ⊂ft Ia,
+the transition map is δJ′
+J : BJ = SJ → SJ′ = BJ′ defined in Proposition 4.1.7.
+Let B = |S| be the geometric realization of B. For u
+J ∈ SJ, we denote by B( u
+J ) the corresponding
+J-facet of B, with interior B( u
+J )◦.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+43
+4.2.5. Remark. For any finite or affine Weyl group W with an automorphism σ preserving the simple
+reflections, we have the notion of σ-twisted J-pieces and one can similarly define B(W; σ). In our setting,
+the normal slice to a facet B( u
+J ) in B is isomorphic to B(WK; u) for the finite Weyl group WK (where
+K = I(J, u)) under the u-twisted conjugation action of WK.
+4.2.6. Functions on B. We have attached three invariants to each element in Tot(S) = �
+J⊂ftIa SJ:
+(1) The length function ℓ : Tot(S) → Z≥0.
+(2) The coarse type τ : Tot(S) → E.
+(3) The enhanced Newton map �ν : Tot(S) → �
+NP ⊂ NP × Ω.
+We may consider these functions as piece-wise continuous functions on the geometric realization B:
+(1) ℓ : B → Z≥0.
+(2) τ : B → E.
+(3) �ν : B → �
+NP.
+such that the value of ℓ, τ and ν on B( u
+J )◦ is ℓ( u
+J ), τ( u
+J ) and νJ( u
+J ) respectively.
+By Lemma 4.1.8, the functions ℓ and τ on B are both lower semicontinuous (non-increasing under
+specialization). By Lemma 4.1.11, the function �ν is locally constant on B. We have decompositions of the
+D◦-set S and the space B:
+S =
+�
+�ν∈ �
+NP
+S�ν,
+B =
+�
+�ν∈ �
+NP
+B�ν
+where S�ν,J is the set of u
+J such that �ν( u
+J ) = �ν, and B�ν = |S�ν|. Note each B�ν ⊂ B is open and closed.
+4.2.7. Essential part. Fix �ν = (ν, ω) ∈ �
+NP. By Lemma 4.1.13, any piece u
+J that appears in B�ν satisfies
+ℓ(u) ≥ ⟨2ρ, ν⟩.
+Let S♥
+�ν,J ⊂ SJ be the subset consisting of
+u
+J with �ν(u) = �ν and ℓ(u) = ⟨2ρ, ν⟩.
+By
+Lemma 4.1.13 and Lemma 4.1.8, we see that the assignment J �→ S♥
+�ν,J defines a D◦-subset S♥
+�ν of S�ν. Let
+B♥
+�ν = |S♥
+�ν | and call it the essential part of B�ν. By Lemma 4.1.13, B♥
+�ν is the closure of the maximal facets
+of B�ν indexed by straight elements.
+4.2.8. Example. For G = SL2, B is a graph. Since G is simply-connected, Ω = 0 and �
+NP = NP. The
+connected components of B are indexed by the possible Newton points ν ∈ Z≥0. Note that W a = ⟨s0, s1⟩.
+Let Tn = (s1s0)n for n ∈ Z.
+For ν > 0, Bν is a cycle with two nodes and two edges:
+T−n
+{s1}
+Tn
+∅
+T−n
+∅
+Tn
+{s0}
+There is only one conjugacy class with Newton point n > 0, namely that of Tn. In this case, we have
+B♥
+ν = Bν.
+For ν = 0, the graph Bν is an infinite linear tree:
+1
+{s1}
+s1
+∅
+1
+∅
+s1
+{s0}
+s0s1s0
+∅
+s0s1s0
+{s1}
+s1s0s1s0s1
+∅
+· · ·
+1
+{s0}
+s0
+∅
+s0
+{s1}
+s1s0s1
+∅
+s1s0s1
+{s0}
+s0s1s0s1s0
+∅
+· · ·
+We draw it in three segments to reflect that there are three conjugacy classes in W a with Newton point
+0: the identity conjugacy class (corresponding to the vertical edge), the conjugacy class of s1 (upper ray)
+and the conjugacy class of s0 (lower ray). In this case, B♥
+ν consists of the edge labelled 1
+∅ and its two end
+points.
+
+44
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+4.2.9. Example. Consider the case G = PGL2. Now �
+W ∼= T Z
+1/2 ⋊ ⟨s1⟩, where T 2
+1/2 = T1 in the notation
+of Example 4.2.8. Let ω = s1T1/2 be the length zero element in �
+W − W a. Then for n ≥ 1 odd, Tn/2 =
+s1s0 · · · s1ω (length n). We have NP = 1
+2Z≥0. The set �
+NP ⊂ NP × Ω ∼= 1
+2Z≥0 × Z/2Z is
+�
+NP = {(n/2, n mod 2)|n ∈ Z≥0} ∪ {(0, 1)}.
+For �ν = (n, 0) ∈ �
+NP, B�ν is the same as Bn described in Example 4.2.8.
+For �ν = (n/2, 1) where n ≥ 1 odd, B�ν is of the form
+T−n/2
+{s1}
+Tn/2
+∅
+Tn/2
+∅
+T−n/2
+{s0}
+There is only one conjugacy class with Newton point n/2 > 0, namely that of Tn/2. In this case we have
+B♥
+�ν = B�ν.
+For �ν = (0, 1), B�ν is an infinite linear tree
+· · ·
+s0ωs0
+{s1}
+s0ωs0
+∅
+ω
+{s0}
+ω
+∅
+ω
+{s1}
+s1ωs1
+∅
+s1ωs1
+{s0}
+· · ·
+There is only one W a-conjugacy class in �
+W with enhanced Newton point �ν = (0, 1), namely the conjugacy
+class of ω. In this case, B♥
+�ν consists of the edge labelled ω
+∅ and its two end points.
+4.2.10. Linear structure on B. We shall describe B as glued from copies of quotients of the apartment A.
+For a relevant affine subspace E ∈ E, let W E ⊂ Aff(E) be the affine transformations of E of the form
+w|E for some w ∈ Stab�
+W (E). Let WE ⊂ W a be the subgroup that fixes E pointwise (this is a parabolic
+subgroup of W a). Note that we have a short exact sequence
+(4.2.1)
+1 → WE → Stab�
+W (E) → W E → 1.
+Consider the following category A. Its objects are pairs (E, w) where E ∈ E and w ∈ W E; its morphisms
+are defined by
+MorA((E, w), (E′, w′)) = {g ∈ W a|gE ⊂ E′, w′(gE) = gE, w′|gE = g ◦ w ◦ g−1 ∈ Aff(gE)}
+with the evident composition.
+For each (E, w) ∈ A, we define a map of D◦-sets
+ϕE,w : Fac(E) → S
+as follows. It sends a J-facet F = xFJ of E (where FJ is the standard J-facet, x ∈ W a/WJ) to σJ(x−1wx)
+(note that x−1wx ∈
+�
+W
+WJ is well-defined, it is the relative position between F and wF).
+4.2.11. Lemma. Suppose g : (E, w) → (E′, w′) is a morphism in A, then
+ϕE,w = (ϕE′,w′|Fac(gE)) ◦ g : Fac(E) → S.
+Proof. We lift w to an element of Stab�
+W (E) and w′ to an element of Stab�
+W (E′) and still denote them by
+w and w′. Let F = xFJ ⊂ E be a J-facet. On the one hand, ϕE,w(F) = σJ(x−1wx) ∈ SJ. On the other
+hand, gF = gxFJ ⊂ E′ is a J-facet of E′, and ϕE′,w′(g(F)) = σJ((gx)−1w′gx) = σJ(x−1(g−1w′g)x).
+Since g : (E, w) → (E′, w′) is a morphism in A, we have g−1w′g ∈ Stab�
+W (E) and it has the same image
+as w in W E. By the exact sequence (4.2.1) we can write g−1w′g = wy for some y ∈ WE. We reduce to
+showing
+(4.2.2)
+σJ(x−1wx) = σJ(x−1wyx).
+For this we use the criterion in Lemma 4.1.6(3). Let E1 = EJ,x−1wx and E2 = EJ,x−1wyx. Then E1 is the
+minimal relevant affine subspace that contains A(J) and stable under x−1wx. Now x−1E contains A(J)
+and is stable under x−1wx, hence E1 ⊂ x−1E. Since y ∈ WE, x−1E is stable under x−1yx, hence also
+stable under x−1wyx. This implies E2 ⊂ x−1E. Moreover, because y fixes E pointwise, the actions of
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+45
+x−1wx and x−1wyx on x−1E are the same. Therefore both E1 and E2 are equal to the minimal relevant
+affine subspace that contains A(J) and contained in x−1E, and stable under x−1wx|x−1E = x−1wyx|x−1E.
+By Lemma 4.1.6(3), (4.2.2) holds.
+□
+By the above lemma, ϕE,w is invariant under AutA(E, w). Using the transition maps g : E → E′ for
+g ∈ MorA((E, w), (E′, w′)), we may form the colimit in the category of D◦-sets
+colim(E,w)∈A Fac(E).
+Lemma 4.2.11 implies that the maps {ϕE,w} together induce a map of D◦-sets
+ϕ : colim(E,w)∈A Fac(E) → S.
+Taking geometric realizations, we get a map
+|ϕ| : colim(E,w)∈A E → B.
+4.2.12. Lemma. The map ϕ is an isomorphism of D◦-sets. Consequently, the map |ϕ| is a homeomorphism
+respecting the facet structures.
+Proof. We need to show for each J ⊂ft Ia, the map on the set of J-facets
+ϕJ : colim(E,w)∈A FacJ(E) → SJ
+is a bijection.
+To see ϕJ is surjective, note that for E = A and w ∈ �
+W = W E, ϕA,w is the composition of �ϕA,w :
+FacJ(A) = W a/WJ →
+�
+W
+WJ (sending x �→ x−1wx) and σJ. As w runs over all of �
+W, the images of �ϕA,w
+cover all of
+�
+W
+WJ , therefore the images of ϕA,w cover all of SJ since σJ is surjective.
+Now we show ϕJ is injective. Let (E1, w1), (E2, w2) ∈ A and Fi = xiFJ ∈ FacJ(Ei) for i = 1, 2 such
+that ϕJ(F1) = ϕJ(F2) ∈ SJ. This means σJ(x−1
+1 w1x1) = σJ(x−1
+2 w2x2). Now we invoke Lemma 4.1.6(3).
+Upon right multiplying x2 by an element in WJ, we may assume EJ,x−1
+1
+w1x1 = EJ,x−1
+2
+w2x2, which we
+denote by E, and that x−1
+1 w1x1|E = x−1
+2 w2x2|E, which we denote by w ∈ W E. Then we have morphisms
+x1 : (E, w) → (E1, w1) and x2 : (E, w) → (E2, w2) in A, and these morphisms send FJ (which is a facet of
+E, because by definition E contains A(J)) to F1 and F2 respectively. This means F1 and F2 are already
+identified in the colimit colim(E,w)∈A FacJ(E). This proves ϕJ is injective.
+□
+4.2.13. Remark. Lemma 4.2.12 tells us that B can be constructed as follows: for each W a-conjugacy
+class in �
+W, take a representative w and form the quotient space A/CW a(w). Then B is obtained from
+the disjoint union of A/CW a(w) (w running over representatives of W a-conjugacy classes in �
+W) by gluing
+along relevant subspaces of dimension less that that of A.
+4.2.14. He-Nie function. Following the idea of He and Nie [HN14, ], we now define a function f : B → R≥0
+as follows. For (E, w) ∈ A, we have the function fE,w : E → R≥0 defined by x �→ ∥x − wx∥2, using a
+fixed W-invariant positive definitive quadratic form ∥ ·∥2 on V = X∗(T )R. For any morphism g : (E, w) →
+(E′, w′) in A, one checks that
+fE,w = fE′,w′ ◦ g : E → R≥0.
+Therefore {fE,w}(E,w)∈A gives a piecewise smooth function on colim(E,w)∈A E ∼= B, which we denote by
+f : B → R≥0.
+Moreover, using the quadratic form ∥ · ∥2 restricted to E, the differential dfE,w turns into a gradient
+vector field ∇fE,w on E. Since fE,w is quadratic, ∇fE,w is a linear vector field. The next lemma shows
+that for varying (E, w) ∈ A, the vector fields ∇fE,w assemble to a piecewise-linear continuous vector field
+∇f on B.
+4.2.15. Lemma. If g : (E, w) → (E′, w′) is a morphism in A, then g∗ takes the vector field ∇fE,w on E
+to the vector field ∇fE′,w′|gE on gE.
+
+46
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Proof. Let (E, w) ∈ A. Let VE ⊂ X∗(T )R be the vector space parallel to E, so that w − 1 is a map
+E → VE. Let w ∈ End(VE) be the linear part of w. Direct calculation shows
+(4.2.3)
+(∇fE,w)(x) = 2(w − 1)∗(w − 1)(x),
+∀x ∈ E.
+Here (w − 1)∗ ∈ End(VE) is the adjoint of (w − 1) ∈ End(VE) under the quadratic form ∥ · ∥2 on VE.
+Replacing (E, w) by (gE, gwg−1), we may assume that E ⊂ E′, w′(E) = E, w′|E = w and g is the
+identity element. Since w′ preserves E, the endomorphism w − 1 of VE′ preserves VE, and so is its adjoint.
+Hence by (4.2.3), ∇fE′,w′|E = ∇fE,w.
+□
+Let Crit(f) ⊂ B be the vanishing locus of ∇f. Let Crit(f)�ν = Crit(f) ∩ B�ν.
+4.2.16. Lemma. For any �ν ∈ �
+NP, the critical locus Crit(f)�ν is contained in the essential part B♥
+�ν .
+Proof. Let x ∈ Crit(f)�ν, then there exists some (E, w) ∈ A (where w ∈ W E) such that x is the image
+of a critical point y of fE,w under |ϕE,w| : E → B. We choose such a (E, w) with E minimal. Choose
+a connected component C of A − ∪E⊂HH (remove all affine root hyperplanes that contain E).
+Let
+�w ∈ Stab�
+W (E) be the lifting of w such that �w(C) = C. Then g = 1 gives a morphism (E, w) → (A, �w) in
+A. We have y ∈ Crit(fE,w) = E ∩ Crit(fA, �
+w). In particular, ν = ν( �w). Apply [HN14, Lemma 2.6] to the
+subspace Crit(fE,w) of Crit(fA, �
+w), and an alcove A ⊂ C that contains y in its closure, we conclude that
+�wA := pos(A, �wA) (relative position, which is in the same W a-conjugacy class of �w) has length ⟨2ρ, ν⟩. The
+image of A under |ϕA, �
+w| : A → B is the maximal facet indexed by �wA ∈ S∅ = �
+W. Since ℓ( �wA) = ⟨2ρ, ν⟩,
+the closure of the maximal facet B( �
+wA
+∅ ) is in the essential part B♥
+�ν . On the other hand, y ∈ A implies x
+is in the closure of B( �
+wA
+∅ ), hence x ∈ B♥
+�ν .
+□
+4.2.17. Remark. It is likely that B♥
+�ν is the smallest D◦-subset of B�ν whose geometric realization contains
+Crit(f)�ν. In other words, it should be true that for any straight element w, the critical locus of fA,w :
+A → R intersects the interior of the fundamental alcove.
+Below we prove a topological property of certain subsets of B�ν. It is a key ingredient in the proof of
+Theorem 4.4.7. Fix �ν = (ν, ω) ∈ �
+NP.
+4.2.18. Definition. Let �ν ∈ �
+NP. A D◦-subset S′ ⊂ S�ν is a called downward if it satisfies:
+• Tot(S′) is finite.
+• S′ contains S♥
+�ν .
+• Let u
+J ∈ S�ν,J, u′
+J′ ∈ S�ν,J′ satisfy ℓ(u) ≤ ℓ(u′) and τ( u
+J ) ≤ τ( u′
+J′ ). If u′
+J′ ∈ S′
+J′, then u
+J ∈ S′
+J.
+A subspace B′ ⊂ B�ν is called downward if it is of the form |S′| for a downward D◦-subset S′ of S�ν.
+4.2.19. Example.
+(1) B♥
+�ν is the smallest downward subspace of B�ν.
+(2) Let n ≥ ⟨2ρ, ν⟩ and [E] ∈ E. Let S�ν,≤(n,[E]) ⊂ S�ν be the D◦-subset consisting of u
+J ∈ S�ν,J such
+that ℓ(u) ≤ n and τ( u
+J ) ≤ [E]. The fact that this is a D◦-subset follows from Lemma 4.1.8.
+When n > ⟨2ρ, ν⟩, S�ν,≤(n,[E]) is downward.
+When n = ⟨2ρ, ν⟩ and [E] = [A], we have
+S�ν,≤(⟨2ρ,ν⟩,[A]) = S♥
+�ν . We denote B�ν,≤(n,[E]) = |S�ν,≤(n,[E])|.
+(3) By definition, any downward subspace B′ ⊂ B�ν is a finite union of the form
+(4.2.4)
+B′ = B♥
+�ν ∪
+��
+i
+B�ν,≤(ni,[Ei])
+�
+where ni > ⟨2ρ, ν⟩.
+By Lemma 4.2.16, for any downward B′
+�ν ⊂ B�ν, we have Crit(f)�ν ⊂ B♥
+�ν ⊂ B′
+�ν.
+4.2.20. Proposition. Fix �ν = (ν, ω) ∈ �
+NP. Let B′ ⊂ B�ν be a downward subspace. Then the inclusion
+Crit(f)�ν ֒→ B′ admits a deformation retract. In particular, both inclusions Crit(f)�ν ⊂ B♥
+�ν ⊂ B′ are
+homotopy equivalences.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+47
+Proof. Abbreviate Crit(f)�ν by C�ν. By Lemma 4.2.15, the gradient flow of f is well-defined as a flow Φt
+on B�ν, for t ∈ R. The deformation retract will be constructed using the flow Φt.
+Claim. The downward subspace B′ is stable under the flow Φt, for t ≤ 0 non-positive.
+Proof of Claim. Since B′ is a union of the form (4.2.4), it suffices to show that B♥
+�ν and B�ν,≤(n,[E]) (where
+n > ⟨2ρ, ν⟩ )are stable under the flow {Φt}t≥0. Since B♥
+�ν = B�ν,≤(⟨2ρ,ν⟩,[A]), we suffices to show that
+B�ν,≤(n,[E]) is stable under the flow {Φt}t≥0 whenever n ≥ ⟨2ρ, ν⟩.
+First consider the case E = A. Since every point of B�ν lies in the image of |ϕA,w| : A → B for some
+w ∈ �
+W with �ν(w) = �ν, and the flow stays inside the image of |ϕA,w|, we only need to prove the same
+statement for A with respect to the flow defined by ∇fA,w. Let A0 ⊂ A be the (closed) fundamental
+alcove. For any alcove A = yA0, let wA = y−1wy. Let Aw,≤n ⊂ A be the union of alcoves A such that
+ℓ(wA) ≤ n. Then Aw,≤n = |ϕA,w|−1(B�ν,≤(n,[A])). We only need to show that Aw,≤n is stable under the
+flow Φt of ∇fA,w for t ≤ 0. Let Z ⊂ A be the union of all H ∩ H′ where H and H′ run over distinct affine
+root hyperplanes. Let U ⊂ Aw,≤n be the subset of x ∈ Aw,≤n such that Φt(x) /∈ Z for all t ≤ 0. Then U
+is dense in Aw,≤n (using that Z has codimension two in A, so ∪t≥0Φt(Z) has codimension at least one in
+A). Therefore it is enough to show that Φt(x) ∈ Aw,≤n for x ∈ U and t ≤ 0. Suppose this is not the case,
+then for some x ∈ U and some t ≤ 0, Φt(x) lies in an alcove A with ℓ(wA) > n. Let t0 be the supremum
+of such t. Then Φt0+ǫ(x) ∈ A with ℓ(wA) ≤ n for small ǫ > 0 and Φt0−ǫ(x) ∈ A′ with ℓ(wA′) > n for small
+ǫ > 0. Moreover, x0 := Φt0(x) is on the common face A ∩ A′ of A and A′, and does not lie on any other
+affine root hyperplane. Let n be a normal vector of H that points to A′. Then
+(4.2.5)
+⟨∇fA,w(x0), n⟩ ≥ 0.
+We have wA′ = swAs where s is the simple reflection determined by hyperplane H = Span(A∩A′). Hence
+ℓ(wA′) = ℓ(wA) + 2. Applying [HN14, Lemma 2.1], we see that ⟨∇fA,w(x0), n⟩ > 0 (note that x0 = Φt0(x)
+is a regular point of A ∩ A′ since x0 /∈ Z). This contradicts (4.2.5).
+Now we consider the case of a general [E] ∈ E. For any x ∈ B�ν,≤(n,[E]) we may find (E, w) ∈ A (with
+E ∈ [E]) such that x lies in the image of |ϕE,w| : E → B. Hence it is enough to prove the analogous
+statement for E with respect to the gradient flow of fE,w. Let �w ∈ �
+W be a lifting of w. Then |ϕE,w| is
+the composition E ֒→ A
+|ϕA, �
+w|
+−−−−→ B. In particular, E ∩ |ϕE,w|−1(B�ν,≤(n,[E])) = A ∩ |ϕA, �
+w|−1(B�ν,≤(n,[E])).
+Moreover, the gradient flow of fE,w is the same as the gradient flow of fA, �
+w restricted to E by Lemma
+4.2.15. Therefore the case [E] = [A] proved above implies the case of a general [E].
+□
+Now for any x ∈ B�ν, limt→−∞ Φt(x) ∈ C�ν. Indeed, we may assume x lies in the image of |ϕA,w| : A →
+B�ν for some w ∈ �
+W, and the corresponding statement is [HN14, Lemma 2.3]. Moreover, the calculation
+in loc.cit. shows that the flow is contracting in a neighborhood of C�ν as t → −∞. This implies that
+the map H : [0, 1) × B′ → B′ given by H(s, x) = Φlog(1−s)(x) can be extended to a continuous function
+H : [0, 1] × B′ → B′ by letting H(1, x) = limt→−∞ Φt(x) ∈ C�ν. Then H gives a deformation retract from
+B′ to C�ν. This proves the proposition.
+□
+4.3. Geometric pieces and sheaves on them. Here we turn to the geometry indexed by the prior
+combinatorics, in particular sheaves on geometric pieces and the natural functors between them.
+4.3.1. Cyclic reduction. Consider the following general situation. Let G be an algebraic group, and P, P ′ ⊂
+G two parabolic subgroups with unipotent radicals P u and P ′u and Levi quotients L and L′ respectively.
+Let x ∈ G.
+Let δ : L′
+∼
+→ L be an isomorphism.
+Let Q′ = Im(P ∩ Ad(x−1)P ′ → L) ⊂ L, Q =
+δ(Im(P ′ ∩ Ad(x)P → L′)) ⊂ L.
+Then Q, Q′ are parabolic subgroups of L.
+Let Qu and Q′u be the
+unipotent radicals of Q and Q′ and M and M ′ be their Levi quotients. Then M ′ = (P ∩ Ad(x−1)P ′)red
+(where (−)red denotes Levi quotient). We have a canonical isomorphism
+δx : M ′ = (P ∩ Ad(x−1)P ′)red
+Ad(x)
+−−−−→ (P ′ ∩ Ad(x)P)red ∼= Im(P ′ ∩ Ad(x)P → L′)
+δ−→ M.
+
+48
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+4.3.2. Lemma (Cyclic reduction). There is a canonical map
+(4.3.1)
+P ′u\(P ′xP)/P u
+Adδ(L′)
+→ Q′u\L/Qu
+Adδx(M ′)
+sending p′xp to pδ(p′) (where p is the image of p under P ։ L; similar for p′). This map is a gerbe for
+the unipotent group P ′u ∩ Ad(x)P u.
+We will refer to the above map (4.3.1) as a cyclic reduction.
+Proof. The left P ′-translation and right P-translation on P ′xP gives a P ′ × P-equivariant isomorphism
+(4.3.2)
+P ′ ×P ′∩Ad(x)P P
+∼
+→ P ′xP,
+(p′, p) �→ p′xp
+where the action of P ′ ∩Ad(x)P on P is by Ad(x−1) : P ′ ∩Ad(x)P
+∼
+→ Ad(x−1)P ′ ∩P and left translation.
+Now quotienting both sides of (4.3.2) by left P ′u, right P u and Adδ(L′)-actions, we get isomorphisms
+(4.3.3)
+L′ ×P ′∩Ad(x)P L
+Adδ(L′)
+∼
+←− P ′u\P ′ ×P ′∩Ad(x)P P/P u
+Adδ(L′)
+∼
+→ P ′u\P ′xP/P u
+Adδ(L′)
+.
+Here the action on P ′∩Ad(x)P on L′ is via the projection P ′∩Ad(x)P → L′, whose image is δ−1(Q) ⊂ L′,
+and right translation on L′. Similarly, the action on P ′ ∩Ad(x)P on L is via the projection P ′∩Ad(x)P
+∼
+→
+Ad(x−1)P ′ ∩ P → Q′ ⊂ L and left translation. The normal subgroup P ′u ∩ Ad(x)P u acts trivially on
+L′ × L, and
+(4.3.4)
+(P ′ ∩ Ad(x)P)/(P ′u ∩ Ad(x)P u) ∼= δ−1(Q) ×M′ Q′
+where the projection δ−1(Q) → M ′ is the composition δ−1(Q)
+δ−→ Q ։ M
+(δx)−1
+−−−−→ M ′.
+Using (4.3.4) we get a P ′u ∩ Ad(x)P u-gerbe
+L′ ×P ′∩Ad(x)P L
+Adδ(L′)
+→ L′ ×δ−1(Q)×M′ Q′ L
+Adδ(L′)
+= (L′/δ−1(Qu)) ×M′ (Q′u\L)
+Adδ(L′)
+.
+Via the map (ℓ′, ℓ) �→ ℓδ(ℓ′) (for ℓ ∈ L, ℓ′ ∈ L′), we have an isomorphism
+(L′/δ−1(Qu)) ×M′ (Q′u\L)
+Adδ(L′)
+∼
+→ Q′u\L/Qu
+Adδx(M ′) .
+Composing all these isomorphisms we get a P ′u ∩ Ad(x)P u-gerbe
+P ′u\(P ′xP)/P u
+Adδ(L′)
+→ Q′u\L/Qu
+Adδx(M ′) .
+□
+Lemma 4.3.2 gives an equivalence of categories by pullback
+(4.3.5)
+Sh(Q′u\L/Qu
+Adδx(M ′) ) ≃ Sh(P ′u\(P ′xP)/P u
+Adδ(L′)
+).
+4.3.3. Nilpotent sheaves under cyclic reduction. Next we show that sheaves with nilpotent singular support
+correspond to each other under the above equivalence.
+To simplify the statements, we introduce the
+following terminology: For a group H acting on a smooth variety X, and F ∈ Sh(X), we say F is H-
+nilpotent if µH(SS(F)) lies in the nilpotent cone of h∗ = (LieH)∗. When there is ambiguity as to how H
+acts on X, we will specify F is H-nilpotent for which H-action.
+Let
+ShN (P ′u\(P ′xP)/P u
+Adδ(L′)
+) ⊂ Sh(P ′u\(P ′xP)/P u
+Adδ(L′)
+)
+be the full subcategory consisting of objects that are L′-nilpotent for left translation by L′ (equivalently,
+L-nilpotent for the right translation). Similarly define the full subcategory
+ShN (Q′u\L/Qu
+Adδx(M ′) ) ⊂ Sh(Q′u\L/Qu
+Adδx(M ′) )
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+49
+using either M ′-nilpotence for the left translation or M-nilpotenve for the right translation.
+4.3.4. Lemma. Under the equivalence (4.3.5), the full subcategories ShN ( P ′u\(P ′xP )/P u
+Adδ(L′)
+) and ShN ( Q′u\L/Qu
+Adδx(M′) )
+correspond. In particular, pullback via the cyclic reduction map induces an equivalence
+ShN (Q′u\L/Qu
+Adδx(M ′) ) ≃ ShN (P ′u\(P ′xP)/P u
+Adδ(L′)
+).
+Proof. Choose a section to P ։ L and realize L as a Levi subgroup of P. Similarly realize L′ as a subgroup
+of P ′, M as a subgroup of Q′ and M as a subgroup of Q.
+We have a commutative diagram
+P ′ × P
+πx
+�
+p′×p
+� L′ × L
+mδ
+� L
+π
+�
+P ′u\(P ′xP )/P u
+Adδ(L′)
+c
+� Q′u\L/Qu
+Adδx(M′) )
+where the bottom arrow is the cyclic reduction map (4.3.1), p, p′ are the projections and mδ(ℓ′, ℓ) = ℓδ(ℓ′),
+πx(g′, g) = g′xg, and π is the quotient map.
+Let F ∈ Sh( Q′u\L/Qu
+Adδx(M′) ), and �F = π∗F be its pullback to L. Let K = c∗F ∈ Sh( P ′u\P ′xP/P u
+Adδ(L′)
+), and
+�K = π∗
+xK be its pullback to P ′ × P. By definition, F ∈ ShN ( Q′u\L/Qu
+Adδx(M′) ) if and only if �F is M ′-nilpotent
+for the left translation of M ′ on L; K ∈ ShN ( P ′u\P ′xP/P u
+Adδ(L′)
+) if and only if �K is L-nilpotent for the right
+translation actions on the P-factor of P ′ × P.
+Therefore, we reduce to prove that the following are
+equivalent:
+(1) �F is M ′-nilpotent for the left translation on L;
+(2) �K = (p′ × p)∗m∗
+δ �F is L-nilpotent for the right translation on the P-factor.
+Since p′ × p is smooth and equivariant for the right translation of L, (2) is equivalent to
+(3) m∗
+δ �F is L-nilpotent for the right translation on the L-factor.
+It is easy to see that on L′×L, L-nilpotence with respect to the left translation is equivalent to L-nilpotence
+with respect to the right translation. Also mδ is equivariant with respect to the left translation action of
+L (but not the the right translation), therefore (3) is equivalent to
+(4) �F is L-nilpotent for the left translation.
+It remains to prove (1) ⇐⇒ (4). Let µL : T ∗L → l∗ be the moment map for the left transaltion of L.
+Recall �
+F is Q′u-equivariant for the left translation action, so µL(SS( �F)) ∈ n⊥
+Q′ (here nQ′ = LieQ′u). Then
+(1) means the image of µL(SS( �F)) under the projection n⊥
+Q′ → m′∗ is nilpotent, which is equivalent to
+saying that µL(SS( �F)) lies in the nilpotent cone of n⊥
+Q′, which is (4).
+□
+4.3.5. Geometric pieces. Recall from Section 2.7.8 that
+YJ = Pu
+J \G/Pu
+J
+LJ
+regarded as an ind-stack over k. For J ⊂ft Ia, Lusztig [Lusa, §3] defined a stratification of YJ indexed by
+SJ: for u
+J ∈ SJ, Y( u
+J ) is the locally closed substack of YJ defined as the image of the projection
+Y( u
+J ) = Im(IuI ⊂ G → YJ).
+We shall call Y( u
+J ) geometric J-pieces. For each w ∈ �
+W and any lifting ˙w ∈ NG(T ), ˙w ∈ Y( u
+J ) if and only
+if σJ( ˙w) = u
+J .
+Below we recall an inductive construction of Y( u
+J ) that leads to a description of it in terms of a twisted
+adjoint quotient, also due to Lusztig.
+
+50
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Let (Jn, J′
+n, un) ∈ SJ corresponding to u ∈ J �
+W. We describe the geometric piece Y( u
+J ) using cyclic
+reduction.
+Choose a lifting ˙un for each un in NLJn−1(T ) (here LJ−1 is understood to be G).
+For n ≥ 0, let
+P ′
+n+1 ⊂ LJn be the standard parabolic subgroup whose Levi is LJ′
+n+1; let Pn+1 ⊂ LJn be the standard
+parabolic subgroup whose Levi is LJn+1. For n ≥ 0 define
+Zn(u0, · · · , un) = P ′u
+n+1\LJn/P u
+n+1
+Ad ˙u0··· ˙un(LJ′
+n+1).
+If we define LJ−1 = G, P ′
+0 = P0 = PJ, then we can define Z−1 using the above formula, and have
+Z−1 = Pu
+J \G/Pu
+J
+LJ
+= YJ.
+We have maps
+Zn(u0, · · · , un)
+Zn(u0, · · · , un+1) := P ′u
+n \P ′
+nun+1Pn/P u
+n
+Ad ˙u0··· ˙un(LJ′
+n+1)
+�
+� �
+�
+Zn+1(u0, · · · , un+1)
+where the second map is the cyclic reduction in Lemma 4.3.2. The geometric piece Y( u
+J ) is defined to be
+the fiber product for n ≫ 0
+(4.3.6)
+Y( u
+J ) := Z−1(u0) ×Z0(u0) Z0(u0, u1) ×Z1(u0,u1) · · · ×Zn−1(u0,··· ,un−1) Zn−1(u0, · · · , un).
+The “shape” of the geometric piece Y( u
+J ) is described by Lusztig:
+4.3.6. Proposition (Lusztig [Lusa, 3.14]). Let (Jn, J′
+n, un)n≥0 ∈ SJ. Choose liftings ˙un ∈ NLJn−1(T ) for
+n ≥ 0 such that ˙un = 1 whenever un = 1 (as usual, we understand LJ−1 = G). Let ˙u = ˙u0 · · · ˙un for
+n ≫ 0. Let K = I(J, u) ⊂ J. Then there is a canonical map
+πJ,u : Y( u
+J ) →
+LK
+Ad ˙u(LK)
+which is an iterated gerbe for (pro-)unipotent groups.
+One can eliminate the choice of ˙u by writing the right side as uLK
+LK .
+Proof. Let n ≫ 0 such that Jn = K. In (4.3.6), projection to the last factor followed by cyclic reduction
+Y( u
+J ) → Zn−1(u0, · · · , un) → Zn =
+LK
+Ad ˙u(LK)
+gives the desired map, which is an iterated gerbe for (pro-)unipotent groups.
+□
+4.3.7. Corollary. For each piece
+u
+J with K = I(J, u), pullback along πJ,u induces an equivalence of
+categories
+(4.3.7)
+π∗
+J,u : Sh(uLK
+LK
+) ≃ Sh(
+LK
+Ad ˙u(LK))
+∼
+→ Sh(Y( u
+J )).
+4.3.8. Partial order. Let ≥ be the Bruhat partial order on �
+W. In [He07, 3.8, 3.9, 3.13], a partial order ≥J
+on J �
+W is defined as follows. For u, u′ ∈ J �
+W, define u ≥J u′ if there is a u′′ in the same WJ-conjugacy
+class of u′ such that u ≥ u′′.
+Let w, w′ ∈ �
+W be in the same WJ-conjugacy class. Recall from loc. cit. that w′ is said to be obtained
+from w by a J-cyclic shift if ℓ(w′) = ℓ(w) and w′ = sws for some simple reflection s ∈ J such that either
+ℓ(sw) = ℓ(w) − 1 or ℓ(ws) = ℓ(w) − 1. Denote w ∼J w′ if w′ can be obtained from w by a sequence of
+J-cyclic shifts. It is shown in loc. cit. that u ≥J u′ if and only if there exists u′′ ∼J u′ such that u ≥ u′′.
+4.3.9. Theorem (He, loc. cit. Theorem 4.5). For u ∈ J �
+W, the closure of Y( u
+J ) is the union of Y( u′
+J ) for
+u ≥J u′.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+51
+4.3.10. The functor chJ′
+J . Let J ⊂ J′ ⊂ft Ia. Then the image of PJ under the projection PJ′ → LJ′ is a
+parabolic subgroup P J′
+J ⊂ LJ′. Consider the diagram
+(4.3.8)
+YJ = Pu
+J \G/Pu
+J
+LJ
+Pu
+J′ \G/Pu
+J′
+P J′
+J
+qJ′
+J
+�
+pJ′
+J
+� Pu
+J′ \G/Pu
+J′
+LJ′
+= YJ′
+Define
+chJ′
+J = (pJ′
+J )!(qJ′
+J )∗ : Sh(YJ) → Sh(YJ′).
+Finally, we recall the following key geometric input we will need to understand the natural transforms
+between sheaves on different geometric pieces. We are grateful to Xuhua He for providing it in a recent
+paper [He]. Note that the sheaves involved in the following theorem need not have nilpotent singular
+support.
+4.3.11. Theorem (He, [He]). Let J ⊂ J′ ⊂ft Ia, u ∈ J �
+W and u′
+J′ = δJ′
+J ( u
+J ). Let K = I(J, u) ⊂ J and
+K′ = I(J′, u′) ⊂ J′. Let iJ,u : Y( u
+J ) ֒→ YJ and iJ′,u′ : Y( u′
+J′ ) ֒→ YJ′ be the inclusions.
+(1) Suppose
+u
+J is not quasi-J′-reduced (namely ℓ(u) > ℓ(u′)).
+Then the image of chJ′
+J ◦ iJ,u! :
+Sh(Y( u
+J )) → Sh(YJ′) is supported on the closed substack YJ′,<ℓ(u) = ∪ℓ(v)<ℓ(u)Y( v
+J′ ).
+(2) Suppose
+u
+J is quasi-J′-reduced (recall this means ℓ(u) = ℓ(u′)).
+Then there is a unique x ∈
+WJ′ ∩ �
+W K′ such that x−1ux = u′ and K1 := x−1(K) ⊂ K′ (note that u′(K1) = K1).
+Let
+Ad(x−1) : Sh(uLK
+LK
+) → Sh(u′LK1
+LK1
+)
+be the functor induced by conjugation by ˙x−1, where ˙x ∈ NLJ′(T ) is any lifting of x. 3
+Consider also the induction functor
+u′IndK′
+K1 = bK′
+K1,!(aK′
+K1)∗ : Sh(u′LK1
+LK1
+) → Sh(u′LK′
+LK′ )
+given by the correspondence
+u′LK1
+LK1
+u′P K′
+K1
+P K′
+K1
+aK′
+K1
+�
+bK′
+K1 � u′LK′
+LK′
+.
+Then the outer square of the following diagram is commutative
+Sh( uLK
+LK )
+Ad(x−1)�
+π∗
+J,u
+≀
+�
+Sh(
+u′LK1
+LK1 )
+u′IndK′
+K1� Sh( u′LK′
+LK′ )
+π∗
+J′,u′
+≀
+�
+Sh(Y( u
+J ))
+γJ′,u′
+J,u
+�❴
+❴
+❴
+❴
+❴
+❴
+❴
+❴
+❴
+❴
+iJ,u!
+�
+Sh(Y( u′
+J′ ))
+iJ′,u′!
+�
+Sh(YJ)
+chJ′
+J
+� Sh(YJ′)
+The same is true when iJ,u! and iJ′,u′! are replaced with iJ,u∗ and iJ′,u′∗ respectively.
+We define the functor γJ′,u′
+J,u
+: Sh(Y( u
+J )) → Sh(Y( u′
+J′ )) as the composition
+(4.3.9)
+γJ′,u′
+J,u
+: Sh(Y( u
+J ))
+(π∗
+J,u)−1
+−−−−−→ Sh(uLK
+LK
+)
+Ad(x−1)
+−−−−−→ Sh(u′LK′
+LK′ )
+π∗
+J′,u′
+−−−−→ Sh(Y( u′
+J′ )).
+(3) In particular, when u
+J is J′-reduced (which implies K1 = K′ in the above notation), then γJ′,u′
+J,u
+is
+an equivalence.
+3Two liftings of x differ by multiplication by t ∈ T ⊂ LK, therefore the resulting functors Sh( uLK
+LK ) → Sh(
+u′LK1
+LK1 ) are
+canonically isomorphic.
+
+52
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Proof. In the following we will quote results from [He] where the author primarily works with a finite
+dimensional reductive group rather than a loop group. However, [He, 4.5] it is explained that the results
+there extend to loop groups in a straightforward way.
+(1) is proved in [He, 4.2(a)].
+(2) From the diagram of the proof of [He, Theorem 4.3], we see that (using notation from (4.3.8))
+(4.3.10)
+pJ′
+J (qJ′
+J )−1(Y( u
+J )) ⊂ Y( u′
+J′ ).
+Using proper base change and the fact that pJ′
+J is proper, (4.3.10) implies that for any F ∈ Sh(Y( u
+J )),
+chJ′
+J iJ,u!F is a !-extension from Y( u′
+J′ ).
+Therefore, to prove the statement, it suffices to show that
+i∗
+J′,u′chJ′
+J iJ,u!F is canonically isomorphic to γJ′,u′
+J,u F. This is exactly the statement of [He, Theorem 4.3].
+For the ∗-version, using (4.3.10) and the fact that qJ′
+J
+is smooth, we see that for any F ∈ Sh(Y( u
+J )),
+chJ′
+J iJ,u∗F is a ∗-extension from Y( u′
+J′ ). Therefore, it suffices to show that i∗
+J′,u′chJ′
+J iJ,u∗F is canonically
+isomorphic to γJ′,u′
+J,u F. This is proved in the same way as the calculations towards the end of the proof of
+[He, Theorem 4.3], using that qJ′
+J is smooth to justify the base change steps.
+(3) follows directly from (2).
+□
+4.3.12. Nilpotent sheaves. For J ⊂ft Ia, recall that HG,J = ShN (YJ). For J ⊂ J′ ⫋ Ia, it is standard to
+check that chJ′
+J sends HG,J to HG,J′.
+For each geometric piece u
+J , we have the equivalence Sh(Y( u
+J )) ≃ Sh( uLK
+LK ) given in Corollary 4.3.7. Con-
+sider the subcategory ShN ( uLK
+LK ) ⊂ Sh( uLK
+LK ) consisting of sheaves whose pullback to uLK is LK-nilpotent
+for the left transaltion (equivalently, LK-nilpotent for the right translation), using the terminology intro-
+duced in Section 4.3.3. We define
+ShN (Y( u
+J )) ⊂ Sh(Y( u
+J ))
+to be the full category corresponding to ShN ( uLK
+LK ) ∼= ShN (
+LK
+Ad ˙u(LK)) under the equivalence (4.3.7).
+4.3.13. Proposition. For u
+J ∈ SJ and iJ,u : Y( u
+J ) ֒→ YJ be the inclusion. Then:
+(1) The category HG,J = ShN (YJ) consists of objects F ∈ Sh(YJ) such that i∗
+J,uF ∈ ShN (Y( u
+J )) for
+all u
+J ∈ SJ. Alternatively, ShN (YJ) consists of objects F ∈ Sh(YJ) such that i∗
+J,uF ∈ ShN (Y( u
+J ))
+for all u
+J ∈ SJ.
+(2) The functors iJ,u! and iJ,u∗ send ShN (Y( u
+J )) (defined above) to ShN (YJ) = HG,J.
+Proof. We prove (1) and (2) follows. For (1), we give the argument for the ∗-pullback statement, and the
+!-version is similar. Recall the notation Zn(u0, · · · , un) and Zn(u0, · · · , un+1) introduced in Section 4.3.5,
+for n ≥ −1, and (u0, u1, · · · ) the sequence of elements in �
+W that appear in any combinatorial piece. Define
+ShN (Zn(u0, · · · , un)) and ShN (Zn(u0, · · · , un+1)) using left LJ′
+n+1-nilpotence. Then HG,J = ShN (Z−1).
+Fix n ≥ −1 and (u0, · · · , un) that appear as the first n + 1 terms of a combinatorial piece (so that
+J0 = J, J1, J′
+1, · · · , Jn+1, J′
+n+1 are defined according the recipe in Section 4.1.1). We have a stratification
+Zn(u0, · · · , un) =
+�
+un+1∈
+J′
+n+1W Jn
+Jn
+Zn(u0, · · · , un, un+1).
+Convention: W−1 := �
+W. We also have the cyclic reduction maps
+cun+1 : Zn(u0, · · · , un, un+1) → Zn+1(u0, · · · , un, un+1).
+Claim. For any n ≥ −1 , F ∈ Sh(Zn(u0, · · · , un)) lies in ShN (Zn(u0, · · · , un)) if and only if for all
+un+1 ∈ J′
+n+1W Jn
+Jn , the sheaf F♭
+un+1 ∈ Sh(Zn+1(u0, · · · , un, un+1)) corresponding to F|Zn(u0,··· ,un,un+1) un-
+der cyclic reduction (i.e., c∗
+un+1F♭
+un+1 ∼= F|Zn(u0,··· ,un,un+1)) lies in ShN (Zn+1(u0, · · · , un+1)).
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+53
+Proof of Claim. Indeed, let �
+Zn(u0, · · · , un) = P ′u
+n+1\LJn/P u
+n+1, and let �
+Zn(u0, · · · , un+1) be the preimage
+of Zn(u0, · · · , un+1) in �
+Zn(u0, · · · , un). Then �
+Zn(u0, · · · , un+1) for various un+1 give a stratification of
+�
+Zn(u0, · · · , un) that is stable under the left translation by LJ′
+n+1. By definition, F ∈ ShN (Zn(u0, · · · , un))
+if and only if its pullback �F on �
+Zn(u0, · · · , un) is left LJ′
+n+1-nilpotent, which happens if and only if
+�F| �
+Zn(u0,··· ,un+1) is left LJ′
+n+1-nilpotent for all un+1, if and only if F|Zn(u0,··· ,un+1) ∈ ShN (Zn(u0, · · · , un+1))
+for all un+1. By Lemma 4.3.4, F|Zn(u0,··· ,un+1) ∈ ShN (Zn(u0, · · · , un+1)) if and only if F♭
+un+1 ∈ ShN (Zn+1(u0, · · · , un+1)).
+This proves the claim.
+□
+We continue with the proof of the proposition. Note that for each combinatorial piece (Jn, J′
+n, un) ∈ SJ,
+Jn will stabilizes for n ≥ r, where r is the semisimple rank of G. Each u
+J ∈ SJ gives an (r + 1)-tuple
+(u0, · · · , ur), and by the above remark, u is determined by (u0, · · · , ur).
+For each u ∈ SJ, we define
+F♭
+u ∈ ShN (Zr(u0, · · · , ur)) to be the result of restricting and apply cyclic reduction successively. In other
+words, under the unipotent gerbe πJ,u : Y( u
+J ) → Zr(u0, · · · , ur), π∗
+J,uF♭
+u ≃ F|Y( u
+J ). Using the Claim in the
+previous paragraph repeatedly for n = −1, 0, · · · , r − 1, we conclude that for F ∈ Sh(YJ), F ∈ ShN (YJ)
+if and only if F♭
+u ∈ Sh(Zr(u0, · · · , ur)) lies in ShN (Zr(u0, · · · , ur)), for all J-pieces u
+J . By definition, the
+latter is the same as saying F|Y( u
+J ) = i∗
+J,uF lies in ShN (Y( u
+J )) for all u
+J ∈ SJ.
+□
+For any locally closed substack Z ⊂ YJ that is a union of geometric J-pieces, we define ShN (Z) ⊂ Sh(Z)
+to be the full subcategory consisting of F ∈ Sh(Z) such that i∗
+J,uF ∈ ShN (Y( u
+J )) whenever u
+J ⊂ Z. By
+Proposition 4.3.13, for Z = YJ, this definition of ShN (YJ) coincides with the old one which is HG,J.
+4.3.14. Corollary. The full subcategories ShN (Z) for closed unions of geometric pieces Z ⊂ YJ give a
+stratification structure on HG,J indexed by the poset (SJ, ≤J) (in the sense recalled in Section A.1.2), such
+that the strata category corresponding to u
+J ∈ SJ is ShN (Y( u
+J )).
+4.4. Semi-orthogonal decomposition of the cocenter. In this section, we prove Theorem 1.3.4 and
+its generalization Theorem 1.4.4 which describes the cocenter hh(HG) up to taking “associated graded”
+indexed by enhanced Newton points.
+4.4.1. Colimit of character sheaves. Let J ⊂ft Ia. By Lemma 4.3.7, we have a canonical equivalence
+Sh(Y( 1
+J )) ≃ Sh(LJ/LJ). By definition, we have
+ShN (Y( 1
+J )) ≃ ShN (LJ/LJ).
+We remark that ShN (LJ/LJ) can be viewed as a version of the category of character sheaves on LJ
+allowing sheaves with infinite-dimensional stalks.
+By Proposition 4.3.13, the closed embedding iJ,1 : Y( 1
+J ) ֒→ YJ gives a fully faithful embedding
+ιJ = iJ,1∗ : ShN (LJ/LJ) ≃ ShN (Y( 1
+J )) ֒→ HG,J.
+For J ⊂ J′ ⊂ft Ia, there is the induction functor of character sheaves defined by Lusztig:
+IndJ′
+J : ShN (LJ/LJ) → ShN (LJ′/LJ′).
+Via the embeddings ιJ and ιJ′, the induction functor is intertwined with chJ′
+J : HG,J → HG,J′. Therefore
+they induce a functor on colimits
+(4.4.1)
+ι : colimD ShN (LJ/LJ) → colimD HG,J
+where D is the groupoid D◦/Ω introduced in Section 2.7.5. Combining the above functor with the equiv-
+alence to hh(HG) given by Corollary 2.7.11, we get
+(4.4.2)
+colimJ∈D ShN (LJ/LJ) → hh(HG).
+4.4.2. Theorem. The functor ι is fully faithful.
+
+54
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+This is the specific statement we seek for this paper; it is an easy consequence of a general theorem
+describing all of hh(HG) up to “taking associated graded”, which we will state and prove next. Below we
+will take the main step towards such a description of hh(HG) by considering hh(HG◦, HG) first.
+4.4.3. Semi-orthogonal decomposition of hh(HG◦, HG). In this section, we arrive at our first approximation
+to the main goal, as encapsulated in Theorem 4.4.4.
+Instead of the cocenter hh(HG), we consider hh(HG◦, HG), which is equivalent to colimJ⊂ftIa HG,J by
+Corollary 2.7.11. By the discussion in Section 2.7.12, HG decomposes into the direct sum of HG◦-bimodules
+Hω
+G according to the connected components of G indexed by ω ∈ Ω, we have a decomposition
+hh(HG◦, HG) =
+�
+ω∈Ω
+hh(HG◦, Hω
+G).
+For �ν ∈ �
+NP, recall the essential part B♥
+�ν ⊂ B�ν, whose J-facets are indexed by the subset S♥
+J,�ν ⊂ SJ
+consisting of u
+J such that �ν(u) = �ν and ℓ(u) = ⟨2ρ, ν⟩. Recall Tot(S♥
+�ν ) = �
+J⊂ftIa S♥
+J,�ν is a poset as defined
+in Section 4.2.1, and the order is opposite to the closure order of facets in B♥
+�ν . If u
+J ≤ u′
+J′ in Tot(S♥
+�ν ), we
+have the functor
+γJ′,u′
+J,u
+: ShN (Y( u
+J )) → ShN (Y( u′
+J′ )).
+defined in Theorem 4.3.11 (see (4.3.9), and here we restrict to sheaves with nilpotent singular support).
+Using the functors γJ′,u′
+J,u
+we may form the colimit
+(4.4.3)
+colim u
+J ∈Tot(S♥
+�ν ) ShN (Y( u
+J )).
+For �ν = (0, 0), Tot(S♥
+�ν ) consists of 1
+J for J ⊂ft Ia, and can be identified with the poset D◦. In this case,
+the above colimit is the same as colimJ⊂ftIa ShN (LJ/LJ) using the induction functors.
+4.4.4. Theorem. Fix ω ∈ Ω.
+Then hh(HG◦, Hω
+G) admits a semi-orthogonal decomposition indexed by
+non-negative integers
+hh(HG◦, Hω
+G)0 ֒→ hh(HG◦, Hω
+G)≤1 ֒→ · · · hh(HG◦, Hω
+G)≤n ֒→ · · · ֒→ hh(HG◦, Hω
+G) =
+�
+n≥0
+hh(HG◦, Hω
+G)≤n.
+In particular, each inclusion hh(HG◦, Hω
+G)≤n ֒→ hh(HG◦, Hω
+G) extends to a recollement.
+For each n ≥ 0, the n-th associated graded category hh(HG◦, Hω
+G)n has the following description: it is
+the direct sum
+hh(HG◦, Hω
+G)n ≃
+�
+�ν=(ν,ω)∈ �
+NP,⟨2ρ,ν⟩=n
+hh(HG◦, Hω
+G)ν,
+and for �ν = (ν, ω) ∈ �
+NP,
+hh(HG◦, Hω
+G)ν = colim u
+J ∈Tot(S♥
+�ν ) ShN (Y( u
+J )).
+Proof. In the argument we shall treat hh(HG◦, HG) as a whole. It is clear that the resulting semi-orthogonal
+decomposition induces a semi-orthogonal decomposition for each summand hh(HG◦, Hω
+G).
+Abbreviate HG,J by CJ. Let YJ,≤n be the union of geometric pieces Y( u
+J ) with ℓ(u) ≤ n. By Theorem
+4.3.9, YJ,≤n is closed in YJ. Let CJ,≤n = ShN (YJ,≤n) (the meaning of ShN is defined in the paragraph
+preceding Corollary 4.3.14). By Theorem 4.3.11(1)(2), for J ⊂ J′ ⊂ft Ia, the functors chJ′
+J send CJ,≤n to
+CJ′,≤n. Therefore we may form the colimit
+C≤n = colimJ⊂ftIa CJ,≤n
+using the functors chJ′
+J .
+For �ν ∈ �
+NP, define YJ,�ν,♥ to be the union of Y( u
+J ) where �ν(u) = �ν and ℓ(u) = ⟨2ρ, ν⟩. Let YJ,≤n,♥ be
+the substack of YJ,≤n
+YJ,≤n,♥ = YJ,≤n−1 ∪ (
+�
+�ν=(ν,ω)∈ �
+NP,⟨2ρ,ν⟩=n
+YJ,ν,♥).
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+55
+Again by Theorem 4.3.9, YJ,≤n,♥ is closed in YJ. Define CJ,≤n,♥ = ShN (YJ,≤n,♥). Theorem 4.3.11(1)(2),
+chJ′
+J sends CJ,≤n,♥ to CJ′,≤n,♥. Therefore we may form the colimit
+C≤n,♥ = colimJ⊂ftIa CJ,≤n,♥
+using the functors chJ′
+J .
+For each J ⊂ft Ia, we have natural inclusions
+CJ,≤0,♥ ֒→ CJ,≤0 ֒→ CJ,≤1,♥ ֒→ · · · ֒→ CJ,≤n−1 ֒→ CJ,≤n,♥ ֒→ CJ,≤n ֒→ · · ·
+These inclusions are compatible with the functors chJ′
+J , hence we get functors between the colimits over
+J:
+C≤0,♥
+κ0
+−→ C≤0
+i0
+−→ C≤1,♥
+κ1
+−→ · · · → C≤n−1
+in
+−→ C≤n,♥
+κn
+−−→ C≤n → · · ·
+For each J, we have CJ ≃ colimn CJ,≤n. Commuting the order of taking colimits, we have
+hh(HG) ≃ colimJ∈D◦ CJ ≃ colimn C≤n.
+The assertion of the theorem will follow from the two claims below:
+(1) For n ≥ 0, the functor in : C≤n−1 → C≤n,♥ is fully faithful and extends to a recollement diagram
+(4.4.4)
+Cn,♥
+jn! �
+jn∗ �
+C≤n,♥
+j!
+n=j∗
+n
+�
+i∗
+n
+�
+i!
+n
+�
+C≤n−1
+in
+�
+where the category Cn,♥ is canonically equivalent to the direct sum of hh(HG◦, Hω
+G)ν defined using
+(4.4.3), for ω ∈ Ω and ⟨2ρ, ν⟩ = n.
+(2) For n ≥ 0, the functor κn : C≤n,♥ → C≤n is an equivalence.
+We first prove Claim (1). Let �
+NPn be the set of �ν = (ν, ω) ∈ �
+NP such that ⟨2ρ, ν⟩ = n. For J ⊂ft Ia,
+let
+YJ,n,♥ =
+�
+�ν∈ �
+NPn
+YJ,�ν =
+�
+�ν∈ �
+NPn
+
+
+
+�
+u
+J ∈S♥
+J,�ν
+Y( u
+J )
+
+
+ .
+Let CJ,n,♥ = ShN (YJ,n,♥). The decomposition of YJ,n,♥ above gives a decomposition
+(4.4.5)
+CJ,n,♥ =
+�
+�ν∈ �
+NPn
+CJ,�ν,♥ =
+�
+�ν∈ �
+NPn
+
+
+
+�
+u
+J ∈S♥
+J,�ν
+ShN (Y( u
+J ))
+
+
+ .
+Then CJ,≤n,♥ carries a recollement structure
+(4.4.6)
+CJ,n,♥
+jJ,n! �
+jJ,n∗ �
+CJ,≤n,♥
+j!
+J,n=j∗
+J,n
+�
+i∗
+J,n �
+i!
+J,n �
+CJ,≤n−1
+iJ,n
+�
+For J ⊂ J′ ⊂ft Ia,
+u
+J ∈ S♥
+�ν,J,
+u
+J is quasi-J′-reduced since ℓ(u) = ℓ(u′) = ⟨2ρ, ν⟩ if
+u′
+J′ = σJ′
+J ( u
+J ).
+By Theorem 4.3.11(2), the functor chJ′
+J
+respects the recollement structure on CJ,≤n,♥ and induces the
+functor ⊕ u
+J ∈S♥
+J,�νγJ′,u′
+J,u
+: CJ,n,♥,�ν) → CJ′,n,♥. By Proposition A.2.1 of Appendix A, we conclude that that
+the colimit C≤n,♥ = colimJ⊂ftIa CJ,≤n,♥ also has a recollement structure by taking termwise colimits of
+(4.4.6). This gives the recollement diagram (4.4.4). Moreover, the category Cn,♥ in (4.4.4) is the direct
+sum over �ν ∈ �
+NPn of C�ν,♥ := colimJ⊂ftIa CJ,�ν,♥. By (4.4.5) and the description of chJ′
+J on CJ,n,♥ in terms
+of γJ′,u′
+J,u , we conclude that Cν,♥ is canonically equivalent to (4.4.3).
+Now we prove Claim (2). We fix n ∈ Z≥0. Let Σ<n ⊂ E× �
+NP be the set of pairs ([E], �ν) ∈ E× �
+NP such
+that ⟨2ρ, ν⟩ < n. Then Σ<n has a partial order ([E], �ν) ≤ ([E′], �ν) if [E] ≤ [E′] in E (no comparison if the
+Newton points are different). We extend this partial order to a total order on Σ<n and denote it by ⪯.
+
+56
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+For J ⊂ft Ia, let SJ,(n,[E],�ν) be the set of combinatorial pieces u
+J ∈ SJ such that ℓ(u) = n, τ( u
+J ) = [E]
+and �ν(u) = �ν. For ([E], �ν) ∈ Σ<n, define a finite subset of SJ by
+SJ,⪯(n,[E],�ν) = SJ,≤n,♥ ∪
+
+
+�
+Σ<n∋([E′],�ν′)⪯([E],�ν)
+SJ,(n,[E′],�ν′)
+
+ .
+By Lemma 4.1.8, for J ⊂ J′ ⊂ft Ia, the map δJ′
+J
+send SJ,⪯(n,[E],�ν) to SJ′,⪯(n,[E],�ν).
+Therefore the
+assignment J �→ SJ,⪯(n,[E],�ν) gives a D◦-subset S⪯(n,[E],�ν) of S�ν. Let
+B⪯(n,[E],�ν) = |S⪯(n,[E],�ν)| ⊂ B�ν.
+Let SJ,≺(n,[E],�ν) = SJ,⪯(n,[E],�ν) \ SJ,(n,[E],�ν). Thus we get a D◦-set S≺(n,[E],�ν) and its geometric realization
+B≺(n,[E],�ν).
+Let YJ,⪯(n,[E],�ν) ⊂ YJ be the union of geometric pieces Y( u
+J ) where u
+J ∈ SJ,⪯(n,[E],�ν). By Theorem
+4.3.9, YJ,⪯(n,[E],�ν) is a closed substack of YJ. Similarly we define the closed substack YJ,≺(n,[E],�ν) ⊂ YJ.
+The category of nilpotent sheaves
+CJ,⪯(n,[E],�ν) := ShN (YJ,⪯(n,[E],�ν)),
+CJ,≺(n,[E],�ν) := ShN (YJ,≺(n,[E],�ν))
+are defined as in the paragraph preceding Corollary 4.3.14.
+Theorem 4.3.11 implies that chJ′
+J
+sends
+CJ,⪯(n,[E],�ν) to CJ′,⪯(n,[E],�ν). We form the colimit
+C⪯(n,[E],�ν) := colimJ⊂ftIa CJ,⪯(n,[E],�ν).
+Similarly define C≺(n,[E],�ν). Since ⪯ is a total order, C≺(n,[E],�ν) = C⪯(n,[E′],�ν′) if ([E′], �ν′) is a predecessor
+of ([E], �ν), or if ([E], �ν) is the minimal element of Σ<n, C≺(n,[E],�ν) ≃ C≤n,♥.
+Since CJ,≤n = colim([E],�ν)∈Σ<n CJ,⪯(n,[E],�ν), taking colimit over J we get
+C≤n ≃ colim([E],�ν)∈Σ<n C⪯(n,[E],�ν).
+To show that C≤n,♥ → C≤n is an equivalence, it suffices to show that for each ([E], �ν) ∈ Σ<n, the natural
+functor
+κ[E],�ν : C≺(n,[E],�ν) → C⪯(n,[E],�ν)
+is an equivalence. For this we apply the contraction principle for cosheaves on posets that we proved in
+Theorem A.5.1, in the form of Corollary A.5.3.
+To set the stage for applying Corollary A.5.3, we define a poset I whose underlying set is
+I := D◦ ⊔ Tot(S(n,[E],�ν)).
+Let f : I → D◦ be the projection that is the identity on D◦ and the natural projection on Tot(S(n,[E],�ν)).
+Let s : D◦ ֒→ I be the inclusion.
+For the partial order, we include all order relations from D◦ and
+Tot(S(n,[E],�ν)), and take the partial order generated by the following extra relations:
+• For any J ∈ D◦ and u
+J ∈ SJ,(n,[E],�ν), we declare s(J) < u
+J .
+• For any J ⊂ J′ in D◦, u
+J ∈ SJ,(n,[E],�ν) such that δJ′
+J ( u
+J ) ∈ SJ′,≺(n,[E],�ν), we declare u
+J < s(J′).
+By construction, f is a coCartesian map of posets, and every fiber f −1(J) has a unique minimal element
+s(J).
+We next define a cosheaf F on I in the sense of Section A.3.2.
+• For J ∈ D◦, let Fs(J) = CJ,≺(n,[E],�ν) = ShN (YJ,≺(n,[E],�ν)).
+• For u
+J ∈ SJ,(n,[E],�ν), let F u
+J = ShN (YJ,≺(n,[E],�ν) ∪ Y( u
+J )). Note that Y≺(n,[E],�ν) ∪ Y( u
+J ) is closed in
+YJ.
+The transition functors are defined as follows:
+• For J ⊂ J′ ∈ D◦, the functor Fs(J) → Fs(J′) is given by chJ′
+J .
+• For J ∈ D◦ and u
+J ∈ SJ,(n,[E],�ν), the functor Fs(J) → F u
+J is ι∗ : ShN (YJ,≺(n,[E],�ν)) ֒→ ShN (YJ,≺(n,[E],�ν)∪
+Y( u
+J )) for the closed embedding ι : YJ,≺(n,[E],�ν) ֒→ YJ,≺(n,[E],�ν) ∪ Y( u
+J ).
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+57
+• For u
+J ≤ u′
+J′ in Tot(S(n,[E],�ν)) (which means δJ′
+J ( u
+J ) = u′
+J′ ), the functor F u
+J → F u′
+J′ is given by chJ′
+J .
+Here we use Theorem 4.3.11(2) to verify that chJ′
+J
+indeed sends sheaves supported on Y( u
+J ) to
+sheaves supported on Y( u′
+J′ ) since ℓ(u) = ℓ(u′) = n.
+• For u
+J < s(J′), where J ⊂ J′, u
+J ∈ SJ,(n,[E],�ν) and δJ′
+J ( u
+J ) ∈ SJ′,≺(n,[E],�ν), the functor F u
+J → Fs(J′)
+is again given by chJ′
+J . Here we again use Theorem 4.3.11(1)(2) to verify that chJ′
+J indeed sends
+sheaves supported on Y( u
+J ) either to sheaves supported on YJ′,≤n−1 ⊂ YJ′,≺(n,[E],�ν) if ℓ(u′) <
+ℓ(u) = n, or to sheaves supported on Y( u′
+J′ ) ⊂ YJ′,≺(n,[E],�ν) if ℓ(u′) = ℓ(u) = n.
+By construction, s∗F is the cosheaf on D◦ whose value at J is CJ,≺(n,[E],�ν) and transition functors are
+given by chJ′
+J . Therefore
+(4.4.7)
+C≺(n,[E],�ν) = colimD◦ s∗F.
+On the other hand, we claim that for any J ∈ D◦, (
+�
+f F)J ≃ CJ,⪯(n,[E],�ν) and this assignment extends to
+an equivalence of cosheaves on D◦ (for the definition of
+�
+f F for a coCartesian map f, see Section A.3.3).
+Indeed, since f −1(J) = {s(J)} ∪ SJ,(n,[E],�ν) as a poset has s(J) as the minimal element and no extra
+relations besides those already in SJ,(n,[E],�ν), (
+�
+f F)J is the pushout of Fs(J) diagonally embedded into
+⊕F u
+J (sum over all u
+J ∈ SJ,(n,[E],�ν)). In other words, (
+�
+f F)J is the pushout of CJ,≺(n,[E],�ν) diagonally
+embedded into ⊕ShN (YJ,≺(n,[E],�ν) ∪ Y( u
+J )) for all
+u
+J ∈ SJ,(n,[E],�ν).
+The latter is precisely CJ,⪯(n,[E],�ν)
+because of the open-closed decomposition
+YJ,⪯(n,[E],�ν) =
+
+
+�
+u
+J ∈SJ,(n,[E],�ν)
+Y( u
+J )
+
+ ⊔ YJ,≺(n,[E],�ν).
+The transition maps (
+�
+f F)J → (
+�
+f F)J′ are given by chJ′
+J , therefore we get
+(4.4.8)
+colimD◦(
+�
+f
+F)J ≃ colimD◦ CJ,⪯(n,[E],�ν) = C⪯(n,[E],�ν).
+Now we would like to apply Corollary A.5.3 to the situation of f : I → D◦ with section s, and the cosheaf
+F on I. The conclusion would be that the natural map colimD◦ s∗F → colimD◦ �
+f F is an equivalence.
+Combining with (4.4.7) and (4.4.8), we conclude that
+κ[E],�ν : C≺(n,[E],�ν) = colimD◦ s∗F → colimD◦
+�
+f
+F = C⪯(n,[E],�ν)
+is an equivalence, proving Claim (2).
+It thus remains to check that the conditions for applying Theorem A.5.1 are satisfied.
+The first condition: we define L to be the cosheaf on Tot(S(n,[E],�ν)) = I \ s(J ) whose value at u
+J is
+ShN (Y( u
+J )), and the transition functors are given by chJ′
+J . We have a natural map β : F|Tot(S(n,[E],�ν)) → L
+termwise given by open restriction ShN (YJ,≺(n,[E],�ν) ∪ Y( u
+J )) → ShN (Y( u
+J )).
+With this definition of L, we need to check that
+L
+F|Tot(S(n,[E],�ν))
+β
+�
+(s!s∗F)|Tot(S(n,[E],�ν))
+α
+�
+is a recollement of cosheaves on Tot(S(n,[E],�ν)). By Lemma A.5.2, s!s∗F ≃ f ∗s∗F, therefore we can replace
+the rightmost term above by (f ∗s∗F)|Tot(S(n,[E],�ν)). Termwise at u
+J ∈ SJ,(n,[E],�ν), they fit into a recollement
+(4.4.9)
+ShN (Y( u
+J ))
+�
+� ShN (YJ,≺(n,[E],�ν) ∪ Y( u
+J ))
+�
+�
+� ShN (YJ,≺(n,[E],�ν))
+�
+by the open-closed decomposition Y( u
+J ) ∪ YJ,≺(n,[E],�ν). All functors above are continuous by Proposi-
+tion B.0.1, therefore (4.4.9) is a recollement in StL
+k . For u
+J ≤ u′
+J′ in Tot(S(n,[E],�ν)), chJ′
+J induces a morphism
+of between the recollement (4.4.9) and its counterpart for u′
+J′ , which follows from Theorem 4.3.11(2). This
+verifies the first condition.
+
+58
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+The second condition: by Theorem 4.3.11(3), L is locally constant.
+The third condition: we need to check that |s(D◦)| ⊂ |I| is a homotopy equivalence.
+Let K :=
+Tot(S⪯(n,[E],�ν)); K′ := Tot(S≺(n,[E],�ν)). Then I \ D◦ = Tot(S(n,[E],�ν)) = K \ K′. For any D◦-set X, let
+D◦ ⊳ X be the poset whose underlying set is D◦ ⊔ X with the added relation J < x for any x ∈ XJ. Then
+|D◦ ⊳ X| is homeomorphic to the mapping cylinder [0, 1]× |X| �
+{0}×|X| |D◦| of the projection |X| → |D◦|.
+Now we can write I as a pushout of posets
+I ∼= (D◦ ⊳ K)
+�
+D◦⊳K′
+D◦
+where the map D◦ ⊳K′ → D◦ is the identity on D◦ and the projection on K′. Under this presentation, the
+inclusion s : D◦ ֒→ I corresponds to the inclusion of the second factor D◦. Taking geometric realizations
+we get
+|I| ∼= ([0, 1] × |K|)
+�
+{0}×|K|∪[0,1]×|K′|
+|D◦|.
+To show |D◦| ֒→ |I| is a homotopy equivalence, it suffice to show {0} × |K| ∪ [0, 1] × |K′| ֒→ [0, 1] × |K| is
+a homotopy equivalence, or |K′| ֒→ |K| is a homotopy equivalence.
+By Remark 4.2.3, |K| is a subdivision of B�ν,⪯(n,[E]), and |K′| is a subdivision of B�ν,≺(n,[E]). Therefore it
+suffices to check that the inclusion B�ν,≺(n,[E]) ֒→ B�ν,⪯(n,[E]) is a homotopy equivalence. Both B�ν,≺(n,[E])
+and B�ν,⪯(n,[E]) are downward subsets of B�ν in the sense of Definition 4.2.18 (see Example 4.2.19). There-
+fore by Proposition 4.2.20, the inclusions Crit(f)�ν ֒→ |B�ν,≺(n,[E])| and Crit(f)�ν ֒→ B�ν,⪯(n,[E]) are both
+homotopy equivalences, hence the inclusion B�ν,≺(n,[E]) ֒→ B⪯(n,[E]) is also a homotopy equivalence. This
+verifies the second condition and completes the proof of the theorem.
+□
+4.4.5. Remark. We expect that hh(HG) has a recollement structure indexed by the stronger poset ( �
+NP, ≤).
+Under Conjecture 1.5.1, this recollement structure should correspond to the Harder-Narasimhan stratifi-
+cation on BunG(E), which is also indexed by �
+NP.
+4.4.6. Semi-orthogonal decomposition of hh(HG). Now consider hh(HG). By the discussion in Section 2.7.12,
+hh(HG) again decomposes into a direct sum hh(HG)ω indexed by ω ∈ Ω.
+Identify Ω with length zero elements in �
+W, then Ω acts on �
+W by conjugation. The length function
+ℓ : �
+W → Z≥0 and the enhanced Newton point function �ν : �
+W → �
+NP are Ω-invariant. Therefore for
+J ⊂ft Ia and ω ∈ Ω, we have a bijection
+dω : SJ
+∼
+→ Sω(J)
+sending u
+J to ωuω−1
+ω(J) .
+For �ν ∈ �
+NP, the map dω induces a bijection
+S♥
+J,�ν
+∼
+→ S♥
+ω(J),�ν.
+This induces an action of Ω on the D◦-set S♥
+�ν , and on the total set Tot(S♥
+�ν ). We can form the groupoid
+[Tot(S♥
+�ν )/Ω] whose objects are Tot(S♥
+�ν ) and morphisms between u
+J and u′
+J′ ∈ Tot(S♥
+�ν ) is the set of ω ∈ Ω
+such that dω( u
+J ) = u′
+J′ .
+The action of cω on HG (see Section 2.7.5) induces an equivalence HG,J → HG,ω(J), which restricts to
+an equivalence ShN (Y( u
+J ))
+∼
+→ ShN (Y(dω( u
+J ))). Therefore, the assignment u
+J �→ ShN (Y( u
+J )) also extends
+to a functor
+ShN (Y(−)) : [Tot(S♥
+�ν )/Ω] → StL,
+u
+J �→ ShN (Y( u
+J )).
+We can then form the colimit
+colim u
+J ∈[Tot(S♥
+�ν )/Ω] ShN (Y( u
+J )).
+For �ν = (0, 0), [Tot(S♥
+�ν )/Ω] can be identified with D.
+In this case, the above colimit is the same as
+colimJ∈D ShN (LJ/LJ) under the induction functors.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+59
+4.4.7. Theorem. For each ω ∈ Ω, the category hh(HG)ω admits a semi-orthogonal decomposition indexed
+by non-negative integers
+hh(HG)ω
+0 ֒→ hh(HG)ω
+≤1 ֒→ · · · hh(HG)ω
+≤n ֒→ · · · ֒→ hh(HG)ω =
+�
+n≥0
+hh(HG)ω
+≤n.
+In particular, each inclusion hh(HG)ω
+≤n ֒→ hh(HG)ω extends to a recollement.
+For n ≥ 0, the n-th associated graded category hh(HG)n has the following description: it is the direct
+sum
+hh(HG)ω
+n ≃
+�
+�ν=(ν,ω)∈ �
+NP,⟨2ρ,ν⟩=n
+hh(HG)�ν
+where, for �ν ∈ �
+NP,
+hh(HG)�ν = colim u
+J ∈[Tot(S♥
+�ν )/Ω] ShN (Y( u
+J )).
+In particular, the functor ι from (4.4.1) is identified with the embedding hh(HG)0 ≃ colimJ∈D ShN (LJ/LJ) ֒→
+hh(HG)0. In particular, ι is fully faithful.
+Proof. Again we will prove the statement for the whole hh(HG), which implies the statement for each sum-
+mand hh(HG)ω. By Corollary 2.7.7 we have hh(HG) ≃ hh(HG◦, HG)Ω. Each filtration step hh(HG◦, HG)≤n
+in hh(HG◦, HG) constructed in Theorem 4.4.4 is stable under Ω. By Proposition A.2.1, taking Ω-coinvariants
+(which is the same as taking colimits over BΩ) gives a semi-orthogonal decomposition of hh(HG) with fil-
+tration pieces
+hh(HG)≤n ≃ (hh(HG◦, HG)≤n)Ω
+and associated graded category
+hh(HG)n ≃ (hh(HG◦, HG)n)Ω
+By the description of hh(HG◦, HG)n given in Theorem 4.4.4, we see hh(HG)n is a direct sum over those
+�ν = (ν, ω) ∈ �
+NP (such that ⟨2ρ, ν⟩ = n) of
+(hh(HG◦, Hω
+G)ν)Ω ≃
+�
+colim u
+J ∈Tot(S♥
+�ν ) ShN (Y( u
+J ))
+�
+Ω .
+The same argument as in the proof of Corollary 2.7.7 allows us to rewrite the right side as a colimit over
+the groupoid [Tot(S♥
+�ν )/Ω].
+□
+5. Endomorphisms of Whittaker functional
+Throughout this section, we assume the coefficient field k is algebraically closed and char(k) = 0.
+5.0.1. Notation. Fix τ a complex number with imaginary part Im(τ) ̸= 0. Let G be a reductive group
+over C with maximal torus T and t = LieT . In this section, we shall adopt the following notations:
+(1)
+• Φ the set of roots of (G, T );
+• �Φ = Z × Φ;
+• �Φ = Z × Z × Φ.
+(2) Each α ∈ Φ, �Φ or �Φ defines an affine linear function on t as follows
+• α ∈ Φ, defines a linear functional α : t → C;
+• α = (n, α) ∈ �Φ defines an affine linear function α := α + n;
+• α = (n, m, α) ∈ �Φ defines an affine linear function α := α + n + mτ.
+Under the above notations, let Hα = {x ∈ t : α(x) = 0}, and sα the reflection about Hα defined
+by sα(x) = x − α(x)α∨.
+For α ∈ Φ, put gα ⊂ g the root space, and for α = (n, α) ∈ �Φ or α = (n, m, α) ∈ �Φ, put
+gα = gα.
+(3) For J ⊂ Φ, �Φ or �Φ, put ǫJ = ∩α∈SHα, and define the following sets of affine subspaces of t
+• S = {ǫJ : J ⊂ Φ};
+
+60
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+• �S = {ǫJ : J ⊂ �Φ} 4;
+• �S = {ǫJ : J ⊂ �Φ}.
+(4)
+• W the Weyl group of (G, T ), it naturally acts t;
+• �
+W = X∗(T ) ⋊ W, it acts on t via (λ, w) · x = λ + w(x);
+• �
+W = (X∗(T ) × X∗(T )) ⋊ W (with the diagonal action of W), it acts on t via (λ1, λ2, w) · x =
+λ1 + τλ2 + w(x).
+(5)
+• For ǫ ∈ S, put Φǫ := {α ∈ Φ : α(ǫ) = 0}, and Wǫ = ⟨sα : α ∈ Φǫ⟩ ⊂ W;
+• For ǫ ∈ �S, put Φǫ := {α ∈ �Φ : α(ǫ) = 0}, and Wǫ = ⟨sα : α ∈ Φǫ⟩ ⊂ �
+W;
+• For ǫ ∈ �S, put Φǫ := {α ∈ �Φ : α(ǫ) = 0}, and Wǫ = ⟨sα : α ∈ Φǫ⟩ ⊂ �
+W.
+For ǫ ∈ S, �S, or �S, put Lǫ the connected reductive subgroup of G containing T with roots Φǫ,
+so that it has Weyl group Wǫ.
+(6)
+• For ǫ ∈ S, put W ǫ = NW (Wǫ)/Wǫ;
+• For ǫ ∈ �S, put �
+W ǫ = N�
+W (Wǫ)/Wǫ;
+• For ǫ ∈ �S, put �
+W ǫ = N�
+W (Wǫ)/Wǫ.
+(7) Put CG the set of isomorphism classes of cuspidal sheaves on NG/G, where NG ⊂ g is the nilpotent
+cone of G. For definition, see [Lus87, §2].
+• C = {(ǫ, B, F) : ǫ ∈ S, F ∈ CLǫ, B ⊃ T a Borel subgroup of Lǫ};
+• �C = {(ǫ, B, F) : ǫ ∈ �S, F ∈ CLǫ, B ⊃ T a Borel subgroup of Lǫ};
+• �C = {(ǫ, B, F) : ǫ ∈ �S, F ∈ CLǫ, B ⊃ T a Borel subgroup of Lǫ}.
+(8) We also use the notation WG, SG, CG, etc, to emphasis the dependence on G.
+(9) For J ⊂ Φ, �Φ or �Φ, put LJ = LǫJ, WJ = WǫJ, W J = W ǫJ, zJ = X∗(Z(LJ)◦) ⊗Z k, CJ = CLJ,
+�CJ = �CLJ, etc.
+5.1. Combinatorial descriptions of character sheaves. In this section, we shall recall the combina-
+torial description of the dg category of character sheaves in [Lia] (for simply-connected group), and [Lib]
+(for reductive group). Using the same idea, we give a combinatorial description of the colimit category
+colimJ∈D ShN (LJ/LJ), the latter may be thought of as an elliptic version of character sheaves. The main
+results and the relationship between various versions of character sheaves is summarized in Corollary 5.1.9.
+5.1.1. Sheaves of categories on affine spaces. Let Λ be a free Z-module of finite rank, and ΛC be com-
+plexified affine space. Let F be a locally finite set of complex affine subspaces of ΛC. For any ǫ ∈ F,
+denote ǫ the linear space parallel to ǫ, and zǫ = (ǫ ∩ Λ) ⊗Z k the finite dimensional affine space over k.
+Put QCohǫ the constant sheaf of categories on ǫ with value QCoh(z∗
+ǫ[−1]). For a map of sets f : G → F,
+denote QCohG = �
+c∈G QCohf(c), viewed as a sheaf of categories on ΛC. Let K be a discrete group of affine
+linear transformations acting properly discontinuously on ΛC, assume that the linear part of K preserves
+Λ, and F is stable under the K-action. Assume also that K acts on G, such that f is K-equivariant, then
+QCohG is naturally a K-equivariant sheaf of categories on ΛC. Therefore, for any K-invariant open subset
+U ⊂ ΛC, we have natural K-action on Γ(U, QCohG). Denote the invariant category by Γ(U, QCohG)K.
+5.1.2. Remark. Let A be an dg-algebra over k, and K be a discrete group acting strictly on A. The
+smash product k[K]#A is by definition a dg algebra whose underlying dg vector space is k[K] ⊗ A, and
+the multiplication is given by (w ⊗ a) · (w′ ⊗ a′) = (ww′ ⊗ w′−1(a)a′), for all w ∈ K and a ∈ A. There is
+natural equivalence of dg categories (see e.g. [Lia, Section 2.2]):
+(k[K]#A)-mod ≃ (A-mod)K
+Using smash product, we can write Γ(ΛC, QCohG)K more concretely as follows: let G � K be the set of
+K-orbits on G, and let [G � K] ⊂ G be a set of representatives of each orbit, and denote Kc the stabilizer
+4Note that by definition �S is in natural bijection with the set E of relevant affine subspaces in the standard apartment
+A = tR, see Section 4.1.4.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+61
+of K at c ∈ G. Then we have equivalence of categories:
+(5.1.1)
+Γ(ΛC, QCohG)K ≃
+�
+c∈[G�K]
+(Sym(zf(c)[1])-mod)Kc ≃
+�
+c∈[G�K]
+k[Kc]#Sym(zf(c)[1])-mod.
+Let K′ ⊂ K be a subgroup, and G′ ⊂ G be a K′-stable subset. We have a pair of adjoint functors:
+(5.1.2)
+IndG/K
+G′/K′ : Γ(ΛC, QCohG′)K′ ⇄ Γ(ΛC, QCohG)K : ResG/K
+G′/K′
+And when the context is clear, we shall simply denote them by Ind, Res.
+Now we take Λ = X∗(T ), and hence ΛC = t. In Notation 5.0.1, the groups W, �
+W and �
+W naturally act
+on S, �S and �S respectively. For any w ∈ W, �
+W or �
+W, denote its image in W by w. Therefore Ad(w)
+gives idenfications Lǫ ≃ Lw(ǫ), and CLǫ ≃ CLw(ǫ). This gives an action of W, �
+W and �
+W on C, �C and �C
+respectively. Let f : C → S, �f : �C → �S and �f : �C → �S be the projections. Therefore the categories
+Γ(t, QCohC)W , Γ(t, QCoh�C)
+�
+W , and Γ(t, QCoh�C)
+�
+W
+are defined.
+We have the following combinatorial description of character sheaves on G and g:
+5.1.3. Theorem. Let G be a connected reductive group over C. There are equivalences of categories:
+(1) Lg : ShN (g/G) ≃ Γ(t, QCohC)W ;
+(2) LG : ShN (G/G) ≃ Γ(t, QCoh�C)�
+W .
+Proof. (1) This is essentially the generalized Springer decomposition. We use notations from Section 1.6.4;
+in particular, I denotes the set of simple roots of G with respect to T and B. Let
+(5.1.3)
+D := {(J, F) : J ⊂ I, F ∈ CJ = CLJ}.
+By [Lia, Theorem 1.2], we have an equivalence of categories:
+(5.1.4)
+ShN (g/G) ≃
+�
+(J,F )∈D
+k[W J]#Sym(zJ[1])-mod.
+Let �f : D → C be the map sending (J, F) to (ǫJ, B∩LJ, F), and let f be the composition D
+�
+f−→ C → C�W.
+Then f is bijective by [Lia, Lemma 3.1] (the content here is: if LJ carries a cuspidal local system, then
+any two parabolics containing LJ as a Levi subgroup are conjugate in G). One can further identify W J
+with Wc, the stabilizer of c = f(J, F) under the W-action on C. Therefore, (1) holds by (5.1.1).
+(2) is proved in [Lib, Theorem 1.1.1].
+□
+In view of Remark 5.1.2, Theorem 5.1.3 implies the following explicit description of the category of
+character sheaves on the group G, analogous to (5.1.4).
+5.1.4. Corollary. For any connected reductive group G over C, there is a canonical equivalence
+(5.1.5)
+ShN(G/G) ≃
+�
+{(J,F ):J⊂ftIa,F ∈CJ}/Ω
+k[�
+W J]#Sym(zJ[1])-mod.
+5.1.5. Remark. Let F ∈ ShN (G/G) be an object whose image on the right side of (5.1.5) lies in a single
+summand indexed by (J, F). Then the support of F is contained in the closed subset of G consisting of
+elements whose semisimple parts lie in Ad(G)(exp(ǫJ)), where the exponential map exp : t → T normalized
+to have kernel X∗(T ).
+Now we explain the relationship between character sheaves on the group and on the Lie algebra. There
+is a natural functor β1 : ShN(G/G) → ShN (g/G) defined as follows: let D be a Ad(G)-invariant star-
+shaped open subset of g such that exp : D → G is an isomorphism onto its image. Let j : D/G ֒→ g/G be
+the open inclusion. Then j∗ : ShN (g/G) → ShN (D/G) is an equivalence of categories, and we put
+(5.1.6)
+β1 = (j∗)−1 ◦ exp |∗
+D/G : ShN (G/G) → ShN (g/G).
+The functor β1 perserves limit, and therefore admits a left adjoint α1.
+
+62
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+5.1.6. Proposition. Under the equivalence in Theorem 5.1.3, the diagrams naturally commute:
+ShN (g/G)
+α1
+�
+≃
+Lg
+� Γ(t, QCohC)W
+Ind �
+ShN(G/G)
+≃
+LG
+�
+β1
+�
+Γ(t, QCoh�C)�
+W
+Res
+�
+Proof. This follows from the proof of [Lib, Theorem 4.3.1]: under the notation loc.
+cit., the functor
+ShN (G/G) → ShN (g/G) is identified with the natural functor limI∈FG ChG → ChG(I = 0), which is
+identified with the functor limI∈FG QCoh
+�
+W
+�C → QCoh
+�
+W
+�C (I = 0), which is then further identified with Res :
+Γ(t, QCoh�C)�
+W → Γ(t, QCohC)W . The other commutative square follows by adjunction.
+□
+Now let P ⊃ T be a parabolic subgroup of G, with Levi subgroup L ⊃ T . Then we have a pair of
+adjoint functors by parabolic induction and restriction:
+IndG
+L⊂P : ShN (L/L) ⇄ ShN (G/G) : ResG
+L⊂P
+5.1.7. Proposition ([Lib],Theorem 1.1.2). Under the equivalence in Theorem 5.1.3(2), the diagrams nat-
+urally commute:
+ShN (L/L)
+IndG
+L⊂P
+�
+≃
+LL
+� Γ(t, QCoh�CL)�
+WL
+Ind
+�
+ShN (G/G)
+≃
+LG
+�
+ResG
+L⊂P
+�
+Γ(t, QCoh�CG)�
+WG
+Res
+�
+Next we will prove a double affine version of Theorem 5.1.3. Instead of considering character sheaves
+on the Lie algebra or the group, we consider the colimit colimJ∈D ShN (LJ/LJ). Recall in Section 4.4.6
+we have identified the initial filtered piece hh(HG)0 of hh(HG) with this colimit. From the point of view of
+Betti geometric Langlands, this colimit can be thought of as an elliptic version of character sheaves, i.e.,
+nilpotent sheaves on the semistable locus of BunG(E) for an elliptic curve.
+5.1.8. Theorem. There is an equivalence of categories
+LG : colimJ∈D ShN (LJ/LJ) ≃ Γ(t, QCoh�C)
+�
+W
+where the colimit is formed using the parabolic induction functors as in Section 4.4.1.
+Proof. We may rewrite the colimit on the left side as a limit
+lim
+J∈Dopp ShN(LJ/LJ)
+using the parabolic restriction functors that are right adjoint to the induction functors.
+We view �
+W as a subgroup of �
+W in two ways: �
+W = (X∗(T ) × 0) ⋊ W, and �
+W τ = (0 × X∗(T )) ⋊ W.
+Denote W a ⊂ �
+W, W a,τ ⊂ �
+W τ the corresponding affine Weyl group. Let A := X∗(T )R. Recall we have
+fixed an imaginary τ so that t = AC = A × Aτ. For a subset X ⊂ A, Xτ denotes the corresponding subset
+of Aτ, which in turn is a subset of t.
+The affine space A is stratified by facets given by connected components of intersections of affine
+roots hyperplanes. For any J ⊂ft Ia, let AJ be the corresponding facet in the fundamental alcove, i.e.,
+AJ = {x ∈ A|α(x) = 0, β(x) > 0, for all α ∈ J, β ∈ Ia \ J}. Let star(AJ) ⊂ A be the star of AJ, namely,
+star(AJ) is the union of facets whose closures contain AJ. Denote the open subset:
+VJ := A × star(AJ)τ ⊂ A × Aτ = AC.
+We have an isomorphism of topological groupoids
+colimJ∈D◦ star(AJ)τ/WJ = Aτ/W a,τ.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+63
+Dividing Ω on both side, we get an isomorphism of topological groupoids
+(5.1.7)
+colimJ∈D star(AJ)τ/WJ ≃ Aτ/�
+W τ.
+Let X∗(T ) ⋊ W τ
+J ⊂ �
+W be the subgroup generated by X∗(T ) × 0 and W τ
+J ⊂ W a,τ.
+Abstractly, it is
+isomorphic to the semi-direct product X∗(T ) ⋊ WJ ≃ �
+WJ, the extended affine Weyl group of LJ. Then
+(5.1.7) induces an isomorphism of topological groupoids
+colimJ∈D VJ/(X∗(T ) ⋊ W τ
+J ) ≃ t/�
+W.
+Since all arrows in the above colimit are open embeddings, it further induces an equivalence
+Γ(t, QCoh�C)
+�
+W ≃ Γ(t/�
+W, QCoh�C)
+≃
+lim
+J∈Dopp Γ(VJ/(X∗(T ) ⋊ W τ
+J ), QCoh�C)
+≃
+lim
+J∈Dopp Γ(VJ, QCoh�C)X∗(T )⋊W τ
+J
+(5.1.8)
+On the other hand, choose any x ∈ AJτ, and put tx : t → t the translation by x. Then there is an
+canonical isomorphism of sheaves on VJ
+(5.1.9)
+tx,∗(QCoh�CJ)|VJ ≃ QCoh�C|VJ .
+This is because affine spaces in �C having nonempty intersections with VJ are precisely those of the form
+tx(ǫ), for ǫ in �CJ. The isomorphism (5.1.9) is naturally X∗(T ) ⋊ W τ
+J ≃ �
+WJ equivariant, and induces an
+equivalence on global sections:
+(5.1.10)
+Γ(VJ, QCoh�C)X∗(T )⋊W τ
+J ≃ Γ(t−1
+x (VJ), QCoh�CJ)
+�
+WJ ≃ Γ(t, QCoh�CJ)
+�
+WJ .
+Here, the second equivalence is because t−1
+x (VJ) is convex and has nonempty intersection with each affine
+space in �CJ (and hence the intersection is contractible). Moreover for J ⊂ J′ ⊂ft Ia, under (5.1.10) the
+restriction map:
+Γ(VJ, QCoh�C)X∗(T )⋊W τ
+J → Γ(VJ′, QCoh�C)X∗(T )⋊W τ
+J′
+is identified with the restriction map defined in (5.1.2):
+Res
+�CJ′/�
+WJ′
+�CJ/�
+WJ
+: Γ(t, QCoh�CJ)
+�
+WJ → Γ(t, QCoh�CJ′ )
+�
+WJ′
+It is also clear that (5.1.10) is equivariant under the action of Ω. Therefore, the functor Dopp → StR
+k
+defined by J �→ Γ(VJ, QCoh�C)X∗(T )⋊W τ
+J can be identified with the functor J �→ Γ(t, QCoh�CJ )�
+WJ (using the
+restriction functors and the Ω-action). In summary, we have
+Γ(t, QCoh�C)
+�
+W ≃
+lim
+J∈Dopp Γ(VJ, QCoh�C)X∗(T )⋊W τ
+J
+by (5.1.8)
+≃
+lim
+J∈Dopp Γ(t, QCoh�CJ)
+�
+WJ
+by (5.1.10)
+≃
+lim
+J∈Dopp ShN(LJ/LJ)
+by Theorem 5.1.3 (2) and Prop 5.1.7
+(5.1.11)
+□
+5.1.9. Corollary. Under Theorem 5.1.3 and Theorem 5.1.8, the diagrams naturally commutes:
+(5.1.12)
+ShN (g/G)
+α1
+�
+≃
+Lg
+�
+ShN (G/G)
+α2
+�
+≃
+LG �
+colimJ∈D ShN(LJ/LJ)
+≃
+LG �
+Γ(t, QCohC)W
+Ind1 � Γ(t, QCoh�C)�
+W
+Ind2
+� Γ(t, QCoh�C)�
+W
+where α2 is the natural map into the colimit corresponding to J = I, and Ind1, Ind2 are the induction
+functors defined in (5.1.2).
+
+64
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Proof. The first square is commutative by Proposition 5.1.6. For the second square, by (5.1.11), the right
+adjoint of α2 corresponds to the restriction map Γ(t, QCoh�C)�
+W → Γ(VI, QCoh�C)�
+W , and the latter can be
+identified with Γ(t, QCoh�C)�
+W . Passing to left adjoints gives the commutativity of the second square.
+□
+5.2. Fourier-Sato transform and Whittaker sheaf. In this section, we study the Lie algebra version
+of the Whittaker sheaf, and identify its image under the equivalence in Proposition 5.1.6. The main tool
+here is the Fourier-Sato transform.
+Consider the diagram
+(5.2.1)
+Ga
+u−/U −
+f
+�
+r− � g/G
+where r− is induced by the inclusion u− ֒→ g, and f = dχ : u− → Ga is the differential of the generic
+character χ : U − → Ga used to define the Whittaker functor on the group, see Section 2.2.1.
+5.2.1. Definition.
+(1) The Whittaker functor on ShN (g/G) is the functor
+Wg/G : ShN (g/G) → k-mod,
+Wg/G(F) = ϕf,0 ◦ r!
+−F.
+(2) The Whittaker sheaf on the Lie algebra is the object Whg/G ∈ ShN (g/G) corepresenting the
+Whittaker functor Wg/G, i.e.,
+Wg/G(F) ≃ HomShN (g/G)(Whg/G, F),
+for all F ∈ ShN (g/G).
+Let
+T : ShN (g/G) ≃ Sh(N/G)
+be the Fourier-Sato transform, normalized to be t-exact with respect to the perverse t-structures.
+5.2.2. Proposition. Let e ∈ u be the regular nilpotent element corresponding to f ∈ (u−)∗ under the
+invariant form on g. Let ie : {e} → N/G be the natural map. Then there is a natural commutative
+diagram:
+ShN (g/G) �
+T
+Wg/G
+�◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+Sh(N/G)
+i∗
+e[r]
+�
+k-mod
+Proof. Since f is linear, we have
+(5.2.2)
+ϕf,0(K) = ϕid,0(f∗(K))
+for any Gm-monodromic sheaf K on u− (where Gm acts by dialition). Moreover, for any Gm-monodromic
+sheaf E on Ga, we have
+(5.2.3)
+ϕid,0(E) ≃ i!
+1(T(E))[1]
+where i1 : {1} ֒→ Ga.
+We have the dual maps to (5.2.1)
+Ga
+i
+� u
+g
+π
+�
+,
+where π is the quotient map to g/b− ∼= u, and i(c) = ce. For any F ∈ Sh(N/G), we have
+Wg ◦ T(F) ≃ ϕid,0 ◦ f∗ ◦ r!
+− ◦ T(F) ≃ ϕid,0 ◦ TGa ◦ i! ◦ π∗(F)[?]
+≃ i!
+1 ◦ i! ◦ π∗(F)[?] ≃ RΓ(i!
+π−1(e)F)[?] = RΓ(i!
+π−1(e)∩N F)[?] ≃ i∗
+e[?]
+where the first isomorphism uses (5.2.2), the second uses the standard property of the Fourier-Sato trans-
+form under linear maps (here TGa is the Fourier-Sato transform for Sh(Ga)), and the third uses (5.2.3).
+The last isomorphism is because π−1(e)∩N = (e+b−)∩N is the free orbit of e under the adjoint action of
+U −, which is contractible and along which F is contant. Although we ignored all the cohomological shifts
+in the above calculation, we still conclude that Wg/G ◦ T ≃ i∗
+e[r], because both functors are t-exact.
+□
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+65
+5.2.3. Proposition. The diagram naturally commutes:
+ShN (G/G)
+β1
+�
+WG/G
+�❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+ShN (g/G)
+Wg/G
+�
+k-mod
+Proof. Let D ⊂ g as in the definition of β1 (c.f (5.1.6)). Put D′ := exp(D) ⊂ G. Let f : U − → Ga and
+f ′ : u− → Ga be the function induced by a non-degenerate character of u−/[u−, u−], then the diagram
+commutes:
+Ga
+=
+�
+u− ∩ D
+exp
+≃
+�
+f
+�
+r0
+� D/G
+exp
+≃
+�
+j
+� g/G
+Ga
+U − ∩ D′
+f ′
+�
+r′
+0
+� D′/G
+j′
+� G/G
+The Whittaker functor Wg/G on the Lie algebra g can be identified with ϕf,− ◦ r!
+0 ◦ j∗, therefore
+Wg/G ◦ β1 = ϕf,0 ◦ r!
+0 ◦ j∗ ◦ (j∗)−1 ◦ exp |∗
+D/G ◦ j′∗ = ϕf ′,1 ◦ r′
+0
+! ◦ j′∗ = WG/G.
+□
+5.2.4. Generalized Springer correspondence. We recall the generalized Springer correspondence due to
+Lusztig [Lus84, Theorem 6.5]. For a nilpotent orbit O, let AG(O) = CG(e)/CG(e)◦ for e ∈ O, which
+is a finite group well-defined up to inner automorphisms. For a finite group Γ, let Irr(Γ) be the set of
+irreducible representations of Γ over k. Then the generalized Springer correspondence is a bijection
+P := {(O, χ) : O a nilpotent orbit of G, χ ∈ Irr(AG(O))}
+GSpr
+1:1
+�
+�D := {(J, F, θ) : J ⊂ I, F ∈ CJ, θ ∈ Irr(W J)}.
+Let Preg ⊂ P be the subset of pairs (O, χ) where O = Oreg is the regular nilpotent orbit. Since AG(Oreg) =
+Z(G)/Z(G)◦, we can identify Preg with Irr(Z(G)/Z(G)◦).
+On the other hand, recall the set D from (5.1.3). Let Dreg ⊂ D be the subset of those (J, F) ∈ D such
+that Supp(F) = NLJ.
+5.2.5. Lemma. Under the map
+(5.2.4)
+P
+GSpr
+−−−→ �D → D
+the subset Preg maps bijectively to Dreg. In particular, it induces a bijection
+(5.2.5)
+Irr(Z(G)/Z(G)◦)
+1:1
+−−→ Dreg
+sending χ to the unique pair (Jχ, Fχ) ∈ Dreg such that GSpr(Oreg, χ) = (Jχ, Fχ, θχ) for some θχ ∈
+Irr(W Jχ).
+Proof. Let (J, F, θ) = GSpr(Oreg, χ). We claim that Supp(F) = NLJ. By the construction of the general-
+ized Springer correspondence, IC(Oreg, χ) is a direct summand of IndG
+LJ(F), where IndG
+LJ is the pull-push
+functor along the diagram (and PJ is a parabolic subgroup of G containing LJ as a Levi subgroup)
+NG/G
+NPJ/PJ
+qJ
+�
+pJ
+�
+NLJ/LJ
+If Supp(F) was properly contained in NLJ, pJ(q−1
+J Supp(F)) would not meet Oreg. Therefore we must
+have Supp(F) = NLJ. This shows that the map (5.2.4) sends Preg to Dreg, and the map (5.2.5) is defined.
+The map (5.2.5) is bijective because for each (J, F) ∈ Dreg, IndG
+LJ(F) has full support on NG, and its
+restriction to Oreg has rank one, hence it has a unique direct summand of the form IC(Oreg, χ) for some
+χ ∈ Irr(Z(G)/Z(G)◦).
+□
+
+66
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+5.2.6. Proposition. Under the equivalence (5.1.4), the Lie algebra Whittaker sheaf
+Whg/G ∈ ShN (g/G)
+is identified with
+�
+(Jχ,Fχ)∈Dreg
+Sym(zJχ[1])[−r] ∈
+�
+(J,F )∈D
+k[W J]#Sym(zJ[1])-mod ≃ ShN (g/G).
+Here we use the bijection in Lemma 5.2.5 to index elements in Dreg by χ ∈ Irr(Z(G)/Z(G)◦).
+Proof. Let AJ = k[W J]#Sym(z∗
+J[−2]). Let ktriv be the 1-dimensional AJ-module in degree 0 where W J
+acts trivially. Then W J-equvariant Koszul duality gives an equivalence
+(5.2.6)
+HomSym(z∗
+J[−2])(ktriv, −) : AJ-perf ≃ k[W J]#Sym(zJ[1])-modfd
+where we identify Sym(zJ[1]) with EndSym(z∗
+J[−2])(ktriv)opp, and -modfd stands for the category of finite
+dimensional modules. Under this equivalence, ktriv corresponds to Sym(zJ[1]).
+Let Shc,N(g/G) ⊂ ShN (g/G) be the full subcategory of constructible sheaves. Using (5.1.4) and (5.2.6),
+we have equivalences
+(5.2.7)
+Shc,N (g/G) ≃ �
+(J,F )∈D k[W J]#Sym(zJ[1])-modfd
+(5.2.6)
+∼
+� �
+(J,F )∈D AJ-perf.
+Moreover, the composition
+Shc(N/G)
+T−→ Shc,N(g/G) ≃
+�
+(J,F )∈D
+AJ-perf
+is identified with the functor HomN/G(⊕(J,F )∈DIndG
+LJ⊂PJ F, −), where we use
+End(IndG
+LJ⊂PJF) ≃ Aopp
+J
+.
+By Prop 5.2.2, T−1Whg/G is isomorphic to (ie)![r] where ie : {e} → N/G is the inclusion of a regular
+nilpotent element e. In particular, T−1Whg/G ∈ Shc(N/G), hence Whg/G corresponds to a collection of
+perfect AJ-modules MJ,F, one for each (J, F) ∈ D.
+For any (J, F) ∈ D, put eJ ⊂ NLJ a regular nilpotent element. Then by Prop 5.2.2
+HomNG/G(T−1Whg/G, IndG
+LJ⊂PJ(F)) ≃ i∗
+e(IndG
+LJ⊂PJ(F))[r] = i∗
+eJ(F)[r] =
+�
+ksign,
+(J, F) ∈ Dreg
+0,
+(J, F) /∈ Dreg
+where ksign is the 1-dimensional End(IndG
+LJ⊂PJ(F))-module where W J acts by the sign representation
+(this is because we identify the Weyl group actions on Springer fibers via Fourier transform, therefore the
+regular Springer fiber corresponds to the sign representation). Therefore we have an isomorphism of right
+AJ-modules (the right AJ-action comes from post-composing with the right AJ-action on itself)
+HomAJ(MJ,F , AJ) ≃
+�
+ksign,
+(J, F) ∈ Dreg
+0,
+(J, F) /∈ Dreg.
+Note also that there is an equivalence of categories between left and right AJ-modules:
+HomAJ(−, AJ) : AJ-perf
+∼
+� Aopp
+J
+-perf
+under which the 1-dimensional trivial module ktriv corresponds to ksign[r] by a Koszul resolution calcula-
+tion. From this we conclude that
+MJ,F ≃
+�
+ktriv[−r],
+(J, F) ∈ Dreg
+0,
+(J, F) /∈ Dreg.
+Therefore, under the equivalence (5.2.7), Whg/G corresponds to
+⊕(J,F )∈Dregktriv[−r] ∈
+�
+(J,F )∈D
+k[W J]#Sym(z∗
+J[−2])-perf,
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+67
+which is further identified under Koszul duality (5.2.6)with
+⊕(J,F )∈DregSym(zJ[1])[−r] ∈
+�
+(J,F )∈D
+k[W J]#Sym(zJ[1])-modfd ⊂
+�
+(J,F )∈D
+k[W J]#Sym(zJ[1])-mod.
+□
+5.3. Calculation of endormorphisms of the Whittaker sheaf. Recall from Section 2.8 the descended
+trace WhG/G ∈ hh(HG) of the universal affine Whittaker sheaf WhG. We can now calculate its endomor-
+phism algebra explicitly.
+For χ ∈ Irr(Z(G)/Z(G)◦), denote Tχ = Z(Lχ)◦, Wχ = W Jχ. Let T ∨
+χ the torus over k dual to Tχ (i.e.,
+X∗(T ∨
+χ ) = X∗(Tχ)), and t∨
+χ = Lie(T ∨
+χ ) = z∗
+Jχ.
+Recall the functor α2 : ShN (G/G) → colimJ∈D ShN(LJ/LJ) corresponding to J = I. Define
+WhG,I := α2(WhG/G) ∈ colimJ∈D ShN (LJ/LJ).
+5.3.1. Theorem. There is a canonical isomorphism of dg algebras:
+EndcolimJ∈D ShN (LJ/LJ)(WhG,I) ≃
+�
+χ∈Irr(Z(G)/Z(G)◦)
+O(T ∨
+χ × T ∨
+χ × t∨
+χ[−1])Wχ
+Proof. By Proposition 5.2.3, we have
+WhG,I = α2(WhG/G) ≃ α2 ◦ α1(Whg/G).
+By Corollary 5.1.9, the image of WhG,I in Γ(t, QCoh�C)�
+W is
+(5.3.1)
+Ind2 ◦ Ind1(Lg(Whg/G)) ∈ Γ(t, QCoh�C)
+�
+W .
+For any ǫ ∈ S, put Λǫ = X∗(T ) ∩ ǫ.
+For c = (ǫ, B, F) ∈ C, the summand corresponds to c
+in Γ(t, QCohC)W is equivalent to the category k[W ǫ]#Sym(zǫ[1])-mod.
+Similarly the summand corre-
+sponds to c in Γ(t, QCoh�C)�
+W (resp.
+in Γ(t, QCoh�C)�
+W ) is equivalent to k[�
+W ǫ]#Sym(zǫ[1])-mod (resp.
+k[�
+W ǫ]#Sym(zǫ[1])-mod). We also have �
+W ǫ = Λǫ ⋊ W ǫ, and �
+W ǫ = (Λǫ × Λǫ) ⋊ W ǫ. Therefore, Ind2 ◦ Ind1
+can be identified with the direct sum of the following induction functors over (J, F) ∈ D:
+Ind
+�
+W ǫ
+W ǫ : k[W J]#Sym(zJ[1])-mod → k[�
+W J]#Sym(zJ[1])-mod.
+By Proposition 5.2.6 and (5.3.1), the image of WhG,I in Γ(t, QCoh�C)�
+W is
+Ind2 ◦ Ind1
+
+
+�
+χ∈Irr(Z(G)/Z(G)◦)
+Sym(zJχ[1])
+
+ =
+�
+χ∈Irr(Z(G)/Z(G)◦)
+Ind
+�
+W Jχ
+W Jχ Sym(zJχ[1]) ∈ Γ(t, QCoh�C)
+�
+W .
+Therefore (all direct sums are over χ ∈ Irr(Z(G)/Z(G)◦))
+End(WhG,I)
+≃
+�
+χ
+Endk[�
+W Jχ ]#Sym(zJχ[1])
+�
+Ind
+�
+W Jχ
+W Jχ Sym(zJχ[1])
+�
+≃
+�
+χ
+Homk[W Jχ ]#Sym(zJχ[1])
+�
+Sym(zJχ[1]), Res
+�
+W Jχ
+W Jχ Ind
+�
+W Jχ
+W Jχ Sym(zJχ[1])
+�
+≃
+�
+χ
+�
+k[ΛJχ × ΛJχ] ⊗ Sym(zJχ[1])
+�W Jχ
+=
+�
+χ
+O(T ∨
+χ × T ∨
+χ × t∨
+χ[−1])Wχ.
+In the last identity we use ΛJχ = X∗(Tχ) = X∗(T ∨
+χ ), t∨
+χ ∼= z∗
+Jχ and Wχ = W Jχ.
+□
+5.3.2. Corollary. There is a canonical isomorphism of dg algebras:
+Endhh(HG)(WhG/G) ≃
+�
+χ∈Irr(Z(G)/Z(G)◦)
+O(T ∨
+χ × T ∨
+χ × t∨
+χ[−1])Wχ.
+
+68
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Proof. By Theorem 2.8.5, we have WhG/G ≃ a(WhG/G) where a is the natural map ShN(G/G) ֒→ HG,I →
+hh(HG). By Theorem 4.4.2, we have a fully-faithful embedding
+(5.3.2)
+colimJ∈D ShN(LJ/LJ) ֒→ hh(HG).
+Under this embedding, we can identify α2 : ShN (G/G) → colimJ∈D ShN (LJ/LJ) (corresponding to the
+term J = I) with a : ShN (G/G) → hh(HG). Hence WhG/G can be identified with the image of WhG,I
+under the embedding (5.3.2). Therefore
+Endhh(HG)(WhG/G) ≃ EndcolimJ∈D ShN (LJ/LJ)(WhG,I).
+The desired statement now follows from Theorem 5.3.1.
+□
+5.3.3. Corollary. Assume Ansatz 1.2.5. Then there is a canonical isomorphism of dg algebras:
+O(Z2
+G∨) ≃
+�
+χ∈Irr(Z(G)/Z(G)◦)
+O(T ∨
+χ × T ∨
+χ × t∨
+χ[−1])Wχ.
+In particular, H0(O(Z2
+G∨)) ≃ �
+χ∈Irr(Z(G)/Z(G)◦) O(T ∨
+χ × T ∨
+χ )Wχ is reduced.
+5.3.4. Remark. Although we primarily focus on the derived structure and reduceness of the commuting
+stack, our theorem also gives new perspectives even for coarse moduli spaces, which we will elaborate in
+Section 6.2.
+5.4. Additional applications. In this section, we use similar method to compute two additional endo-
+morphism algebras of interest: (i) the derived spherical Hecke algebra and (ii) endomorphisms of parabolic
+inductions (for simplicity, we will restrict to inductions from a torus). First, we will make the automorphic
+calculations, then explain their spectral implications.
+5.4.1. Automorphic calculations. For each of the categories Γ(t, QCohC)W , Γ(t, QCoh�C)�
+W and Γ(t, QCoh�C)�
+W ,
+there is a “principal block” (direct summand) corresponding to the affine space ǫ = t and the skyscraper
+sheaf on T . They are described as follows:
+k[W]#Sym((t∨)∗[1])-mod ≃ k[W]#O(t∨[−1])-mod ⊂ Γ(t, QCohC)W ,
+k[�
+W]#Sym((t∨)∗[1])-mod ≃ k[W]#O(T ∨ × t∨[−1])-mod ⊂ Γ(t, QCoh�C)
+�
+W ,
+k[�
+W]#Sym((t∨)∗[1])-mod ≃ k[W]#O(T ∨ × T ∨ × t∨[−1])-mod ⊂ Γ(t, QCoh�C)
+�
+W .
+5.4.2. Lemma. A simple perverse sheaf F ∈ ShN (G/G) lies in the principal block if and only if the
+following two conditions hold:
+(1) The support of F (which is closed by definition) contains 1 ∈ G;
+(2) Every simple constituent of the Fourier transform T−1(β1F) ∈ Sh(N/G) is a summand of the
+Springer sheaf Spr ∈ Sh(N/G) (normalized to be perverse, i.e., stalks along regular nilpotent
+elements lie in degree r).
+Moreover, when F ∈ ShN (G/G) is in the principal block, Lg(β1F) ∈ k[W]#Sym((t∨)∗[1])-mod is Koszul
+dual to
+HomN/G(Spr, T−1(β1F)) ∈ End(Spr)opp-mod ≃ k[W]#Sym(t∨[−2])-mod.
+Proof. First, assume F lies in the principal block. By Proposition 5.1.6, β1F lies in the principal block and
+is nonzero because the restriction functor Res : k[�
+W]#Sym((t∨)∗[1])-mod → k[W]#Sym((t∨)∗[1])-mod is
+faithful. Being in the principal block, all simple constituents of β1F are summands of the Grothendieck-
+Springer sheaf (direct image of constant sheaf under b/B → g/G).
+This implies that all simple con-
+stituents of T−1(β1F) are summands of Spr, which verifies condition (2). Since all simple summands of
+the Grothendieck-Springer sheaf have full support, and β1F ̸= 0, its support contains 0. In particular, the
+support of F contains 1, which verifies condition (1).
+Conversely, assume the conditions (1) and (2) hold. Using Remark 5.1.5 and condition (1), we see that
+LG(F) lies in a summand in (5.1.5) indexed by (J, F) such that 1 ∈ exp(ǫJ). This implies that, up to
+changing J by the action of Ω, we may arrange J ⊂ I. Under the restriction functor Res : Γ(t, QCoh�C)�
+W →
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+69
+Γ(t, QCohC)W , the block indexed by (J, F) with J ⊂ I on the source maps into the block indexed by the
+same (J, F) on the target, under the decomposition (5.1.4). Therefore by Proposition 5.1.6, LG(F) lies
+in the principal block if and only if Lg(β1F) lies in the principal block, which is guaranteed by condition
+(2).
+□
+Recall the natural monoidal embedding i! : HG → HG, and let a : ShN (G/G) ≃ hh(HG) → hh(HG)
+denote the induced functor.
+5.4.3. Lemma. Suppose F ∈ ShN (G/G) lies in the principal block, with the corresponding k[W]#O(T ∨ ×
+t∨[−1])-module LG(F). Then there is a canonical equivalence of dg algebras
+Endhh(HG)(a(F)) ≃
+�
+O(T ∨) ⊗ EndO(T ∨×t∨[−1])(LG(F))
+�W .
+Proof. Note that a(F) can be identified with α2(F) ∈ colimJ∈D ShN (LJ/LJ), the latter identified with
+the full subcategory hh(HG)0 by Theorem 4.4.2. By Corollary 5.1.9, we have
+(5.4.1)
+LG(α2(F)) ≃ Ind
+�
+W
+�
+W (LG(F)) ∼= O(T ∨) ⊗ LG(F) ∈ k[W]#O(T ∨ × T ∨ × t∨[−1])-mod.
+From this we see
+Endhh(HG)(a(F))
+≃
+Endk[W]#O(T ∨×T ∨×t∨[−1])(LG(α2(F)))
+≃
+Endk[W]#O(T ∨×T ∨×t∨[−1])(O(T ∨) ⊗ LG(F))
+≃
+�
+O(T ∨) ⊗ EndO(T ∨×t∨[−1])(LG(F))
+�W .
+□
+Write trG : HG → hh(HG), trG : HG → hh(HG) for the trace maps, and recall the natural isomorphism
+trG|HG ≃ a ◦ trG.
+5.4.4. Theorem.
+(1) Let kG/G denote the constant sheaf on G/G. There is a canonical equivalence
+of dg algebras
+Endhh(HG)(a(kG/G)) ≃ O(T ∨ × (t∨)∗[1] × (t∨)∗[2])W .
+(2) Let eG be the monoidal unit of the universal affine Hecke category HG.
+There is a canonical
+equivalence of dg algebras
+Endhh(HG)(trG(eG)) ≃ k[W]#O(T ∨ × T ∨ × t∨[−1]).
+Proof. In both calculations, we are computing End(a(F)) for some F ∈ ShN (G/G). We shall show in both
+cases F lies in the principal block of ShN(G/G), and identify the corresponding k[W]#O(T ∨ × t∨[−1])-
+module LG(F). Then we conclude the calculation by invoking Lemma 5.4.3.
+(1) Note that T−1(β1kG/G) = T−1(kg/G) ≃ k0/G[− dim G], which is a direct summand of Spr. By
+the criterion in Lemma 5.4.2, LG(kG/G) lies in the principal direct summand.
+Under Koszul duality,
+Lg(β1kG/G) = Lg(kg/G) ∈ k[W]#Sym(t[1])-mod corresponds to
+HomN/G(Spr, k0/G[− dim G]) ∈ k[W]#Sym(t∨[−2])-mod.
+Using the Cartesian diagram
+{0}/B
+ν0
+�
+� �
+i′
+0
+� u/B
+ν
+�
+{0}/G� �
+i0
+� N/G
+
+70
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+we have by adjunction and proper base change
+HomN/G(Spr, i0∗k0/G[− dim G])
+=
+Hom(ν!ku/B[−r], k0/G[− dim G])
+≃
+Hom(ku/B[−r], ν!i0∗k0/G[− dim G])
+≃
+Hom(ku/B[−r], i′
+0∗ν!
+0k0/G[− dim G])
+=
+Hom(ku/B, i′
+0∗k0/B)
+≃
+H∗
+B(pt) ≃ Sym(t∨[−2]).
+This shows that the Koszul dual of Lg(β1kG/G) is Sym(t∨[−2]) as a natural k[W]#Sym(t∨[−2])-module,
+hence Lg(β1kG/G) ≃ ktriv as a k[W]#Sym((t∨)∗[1])-module.
+In other words, LG(kG/G) ∈ k[�
+W]#Sym((t∨)∗[1]) is one-dimensional over k with the trivial action of
+k[W]#Sym((t∨)∗[1]). Using the fact that kG/G appears as a direct summand of IndG
+T ⊂B(kT/T ), and the
+compatibility of LG with parabolic induction in Proposition 5.1.7, we see that the action of the lattice
+part X∗(T ) ⊂ �
+W on LG(kG/G) is also trivial.
+We conclude that LG(kG/G) ≃ ktriv as an object in
+k[�
+W]#Sym((t∨)∗[1])-mod = k[W]#O(T ∨ × t∨[−1])-mod. Therefore we have a W-equivariant equivalence
+of dg algebras
+EndO(T ∨×t∨[−1])(LG(kG/G)) ≃ O((t∨)∗[1] × (t∨)∗[2])
+By Lemma 5.4.3, we get
+Endhh(HG)(a(kG/G)) ≃ (O(T ∨) ⊗ EndO(T ∨×t∨[−1])(LG(kG/G)))W ≃ O(T ∨×(t∨)∗[1] × (t∨)∗[2])W .
+(2) Let eG be the monoidal unit of HG, then trG(eG) ≃ a(trG(eG)). We claim that trG(eG) is in fact
+the “universal Grothendieck-Springer sheaf”.
+Indeed, consider the following commutative diagram with
+a Cartesian square on the left
+H
+pH
+�
+U\B/U
+qH
+�
+B
+U
+δ
+�
+�
+π
+� G
+G
+H
+H
+B
+B
+q′
+H
+�
+π′
+� G
+G
+By Theorem 2.7.2, trG = γ is the horocycle functor. Therefore
+trG(eG) = π!δ∗(eG) = π!δ∗q∗
+H(exp! kh)[r].
+By proper base change, we can identify it with
+(5.4.2)
+trG(eG) ≃ π′
+!q′∗
+H(pH! exp! kh)[r] ≃ IndG
+T ⊂B(pH! exp! kt[r]).
+Under the equivalence ShN (T/Ad(T )) = Sh0(T/Ad(T )) ≃ O(T ∨×t∨[−1])-mod, pH! exp! kt[r] corresponds
+to the free rank one module O(T ∨ ×t∨[−1]).
+By Proposition 5.1.7, LG(trG(eG)) ∼= k[W]#O(T ∨ ×t∨[−1])
+is also the free rank one k[W]#O(T ∨ × t∨[−1])-module. By (5.4.1), LG(a(trG(eG))) is again the free rank
+one k[W]#O(T ∨ × t∨[−1])-module. Therefore End(trG(eG)) ≃ End(a(trG(eG))) ≃ End(LG(a(trG(eG))))
+is the dg algebra k[W]#O(T ∨ × T ∨ × t∨[−1]).
+□
+5.4.5. Remark. The functor a is analogous to the compact induction functor for p-adic representations
+c-ind : G(OK)-rep → G(K)-rep.
+Here K is a local non-archimedean field with valuation ring OK, and G
+is a split reductive group over K. In particular, Endhh(HG)(a(kG/G)) is analogous to the spherical Hecke
+algebra EndG(K)(c-ind(k)).
+5.4.6. Spectral consequences. Now let us introduce the objects corresponding to a(kG/G), trG(eG) ∈ hh(HG)
+on the spectral side.
+Introduce the natural closed embedding i2 : G∨/G∨ → Z2
+G∨ induced by g �→ (g, 1), and the coherent
+sheaf i2∗OG∨/G∨ on Z2
+G∨. Introduce as well the natural projection p : Z2
+B∨ → Z2
+G∨, and the coherent sheaf
+p∗OZ2
+B∨ on Z2
+G∨.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+71
+Consider the closed embedding of derived stacks
+σ : B∨\G∨/B∨ ≃ BB∨ ×BG∨ BB∨� �
+� StG∨.
+Taking direct image of ind-coherent sheaves gives a monoidal functor
+(5.4.3)
+σ∗ : IndCoh(B∨\G∨/B∨)
+� IndCoh(StG∨).
+Let 2ρ∨ ∈ X∗(T ∨) be the sum of positive roots of G∨ with respect to B∨. Let N be the number of
+positive roots of G∨. When ρ∨ ∈ X∗(T ∨), we consider the sheaf
+A := OB∨\G∨/B∨(−ρ∨, −ρ∨)[N] ∈ IndCoh(B∨\G∨/B∨)
+obtained as the tensor product of pullbacks of O(−ρ∨) from both factors BB∨. When ρ∨ is not a character
+of T ∨, let ν : G∨
+1 → G∨ be the double cover for which ρ∨ is a character of T ∨
+1 = ν−1(T ∨), then we have
+B∨\G∨/B∨ = G∨
+1 /(B∨
+1 × B∨
+1 /∆(ker(ν))) (where B∨
+1 = ν−1(B∨), two factors of B∨
+1 act on G∨
+1 by left and
+right translations). Since (−ρ∨, ρ∨) is always character of B∨
+1 × B∨
+1 /∆(ker(ν)), it defines a line bundle
+on G∨
+1 /(B∨
+1 × B∨
+1 /∆(ker(ν))) and hence on B∨\G∨/B∨, which we still denote by OB∨\G∨/B∨(−ρ∨, −ρ∨).
+In any case, we have the object A := OB∨\G∨/B∨(−ρ∨, −ρ∨)[N] ∈ IndCoh(B∨\G∨/B∨), and the object
+σ∗A ∈ IndCoh(StG∨).
+5.4.7. Lemma. The object σ∗A ∈ IndCoh(StG∨) carries a canonical algebra structure for which its de-
+scended trace is
+(5.4.4)
+tr(σ∗A) ≃ i2∗OG∨/G∨ ∈ IndCoh(Z2
+G∨).
+Proof. We prove the lemma in the case ρ∨ ∈ X∗(T ∨); the general case can be deduced by passing to double
+cover G∨
+1 .
+We construct an algebra structure on A. Let VG∨ = B∨/B∨×G∨/G∨ BG∨. Consider the correspondence
+StG∨ = B∨/B∨ ×G∨/G∨ B∨/B∨
+B∨/B∨ ×G∨/G∨ BB∨
+c
+�
+d
+� B∨/B∨ ×G∨/G∨ BG∨ = VG∨
+where the maps are induced by the natural maps B∨/B∨ ←֓ BB∨ → BG∨ on the second factors. Consider
+the adjoint functors
+ΠL = d∗(c∗ ⊗ p∗
+2O(−ρ∨)[N]) : IndCoh(StG∨)
+� IndCoh(VG∨) : Π = c∗(d∗ ⊗ p∗
+2O(−ρ∨))
+�
+From these we get a natural isomorphism of endo-functors of IndCoh(StG∨):
+(5.4.5)
+Π ◦ ΠL ≃ (−) ⋆ σ∗A.
+The monad structure on Π ◦ ΠL then gives an algebra structure on A.
+From (5.4.5) and the Barr-Beck-Lurie theorem, we conclude that the category RModσ∗A(IndCoh(StG∨))
+of right σ∗A-modules in IndCoh(StG∨) can be identified with IndCoh(VG∨). Similarly, BimodA(IndCoh(StG∨))
+can be identified with IndCoh(WG∨), where
+WG∨ = BG∨ ×R
+G∨/G∨ BG∨
+is the spectral stack hosting the derived spherical Hecke category. Under this equivalence, the regular
+bimodule A itself corresponds to the monoidal unit ∆∗OBG∨ ∈ IndCoh(WG∨). By Definition 2.5.3, we
+have
+(5.4.6)
+tr(σ∗A) ≃ tr(∆∗OBG∨) ∈ IndCoh(Z2
+G∨).
+Here the right side a priori lies in hh(IndCoh(WG∨)), which is a full subcategory of IndCoh(Z2
+G∨).
+To compute tr(∆∗OBG∨), we apply Example 2.6.5 to the map f : X = BG∨ → G∨/G∨ = Y . The
+induced map on the loop spaces Lf : LX = G∨/G∨ → Z2
+G∨ = LY can be identified with i2. Therefore
+Example 2.6.5 gives
+tr(∆∗OBG∨) ≃ i2∗OG∨/G∨.
+Combined with (5.4.6) we get (5.4.4).
+□
+
+72
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+To state the next result, we need an variant of Ansatz 2.3.4.
+5.4.8. Ansatz. The monoidal equivalence (2.3.1) in Ansatz 2.3.4 holds:
+Φ : IndCoh(StG∨)
+∼
+� HG,
+Moreover, Φ takes σ∗A to i!kU\G/U ∈ HG (where kU\G/U ∈ HG is the constant sheaf and i! : HG ֒→ HG),
+compatibly with their respective algebra structures.
+Recall assuming equivalence (2.3.1), we have the equivalence (see (2.8.6))
+(5.4.7)
+IndCohN (Z2
+G∨)
+∼
+� hh(HG) .
+5.4.9. Proposition.
+(1) Assume Ansatz 5.4.8 holds. Then under the equivalence (5.4.7), i2∗OG∨/G∨
+corresponds to a(kG/G).
+(2) Assume the monoidal equivalence (2.3.1) holds. Under the equivalence (5.4.7), p∗OZ2
+B∨ corresponds
+to tr(eG).
+Proof. (1) To see the first identification, we first have an analogous (but simpler) version of Theorem 2.8.5.
+In place of the universal finite Whittaker sheaf WhG ∈ HG, we consider the constant sheaf kU\G/U ∈ HG
+as an algebra object. Then it is straightforward to check its descended trace trG(kU\G/U) ∈ hh(HG) ≃
+ShN (G/G) (as a bimodule over itself) is the constant sheaf kG/G ∈ ShN (G/G).
+By the functoriality of descended trace in Lemma 2.5.5 applied to i! : HG → HG, we have
+(5.4.8)
+trG(i!kU\G/U) ≃ a(kG/G) ∈ hh(HG).
+Comparing (5.4.8) with (5.4.4), we conclude that i2∗OG∨/G∨ matches with a(kG/G) under (5.4.7).
+(2) Under the monoidal equivalence (2.3.1), the monoidal unit ∆∗OB∨/B∨ ∈ IndCoh(StG∨) corresponds
+to the monoidal unit eG ∈ HG. Therefore tr(eG) corresponds to tr(∆∗OB∨/B∨). By Example 2.6.5 applied
+to the map f : X = B∨/B∨ → G∨/G∨ = Y , we conclude that tr(∆∗OB∨/B∨) ≃ p∗OZ2
+B∨ , as desired.
+□
+5.4.10. Corollary.
+(1) Assume Ansatz 5.4.8 holds. Then there is a canonical equivalence of dg alge-
+bras
+EndZ2
+G∨ (i2∗OG∨/G∨) ≃ O(T ∨ × (t∨)∗[1] × (t∨)∗[2])W .
+(2) Assume the monoidal equivalence (2.3.1) holds. Then there is a canonical equivalence of dg algebras
+EndZ2
+G∨ (p∗OZ2
+B∨ ) ≃ k[W]#O(T ∨ × T ∨ × t∨[−1]).
+6. Functions on the commuting stack
+The goal of this section is to prove Theorem 1.1.2.
+6.1. Almost commuting pairs of semisimple elements. Let G∨ be a connected reductive group over
+C. We first recall some results of Borel, Friedman and Morgan [BFM, §4] concerning almost commuting
+pairs with compact simple and simply-connected Lie groups. We will state an extension of their results
+to almost commuting pairs of semisimple elements in complex reductive groups. It is easy to adapt their
+proof to this situation.
+Let G∨,sc be the simply-connected cover of G∨,der. Let �
+G∨ = G∨,sc × (ZG∨)◦. Then the natural map
+π : �
+G∨ → G∨ is a finite central isogeny whose kernel contains ker(G∨,sc → G∨,der) = π1(G∨,der). For any
+c ∈ π1(G∨,der), let Z2
+G∨(c) ⊂ Z2
+G∨ be the open and closed substack of pairs (x, y) in G∨ such that, for
+some (any) liftings �x, �y of x and y to �
+G∨, [�x, �y] = c, all up to G∨-conjugation. An almost commuting pair
+in �
+G∨ refers to a pair (�x, �y) ∈ �
+G∨2 as above. We have a decomposition
+Z2
+G∨ =
+�
+c∈π1(G∨,der)
+Z2
+G∨(c).
+Let (x, y) ∈ Z2
+G∨(c) with x, y semisimple. Consider the simulteneous centralizer CG∨(x, y), which is a
+reductive (possibly disconnected) subgroup of G∨. Let S ⊂ CG∨(x, y) be a maximal torus. It is shown in
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+73
+[BFM, Proposition 4.2.1] that the G∨-conjugacy class of S is independent of the choice of (x, y). Let us
+fix such a torus S and denote it by Sc.
+Let L∨
+c = CG∨(Sc), a Levi subgroup of G∨. It is known that Sc = Z(L∨
+c )◦. Let T ∨
+c = L∨
+c /L∨,der
+c
+be
+the quotient torus. We have T ∨
+c = Sc/(Sc ∩ L∨,der
+c
+) (T ∨
+c is denoted Sc in [BFM]). Let Wc = W(Sc, G∨) =
+NG∨(Sc)/L∨
+c be the relative Weyl group of G∨ with respect to Sc. Then Wc also acts on T ∨
+c . Fix a pair
+(xc, yc) ∈ (L∨,der
+c
+)2 such that (xc, yc) ∈ Z2
+G∨(c). We have a map
+�ιc : Z2
+Sc → Z2
+G∨(c)
+sending (t1, t2) ∈ Z2
+T ∨
+c to (t1xc, t2yc).
+Now we state the generalized form of results in [BFM] replacing elements in compact groups to semisim-
+ple elements in reductive groups; the proofs in loc.cit. works without change.
+6.1.1. Proposition ([BFM, Proposition 4.2.1, Corollary 4.2.2]).
+(1) Any other choice of (x′
+c, y′
+c) ∈
+(L∨
+c )2 such that (x′
+c, y′
+c) ∈ Z2
+G∨(c) is L∨
+c -conjugate to (xc, yc). Moreover, CL∨,der
+c
+(xc, yc) is finite.
+(2) The map �ιc factors through Z2
+T ∨
+c and is Wc-invariant for the diagonal action, inducing a map of
+derived stacks
+ιc : Z2
+T ∨
+c /Wc → Z2
+G∨(c).
+(3) The map ιc restricts to a bijection on the set of semisimple closed points:
+(T ∨
+c (C) × T ∨
+c (C))/Wc ↔ |Z2
+G∨(c)(C)ss|.
+Here |Z2
+G∨(c)(C)ss| denotes the set of G∨-conjugacy classes of semisimple pairs (x, y) ∈ Z2
+G∨(c)(C).
+6.2. Chevalley restriction theorem for the commuting stack. We finish the proof of Theorem 1.1.2,
+which is restated as follows.
+6.2.1. Theorem. Let c ∈ π1(G∨,der).
+(1) Z2
+G∨(c) ̸= ∅. 5
+(2) The map ιc defined in Proposition 6.1.1(2) induces an isomorphism on classical functions:
+ι∗
+c : H0O(Z2
+G∨(c))
+∼
+→ H0O(Z2
+T ∨
+c )Wc = O(T ∨
+c × T ∨
+c )Wc.
+Proof. (1) Let Xc = Spec H0O(Z2
+G∨(c)), which is understood to be empty if Z2
+G∨(c) is empty. First, Xc is
+connected if non-empty. Indeed, assuming Z2
+G∨(c) ̸= ∅, and suppose for the contrary that Xc decomposes
+into a disjoint union of open and closed non-empty subschemes X′
+c
+� X′′
+c .
+Let C2
+G∨(c) ⊂ C2
+G∨ be the
+preimage of Z2
+G∨(c) in the commuting scheme C2
+G∨. Let Z2
+G∨(c) = Z′
+c
+� Z′′
+c and C2
+G∨(c) = C′
+c
+� C′′
+c be the
+corresponding decompositions by taking preimages of X′
+c and X′′
+c . Now C′
+c contains a closed G∨-orbit,
+which then consists of semisimple pairs (x, y) ∈ C2
+G∨(c). This implies Z′
+c contains a semisimple pair (x, y).
+By Proposition 6.1.1(2)(3), Z′
+c meets the image of ιc : (T ∨
+c ×T ∨
+c )/W → Z2
+G∨(c). Similarly, Z′′
+c also contains
+a semisimple pair, hence it also meets the image of ιc. But T ∨
+c × T ∨
+c is irreducible, we get a contraction.
+This proves that Xc is connected if non-empty.
+Corollary 5.3.3 shows that Spec H0O(Z2
+G∨) = �
+c∈π1(G∨,der) Xc has exactly #Irr(Z(G)/Z(G)◦) con-
+nected components. Since #Irr(Z(G)/Z(G)◦) = #π1(G∨,der), and each Xc is connected if non-empty, all
+of them must be non-empty and connected.
+(2) We claim that Xc is irreducible, reduced and normal. Above we have shown that Xc is a connected
+component of Spec H0O(Z2
+G∨), hence isomorphic to Spec O(T ∨
+χ ×T ∨
+χ )Wχ for a unique χ ∈ Irr(Z(G)/Z(G)◦).
+In particular, Xc is irreducible, reduced and normal.
+The map ιc induces a map on coarse moduli spaces
+ι′
+c : (T ∨
+c × T ∨
+c ) � Wc → Xc.
+We next claim that ι′
+c induces a bijection on closed points. Indeed, by a result of Richardson [Ric88],
+the closed points in Xc are in bijection with G∨-orbits of (x, y) ∈ Z2
+G∨(c) such that the Zariski closure
+A(x, y) ⊂ G∨ of the subgroup generated by x and y is reductive. Since x commutes with y, A(x, y) is a
+5Although implicit in [BFM] but we did not find an explicit statement of the non-emptiness of Z2
+G∨(c).
+
+74
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+commutative; it is reductive if and only if x and y are both semisimple. Therefore by Proposition 6.1.1(3),
+ι′
+c induces a bijection on closed points.
+Since (T ∨
+c × T ∨
+c ) � W and Xc are reduced and irreducible of finite type over C, and ι′
+c is a bijection on
+closed points, to show ι′
+c is an isomorphism if suffices to show that ι′
+c is finite.
+We have a map �s = (�s1, �s2) : Z2
+G∨(c) → (G∨ � G∨)2 = (T ∨ � W)2 by recording the G∨-invariants of x
+and y in a commuting pair (x, y). This induces a map s = (s1, s2) : Xc → (T ∨ � W)2. We have maps of
+affine schemes
+Sc × Sc
+p−→ (T ∨
+c × T ∨
+c ) � Wc
+ι′
+c
+−→ Xc
+s−→ (T ∨ � W)2
+where the composition is the square of the natural projection Sc → T ∨ � W, hence finite. Since p is
+surjective, s ◦ ι′
+c : (T ∨
+c × T ∨
+c ) � Wc → (T ∨ � W)2 is also finite, therefore ι′
+c is finite as well. This finishes
+the proof.
+□
+6.2.2. Remark. Let G∨
+Q be the split form of G∨ over Q. Then Z2
+G∨ has a Q-form Z2
+G∨,Q. It is easy to
+deduce from Theorem 1.1.2 a description of global functions on Z2
+G∨,Q. Indeed, we have a decomposition
+of Z2
+G∨,Q into open and closed substacks Z2
+G∨,Q([c]) indexed by Galois orbits [c] ∈ π1(G∨,der)/Gal(Q/Q).
+If Q[c] is the fixed field of the stabilizer of any c ∈ [c] under Gal(Q/Q) (so Q[c] is the smallest cyclotomic
+extension of Q over which Z2
+G∨(c) is defined), we have an isomorphism Z2
+G∨,Q([c]) ∼= Z2
+G∨,Q[c](c), the latter
+being the descent of Z2
+G∨(c) to Q[c]. For any c ∈ [c], T ∨
+c has a Q[c]-form T ∨
+c,Q[c]. We conclude
+H0O(Z2
+G∨,Q)
+∼
+→
+�
+[c]∈π1(G∨,der)/Gal(Q/Q)
+O(T ∨
+c,Q[c] × T ∨
+c,Q[c])Wc.
+6.2.3. Identifying χ and c. There is a canonical isomorphism
+δ : Irr(Z(G)/Z(G)◦) ≃ π1(G∨,der)
+because both sides are canonically dual to the torsion part of X∗(T )/root lattice of G.
+On the other hand, by Corollary 5.3.3 and Theorem 6.2.1, we have two expressions of the ring H0(O(Z2
+G∨)):
+(6.2.1)
+�
+χ∈Irr(Z(G)/Z(G)◦)
+O(T ∨
+χ × T ∨
+χ )Wχ ≃ H0(O(Z2
+G∨)) ≃
+�
+c∈π1(G∨,der)
+O(T ∨
+c × T ∨
+c )Wc.
+Fratila [Fra16] and Bonnaf´e [Bon04] observe a mysterious coincidence that the pairs (T ∨
+c , Wc) that
+appear on the right side of (6.2.1) are exactly those coming from Levi subgroups of G that support
+cuspidal local systems on the regular nilpotent orbit, i.e., those pairs (T ∨
+χ , Wχ) that appear on the left
+side of (6.2.1). They verified this fact by a case-by-case explicit computation. Below we give a uniform
+conceptual proof of this combinatorial observation.
+6.2.4. Proposition. Let χ ∈ Irr(Z(G)/Z(G)◦) and c = δ(χ) ∈ π1(G∨,der). Then the following hold:
+(1) Put Lχ := LJχ as in Section 5.2.4, then L∨
+c (defined in Section 6.1) is the dual Levi of Lχ (up to
+G∨-conjugacy).
+(2) There is an isomorphism of pairs (T ∨
+χ , Wχ) ≃ (T ∨
+c , Wc) compatible with their respective actions.
+Moreover, such an isomorphism is canonical up to conjugation by Wχ.
+Proof. (1) Let L∨
+χ ⊂ G∨ be the dual Levi to Lχ. Note that the canonical map π1(L∨,der
+χ
+) → π1(G∨,der) is
+always injective, and c = δ(χ) lies in its image. Therefore it makes sense to form Z2
+L∨
+χ(c) ⊂ Z2
+L∨
+χ, which is
+non-empty by Theorem 6.2.1(1). Viewing c as an element in π1(G∨,der), we can form the Levi L∨
+c of G∨
+by taking a maximal torus Sc ⊂ CG∨(x, y) and take its centralizer. Now Z(L∨
+χ)◦ is a torus in CG∨(x, y),
+hence we may take Sc to contain Z(L∨
+χ)◦. This implies L∨
+c = ZG∨(Sc) ⊂ ZG∨(Z(L∨
+χ)◦) = L∨
+χ. Since T ∨
+c
+and T ∨
+χ are the abelianizations of L∨
+c and L∨
+χ respectively, we have a surjection T ∨
+c ։ T ∨
+χ . In particular,
+we have the inequality
+(6.2.2)
+dim T ∨
+c ≥ dim T ∨
+χ , for all χ ∈ Irr(Z(G)/Z(G)◦) and c = δ(χ).
+We remark that, since L∨
+c is a Levi subgroup of L∨
+χ up to conjugation, L∨
+c is conjugate to L∨
+χ if and only
+if dim T ∨
+c = dim T ∨
+χ .
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+75
+On the other hand, by taking the Krull dimensions of all direct summands on both sides of (6.2.1), we
+get
+�
+χ∈Irr(Z(G)/Z(G)◦)
+2 dim T ∨
+χ =
+�
+c∈π1(G∨,der)
+2 dim T ∨
+c .
+Combined with the inequality (6.2.2), we are forced to have an equality dim T ∨
+χ = dim T ∨
+δ(χ) for every χ.
+As we remarked earlier, this implies that L∨
+δ(χ) is conjugate to L∨
+χ in G∨.
+(2) Let c = δ(χ). Note that Wχ = NG∨(L∨
+χ)/L∨
+χ, T ∨
+χ is the abelianization of L∨
+χ, and the same is
+true if χ is replaced with c. Any g ∈ G∨ that conjugates L∨
+χ to L∨
+c induces an isomorphism of pairs
+ιg : (T ∨
+χ , Wχ) ≃ (T ∨
+c , Wc) compatible with the actions. Different choices of g differ by right multiplication
+by NG∨(L∨
+χ), therefore the resulting ιg differ by Wχ-conjuation.
+□
+Combining Corollary 5.3.3 and Proposition 6.2.4, we get the following description of the dg ring of
+derived functions on Z2
+G∨ promised in Theorem 1.1.6.
+6.2.5. Corollary. Assume Ansatz 1.2.5. Then there is a canonical isomorphism of dg algebras:
+O(Z2
+G∨) ≃
+�
+c∈π1(G∨,der)
+O(T ∨
+c × T ∨
+c × t∨
+c [−1])Wc.
+6.3. From groups to Lie algebras. We will deduce the Lie algebra analogue of the Chevalley restriction
+theorem from the group case.
+Let G∨ be a connected reductive group with Lie algebra g∨. Let C2
+g∨ be the commuting scheme for g∨,
+i.e., it is the (classical) fiber over 0 of the Lie bracket map [·, ·] : g × g → g. Then G∨ acts on C2
+g∨ by
+diagonal adjoint action. We are interested in calculating the C-algebra O(C2
+g∨)G∨.
+Let T ∨ ⊂ G∨ be a maximal torus and t∨ = LieT ∨. The embedding t∨ × t∨ ֒→ C2
+g∨ induces a map of
+stacks
+ιg∨ : (t∨ × t∨)/W → C2
+g∨/G∨
+6.3.1. Theorem. The map ιg∨ induces an isomorphism on global (i.e. invariant) functions
+(6.3.1)
+ι∗
+g∨ : O(C2
+g∨)G∨
+∼
+→ O(t∨ × t∨)W .
+Proof. For the statement of the theorem we may replace G∨ by another group isogenous to it. In particular,
+we may assume the derived group G∨,der is simply-connected.
+We have the natural map
+s : C2
+g∨ → (t∨ � W)2
+by taking the invariants of both elements in the commuting pair. Let �C2
+g∨ be the formal completion of C2
+g∨
+along the fiber s−1(0, 0) (which classifies commuting pairs of nilpotent elements ).
+Analogously, let C2
+G∨ ⊂ G∨ × G∨ be the commuting scheme for G∨. We have the natural map
+s : C2
+G∨ → (T ∨ � W)2
+by taking the invariants of both elements in the commuting pair. Let �C2
+G∨ be the formal completion of
+C2
+G∨ along the fiber s−1(1, 1) (which classifies commuting pairs of unipotent elements ).
+By Lemma 6.3.2 below, the exponential map gives a G∨-equivariant isomorphism
+exp : �C2
+g∨
+∼
+→ �C2
+G∨.
+Moreover, letting �
+t∨ × t∨ and
+�
+T ∨ × T ∨ be the formal completions at (0, 0) and (1, 1) respectively, we have
+a commutative diagram of formal schemes
+�
+t∨ × t∨
+ιg∨
+�
+exp
+�
+�C2
+g∨
+exp
+�
+�
+T ∨ × T ∨
+ιG∨ � �C2
+T ∨
+
+76
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+where ιG∨ is the obvious analogue of ιg∨. The vertical maps are isomorphism. This gives a commutative
+diagram of algebras
+(6.3.2)
+O(�C2
+g∨)G∨
+�ι∗
+g∨
+� O( �
+t∨ × t∨)W
+O(�C2
+G∨)G∨
+�ι∗
+G∨ �
+∼
+=
+�
+O( �
+T ∨ × T ∨)W
+∼
+=
+�
+Now the top arrow is obtained from the map of S = O(t∨)W ⊗ O(t∨)W -algebras (6.3.1) by completing
+along the maximal ideal of O(t∨)W ⊗O(t∨)W corresponding to (0, 0). Similarly, the bottom row is obtained
+from the map of S = O(T ∨)W ⊗ O(T ∨)W -algebras
+ι∗
+G∨ : O(C2
+G∨)G∨ → O(T ∨ × T ∨)W
+by completing along the maximal ideal of O(T ∨)W ⊗ O(T ∨)W corresponding to (1, 1).
+By Theorem 6.2.1, ι∗
+G∨ is an isomorphism (using that G∨,der is simply-connected this follows directly,
+but in fact this is true without assuming G∨,der is simply-connected), hence so is its completed version
+�ι∗
+G∨. By the commutative diagram (6.3.2) where both vertical arrows are isomorphisms, we conclude that
+�ι∗
+g∨ is an isomorphism.
+We would like to deduce that ι∗
+g∨ is an isomorphism from the fact that �ι∗
+g∨ is an isomorphism. Note
+that we have a Gm-action on C2
+g∨ by simultaneously scaling the elements in the commuting pair. This
+gives a grading R := O(C2
+g∨)G∨ = ⊕n≥0Rn. Similarly, T = O(t∨ × t∨)W has a grading T = ⊕n≥0Tn as
+well as S = ⊕n≥0Sn. The map ι∗
+g∨ is a map of graded S-algebras:
+ι∗
+g∨ : R = ⊕n≥0Rn → T = ⊕n≥0Tn.
+The map �ι∗
+g∨ is obtained from ι∗
+g∨ by S+-adic completion. It is clear that the S+-adic completion of T is
+given by �T = �
+n≥0 Tn.
+Note that R/S+R = O(C2
+N ∨)G∨ where C2
+N ∨ = s−1(0, 0) ⊂ N ∨,2 is the scheme of commuting nilpotent
+elements in g∨. Similarly, for R = O(C2
+G∨)G∨ and S+ the augmentation ideal of S, R/S+R = O(C2
+U ∨)G∨
+where C2
+U ∨ = s−1(1, 1) ⊂ U∨,2 is the the scheme of commuting unipotent elements in G∨.
+Lemma
+6.3.2 below (or rather a simpler version of the same argument) shows that R/S+R ∼= R/S+R.
+By
+Theorem 6.2.1, R/S+R is the coordinate ring of the central fiber of (T ∨ × T ∨) � W → (T ∨/ � W)2, hence
+finite dimemsional. We conclude that R/S+R is finite-dimensional over C. This implies that S+R and
+R+ = ⊕n>0Rn define the same topology on R. Therefore the S+-adic completion �R is �
+n≥0 Rn. The
+fact that
+ι∗
+g∨ : �R =
+�
+n≥0
+Rn → �T =
+�
+n≥0
+Tn
+is an isomorphism implies its restriction to each degree Rn → Tn is bijective, hence ι∗
+g∨ is an isomorphism.
+This finishes the proof.
+□
+6.3.2. Lemma. The exponential map (x, y) �→ (exp(x), exp(y)) induces a G∨-equivariant isomorphism of
+formal schemes
+exp2 : �C2
+g∨
+∼
+→ �C2
+G∨.
+Proof. Let Nil be the category of pairs (R, I) where R is a C-algebra with a nilpotent ideal I (i.e. IN = 0
+for some N > 0). We write R = R/I. Then the category of formal schemes over C full faithfully embeds
+into the category of functors Fun(Nil, Sets).
+The formal completion �
+g∨ of g∨ along the nilpotent cone N ∨ is the functor that sends (R, I) to the set
+of x ∈ g∨ ⊗ R such that x (the image of x in g∨ ⊗ R) lies in N ∨(R). Similarly, the formal completion �
+G∨
+of G∨ along the unipotent variety U∨ sends (R, I) to the set of g ∈ G∨(R) such that g ∈ U∨(R).
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+77
+Similarly, the completion �C2
+g∨ sends (R, I) to the set of commuting pairs (x, y) ∈ C2
+g∨(R) such that
+x, y ∈ N ∨(R). The completion �C2
+G∨ sends (R, I) to the set of commuting pairs (g, h) ∈ C2
+G∨(R) such that
+g, h ∈ U∨(R).
+We first check that the exponenital map and logarithm map give inverse isomorphisms between the
+functors �
+g∨ and �
+G∨. To construct these maps, we choose a faithful representation V of G∨ and embed G∨
+into GL(V ). We define �gl(V ) and �
+GL(V ) as the formal completions of gl(V ) and GL(V ) along the closed
+subsets of nilpotent and unipotent matrices, viewed as functors on Nil. Note that any x ∈ �gl(V )(R, I)
+is nilpotent (because xn has entries in I, hence nilpotent). Similarly, any g ∈ �
+GL(V )(R, I) is unipotent
+(i.e., g − 1 is nilpotent). The exponential map exp : �gl(V )(R, I) → �
+GL(V )(R, I) is defined using the
+usual formula exp(x) = �
+n≥0
+xn
+n! which is a finite sum since x is nilpotent. The logarithm map log :
+�
+GL(V )(R, I) → �gl(V )(R, I) is defined using the usual formula log(g) = − �
+n≥1
+(1−g)n
+n
+which is a finite
+sum since g − 1 is nilpotent. Clearly exp and log give inverse isomorphisms between the functors �gl(V )
+and �
+GL(V ).
+We claim that exp(�
+g∨(R, I)) ⊂ �
+G∨(R, I) and log( �
+G∨(R, I)) ⊂ �
+g∨(R, I). Once this is shown, exp and
+log then restrict to give inverse bijections between �
+g∨(R, I) and �
+G∨(R, I) functorially for (R, I) ∈ Nil,
+hence giving inverse isomorphisms (clearly G∨-equivariant) between the formal schemes �
+g∨ and �
+G∨. To
+check exp(�
+g∨(R, I)) ⊂ �
+G∨(R, I), let {vi} be a collection of tensors on V such that G∨ is defined as
+the simultaneous stabilizer of {vi} in GL(V ). Then g∨ is the simultaneous annihilator of {vi} in gl(V ).
+If x ∈ �
+g∨(R, I), then xvi = vi, hence exp(x)vi = vi and therefore exp(x) ∈ �
+G∨(R, I).
+This proves
+exp(�
+g∨(R, I)) ⊂ �
+G∨(R, I). The other inclusion log( �
+G∨(R, I)) ⊂ �
+g∨(R, I) is proved in the same way.
+Now we have a G∨ × G∨-equivariant isomorphism exp2 : �
+g∨ × �
+g∨
+∼
+→ �
+G∨ × �
+G∨ with inverse log2. We
+claim that for any (R, I) ∈ Nil, exp2 sends �C2
+g∨(R, I) to �C2
+G∨(R, I), and log2 sends �C2
+G∨(R, I) to �C2
+g∨(R, I).
+Once this is checked, exp2 and log2 then restrict to give inverse bijections between �C2
+g∨(R, I) and �C2
+G∨(R, I)
+functorially for (R, I) ∈ Nil, hence giving inverse isomorphisms between the formal schemes �C2
+g∨ and �C2
+G∨,
+as desired. Now suppose (x, y) ∈ �C2
+g∨(R, I). Since exp(x) and exp(y) are polynomials in x and y, and
+[x, y] = 0, we see that exp(x) commutes with exp(y) as elements in G∨(R), hence a fortiori in �
+G∨(R, I).
+This verifies that exp2 sends �C2
+g∨(R, I) to �C2
+G∨(R, I). The statement about log2 is proved similarly, using
+that log(g) and log(h) are polynomials in g and h. This finishes the proof.
+□
+Same argument as in the proof of Theorem 6.3.1 proves also the Chevalley restriction theorem for
+the Lie algebra-group commuting scheme. Let Cg∨,G∨ be the subscheme of (x, g) ∈ g∨ × G∨ such that
+Adg(x) = x. We have the obvious map of stacks
+ιg∨,G∨ : (t∨ × T ∨)/W → Cg∨,G∨/G∨.
+6.3.3. Theorem. Let G∨ be a reductive group. The map ιg∨,G∨ induces an isomorphism on global (i.e. in-
+variant) functions
+ι∗
+g∨,G∨ : O(Cg∨,G∨)G∨
+∼
+→ O(t∨ × T ∨)W .
+Sketch of proof. We let S = O(t∨)W ⊗ O(T ∨)W equipped with the grading coming from the scaling
+action on t∨. Then S/S+ ∼= O(T ∨)W . Then R = O(Cg∨,G∨)G∨ = ⊕n≥0Rn and T = O(t∨ × T ∨)W =
+⊕n≥0Tn are both graded S-algebras with gradings coming from scaling actions on g∨ and t∨. The same
+argument as in Theorem 6.3.1 shows that the map induced by ι∗
+g∨,G∨ on the S+-adic completions �R → �T
+is an isomorphism. It is easy to see that �T = �
+n≥0 Tn. We claim that �R = �
+n≥0 Rn. Indeed, we
+have R/S+R = O(CN ∨,G∨)G∨ where, CN ∨,G∨ is the scheme of commuting pairs in N ∨ × G∨.
+The
+exponenital map gives a G∨-equivariant isomorphism CN ∨,G∨
+∼
+→ CU ∨,G∨ where the nilpotent cone is
+replaced with the unipotent variety U∨. By Theorem 6.2.1, O(CU ∨,G∨)G∨ ∼= O(T ∨)W is free of rank one
+over S/S+ = O(T ∨)W . Therefore R/S+R ∼= O(CN ∨,G∨)G∨ ∼= O(T ∨)W ∼= R0 = O(G∨)G∨. This implies
+
+78
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+S+R = R+ = ⊕n>0Rn, which implies �R = �
+n≥0 Rn. Therefore the isomorphism �R
+∼
+→ �T implies the
+isomorphism before completion.
+□
+Appendix A. Calculating some colimits
+This appendix collects some results about colimits of categories called upon in Section 4.4.We will
+consider colimits of stable presentable ∞-categories viewed as objects of the ∞-category StL of stable
+presentable ∞-categories with morphisms given by left adjoints. (The discussion extends verbatim if we
+additionally work in the k-linear context.)
+The main new result here is a categorical analog of the contraction principle for sheaf cohomology. It
+allows us to reduce the calculation of a colimit indexed by a complicated diagram to a simpler one, assuming
+a certain contractibility condition. This is a key categorical ingredient in the proof of Theorem 4.4.4.
+A.1. Recollements and stratifications.
+A.1.1. Recollements. For recollement in the setting of ∞-categories, we recommend the reference [Lur12,
+A.8]. In our setting of stable presentable ∞-categories, we will largely follow the reference [AMR, 1.1].
+Recall a recollement [AMR, Definition 1.1.1] of stable presentable ∞-categories is a diagram of adjoint
+triples
+C0
+j!
+�
+j∗
+�
+C
+j!=j∗
+�
+i∗
+�
+i!
+�
+C1
+i∗=i!
+�
+where j!, j∗, and i∗ = i! are fully faithful with Im(j!) = Ker(i∗), Im(j∗) = Ker(i!), and Im(i∗ = i!) =
+Ker(j! = j∗). Note the recollement is determined by the subdiagram
+C0
+C
+j!=j∗
+�
+C1
+i∗=i!
+�
+with the rest of the notion comprising properties to be verified.
+In our topological setting, we call C0 the open subcategory and C1 the closed subcategory of the rec-
+ollement.6 We refer to C0 and C1 as the recollement complements of each other.
+Suppose we have a commutative diagram in StL
+(A.1.1)
+C0
+f0
+�
+C
+j!=j∗
+�
+f
+�
+C1
+i∗=i!
+�
+f1
+�
+D0
+D
+j!=j∗
+�
+D1
+i∗=i!
+�
+such that both rows extend to recollements. We say the above diagram (or the functors (f0, f, f1)) is a
+morphism of recollements if the square on the left is both left and right adjointable with respect to the
+horizontal arrows j! = j∗ (equivalently, the square on the right is both left and right adjointable with
+respect to the horizontal arrows i! = i∗).
+If in a morphism of recollement as in (A.1.1), both f0 and f1 are equivalences, then f is also an
+equivalence (this follows from [Lur12, Prop. A.8.14]).
+A.1.2. Stratifications. We will use the more general notion of a recollement of stable presentable ∞-
+categories indexed by a poset P as introduced in [AMR, 1.3] under the name stratification. Since we
+work in the topological rather than algebraic setting, we will adapt the notation and terminology in the
+definitions to reflect this. For a finite totally ordered poset, a stratification also goes by the name semi-
+orthogonal decomposition [BK90]. When we apply the below definitions, our poset will be the opposite of
+the non-negative natural numbers Z≥0 with their usual total order.
+6The terminology is intended to model sheaves on a space with an open-closed decomposition. Note this is opposite to the
+convention of [AMR] where C0 is called a closed subcategory since it models quasi-coherent sheaves on a scheme supported
+along a closed subscheme.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+79
+Given a stable presentable ∞-category, an open subcategory (called a closed subcategory in [AMR,
+Definition 1.3.1]) is an adjoint triple
+C0
+j!
+�
+j∗
+�
+C
+j!=j∗
+�
+where j!, j∗ are fully faithful. We write Copen for the poset of open subcategories of C with respect to
+inclusion under j!.
+Given a poset P, a P-stratification (following [AMR, Definition 1.3.2]) of a stable presentable ∞-category
+C is a functor
+Z• : P
+� Copen
+such that C = �
+p∈P Zp and for any p, q ∈ P, there exists a factorization
+�
+r≤p,q Zr
+j!
+� Zp
+Zq
+�✤
+✤
+✤
+j!
+� Z
+j!=j∗
+�
+Given a P-stratification Z•, its pth stratum [AMR, Definition 1.3.3] is the quotient
+Cp = Zp/
+�
+q<p
+Zq
+By construction, there is a recollement
+�
+q<p Zq
+j!
+�
+j∗
+�
+Zp
+j!=j∗
+�
+i∗
+�
+i!
+�
+Cp
+i∗=i!
+�
+A.1.3. Example. Suppose X = ∪p∈Z≥0Xp is a topological space written as the increasing union of closed
+subspaces. Suppose C = colimp∈Z≥0 Sh(Xp) where the transition maps are given by extension by zero.
+This fits into the above formalism as follows. For p ∈ Z≥0, set Up = X \ Xp−1 (convention: X−1 = ∅)
+viewed as the increasing union of closed subspaces Up = ∪q∈Z≥0Up ∩Xq. Set Zp = colimq∈Z≥0 Sh(Up ∩Xq)
+where the transition maps are given by extension by zero.
+Then we have a stratification C = Z0 ←֓ Z1 ←֓ Z2 · · · with respect to the poset P = (Z≥0)op with open
+subcategories
+Zp
+j!
+�
+j∗
+�
+C
+j!=j∗
+�
+with inclusions given by !-extensions. The strata are given by Cp = Zp/Zp+1 = Sh(Xp \ Xp−1) noting that
+Up \ Up+1 = Xp \ Xp−1. By construction, there are recollements
+Zp+1
+j!
+�
+j∗
+�
+Zp
+j!=j∗
+�
+i∗
+�
+i!
+�
+Cp
+i∗=i!
+�
+The structure is equivalent to giving the inclusions Y0 ֒→ Y1 ֒→ Y2 · · · of the complementary closed
+categories Yp = Sh(Xp) with their recollements
+Cp
+j!
+�
+j∗
+�
+Yp
+j!=j∗
+�
+i∗ �
+i! �
+Yp−1
+i∗=i!
+�
+
+80
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+A.2. Interaction of colimits and recollement. The following gives natural hypotheses for when a
+colimit of stable presentable ∞-categories with recollement inherits a recollement.
+A.2.1. Proposition. Let I be a small category, C′
+•, C• : I → StL functors to stable presentable ∞-categories
+(with morphisms given by left adjoints), and i• : C′
+• → C• a map of functors.
+(1) Assume the following:
+(a) For each J ∈ I, the functor iJ extends to a recollement in StL:
+C′′
+J
+jJ! �
+jJ∗ �
+CJ
+jJ
+�
+i∗
+J
+�
+i!
+J
+�
+C′
+J
+iJ
+�
+In particular, iJ is fully faithful and C′′
+J = CJ/iJ(C′
+J) (hence the assignment J �→ C′′
+J extends
+naturally to a functor C′′
+• : I → StL).
+(b) For all maps I → J in I, the natural commutative diagram
+(A.2.1)
+C′′
+I
+�
+CI
+jI
+�
+�
+C′
+I
+iI
+�
+�
+C′′
+J
+CJ
+jJ
+�
+C′
+J
+iJ
+�
+is a morphism of recollements. Here the vertical morphisms are given as part of the data of
+C′′
+• , C• and C′
+• as functors.
+Then the map
+i = colim i• : colimI C′
+•
+� colimI C•
+extends to a recollement
+colim C′′
+•
+j!
+�
+j∗
+�
+colim C•
+j
+�
+i∗
+�
+i!
+�
+colim C′
+•
+i
+�
+with i! ≃ colimI i!
+J, i∗ ≃ colimI i∗
+J, j = colimI jJ, j! = colimI jJ! and j∗ = colimI jJ∗.
+(2) Assume further we are given a functor f : I0 → I, denote by C′
+0• = C′
+• ◦ f, C0• = C• ◦ f and
+C′′
+0• = C′′
+• ◦ f. Then the natural commutative diagram is a morphism of recollements:
+colim C′′
+0•
+�
+colim C0•
+�
+�
+colim C′
+0•
+�
+�
+colim C′′
+•
+colim C•
+�
+colim C′
+•
+�
+Proof. (1) By assumption (a), we have adjoint triples (i∗
+J, iJ, i!
+J) in StL, so that the unit u : id → i!
+JiJ
+and counit ǫ : i∗
+JiJ → id are isomorphisms. Assumption (b) implies (i∗ := colim i∗
+J, i = colim iJ, i! :=
+colim i!
+J) is also an adjoint triple, with counit i∗i → id an isomorphism.
+Therefore i is fully faithful,
+and colim CJ is a recollement of colim C′
+J and colim CJ/ colim C′
+J ≃ colim C′′
+J. Moreover, the natural map
+j : colim C• → colim C′′
+• is naturally isomorphic to colimI jJ, by the functoriality of cokernel (cokernel of
+a functor f : A → B is by definition the pushout of 0 ← A
+f−→ B). Since taking colimit preserves adjoint
+pairs, so we also have j! = colimI jJ! and j∗ = colimI jJ∗.
+(2) Follows directly from the functoriality of colimits.
+□
+A.2.2. Corollary. Suppose we have a morphism of recollement as in (A.1.1) in which f0 is an equivalence,
+then the functor
+f
+�
+i∗ : C
+�
+C1
+D1 → D
+is an equivalence.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+81
+Proof. Applying Prop. A.2.1 to the pushout of the recollements (C0 ← C ← C1) and (0 ← D1 ← D1) along
+(0 ← C1 ← C1), we get a recollement (C0 ← C �
+C1 D1 ← D1) that fit into a commutative diagram
+C0
+f0
+�
+C �
+C1 D1
+�
+f � i∗
+�
+D1
+�
+D0
+D
+�
+D1
+�
+It is easy to check the left square is both left and right adjointable, hence the above diagram is a morphism
+of recollements. Since both of the vertical arrows at the two ends are equivalences, so is the middle vertical
+arrow.
+□
+A.3. Posets and cosheaves.
+A.3.1. Generalities on posets. For a poset (partially ordered set) I, let N(I) denote its nerve, viewed as
+a simplicial set and hence a quasi-category. We say a subset J ⊂ I is up-closed if i ∈ J , i ≤ i′ implies
+i′ ∈ J ; similarly, we say a subset J ⊂ I is down-closed if i ∈ J , i′ ≤ i implies i′ ∈ J .
+For a poset I, let |I| be the geometric realization of I, defined as the usual geometric realization of
+the simplicial set N(I) (hence a simplicial complex). In particular, the r-dimensional faces of |I| are
+in bijection with non-degenerate r-dimensional simplices in N(I), which are parametrized by strictly
+increasing chains i0 < i1 < · · · < ir in I.
+A poset I is called acyclic if |I| is weakly contractible.
+An order-preserving map of posets f : I → J is called coCartesian, if for any pair j ≤ j′ in J , and any
+i ∈ f −1(j), the set {i′ ∈ I|i ≤ i′, j′ ≤ f(i′)} has a unique minimal element ∂j→j′(i) that maps to j′.
+An order-preserving map of posets f : I → J is called a left covering map, if for each pair j ≤ j′
+in J , and any i ∈ f −1(j), there is a unique i′ ∈ f −1(j′) such that i ≤ i′. A left covering map is in
+particular coCartesian. If f is a left covering map, then each fiber f −1(j) is a discrete poset (any pair
+of distinct elements are incomparable), and ∂j→j′ gives a map f −1(j) → f −1(j′) for each pair j ≤ j′
+in J , compatible with compositions. Such a structure is simply a functor J → Sets (called a J -set in
+Section 4.2.1). Conversely, given a functor X : J → Sets, if we let I = �
+j∈J Xj and define the partial
+order on I to be generated by i ≤ Xj→j′(i) for all j ≤ j′ ∈ J and i ∈ Xj, then the natural projection
+f : I → J is a left covering map of posets. We conclude that a left covering map of posets f : I → J is
+the same datum as a J -set, i.e., a functor J → Sets.
+A.3.2. Generalities on cosheaves. Let I be a poset with nerve N(I). We call a functor F : I → StL a
+cosheaf on I. In other words, for each i ∈ I, F assigns a stable presentable ∞-category Fi, and for each
+arrow ϕ : i → j in I, there is a functor Fϕ : Fi → Fj, together with higher compatibilities. We call
+colimI F ∈ StL the (category of) cosections of F.
+Let f : I → J be an order-preserving map of posets. Let G be a cosheaf on J , then we define f ∗G to
+be the cosheaf on I given by the composition I
+f−→ J
+G−→ StL. If f is injective, then we write f ∗F as F|I.
+In the following, we gather some useful tools for calculating colimits. Most of them can be found in
+Lurie [Lur09, Section 4].
+A.3.3. Pushforward. Let f : I → J be an order-preserving map of posets and F be a cosheaf on I. We
+denote by f!F : J → StL a left Kan extension of F along the map f. Then colimI F
+∼
+→ colimJ f!F by
+the general properties of left Kan extensions.
+Suppose f : I → J is coCartesian, then for j ≤ j′ in J , there is a natural functor
+colimi∈f −1(j) Fi → colimi′∈f −1(j′) Fi′
+induced by the functors (for i ∈ f −1(j))
+Fi → F∂j→j′ (i) → colimi′∈f −1(j′) Fi′
+where the second map uses that ∂j→j′(i) ∈ f −1(j′). These maps give a cosheaf
+�
+f F on J whose value at
+j is colimi∈f −1(j) Fi.
+
+82
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+A.3.4. Proposition (Lurie, special case of [Lur09, Proposition 4.3.3.10]). Suppose f : I → J is a co-
+Cartesian map of posets and F is a cosheaf on I. Then
+�
+f F is a left Kan extension of F along f. In
+particular, there is an equivalence colimI F ≃ colimJ
+�
+f F.
+Another instance of pushforward is extension by zero. Let ı : I1 ⊂ I be the inclusion of an up-closed
+subset. Then for a cosheaf F1 on I1, the cosheaf ı!F1 on I is obtained by extension by zero: it assigns to
+each i ∈ I \ I1 the zero category.
+Dually, let  : I0 ⊂ I be the inclusion of a down-closed subset. Let F0 be a cosheaf on I0. We define
+j∗F0 to be extension by zero of F0 as a cosheaf on I. Then j∗ : Fun(I0, StL) → Fun(I, StL) is a right
+adjoint to the restriction j∗ : Fun(I, StL) → Fun(I0, StL).
+A.3.5. Subdivision. Let sd(I) be the set of non-degenerate simplices in the nerve N(I): they are strictly
+increasing non-empty chains i = (i0 < · · · < ir) in I. Equip sd(I) with a partial order by defining i ≤ i′
+if and only if i′ is part of i. We have a map of posets π : sd(I) → I sending i = (i0 < · · · < ir) to i0.
+For a cosheaf F on I, let sd(F) = π∗F be the pullback cosheaf on sd(I).
+A.3.6. Lemma. The map π : sd(I) → I is cofinal. In particular, for any cosheaf F on I, there is a
+canonical equivalence
+colimsd(I) sd(F) ≃ colimI F.
+Proof. To check π is cofinal, we apply Quillen’s Theorem A as in [Lur09, Theorem 4.1.3.3]. For i ∈ I,
+the category M = sd(I) ×I Ii/ consists of strictly increasing non-empty chains i0 < · · · < ir in I such
+that i ≤ i0. We need to show that the geometric realization |M| of M is contractible. Let L ⊂ M be
+the subposet where i0 = i; let K ⊂ M be the subposet where i < i0. Then M = L ⊔ K, and we can
+identify L ∼= K⊲ (the right cone of K) because L has a maximal element that is i itself as a chain, and its
+complement is isomorphic to K via i = (i0 < · · · < ir) → i′ = (i1 < · · · < ir). Also we have i ≤ i′ (where
+i′ ∈ K and i ∈ L is obtained from i′ by adding i), and all the other order relations are consequences of
+it and the fact that L and K are subposets. We conclude that M ∼= (K × [1]) �
+K×{0} K⊲ (where [1]
+is the poset {0, 1}). The geometric realization |M| is thus homeomorphic to the cone over |K|, hence
+contractible.
+□
+A.3.7. Double colimits. In some situations we want to rewrite colimI F as a colimit of colimits, where the
+inner colimit is taken over a subposet of I. Lurie’s theorem [Lur09, Corollary 4.2.3.10] gives a general
+framework for doing this. We recall the statement of loc.cit. in the simplified situation where the colimits
+are parametrized by posets. Suppose K is a poset and F : K → StL is a cosheaf on K. Let J be another
+poset with an assignment J ∋ j �→ Kj ⊂ K such that j ≤ j′ in J implies Kj ⊂ Kj′.
+A.3.8. Proposition (Lurie, special case of [Lur09, Corollary 4.2.3.10]). Under the above notation, assume:
+Kj is up-closed for each j ∈ J , and for each k ∈ K, the subposet Jk := {j ∈ J |k ∈ Kj} is acyclic. Then
+the natural maps colimKj F → colimK F give an equivalence
+colimj∈J (colimKj F) ≃ colimK F.
+As an immediate application, we get a Mayer-Vietoris type theorem for colimits.
+A.3.9. Corollary. Let I be a poset. Suppose I = I0 ∪ I1, where I0 and I1 are both up-closed. Let F be a
+cosheaf on I. Let I01 = I0 ∩ I1. Then the natural maps give an equivalence
+colimI0 F|I0
+�
+colimI01 F|I01
+colimI1 F|I1
+∼
+→ colimI F.
+Proof. Let D1 be the poset of proper subsets of {0, 1} under inclusion. Apply Proposition A.3.8 to the
+assignment ∅ �→ I01, {0} �→ I0 and {1} �→ I1.
+□
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+83
+A.3.10. Locally constant cosheaves. A locally constant cosheaf on I is one such that Fϕ is an equivalence
+for all arrows ϕ in I.
+A.3.11. Proposition (Lurie, [Lur09, Corollary 4.4.4.9]). Let f : I ֒→ J be an order-preserving map of
+posets and F a locally constant cosheaf on J . If |I| ֒→ |J | is a weak homotopy equivalence, then the
+natural functor
+colimI f ∗F → colimJ F
+is an equivalence.
+In particular, if F is a cosheaf on an acyclic poset I, then for any i ∈ I, the natural map Fi → colimI F
+is an equivalence.
+A.3.12. Recollement. Given maps of cosheaves on a poset I
+F′′
+F
+β
+�
+F′
+α
+�
+we say it is a recollement of cosheaves, if
+(1) For each i ∈ I, evaluating α and β at i gives a recollement in StL:
+F′′
+i
+βi!
+�
+βi∗ �
+Fi
+βi
+�
+α∗
+i
+�
+α!
+i
+�
+F′
+i
+αi
+�
+(2) For i ≤ i′ in I, the commutative diagram where the vertical maps are the transition functors for
+F, F′ and F′′
+F′′
+i
+�
+Fi
+βi
+�
+�
+F′
+i
+αi
+�
+�
+F′′
+i′
+Fi′
+βi′
+�
+F′
+i′
+αi′
+�
+is a morphism of recollements.
+In this situation, Proposition A.2.1 is applicable to conclude that
+colimI F′′
+colimI F
+colim β
+�
+colimI F′
+colim α
+�
+extends to a recollement.
+A.3.13. Remark. A caution to those who bring intuition of cosheaves with values in a stable category
+(the ∞-category StL is itself not stable): one needs the hypothesis of a recollement of cosheaves in the
+statement of Proposition A.2.1. Else a “triangle of cosheaves” need not lead to a “triangle of cosections”.
+For a toy example of what could go wrong, consider the diagram of cosheaves
+Q
+F
+�
+K
+�
+on the poset ∆opp
+1
+= {0 ← ∅ → 1}, where each cosheaf is given by a horizontal row of the diagram
+(A.3.1)
+0
+�
+0
+�
+�
+� Vect ⊕ 0
+i
+�
+0
+�
+Vect
+�
+=
+�
+d
+� Vect ⊕ Vect
+p
+�
+0
+Vect
+�
+∼
+� 0 ⊕ Vect
+Here d is the diagonal embedding, i is inclusion to the first factor, and p is projection to the second factor.
+Note at each element of ∆opp
+1
+, the corresponding vertical sequence in (A.3.1) is a recollement. But note
+the right squares are not adjointable with respect to the vertical maps, so we do not have a recollement of
+cosheaves.
+
+84
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Then we have
+colim∆opp
+1
+Q
+colim∆opp
+1
+F
+�
+colim∆opp
+1
+K
+�
+0
+0
+�
+Vect
+�
+So while colim∆opp
+1
+Q is the cokernel of colim∆opp
+1
+F ← colim∆opp
+1
+K, we see K is not the kernel of
+colim∆opp
+1
+Q ← colim∆opp
+1
+F. Thus we do not obtain a recollement on cosections.
+A.4. Expansion of cosheaves. In this section, we prove the invariance of colimits of cosheaves under a
+certain modification called expansion.
+A.4.1. Expansion of posets.
+A.4.2. Definition. Let I be a poset and I′ ⊂ I.
+(1) We call I a weak elementary expansion of I′ if:
+• I \ I′ contains a unique minimal element, which we denote by 0.
+• Let I′
+>0 = {i ∈ I′|0 < i}. Then I′
+>0 is up-closed in I and acyclic (in particular non-empty).
+We denote the above situation by I′ ր I or I′ ր0 I if want to indicate the minimal element in
+I \ I′.
+(2) We call I an expansion of I′ if there is a finite sequence of up-closed subsets In ⊂ I (1 ≤ n ≤ N)
+such that I0 = I′, IN = I and In is a weak elementary expansion of In−1:
+(A.4.1)
+I′ = I0 ր I1 ր I2 ր · · · ր IN = I.
+Note that a weak elementary expansion may not be an expansion; it is an expansion if I′ is up-closed
+in I.
+We give an example of a weak elementary expansion that we use in the paper. Fix n ≥ 0, and set
+[n] = {0, . . ., n}. Let Dn be the partially ordered set of proper subsets J ⫋ [n] with respect to inclusion.
+A.4.3. Lemma. Fix a nonempty J ⊂ [n]. Set DJ
+n ⊂ Dn to be the subsets of [n] not contained in J.
+(1) The inclusion DJ
+n ⊂ Dn is a weak elementary expansion.
+(2) More generally, for any poset I with an embedding Dn ֒→ I whose image is up-closed, I is a weak
+elementary expansion of I′ := I \ (Dn \ DJ
+n).
+Proof. (1) Clearly DJ
+n is up-closed in Dn and Dn \ DJ
+n has the minimal element ∅. We need to show that
+DJ
+n is acyclic. We can identify the geometric realization of the nerve N(Dn) with the standard n-simplex
+∆n with vertices �0, · · · , �n in such a way that J ∈ Dn corresponds to the barycenter of the face σJ whose
+vertices are {�i|i ∈ [n]\J} (in particular, σ∅ is the maximal face of ∆n, and J1 ⊂ J2 if and only if σJ1 ⊃ σJ2).
+Under this identification, the geometric realization of N(DJ
+n) is ∆J
+n := ∪J′∈DJ
+nσJ′ = ∪J′⫋[n],J′̸⊂JσJ′. We
+need to show that ∆J
+n is contractible.
+Without loss of generality, we may assume J = {0, 1, · · · , j}, for j ≥ 1. Let ∆j ⊂ ∆n be the face
+with vertices �0, · · · ,�j. Then ∆J
+n is the union of faces whose vertices do not contain all vertices �i with
+j + 1 ≤ i ≤ n. Define a deformation retraction ∆n → ∆j by linearly extending the map on vertices given
+by �i �→ �i, if i ≤ j, and �i �→ �j, if i > j. Restriction of this deformation retraction to ∆J
+n ⊂ ∆n gives a
+deformation retraction ∆J
+n → ∆j, proving that ∆J
+n is contractible.
+(2) Let ∅ ∈ Dn be the unique minimal element. Then I′
+>∅ = DJ
+n is up-closed in Dn hence also in I by
+assumption, and is acyclic by (1).
+□
+A.4.4. Expansion of cosheaves.
+A.4.5. Definition. Let I be a poset and I′ ⊂ I. Let ϕ : E → F be a map of cosheaves on I.
+(1) If I′ ր0 I, we say that ϕ is a weak elementary expansion along I′ ր0 I if
+• ϕ|I′ : E|I′ → F|I′ is an equivalence.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+85
+• For i ∈ Inew = I \ I′, the following square is a pushout
+E0
+�
+ϕ0 � F0
+�
+Ei
+ϕi
+� Fi
+(2) Suppose I is an expansion of I′. We call ϕ an expansion along I′ ⊂ I if there exists a sequence
+of elementary expansions as in (A.4.1), where In are up-closed, such that:
+• ϕ|I′ : E|I′ → F|I′ is an equivalence.
+• For each 1 ≤ n ≤ N, let 0n ∈ In \ In−1 be the unique minimal element. Then for any
+i ∈ In \ In−1, the following square is a pushout
+E0n
+�
+ϕ0n � F0n
+�
+Ei
+ϕi
+� Fi
+The key property of an expansion of cosheaves is the invariance of colimit.
+A.4.6. Proposition.
+(1) Let I′ ր0 I be a weak elementary expansion of posets, and let ϕ : E → F
+be a weak elementary expansion of cosheaves on I along I′ ր0 I. Then ϕ induces an equivalence
+on colimits
+(A.4.2)
+colim ϕ : colimI E
+∼
+→ colimI F.
+(2) The same holds if I′ ⊂ I is an expansion and ϕ is an expansion of cosheaves on I along I′ ⊂ I.
+Proof. (1) Let Iold = I′ and Inew = I \ Iold. Let I+ = I ⊔ {0′}, and extend the partial order from I to
+I+ by adding the relations 0 < 0′ < i for all i ∈ Iold
+>0 .
+We first claim that the map of posets I → I+ is cofinal. Indeed, we apply Quillen’s Theorem A to the
+map i, and note that for any i ∈ I+, I ×I+ I+
+≥i either has a unique minimal element i if i ∈ I, or when
+i = 0′, I ×I+ I+
+≥0′ = {i ∈ Iold|0 ≤ i} = Iold
+>0 which is acyclic by assumption. Therefore the assumption
+for applying Quillen’s Theorem A is satisfied.
+Extend E to a cosheaf E+ on I+ by assigning E+
+0′ := F0 and the functors E+
+0′ → E+
+i
+for i ∈ Iold are
+given by the composition F0 → Fi
+ϕ−1
+i
+−−→ Ei. The cofinality of I in I+ implies
+(A.4.3)
+colimI E
+∼
+→ colimI+ E+.
+Next we would like to apply Proposition A.3.8 to write colimI+ E+ as a double colimit where the
+outer colimit is parametrized by I again. We apply it to K = I+, J = I and the following assignment
+I ∋ i �→ I+
+i : if 0 ≤ i, let I+
+i = {j ∈ I|j ≤ i} ∪ {0′}; otherwise let I+
+i = {j ∈ I|j ≤ i}. We check that the
+acyclicity condition holds. Let k ∈ I+. If k ∈ I, then {i ∈ I|k ∈ I+
+i } contains k as the unique minimal
+element hence is acyclic. If k = 0′, then {i ∈ I|0′ ∈ I+
+i } contains 0′ as the unique minimal element, hence
+again is acyclic. Proposition A.3.8 gives a natural equivalence
+(A.4.4)
+colimi∈I(colimI+
+i E+) ≃ colimI+ E+.
+For each i ∈ Iold, I+
+i
+has a unique maximal element i, hence colimI+
+i E+ ≃ E+
+i
+= Ei ≃ Fi. For i ∈
+Inew, I+
+i
+does not intersect Iold
+>0 since Iold
+>0 is up-closed. This implies {0, 0′, i} is cofinal in I+
+i , hence
+colimI+
+i E+ ≃ E0′ �
+E0 Ei ≃ Fi by assumption. Combining these calculations we see that the right side of
+(A.4.4) is canonically equivalent to colimI F. Combined with (A.4.3) we conclude that (A.4.2) holds.
+(2) We would like to reduce to the situation of (1). Let I′ = I0 ր01 I1 ր02 I2 ր · · · ր0n IN = I be
+a sequence of elementary expansions, with each In up-closed in I such that ϕ satisfies the conditions in
+Definition A.4.5(2). For 1 ≤ n ≤ N, let En be the cosheaf on I that is F|In on In and is E|I\In on I \ In.
+For I \ In ∋ i ≤ i′ ∈ In, the transition functor (En)i = Ei → (En)i′ = Fi′ is defined to be the composition
+
+86
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+Ei
+ϕi
+−→ Fi → Fi′. Then E0 = E and EN = F and there is a natural map ϕn : En−1 → En. It suffices to
+show that ϕn induces an equivalence on colimits.
+Let Inew
+n
+= In \In−1 and Iold = I \Inew
+n
+. We claim that I is an elementary expansion of Iold. Indeed,
+Inew
+n
+has a unique minimal element 0n by assumption, and Iold
+>0n = (In−1)>0n because In−1 is up-closed
+in I. By construction, ϕn : En−1 → En is a weak elementary expansion along Iold ր I in the sense of
+Definition A.4.5(1). Therefore we reduce to the case of (1).
+□
+A.5. A categorical contraction principle. If Y is a manifold with a flow {Φt}t>0 that contracts Y
+to a subspace Z ⊂ Y as t → 0, (i.e., the flow {Φt}t>0, viewed as an R>0-action, extends to an action
+{Φt}t≥0 of the monoid R≥0 with Z = Φ0(Y )) and if F is a constructible sheaf on Y locally-constant (hence
+constant) on flow lines, then the restriction map gives an isomorphism RΓ(Y, F) ∼= RΓ(Z, F|Z). This is
+usually called the contraction principle for sheaves. Below we formulate and prove a categorical analog of
+this principle.
+A.5.1. Theorem. Let J ⊂ I be a subposet.
+Denote s : J → I the inclusion and I0 = I\J .
+Let
+F : I → StL be a functor, denote by L : I0 → StL the cokernel of the natural map (s!s∗F)|I0 → F|I0.
+Assume that
+(1) The maps of cosheaves on I0
+L
+F|I0
+β
+�
+(s!s∗F)|I0
+α
+�
+is a recollement of cosheaves in the sense of Section A.3.12.
+(2) L is a locally constant cosheaf on I0.
+(3) The inclusion |s| : |J | ֒→ |I| is a homotopy equivalence.
+Then the natural map is an equivalence:
+colimJ s∗F ≃ colimI F.
+Before proving the theorem, we give a situation where it is easy to compute the left Kan extension
+s!s∗F of s∗F. Let f : I → J be a coCartesian map of finite posets such that each fiber f −1(j) contains a
+unique minimal element s(j). Then s : J → I is a section to f. Let I0 = I \ s(J ). Let F be a cosheaf
+on I.
+A.5.2. Lemma. The pullback f ∗s∗F is the left Kan extension s!(s∗F) of s∗F along s : J → I.
+Proof. For each i ∈ I, {j ∈ J |s(j) ≤ i} has a unique maximal element f(i). Therefore the value of s!(s∗F)
+at i is (s∗F)f(i) = Fs(f(i)) = (f ∗s∗F)i.
+□
+Since the map f is coCartesian, f!F is equivalent to
+�
+f F by Proposition A.3.4. We get the following
+corollary from Theorem A.5.1.
+A.5.3. Corollary. Under the above notations, assume the conditions in Theorem A.5.1 hold. Then the
+natural maps are equivalences
+colimJ s∗F ≃ colimI F ≃ colimI
+�
+f
+F.
+The proof of Theorem A.5.1 will be given after recalling some facts about simplicial complexes.
+A.5.4. Simplicial complexes and exit poset. For a poset I we have a simplicial complex |I|. For a simplicial
+complex Y we define its exit poset P(Y ) to be the set of (closed) faces σ of Y with the partial order given
+by inclusions. From the definition, faces in |I| are parametrized by non-degenerate simplices in N(I),
+therefore we get a canonical isomorphism of posets
+(A.5.1)
+sd(I) ∼= P(|I|)opp.
+For any subset I′ ⊂ I with the inherited poset structure, |I′| ⊂ |I| is a closed subcomplex.
+The
+complement |I| \ |I′| is the open star star ◦(|I \ I′|).
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+87
+For a simplicial complex Y , let sd(Y ) be its barycentric subdivision. Then sd(|I|) can be identified
+with |sd(I)| as simplicial complexes.
+For a simplicial complex X, by a cosheaf F on X we mean a cosheaf F on P(X)opp, i.e., a functor
+F : P(X)opp → StL (equivalently, by passing to right adjoints, it is the same datum as a functor P(X) →
+StR). We will write
+�
+X F for colimP(X)opp F. We write FZ = F|P(Z)opp for the restriction of a cosheaf to
+a subcomplex Z ⊂ Y .
+Let F be a cosheaf on a poset I. By (A.5.1), sd(F) is a cosheaf on P(|I|)opp, hence a cosheaf on |I|.
+By Lemma A.3.6, there is a canonical equivalence
+�
+|I|
+sd(F) ≃ colimI F.
+A.5.5. Simplicial collapse. Recall a simplex τ ⊂ K in a simplicial complex K is called a free face if there
+exists a unique maximal simplex σ ⊂ K such that τ ⫋ σ. In this case, the simplicial collapse of K along
+the free face τ ⊂ K is defined to be the subcomplex K′ = K \ (∪τ⊂γ⊂σγ◦) ⊂ K. Note the pair K′ ⊂ K
+implicitly knows the free face τ as the unique minimal face of K removed to obtain K′. Given general
+simplicial complexes K0 ⊂ K, by a simplicial collapse of K to K0, we will mean a finite sequence of
+subcomplexes K0 ⊂ · · · ⊂ Kn−1 ⊂ Kn ⊂ · · · ⊂ KN = K such that Kn−1 results from Kn via a simplicial
+collapse along a free face τn ⊂ Kn.
+A.5.6. Lemma (Whitehead [Whi39, Corollary on page 252]). For an inclusion of finite simplicial complexes
+Z0 ⊂ Y0, after the second barycentric subdivision Z = sd2(Z0) ⊂ sd2(Y0) = Y , there is always a simplicial
+collapse of the closed star U = star(Z) ⊂ Y to Z.
+Proof. We give a sketch here and refer to [Whi39] for more details.
+After the first barycentric subdivision, Z1 = sd(Y0) will be a full subcomplex of Y1 = sd(Y0), i.e. if the
+vertices of a simplex of Y1 lie in Z1, then the simplex itself lies in Z1. To see this, recall the k-simplices
+σ ⊂ Y1 are given by k-flags of simplices σ0 ⊂ σ1 ⊂ · · · ⊂ σk ⊂ Y0. The vertices of σ are the 0-flags
+σ0, σ1, . . . , σk ⊂ Y0 that occur in the flag. If these 0-flags satisfy σ0, σ1, . . . , σk ⊂ Z1, then the simplices
+satisfy σ0, σ1, . . . , σk ⊂ Z0. Thus the k-flag representing σ indeed satisfies σ0 ⊂ σ1 ⊂ · · · ⊂ σk ⊂ Z0, and
+hence σ ⊂ Z1.
+Now starting from the full subcomplex Z1 ⊂ Y1, after a barycentric subdivision Z = sd(Z1) ⊂ sd(Y1) =
+Y , we can always find a simplicial collapse of the closed star U = star(Z) ⊂ Y to Z as follows. Such a
+simplicial collapse will be determined by a total order on the simplices of U \ Z. We can take any total
+order compatible with the following partial order.
+First, consider the prior open star U ◦
+1 = star ◦(Z1) ⊂ Y1, and the complement U ◦
+1 \ Z1. Note every
+relatively open simplex σ◦ ⊂ U \ Z lies in a unique relatively open simplex ˆσ◦ ⊂ U ◦
+1 \ Z1.
+Partially order the simplices of U ◦
+1 \Z1 opposite to their natural order by closure inclusion; so the partial
+order begins with the maximal simplices of U ◦
+1 \ Z1 and proceeds to the minimal. (Note U ◦
+1 \ Z1 has no
+vertices – its simplex closures must contain vertices from both Z1 and its complement – so its minimal
+simplices will at least be edges).
+Partially order the simplices σ◦ ⊂ U \ Z by the above partial order on the simplices ˆσ◦ ⊂ U ◦
+1 \ Z1
+containing them. From here, it suffices to independently define for each τ ◦ ⊂ U ◦
+1 \ Z1 with partial closure
+τ = τ ◦ ⊂ U ◦
+1 , a simplicial collapse of the subcomplex τ ∩ U to the subcomplex ∂τ ∩ U.
+Thus it remains to solve the following general local problem: given a closed simplex τ1 and a proper
+closed facet ρ1 ⊂ τ1, consider the barycentric subdivision τ = sd(τ1) and its subcomplex ρ = sd(ρ1) ⊂ τ.
+Let U = star(ρ) ⊂ τ be the closed star of ρ ⊂ τ. Then we need a simplicial collapse of U to U ∩ ∂τ. To
+achieve this, we can take any total order on the simplices of U \ (U ∩ ∂τ) compatible with the following
+partial order. First, partially order the simplices of U \ (U ∩ ∂τ) opposite to their natural order by closure
+inclusion; so the partial order begins with the maximal simplices of U \ (U ∩ ∂τ) and proceeds to the
+minimal. Then refine this partial ordering, by further ordering the simplices of U \ (U ∩ ∂τ) starting with
+those with the maximal number of vertices not in ρ and ending with those with the minimal number of
+vertices not in ρ. It is elementary to check this indeed gives a simplicial collapse of U to U ∩ ∂τ.
+□
+
+88
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+A.5.7. Proof of Theorem A.5.1. Let E = s!s∗F. We need to show that the adjunction map α : E → F
+induces an equivalence colimI E ≃ colimI F.
+Let Y0 = |I| and Z0 = |I1| = |s(J )|, which are finite simplicial complexes by assumption, and Z0 ⊂ Y0
+is a closed subcomplex. We denote Y = sd2(Y0) and Z = sd2(Z0). Then sd(F) and sd(E) define cosheaves
+on Y0 and Z0. We abuse the notation to denote the pullback of sd(F), sd(E) to Y and Z still by F and E.
+By Lemma A.3.6, it suffices to show the natural map
+�
+Y E →
+�
+Y F is an equivalence.
+The locally constant cosheaf L on I0 defines a locally constant cosheaf sd(L) on sd(I) \ sd(s(J )) via
+pullback by the first vertex map sd(I) \ sd(s(J )) → I \ s(J ) = I0. In other words, L pullbacks to a
+locally constant cosheaf sd(L) on P(Y0)opp \ P(Z0)opp. Iterating this procedure, we get a locally constant
+cosheaf sd3(L) on P(Y )opp \ P(Z)opp. For simplicity we still denote it by L. For a cosheaf F′ on Y , we
+denote its restriction to P(Y )opp \ P(Z)opp by F′
+Y \Z. The assumptions on the original F imply the maps
+α and β restricted to Y \ Z extend to a recollement of cosheaves on P(Y )opp \ P(Z)opp:
+L
+�
+� FY \Z
+β
+�
+�
+� EY \Z
+α
+�
+Let U ◦ = star ◦(Z) ⊂ Y be the open star of Z, and U = star(Z) ⊂ Y the closed star of Z. Set
+V = Y \ U ◦ and W = U ∩ V . By Corollary A.3.9, we have
+(A.5.2)
+�
+U
+FU
+�
+�
+W FW
+�
+V
+FV
+∼
+→
+�
+Y
+F.
+The same applies to E in place of F. We will prove the following claims:
+(1) The diagram
+(A.5.3)
+�
+W EW
+�
+�
+�
+W FW
+�
+�
+V EV
+� �
+V FV
+is a pushout square.
+(2) The natural map
+�
+U EU →
+�
+U FU is an equivalence.
+We first show that these two claims imply the proposition. Let D1 be the poset of proper subsets of {0, 1}.
+We define a cosheaf A on D1 by
+A∅ =
+�
+W
+EW ,
+A{1} =
+�
+U
+EU,
+A{0} =
+�
+V
+EV .
+Define another cosheaf A′ on D1 by
+A′
+∅ =
+�
+W
+FW ,
+A′
+{1} =
+�
+U
+FU,
+A′
+{0} =
+�
+V
+FV .
+Then α induces a map of cosheaves A → A′. The inclusion {{0}} ⊂ D1 is a weak elementary expansion
+in the sense of Definition A.4.2(1), and A → A′ is a weak elementary expansion along {{0}} ր D1. Then
+Proposition A.4.6(1) applies to the situation to conclude that colimD1 A
+∼
+→ colimD1 A′. By (A.5.2) and
+its analog for E, we conclude that
+�
+Y E
+∼
+→
+�
+Y F.
+We prove (1). By the second assumption, over V , the termwise recollement
+(A.5.4)
+LV
+FV
+βV
+�
+EV
+αV
+�
+extends to a recollement of cosheaves, i.e., all six-term diagrams relating the values of (A.5.4) at two faces
+σ ⊂ τ in V are morphisms of recollements. By Proposition A.2.1(1), we conclude that the colimits also fit
+into recollements
+�
+V LV
+�
+�
+�
+V FV
+βV
+�
+�
+�
+�
+V EV
+αV
+�
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+89
+The same holds for W in place of W, so we have a diagram where the two rows extend to recollements
+(A.5.5)
+�
+W LW
+�
+�
+W FW
+βW
+�
+�
+�
+W EW
+αW
+�
+�
+�
+V LV
+�
+V FV
+βV
+�
+�
+V EV
+αV
+�
+Moreover, by Proposition A.2.1(2) this is a morphism of recollements. Now consider the map
+(A.5.6)
+�
+W
+LW →
+�
+V
+LV .
+We claim that W ֒→ V is a homotopy equivalence. Indeed, since the inclusions Z ⊂ Y , Z ⊂ U are
+homotopy equivalences, the inclusion U ⊂ Y is as well. Therefore there is a deformation retraction of
+Y to U. (See for example [Hat02, Chapter 0].) Removing U ◦ then gives a deformation retraction of
+V = Y \ U ◦ to W = U \ U ◦; in particular, the inclusion W ֒→ V is a homotopy equivalence. Since LV is
+locally constant, we can then apply Proposition A.3.11 to conclude that (A.5.6) is an equivalence. Then
+by Corollary A.2.2 applied to the morphism of recollements (A.5.5), we conclude that (A.5.3) is a pushout
+diagram.
+We prove (2). By Lemma A.5.6, we have a sequence of subcomplexes Z = K0 ⊂ K1 ⊂ · · · KN = U
+such that Kn−1 is a simplicial collapse of Kn along a free face τn, 1 ≤ n ≤ N. We claim P(Kn)opp
+is a weak elementary expansion of P(Kn−1)opp with minimal element in P(Kn)opp \ P(Kn−1)opp given
+by the unique maximal face σn in Kn containing τn. Indeed, we use notations from Lemma A.4.3. Let
+d = dim σn, we can identify P(σn) with Dd after choosing an identification of the vertices of σn with
+[d], and then P(Kn−1 ∩ σn)opp is identified with DJ
+d for some proper subset J ⊂ [d] corresponding to τn.
+Then the partition P(Kn)opp = P(Kn−1)opp ⊔(Dd \DJ
+d ) satisfies the conditions in Lemma A.4.3(2), which
+verifies that P(Kn−1)opp րσn P(Kn)opp. Therefore P(Z)opp ⊂ P(U)opp is an expansion in the sense of
+Definition A.4.2.
+We claim that αU : EU → FU is an expansion of cosheaves along P(Z)opp ⊂ P(U)opp in the sense of
+Definition A.4.5. Indeed, by construction, αU is an equivalence when restricted on Z. For any two faces
+τ ⊂ σ of U not contained in Z, we have a morphism of recollements
+Lσ
+�
+Fσ
+βσ
+�
+�
+Eσ
+ασ
+�
+�
+Lτ
+Fτ
+βτ
+�
+Eτ
+ατ
+�
+where the left vertical map is an equivalence since L is locally constant. By Corollary A.2.2, the right
+square above is a pushout square. This verifies the second condition for an expansion of cosheaves. Claim
+(2) now follows from Proposition A.4.6.
+□
+Appendix B. Functoriality of sheaves with Lagrangian singular support
+Let k be a field of characteristic 0, and let X be a smooth algebraic stack over C of finite type, L ⊂ T ∗X
+be a closed (C×-)conic Lagrangian substack. Denote Sh(X) the ∞-category of sheaves of k-modules on
+X, and ShL(X) ⊂ Sh(X) the full subcategory of sheaves with singular support contained in L. It is
+known that ShL(X) is compactly generated [AGK+, Cor. G.7.8], Sh(X) is presentable, and the inclusion
+ShL(X) → Sh(X) preserves both limits and colimits [AGK+, Cor. G.7.5] The goal of this section is to
+prove the following:
+B.0.1. Proposition. Let f : X → Y be a representable map between finite type smooth algebraic stacks
+over C.
+(1) Let LY ⊂ T ∗Y be a closed conic Lagrangian substack, then the functors f ∗, f ! : ShLY (Y ) → Sh(X)
+preserve both limits and colimits.
+
+90
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+(2) Let LX ⊂ T ∗X be a closed conic Lagrangian substack, then the functors f!, f∗ : ShLX(X) → Sh(Y )
+preserve both limits and colimits.
+Proof. (1) We show the statement for f ∗, the one for f ! follows from similar argument. Since f ∗ : Sh(Y ) →
+Sh(X) is a left adjoint, it preserves colimits; hence f ∗|ShLY (Y ) also preserves colimits since the inclusion
+ιLY : ShLY (Y ) ֒→ Sh(Y ) does.
+Let F : I → ShLY (Y ) be a diagram of sheaves. We have the natural map
+(B.0.1)
+φ : f ∗(lim
+i∈I Fi) → lim
+i∈I f ∗Fi
+in Sh(X). To show it is an equivalence, it suffices to show the same after pulling back to a smooth cover
+pX : �
+X → X, since p∗
+X is conservative. Let pY : �Y → Y be a smooth surjective map from a scheme �Y ,
+and form the Cartesian diagram
+�
+X
+pX
+�
+�
+f
+� �Y
+pY
+�
+X
+f
+� Y
+Then p∗
+Xφ can be factored as
+p∗
+Xf ∗(lim Fi) = �f ∗p∗
+Y (lim Fi) ≃ �f ∗ lim(p∗
+Y Fi) → lim �f ∗p∗
+Y Fi = lim p∗
+Xf ∗Fi ≃ p∗
+X lim f ∗Fi.
+Here we use that p∗
+X and p∗
+Y preserves limits (being smooth, they agree with p!
+X and p!
+Y up to a shift). To
+show the above composition is an equivalence, it suffices to show the middle arrow is an equivalence, i.e.,
+�f ∗ : ShL �
+Y (�Y ) → Sh( �
+X) preserves limits (here L�Y ⊂ T ∗ �Y is the transport of LY under the Lagrangian
+correspondence between T ∗ �Y and T ∗Y ).
+Henceforth we assume both X, Y are smooth schemes. For any closed conic Lagrangian LY ⊂ T ∗Y ,
+there is a Whitney stratification {Yα}α∈S of Y , so that LY ⊂ ∪αT ∗
+XαX. So suffices to prove the claim for
+LY is conormal to a Whitney stratification. Now pick a Whitney stratification {Xβ}β∈T of X, so that the
+images of f ∗ land in ShM(X) ⊂ Sh(X), for M = ∪βT ∗
+XβX. Now let ix : x → X be the inclusion of a point.
+By [NYa, Lemma A.1.11], the functor i∗
+x : ShM(X) → k-mod preserves limits. Now let F : I → ShLY (Y )
+be a diagram of sheaves in ShLY (Y ), we need to show the natural map φ in (B.0.1) is an equivalence.
+Equivalently, we need to check that φ induces an equivalence on each stalk. Taking stalks at x ∈ X, φx
+can be identified with the composition of equivalences
+φx : (f ∗(lim Fi))x ≃ (lim Fi)f(x) ≃ lim(Fi)f(x) ≃ lim f ∗(Fi)x ≃ (lim f ∗(Fi))x
+Here the second (resp. last) equivalence uses the fact that i∗
+f(x) : ShLY (Y ) → k-mod (resp. ix : ShM(X) →
+k-mod) preserves limits. This show φ is an equivalence.
+(2) We only prove that f! preserves limits, and the case of f∗ follows from a similar argument. By the
+same reduction using smooth covers as in (1), we can assume X, Y are smooth schemes. We can further
+assume LX is conormal to a Whitney stratification X = ∪α∈SXα. Now pick a relative compactification
+X of f, i.e, f factors as X
+j−→ X
+p−→ Y , where j is an open embedding, and p is proper. Now suffices to
+show that j! : ShXα(X) → Sh(X) preserves limits, since p! = p∗ preserves limits. Let X = ∪α′∈S′Xα′ be
+a Whitney stratification on X that refines the partition X = (∪Xα) ∪ (X\X). Define L′
+X and L′
+X to be
+the conormals to the new stratifications of X and X. Now suffices to show that j! : ShL′
+X(X) → ShL′
+X(X)
+preserves limits. Let F : I → ShL′
+X(X) be a diagram of sheaves. Consider the natural map ϕ : j!(lim Fi) →
+lim j!(Fi). We only need to check that ϕ is an equivalence on each stalk. For x ∈ X, it is clear that ϕx is
+an isomorphism. For x ∈ X \ X, we have (j!(lim Fi))x = 0 and (lim j!(Fi))x = lim(j!(Fi))x = 0 since i∗
+x
+preserves limits. Therefore ϕx is also isomorphism, and hence ϕ is an isomorphism.
+□
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+91
+Appendix C. Universal version of Tao-Travkin Theorem
+The method of [TT] can be applied to deduce the following theorem:
+C.0.1. Theorem. Let G◦ be the neutral component of the loop group G = G((t)) of a connected reductive
+group G over C. Let HG◦ ⊂ HG be the full monoidal subcategory of the universal affine Hecke category
+consisting of sheaves supported on Iu\G◦/Iu.
+Then the natural maps induce an equivalence of stable
+presentable monoidal categories:
+colim⊗
+J⊂ftIa HLJ
+∼
+� HG◦
+where the colimit is of stable presentable monoidal categories.
+In this section, we shall adopt the notations in loc.cit.. Denote by I the convolution Schubert 1-category
+for the affine Weyl group W a, which is denoted Word1
+fr and defined in [TT, Definition 4.1.1]. Objects in
+I are sequences w = (w1, · · · , wn) where each wi ∈ W a lies in a finite standard parabolic subgroup.
+Recall G≤w = IwI ⊂ G. For w = (w1, ..., wn) ∈ I, put Xw = Iu\G≤w1 ×Iu ... ×Iu G≤wn/Iu. Let
+X◦
+w ⊂ Xw be the open stratum where G≤wi are replaced with Gwi = IwiI, and let ∂Xw = Xw \ X◦
+w.
+Then Xw is stratified into the union of Xw′, for w′ = (w′
+1, · · · , w′
+n) ∈ I such that w′
+i ≤ wi for each i.
+Define Sh′(X◦
+w) ⊂ Sh(X◦
+w) to be the full subcategory of locally constant sheaves. Denote by Sh′(Xw) ⊂
+Sh(Xw) the full subcategory consisting of sheaves that are locally constant on each stratum Xw′. Similarly
+define Sh′(∂Xw).
+We have a functor H : I → StL
+k , which assigns to an object w ∈ I the category Sh′(Xw), and to
+a morphism w1 → w2 in I, the functor H(w1) → H(w2) given by convolution. This is the universal
+monodromic analogue of [TT, Corollary 5.3.3].
+Sketch of proof of Theorem C.0.1. The proof of [TT, Theorem 5.4.3] goes through, except we need to
+check the analogous statement for [TT, Corollary 5.4.2]: for every birational map w1 → w2 in I, the
+diagram
+colimw′
+1→w1 strict emb. H(w′
+1)
+�
+�
+H(w1)
+�
+colimw′
+2→w2 strict emb. H(w′
+2)
+� H(w2)
+is coCartesian. Now by same proof as in [TT, Corollary 5.4.2], we have a natural equivalence for every
+w ∈ I:
+colimw′→w strict emb. H(w′) ≃ Sh′(∂Xw).
+Therefore we are left to check the following Lemma, analogous to [TT, Lemma 5.4.1].
+□
+C.0.2. Lemma. For any birational map w1 → w2 in I, the natural diagram is coCartesian:
+(C.0.1)
+Sh′(∂Xw1)
+�
+�
+Sh′(Xw1)
+�
+Sh′(∂Xw2)
+� Sh′(Xw2)
+Proof. We have a morphism of recollements (see Section A.1.1)
+Sh′(X◦
+w1)
+�
+�
+� Sh′(Xw1)
+�
+�
+�
+� Sh′(∂Xwl)
+�
+�
+Sh′(X◦
+w2)
+�
+� Sh′(Xw2)
+�
+�
+� Sh′(∂Xw2)
+�
+Since w1 → w2 is birational, X◦
+w1 ∼= X◦
+w2, the left vertical arrow above is an equivalence. The desired
+statement follows from Corollary A.2.2.
+□
+
+92
+PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN
+References
+[AGK+] D. Arinkin, D. Gaitsgory, D. Kazhdan, S. Raskin, N. Rozenblyum, and Y. Varshavsky. The stack of local systems
+with restricted variation and geometric Langlands theory with nilpotent singular support. arXiv:2010.01906.
+[AMR]
+David Ayala, Aaron Mazel-Gee, and Nick Rozenblyum. Stratified noncommutative geometry. arXiv:1910.14602.
+[Bez16]
+Roman Bezrukavnikov. On two geometric realizations of an affine Hecke algebra. Publications math´ematiques de
+l’IH ´ES, 123(1):1–67, 2016.
+[BFM]
+Armand Borel, Robert Friedman, and John W. Morgan. Almost commuting elements in compact Lie groups.
+arXiv:math/9907007.
+[BFN10] David Ben-Zvi, John Francis, and David Nadler. Integral transforms and Drinfeld centers in derived algebraic
+geometry. Journal of the American Mathematical Society, 23(4):909–966, 2010.
+[BFO12] Roman Bezrukavnikov, Michael Finkelberg, and Victor Ostrik. Character D-modules via Drinfeld center of Harish-
+Chandra bimodules. Inventiones mathematicae, 188(3):589–620, 2012.
+[BK90]
+A.I. Bondal and M.M. Kapranov. Representable functors, Serre functors, and mutations. Mathematics of the USSR-
+Izvestiya, 35(3):519–541, 1990.
+[BNa]
+David Ben-Zvi and David Nadler. The character theory of a complex group. arXiv:0904.1247.
+[BNb]
+David Ben-Zvi and David Nadler. Nonlinear Traces. arXiv:1305.7175.
+[BNP17] David Ben-Zvi, David Nadler, and Anatoly Preygel. A spectral incarnation of affine character sheaves. Compositio
+Mathematica, 153(9):1908–1944, 2017.
+[Bon04]
+C´edric Bonnaf´e. ´El´ements unipotents r´eguliers des sous-groupes de Levi. Canadian Journal of Mathematics,
+56(2):246–276, 2004.
+[BRY22] Yuri Berest, Ajay C Ramadoss, and Wai-Kit Yeung. Representation Homology of Topological Spaces. International
+Mathematics Research Notices, 2022(6):4093–4180, 2022.
+[Fra16]
+Dragos Fratila. On the stack of semistable G-bundles over an elliptic curve. Mathematische Annalen, 365(1-2):401–
+421, 2016.
+[GR19]
+Dennis Gaitsgory and Nick Rozenblyum. A Study in Derived Algebraic Geometry Volume I: Correspondences and
+Duality. American Mathematical Society, 2019.
+[Hat02]
+Allen Hatcher. Algebraic Topology. Cambridge University Press, Cambridge ; New York, 2002.
+[He]
+Xuhua He. A generalization of cyclic shift classes. in preparation.
+[He07]
+Xuhua He. The G-stable pieces of the wonderful compactification. Transactions of the American Mathematical
+Society, 359(7):3005–3024, 2007.
+[He18]
+Xuhua He. Cocenter of p-adic groups, I: Newton decomposition. Forum of Mathematics, Pi, 6:e2, 2018.
+[HN14]
+Xuhua He and Sian Nie. Minimal length elements of extended affine Weyl groups. Compositio Mathematica,
+150(11):1903–1927, 2014.
+[Kra09]
+Daan Krammer. The conjugacy problem for Coxeter groups. Groups, Geometry, and Dynamics, pages 71–171,
+2009.
+[KS90]
+Masaki Kashiwara and Pierre Schapira. Sheaves on Manifolds, Volume 292 of Grundlehren Der Mathematischen
+Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1990.
+[Lia]
+Penghui Li. Derived categories of character sheaves. arXiv:1803.04289 [math].
+[Lib]
+Penghui Li. Derived categories of character sheaves II: Canonical induction/restriction functors. in preparation.
+[LN21]
+Penghui Li and David Nadler. Uniformization of semistable bundles on elliptic curves. Advances in Mathematics,
+380:107572, 2021.
+[LNY]
+Penghui Li, David Nadler, and Zhiwei Yun. On the cocenter of the affine Hecke category and Betti geometric
+Langlands in genus one. in preparation.
+[Lur09]
+Jacob Lurie. Higher Topos Theory. Number 170 in Annals of Mathematics Studies. Princeton University Press,
+Princeton, N.J, 2009.
+[Lur12]
+Jacob Lurie. Higher Algebra. 2012.
+[Lusa]
+G. Lusztig. Parabolic character sheaves, I. arXiv:math/0302151.
+[Lusb]
+G. Lusztig. Truncated convolution of character sheaves. arXiv:1308.1082.
+[Lus84]
+G. Lusztig. Intersection cohomology complexes on a reductive group. Inventiones Mathematicae, 75(2):205–272,
+1984.
+[Lus87]
+George Lusztig. Fourier transforms on a semisimple Lie algebra over Fq. In Arjeh M. Cohen, Wim H. Hesselink,
+Wilberd L. J. van der Kallen, and Jan R. Strooker, editors, Algebraic Groups Utrecht 1986, volume 1271, pages
+177–188. Springer Berlin Heidelberg, Berlin, Heidelberg, 1987.
+[MV88]
+I. Mirkovi´c and K. Vilonen. Characteristic varieties of character sheaves. Inventiones Mathematicae, 93(2):405–418,
+1988.
+[NT]
+David Nadler and Jeremy Taylor. The Whittaker functional is a shifted microstalk. arXiv:2206.09216.
+[NYa]
+David Nadler and Zhiwei Yun. Automorphic gluing functor in Betti Geometric Langlands. arXiv:2105.12318.
+[NYb]
+David Nadler and Zhiwei Yun. Compatibilities of automorphic gluing functor. in preparation.
+[Ric88]
+R. W. Richardson. Conjugacy classes of n-tuples in Lie algebras and algebraic groups. Duke Mathematical Journal,
+57(1), 1988.
+[TT]
+James Tao and Roman Travkin. The affine Hecke category is a monoidal colimit. arXiv:2009.10998.
+
+FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY
+93
+[Whi39]
+J. H. C. Whitehead. Simplicial Spaces, Nuclei and m-Groups. Proceedings of the London Mathematical Society,
+s2-45(1):243–327, 1939.
+Yau Mathematical Sciences Center, Tsinghua University, Beijing, China
+Email address: lipenghui@mail.tsinghua.edu.cn
+Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720-3840
+Email address: nadler@math.berkeley.edu
+Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, MA 02139
+Email address: zyun@mit.edu
+
diff --git a/49E0T4oBgHgl3EQfvgE1/content/tmp_files/load_file.txt b/49E0T4oBgHgl3EQfvgE1/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..47db96ef2b0c8ba8b04679a6f6791ab2710f48cd
--- /dev/null
+++ b/49E0T4oBgHgl3EQfvgE1/content/tmp_files/load_file.txt
@@ -0,0 +1,5376 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf,len=5375
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='02618v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='RT]  6 Jan 2023  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We calculate the dg algebra of global functions on commuting stacks of complex reductive  groups using tools from Betti Geometric Langlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, we prove that the ring of invariant  functions on the commuting scheme is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Our main technical results include: a semi-orthogonal  decomposition of the cocenter of the affine Hecke category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' and the calculation of endomorphisms of a  Whittaker sheaf in a diagram organizing parabolic induction of character sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Introduction  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Application to commuting stacks  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Background: universal affine Hecke category and its cocenter  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Main automorphic results  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Parabolic character sheaves and semi-orthogonal decomposition of the cocenter  7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Further results  8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Conventions  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Acknowledgements  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Universal affine Hecke category  10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Hecke categories  10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Whittaker objects  12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Langlands duality  14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Coxeter presentation  15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Hochschild homology and cocenters  15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Spectral realization  18  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Automorphic realization  21  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Descended trace of Whittaker object  25  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Horocycle descent  29  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Preliminaries  29  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Descent for smooth stacks  30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Singular and ind-version  36  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Harder-Narasimhan subcategories  37  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Combinatorial pieces  37  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The space B  42  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Geometric pieces and sheaves on them  47  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Semi-orthogonal decomposition of the cocenter  53  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Endomorphisms of Whittaker functional  59  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Combinatorial descriptions of character sheaves  60  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Fourier-Sato transform and Whittaker sheaf  64  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Calculation of endormorphisms of the Whittaker sheaf  67  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Additional applications  68  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Functions on the commuting stack  72  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Almost commuting pairs of semisimple elements  72  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Chevalley restriction theorem for the commuting stack  73  Date: January 9, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1  2  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  From groups to Lie algebras  75  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Calculating some colimits  78  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recollements and stratifications  78  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Interaction of colimits and recollement  80  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Posets and cosheaves  81  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Expansion of cosheaves  84  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A categorical contraction principle  86  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Functoriality of sheaves with Lagrangian singular support  89  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Universal version of Tao-Travkin Theorem  91  References  92  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Introduction  This paper is part of a broader study of the cocenter of the universal affine Hecke category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Its main  results include (i) a semi-orthogonal decomposition of the cocenter, as found in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4, whose  distinguished case spelled out in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4 proves a conjecture of [LN21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' and (ii) the calculation of  endomorphisms of a distinguished Whittaker object in the cocenter, as found in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6, which  builds on work of [Lia].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In a sequel, we will combine the results of this paper with those of [NYa] to  identify the cocenter of the universal affine Hecke category with the genus one automorphic category in  Betti Geometric Langlands, proving the Betti Geometric Langlands Conjecture in genus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' But already  a concrete output of the results proved here is a calculation of the dg algebra of global functions on the  commuting stack of a reductive group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will first explain this application, then turn to more details of  the main results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Application to commuting stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Classical commuting stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let us start with the underived statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let G∨ be a connected reductive group over the complex numbers C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let C2  G∨ be the commuting  scheme of G∨, the closed subscheme of G∨ × G∨ consisting of (g1, g2) ∈ G∨ × G∨ satisfying the equation  g1g2g−1  1 g−1  2  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The group G∨ acts on C2  G∨ by simultaneous conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Following [BFM], C2  G∨ decomposes into open and closed subschemes indexed by π1(G∨,der), the funda-  mental group of the derived group G∨,der of G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For each c ∈ π1(G∨,der), one defines a torus T ∨  c , as the  abelianization of a Levi subgroup L∨  c of G∨, and a map of stacks  ιc : (T ∨  c × T ∨  c )/Wc → C2  G∨/G∨  where Wc = NG∨(L∨  c )/L∨  c is the relative Weyl group of L∨  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' When c = 1 ∈ π1(G∨,der) is the identity,  T ∨  c = T ∨ is a maximal torus of G∨, and Wc is the usual Weyl group W = W(G∨, T ∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For more details  of this construction, we refer to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem (See Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G∨ be a connected reductive group over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assume Ansatz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then the maps ιc for c ∈ π1(G∨,der) induce an isomorphism on rings of invariant functions  O(C2  G∨)G∨ ≃  �  c∈π1(G∨,der)  O(T ∨  c × T ∨  c )Wc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In particular, O(C2  G∨)G∨ is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The Ansatz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5 that we assume in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2 is a universal monodromic version of  Bezrukavnikov’s equivalence [Bez16] in the Betti setting, which we expect to be proved by similar methods  to [Bez16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2, we also deduce the following description of invariant functions on the Lie algebra  commuting scheme and Lie algebra-Lie group commuting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem (See Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G∨ be a connected reductive group over C  with maximal torus T ∨ ⊂ G∨, and respective Lie algebras t∨ ⊂ g∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assume Ansatz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let C2  g∨ be the Lie algebra commuting scheme of pairs (X1, X2) ∈ g∨ × g∨ satisfying adX1(X2) = 0, and  Cg∨,G∨ the Lie algebra-Lie group commuting scheme of pairs (X, g) ∈ g∨ × G∨ satisfying Adg(X) = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then restrictions to t∨ × t∨ and t∨ × T ∨ respectively give isomorphisms on rings of invariant functions  O(C2  g∨)G∨  ∼  → O(t∨ × t∨)W ,  O(Cg∨,G∨)G∨  ∼  → O(t∨ × T ∨)W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In fact, it is possible to adapt our methods and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4 directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' With this approach, we  do not need to assume Ansatz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5, but can simply appeal to Bezrukavnikov’s equivalence [Bez16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since  we do not take this approach in this paper, we include Ansatz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5 in the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Derived commuting stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2 from a derived statement which we present  next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The derived commuting stack Z2  G∨ is the derived moduli of pairs g1, g2 ∈ G∨ with g1g2g−1  1 g−1  2  = 1 up  to conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' More formally, it is the adjoint quotient of the derived fiber product  Z2  G∨ ≃ ((G∨ × G∨) ×R  G∨ {1})/G∨  with respect to the commutator map G∨ × G∨ → G∨, (g1, g2) �→ g1g2g−1  1 g−1  2  and unit element 1 ∈ G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The underlying classical stack of Z2  G∨ is C2  G∨/G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' More geometrically, it is the moduli  Z2  G∨ ≃ LocG∨(T 2)  of G∨-local systems on the two-torus T 2, where g1, g2 are the monodromies around the two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We describe the entire dg algebra of derived global functions on Z2  G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem (Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G∨ be a connected reductive group over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assume Ansatz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then there is an equivalence of dg algebras  O(Z2  G∨) ≃  �  c∈π1(G∨,der)  O(Z2  T ∨  c )Wc ≃  �  c∈π1(G∨,der)  O(T ∨  c × T ∨  c × t∨  c [−1])Wc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We explain some notation in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Here t∨  c = LieT ∨  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The notation t∨  c [−1] denotes the affine  derived scheme with coordinate ring O(t∨  c [−1]) equal to the exterior algebra Sym∗((t∨  c )∗[1]) generated in  degree −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In particular, when G∨,der is simply-connected, we have a simple equality  O(Z2  G∨) ≃ O(T ∨ × T ∨ × t∨[−1])W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In this case, the above isomorphism was conjectured in [BRY22, Conjecture 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Background: universal affine Hecke category and its cocenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Although Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6 only  involves G∨, our proof focuses on the Langlands dual group G and in particular its loop group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Throughout the remainder of the introduction, we will make the simplifying assumption that G is  almost simple and simply-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Most results we state here are proved for general reductive G either  in the literature or in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Below we summarize known results about the affine Hecke category (or variants thereof) that will be  used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Universal affine Hecke category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let B ⊂ G be a Borel subgroup, U ⊂ B its unipotent radical, and  H = B/U the universal Cartan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G = G((t)) be the loop group, I ⊂ G the standard Iwahori subgroup  corresponding to B, Iu ⊂ I its pro-unipotent radical, and note H ≃ I/Iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Fix a maximal torus T ⊂ B ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let W = NG(T )/T be the Weyl group of G, X∗(T ) = Hom(Gm, T )  the coweight lattice of T , and �  W = NG(T )/T [[t]] ≃ X∗(H) ⋊ W the extended affine Weyl group of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let  W a ⊂ �  W be the affine Weyl group generated by affine simple reflections, and set Ω = �  W/W a ∼= NG(I)/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  The universal finite Hecke category of G (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' universal affine Hecke category of G) is the convolution  monoidal dg derived category of H-bimonodromic complexes of sheaves of C-modules  HG = Shbimon(U\\G/U)  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' HG = colimw∈�  W Shbimon(Iu\\G≤w/Iu) )  where the latter is with respect to the natural ind-scheme structure G = colimw∈�  W G≤w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Cocenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will regard HG and HG as algebra objects in C-linear stable presentable ∞-categories  StL  C (where morphisms are left adjoints) and make constructions therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is also sometimes useful to re-  gard HG and HG as algebra objects in the C-linear stable presentable bimodule ∞-category BimodHH(StL  C)  where the finite Hecke category HH = Shmon(H) for the universal Cartan naturally acts by left and right  convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will formulate our results in the absolute setting, but most hold in this relative setting  as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' in fact, certain proofs are made easier by shifting to the relative setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall the cocenter of a monoidal category A is the Hochschild homology category  hh(A) = A ⊗A⊗Aop A  The trace map is the natural projection  tr : A  � A ⊗A⊗Aop A = hh(A)  induced by the unit of A in the first factor of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The closed embedding G ⊂ G induces a natural fully faithful monoidal functor HG → HG, and in turn  a natural map on cocenters we will denote by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  a : hh(HG)  � hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  As a special case of a more general coequalization result in Section 3, we show for the universal finite  Hecke category HG, there is a natural identification of its cocenter (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  hh(HG) ≃ ShN (G/G)  where ShN (G/G) is the dg derived category of complexes of sheaves of C-modules with nilpotent singular  support on the adjoint-quotient G/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' One can think of ShN (G/G) as a version of character sheaves on  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A guiding goal is to arrive at a similar geometric description of the cocenter hh(HG) of the universal  affine Hecke category HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' As a starting point, we will use a universal monodromic version of a result of  Tao-Travkin [TT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In Appendix C, we provide the necessary extensions to apply their arguments to the  universal monodromic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let I be the set of simple roots of G and Ia be the set of affine simple roots of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For each proper  subset J ⫋ Ia, let LJ ⊂ G be the corresponding Levihoric, and HLJ ⊂ HG the universal finite Hecke  category of LJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For J ⊂ J′ ⫋ Ia, we have a natural diagram of monoidal inclusions HLJ ⊂ HLJ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem (Tao-Travkin [TT], see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assume G is simply-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the natural  maps induce a monoidal equivalence  colim⊗  J⫋Ia HLJ  ∼  � HG  where the colimit is of monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Spectral realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To deduce spectral consequences of our automorphic results, we will use a  universal version of a result of Bezrukavnikov [Bez16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since it is not yet in the literature, we state it here  as an Ansatz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' we will provide a proof in a sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Given a Borel subgroup B∨ ⊂ G∨, the universal Steinberg stack is the fiber product of adjoint quotients  StG∨ = B∨/B∨ ×G∨/G∨ B∨/B∨  (Here the derived and naive fiber product agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=') More geometrically, it is the moduli  StG∨ ≃ LocG∨,B∨(S1 × [0, 1], S1 × {0, 1})  of G∨-local systems on the cylinder S1 × [0, 1] with B∨-reductions along the boundary S1 × {0, 1}  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  5  The universal spectral affine Hecke category is the monoidal convolution category IndCoh(StG∨) of  ind-coherent sheaves on the universal Steinberg stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is a monoidal equivalence of universal affine Hecke categories  Φ : IndCoh(StG∨)  ∼  � HG  with the following properties:  (1) Φ identifies the structure sheaf OStG∨ with the universal affine Whittaker object WhG as coalgebra  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6 for the coalgebra structure on OStG∨ , and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7 for the definition  of WhG and its coalgebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=')  (2) Φ is naturally an equivalence of algebras in bimodules over QC(H∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (Here we regard the HH-  bimodule HG as a QC(H∨)-bimodule using the canonical monoidal equivalence HH ≃ QC(H∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=')  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Ansatz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5 (2) is not used in this paper, we only record it for conceptual completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  With this in hand, we can invoke the spectral calculations of [BNP17] to conclude:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assuming Ansatz (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5), there is a canonical equivalence  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  IndCohN (Z2  G∨) ≃ hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Main automorphic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now we formulate our main results regarding the cocenter hh(HG)  that will lead to a proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Whittaker objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By the Whittaker functional on ShN (G/G), we mean the functor  WG/G : ShN (G/G)  � ModC  WG(F) = ϕ0χ∗r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (F)  given by the right-adjoint transport χ∗r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' across the Whittaker correspondence  G/G  U/U  r  �  χ  � A1  followed by vanishing cycles ϕ0 for the coordinate function on A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here χ is induced from a generic  character U → U/[U, U] ≃ ⊕i∈IA1  i → A1  By the Whittaker object WhG/G ∈ ShN (G/G), we mean the object corepresenting WG/G in the sense  of a natural equivalence WG/G(F) ≃ Hom(WhG/G, F) for F ∈ ShN (G/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall the natural maps a : ShN (G/G) ≃ hh(HG) → hh(HG) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For the purpose of  introduction, we define the cocenter Whittaker object WhG/G as  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  WhG/G := a(WhG/G) ∈ hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The actual definition of WhG/G in the main text uses descended trace, which makes the above identity a  nontrivial theorem (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem (See Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assume Ansatz (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then under the equivalence (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3), the  structure sheaf OZ2  G∨ corresponds to the cocenter Whittaker object WhG/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Using this theorem, to calculate the dg algebra O(Z2  G∨), which is the derived endomorphism ring of  OZ2  G∨ , it suffices to calculate the derived endomorphism ring of WhG/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Fully faithful embedding from colimit of character sheaves to affine cocenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To calculate the derived  endomorphism ring of WhG/G), we will identify a full subcategory of hh(HG) containing WhG/G, in which  the endomorphism ring is easier to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall the notion of the cocenter of a monoidal category A generalizes to the Hochschild homology  category of any A-bimodule M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' One defines  hh(A, M) = A ⊗A⊗Aop M  with trace map the natural projection  tr : M  � A ⊗A⊗Aop M = hh(A, M)  6  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  induced by the unit of A in the first factor of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' So the cocenter hh(A) is the Hochschild homology  category of the regular A-bimodule category A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To begin, from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, for the bimodule category HG, we obtain an equivalence after passing  to Hochschild homology categories  colimJ⫋Ia hh(HLJ, HG)  ∼  � hh(HG, HG) = hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We can restrict the domain of this equivalence to obtain a functor relating cocenters  colimJ⫋Ia hh(HLJ) = colimJ⫋Ia hh(HLJ, HLJ)  � hh(HG, HG) = hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Our first main result is the following theorem conjectured in [LN21, see Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall G is assumed to be almost simple and simply-connected for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The natural  map is a fully faithful embedding  colimJ⫋Ia hh(HLJ) = colimJ⫋Ia hh(HLJ, HLJ)� �  � hh(HG, HG) = hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In fact, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4 is the “semistable” part of a more general result that describes a semi-orthogonal  decomposition of hh(HG) indexed by “Harder-Narasimhan” strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The generalization is not needed for  the specific applications of this paper, but is a key input to the work in progress [LNY].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will give a  precise statement of the semi-orthogonal decomposition in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall WhG/G is the image of WhG/G ∈ ShN (G/G) ≃ hh(HG) under the map a defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note  G = LI, for I ⊂ Ia the finite simple roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let us write WhG,I for the image of WhG/G under the natural  map  ShN (G/G) ≃ hh(HG) = hh(HLI)  � colimJ⫋Ia hh(HLJ) ≃ colimJ⫋Ia ShN (LJ/LJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4, to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6, remains to calculate the derived endomorphism ring of  WhG,I as an object in colimJ⫋Ia ShN (LJ/LJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Calculation of endomorphisms of WhG,I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To state the result of the calculation of End(WhG,I), we  need to recall some constructions from generalized Springer theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Z(G) be the center of G, and Irr(Z(G)) be the group of irreducible complex characters of Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To each χ ∈ Irr(Z(G)/Z0(G)), under the generalized Springer correspondence, the local system on the  regular nilpotent orbit of G with central character χ appears in the parabolic induction of a cuspidal local  system on a Levi subgroup Lχ ⊂ G, unique up to conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Set Tχ to be the connected center of Lχ  with Lie algebra tχ, and let T ∨  χ denote the dual torus with Lie algebra t∨  χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Set Wχ = NG(Lχ)/Lχ to be  the relative Weyl group which acts on Tχ and tχ, hence also on T ∨  χ and t∨  χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem (See Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is a canonical equivalence of dg algebras  End(WhG,I) ≃ ⊕χ∈Irr(Z(G))O(T ∨  χ × T ∨  χ × t∨  χ[−1])Wχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Note that Irr(Z(G)) can be canonically identified with π1(G∨), therefore the index set of the above  decomposition can be replaced with π1(G∨), which is what appeared in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The proof of the theorem uses generalized Springer theory and the decomposition of character sheaves  of [Lia].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Theorems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6 together imply an unconditional description of the derived endomorphism  ring of the cocenter Whittaker object WhG/G ∈ hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem (See Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is an equivalence of dg algebras  End(WhG/G) ≃ ⊕χ∈Irr(Z(G))O(T ∨  χ × T ∨  χ × t∨  χ[−1])Wχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Parabolic character sheaves and semi-orthogonal decomposition of the cocenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We give  more details that lead to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The results we are about to present here are not used  in full strength in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4, but they will be used in future work to identify the affine  cocenter with the genus one Betti geometric Langlands automorphic category, and in doing so, establish  Betti geometric Langlands in genus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For simplicity, we will continue to assume here G is almost simple and simply-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Parabolic character sheaves and Hochschild homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In Section 3, we give a geometric realization  of the Hochschild homology hh(HLJ, HG) in terms of Lusztig’s theory of parabolic (or rather parahoric)  character sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For J ⫋ Ia, let PJ ⊂ G be the standard parahoric subgroup with Levi quotient LJ, and Pu  J ⊂ PJ its  unipotent radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let HG,J be the full subcategory  HG,J = ShN  �Pu  J \\G/Pu  J  Ad(LJ)  �  where ShN (−) means sheaves whose singular support is nilpotent under the moment map for the left (or  right) LJ-action when pulled back to Pu  J \\G/Pu  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then HG,J can be viewed as a Betti version of Lusztig’s  parabolic character sheaves for loop groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For J ⫋ Ia, there is a canonical equivalence  hh(HLJ, HG) ≃ HG,J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Moreover, for J ⊂ J′ ⫋ Ia, the natural functor hh(HLJ, HG) → hh(HL′  J, HG) gets transported under  the above equivalence to the functor HG,J → HG,J′ given by a natural horocycle correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Under the identifications in the theorem, the diagram of full subcategories  J ⫋ Ia �→ hh(HLJ, HLJ) = ShN (LJ/LJ)  is identified with the full subcategory of HG,J of sheaves supported on Pu  J \\PJ/Pu  J  LJ  , which is essentially the  adjoint quotient LJ/LJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The transition functors for J ⊂ J′ are given by parabolic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2 is a consequence of a very general result we prove in Section 3, which says that for  very general HL-bimodules M coming from geometry (where L is a reductive group), the trace map  M → hh(HL, M) can be geometrically realized as the pull-push functor along a horocycle correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Semi-orthogonal decomposition of the cocenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To state the semi-orthogonal decomposition of hh(HG),  we need the notion of Newton points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' First, there is a Newton point map ν : W a → X∗(T )+  Q from the  affine Weyl group W a to rational dominant coweights: for any w ∈ W a, and sufficiently divisible n, we  have wn ∈ X∗(T ), and set ν(w) ∈ X∗(T )+  Q to be the rational dominant coweight so that nν(w) and wn  are in the same W-orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The Newton point map ν is invariant under conjugation by W a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' we denote by  NP ⊂ X∗(T )+  Q its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note NP has a natural positive coroot partial ordering but we will work with a  coarser linear order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Next, to each Newton point ν ∈ NP, we associate a finite simplicial complex1 B♥  ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For example, when  ν = 0, B♥  0 recovers the fundamental alcove of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For each facet σ of B♥  ν , we attach a category of (possibly  twisted) character sheaves ShN (Yν,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Together they form a cosheaf of categories on the poset opposite  to the set of facets of B♥  ν , where the transition functors are given by (twisted) parabolic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For  example, over B♥  0 , we recover the cosheaf of categories J �→ ShN (LJ/LJ) for J ⫋ Ia, with transition  maps given by the usual parabolic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The cocenter category hh(HG) has a semi-orthogonal decomposition indexed by non-  negative integers n ≥ 0 with the n-th associated graded category of the form  hh(HG)n =  �  ν∈NP,⟨2ρ,ν⟩=n  hh(HG)ν  1Strictly speaking, B♥  ν may only be a simplicial complex after a barycentric subdivision: as naturally constructed, the  intersection of two simplices in B♥  ν may be a union of more than one simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  8  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  and  hh(HG)ν ≃ colimσ⊂B♥  ν ShN (Yν,σ)  In particular, the first filtered piece hh(HG)0 is a full subcategory of hh(HG), and the theorem identifies  it with the colimit colimJ⫋Ia ShN (LJ/LJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This leads to the fully faithfulness asserted in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4 is a categorification of a result of Xuhua He [He18] where, among other things, he gave  a decomposition of the cocenter of the affine Hecke algebra indexed by Newton points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Idea of proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The essential idea of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4 is to perform a  categorical version of Morse theory on a cosheaf on a certain topological space Bν (or rather the poset of  its facets) that encodes the combinatorics of conjugacy classes in the affine Weyl group W a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We sketch the construction of Bν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let A be the apartment associated to T in the building of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We  regard A as a labelled simplicial complex, with each facet labelled by its type J ⊊ Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We view the  labelling as a simplicial map A → A/W a ≃ ∆ where ∆ is the fundamental alcove whose facets are in  bijection with J ⫋ Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Given a conjugacy class [w] ⊂ W a of an element w ∈ W a, with centralizer Cw ⊂ W a, we may identify  [w] ≃ W a/Cw with the open alcoves (open facets, or equivalently, J-facets with J = ∅) in the quotient  space X[w] = A/Cw which depends only on the conjugacy class [w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In general, the J-facets of Xw, for any  J ⫋ Ia, index the image of O[w] under the natural projection to the adjoint quotient W a → W a/Ad(WJ)  For each ν ∈ NP, we glue together the X[w], for conjugacy classes of [w] with Newton point ν, into a  simplicial complex 2 Bν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The gluing procedure relies on the combinatorics called pieces for the affine Weyl  group (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The space B♥  ν mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 is a subspace of Bν which we call the  essential part of Bν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  There is a natural function fν : Bν → R obtained by gluing a certain quadratic function on X[w]  introduced by He and Nie [HN14] in their work on minimal length elements in conjugacy classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The  critical locus Crit(fν) ⊂ Bν is contained in the subspace B♥  ν ⊂ Bν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For example, when ν = 0, B0 is obtained by gluing X[w] for all conjugacy classes [w] of finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The critical locus of f0, which coincides with B♥  0 in this case, is exactly the image of X[1] → B0, which  can be identified with the fundamental alcove ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For each J, the category of parahoric character sheaves HG,J has a semi-orthogonal decomposition given  by the stratification of Pu  J \\G/Pu  J  LJ  by geometric pieces, which were introduced by Lusztig [Lusa].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The strata  in HG,J are indexed by J-facets of B = �  ν∈NP Bν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' When we try to compute the colimit colimJ HG,J, the  complication is that the transition functors do not respect the semi-orthogonal decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' However,  by performing a categorical version of Morse theory on Bν, we are able to show that only the pieces of  HG,J indexed by the essential part B♥  ν ⊂ Bν contribute to the cocenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  There are two underlying reasons we are able to implement categorical Morse theory (and specifically,  a contraction principle) in our situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' One is that the strata categories of parahoric character sheaves  attached to facets of Bν have a certain local constancy property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This is a consequence of a geometric  result proved by He [He].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Another is that the embedding B♥  ν ֒→ Bν is a homotopy equivalence, which uses  the gradient flow of the function fν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In Appendix A, we collect general methods of calculating colimits  indexed by posets, and prove a general contraction principle for cosheaves of categories (see Theorem  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Further results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4, we use similar techniques to calculate the derived endomorphism  rings of two other natural objects in hh(HG) in terms of spectral data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' One of the calculations can be  interpreted as a derived spherical Hecke algebra, and the other one is the endomorphism ring of the  universal Eisenstein series in the genus one automorphic category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The methods of this paper lead not only to a calculation of the endomorphisms of objects in the cocenter  but to a full description of the cocenter of the universal affine Hecke category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The primary additional  inputs are:  2Same comment as above: Bν may only be a simplicial complex after a barycentric subdivision: as naturally constructed,  the intersection of two simplices in Bν may be a union of more than one simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  9  (1) The automorphic gluing construction under nodal degenerations of curves [NYa].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The description of nilpotent sheaves on degree zero semistable G-bundles on a genus one curve [LN21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Combined with the methods of this paper, we are able to prove the following to appear in a sequel [LNY].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For X a smooth projective curve, its Betti automorphic category ShN (BunG(X)) is the dg derived  category of complexes of sheaves of C-modules on the moduli of G-bundles on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For E a smooth projective genus one curve, there is a natural equivalence from the  cocenter of the universal affine Hecke category to the Betti automorphic category  hh(HG)  ∼  � ShN (BunG(E))  We can invoke the spectral calculations of [BNP17] to deduce the Betti geometric Langlands conjecture  in genus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall for X be a smooth projective curve, there is a natural spectral action on its Betti au-  tomorphic category ShN (BunG(X)) by the tensor category QCoh(LocG∨(E)) of quasi-coherent complexes  on the moduli of G∨-local systems on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary (of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For E a smooth projective genus one curve, the Betti geometric  Langlands conjecture holds: there is an equivalence of QCoh(LocG∨(E))-module categories  IndCohN (LocG∨(E))  ∼  � ShN (BunG(E))  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In the rest of the paper, we will use the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We fix a field k of characteristic 0 as the coefficient field for our sheaves and categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We denote by StL (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' StR) the ∞-category of stable presentable categories with morphisms left  adjoints (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' right adjoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We denote by StL  k (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' StR  k ) the ∞-category of stable presentable k-linear categories with morphisms  left adjoints (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' right adjoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In the many body of the paper, we will be working primarily with StL  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By a monoidal category, we will typically mean an algebra object in StL  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For a complex algebraic stack X, we denote by Sh(X) the k-linear dg derived category of complexes  of sheaves in k-vector spaces on X under the analytic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will refer to objects in Sh(X) as sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  If X is smooth and Λ ⊂ T ∗X is a conical closed subset, we denote by ShΛ(X) the full subcategory of  Sh(X) consisting of sheaves with singular support in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, using 0 to denote the zero section,  Sh0(X) is the full subcategory of sheaves with locally constant cohomology sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let G be a connected reductive group over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' When needed, we will choose a maximal torus T  and a pair of opposite Borel subgroups B and B− containing T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let U and U − be the unipotent radicals  of B and B−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The quotient H = B/U (the universal Cartan) is canonically independent of the choice of  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let r = dim H be the rank of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let W = W(G, T ) be the Weyl group of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let g, t, b, u, · · · denote  the Lie algebras of G, T, B, U, · · ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We will fix an Ad(G)-invariant non-degenerate symmetric bilinear form on g to identify g and g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For an affine algebraic group L over C, let BL = [(Spec C)/L] be its classifying space, regarded as  an Artin stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We thank David Ben-Zvi and Quoc P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Ho for inspiring discussions, and Tsao-  Hsien Chen, Peter Haine and James Tao for generous technical help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We thank Xuhua He especially for  providing a key geometric ingredient needed in this paper in the form of [He].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  PL was partially supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' 12101348).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  DN was partially supported by NSF grant DMS-2101466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ZY was partially supported by the Simons  Investigatorship and the Packard Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  10  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Universal affine Hecke category  In this section, we provide more details about the universal affine Hecke category as introduced in  Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  But here and in the rest of the paper, unless otherwise stated, we follow the setup of  Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4, and work with G a general connected complex reductive group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Hecke categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Finite Hecke categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the quotient stack U\\G/U, and its convolution diagram  U\\G/U × U\\G/U  p1  �♥♥♥♥♥♥♥♥♥♥♥♥  p2  �❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  U\\G ×U G/U  δ  �  π  � U\\G/U  U\\G/U  U\\G/U  Note δ is smooth (a base change of the diagonal BU → BU × BU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The map π is given by the  multiplication on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is a base change of the projection BU → BG with fibers isomorphic to G/U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It  behaves like a proper map for H-monodromic sheaves on the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' More precisely, π has a factorization  π : U\\G ×U G/U  h  � U\\G ×B G/U  π′  � U\\G/U  where h is an H-torsor (a base change of BU → BB with fibers isomorphic to H = B/U), and π′ is proper  (a base change of the projection BB → BG with fibers isomorphic to G/B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In general, for any H-torsor  p : E → X, and a sheaf F ∈ Sh(E) that is H-monodromic, so locally constant along the fibers of p, we  have a canonical isomorphism  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F ≃ p∗F[−r]  where as usual r = dim H is the rank of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, we write H(C) = H>0Hc where Hc is the compact  real form of H and H>0 the neutral component of the split real form of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then p factors as  p : E  p0 � E/H>0  pc  � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now pc is proper and H>0 is contractible so F descends to F ∈ Sh(E/H>0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F = pc!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='p0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F =  pc∗p0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (p∗  0F) ≃ pc∗(F ⊗ p0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='k), and p∗F ≃ pc∗F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The relative fundamental class of p0 gives a canonical  isomorphism p0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='k ≃ k[−r] ∈ Sh(E/H>0), hence a natural isomorphism p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F ≃ p∗F[−r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Returning to the convolution diagram, for any sheaf F ∈ Sh(U\\G ×U G/U) that is H-monodromic for  the action t · (g1, g2) = (g1t−1, tg2) (for t ∈ H, g1, g2 ∈ G), so locally constant along the fibers of h, we  have canonical isomorphisms  π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F ≃ π′  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='h!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F ≃ π′  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='h∗F[−r] ≃ π′  ∗h∗F[−r] ≃ π∗F[−r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Thus for such sheaves, the pushforward π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' satisfies all the base change identities of a proper map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Consider the closed embedding of the unit coset  U\\B/U  u  � U\\G/U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let qH : U\\B/U → H be the natural projection, which is a U-gerbe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let exp : h → H be the universal  cover, and introduce the universal local system  Luniv = q∗  H exp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Dh ∈ Sh0(U\\B/U)  where Dh ≃ kh[r] ∈ Sh0(h) is the Verdier dualizing sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that Luniv is concentrated in degree −r  with stalks isomorphic to the group algebra k[X∗(H)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The universal finite Hecke category of G is the monoidal category of H-bimonodromic  sheaves on U\\G/U (under the left and right translations of H)  HG = Shbimon(U\\G/U)  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  11  equipped with convolution  ⋆ : HG ⊗ HG  � HG  F1 ⋆ F2 = π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗(p∗  1F1 ⊠ p∗  2F2)  and unit object e = u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='Luniv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  It is easy to see that H-bimonodromic sheaves on U\\G/U are exactly U × U-equivariant sheaves  with nilpotent singular support when pulled back to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore we also denote Shbimon(U\\G/U) by  ShN (U\\G/U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For the torus H, we have HH = Sh0(H) the dg derived category of locally constant  sheaves on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Convolution is simply the pushforward L1 ⋆L2 = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (L1 ⊠L2) along the multiplication map  m : H × H → H, and the unit object is the universal local system e = Luniv = exp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We can also naturally regard HG as a monoidal category in HH-bimodule categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Loop group and parahorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G = G((t)) be the loop group of G, I ⊂ G the Iwahori subgroup  given by the preimage of B under the reduction mod t map G[[t]] → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Ia be the set of simple (affine)  roots of G with respect to I, and I ⊂ Ia the subset of simple roots of G with respect to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let �  W = X∗(H) ⋊ W be the extended affine Weyl group of G and W a ⊂ �  W the affine Weyl group  generated by affine simple reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We think of �  W as acting on the standard apartment A = X∗(T )R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For a simple reflection s ∈ W a, let αs ∈ Ia denote the corresponding affine simple root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A subset J ⊂ Ia is of finite type if the subgroup WJ generated by simple reflections s for αs ∈ J is  finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We use the notation J ⊂ft Ia to mean that J is a finite type subset of Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' When G is almost  simple, J ⊂ft Ia simply means that J ⫋ Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Given J ⊂ft Ia, let PJ ⊂ G be the standard parahoric subgroup containing I of type J, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', if Pu  J ⊂ PJ  denotes its pro-unipotent radical, and LJ = PJ/Pu  J its Levi quotient, then J are the simple roots of LJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Note that BJ = I/(I ∩ Pu  J ) is a Borel subgroup of LJ, with unipotent radical UJ = Iu/(Iu ∩ Pu  J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  When J = ∅, we have I = P∅ and H = L∅, and we write Iu = Pu  ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' When J = I, we have PI = G[[t]],  LI = G, B = BI and U = UI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We identify �  W with NG(T )/T [[t]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For w ∈ �  W and any lift ˙w ∈ NG(T ), the subspace I ˙wI ⊂ G is  independent of the choices of T and ˙w, and we denote it by Gw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let ≤ denote the Bruhat order on  W a extended to �  W by declaring w1 and w2 are incomparable if they are in different cosets of W a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let  G≤w = ∪w′≤wG(w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is well-known that Gw/I ⊂ G/I is isomorphic to an affine space of dimension ℓ(w),  and its closure is G≤w/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Affine Hecke categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' As in the finite-dimensional case, consider the quotient stack Iu\\G/Iu,  and its convolution diagram  Iu\\G/Iu × Iu\\G/Iu  p1  �♠♠♠♠♠♠♠♠♠♠♠♠♠  p2  �❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  Iu\\G ×Iu G/Iu  δ  �  π  � Iu\\G/Iu  Iu\\G/Iu  Iu\\G/Iu  Note δ is pro-smooth (a base change of the diagonal BIu → BIu ×BIu), and π has the similar “almost”  ind-proper property as in the finite-dimensional case for H-monodromic sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' More precisely, consider  the factorization  π : Iu\\G ×Iu G/Iu  h  � Iu\\G ×B G/Iu  π′  � Iu\\G/Iu  where h is an H-torsor (a base change of BIu → BI with fibers isomorphic to H ≃ I/Iu), and π′ is ind-  proper (a base change of the projection BI → BG with fibers isomorphic to the affine flag variety G/I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For  any sheaf F ∈ Sh(Iu\\G×IuG/Iu) that is H-monodromic with respect to the action t·(g1, g2) = (g1t−1, tg2),  so locally constant along the fibers of h, we have a canonical isomorphism by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  h!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F ≃ h∗F[−r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  12  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  On the other hand, if in addition F is supported on Iu\\G≤w1 ×Iu G≤w2/Iu for some w1, w2 ∈ �  W, then  since π′ is proper when restricted to Iu\\G≤w1 ×I G≤w2/Iu, we have canonical isomorphisms  π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F ≃ π′  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='h!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F ≃ π′  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='h∗F[−r] ≃ π′  ∗h∗F[−r] ≃ π∗F[−r]  Thus for H-monodromic sheaves F suppored on some Iu\\G≤w1 ×Iu G≤w2/Iu, the pushforward π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' satisfies  all the base change identities of a proper map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Consider the closed embedding of the unit coset  Iu\\I/Iu  u  � Iu\\G/Iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let qH : Iu\\I/Iu → H be the natural projection, which is a Iu-gerbe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Introduce the universal local  system  Luniv = q∗  H exp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Dh ∈ Sh0(Iu\\I/Iu)  where Dh ≃ kh[r] ∈ Sh0(h) is the Verdier dualizing sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The universal affine Hecke category of G is the colimit of H-bimonodromic sheaves on  Iu\\G≤w/Iu for w ∈ �  W  HG = colimw∈�  W Shbimon(Iu\\G≤w/Iu)  with respect to the full embeddings Shbimon(Iu\\G≤w1/Iu) ֒→ Shbimon(Iu\\G≤w2/Iu) whenever w1 ≤ w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  It is equipped with a monoidal structure given by convolution  ⋆ : HG ⊗ HG  � HG  F1 ⋆ F2 = π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗(p∗  1F1 ⊠ p∗  2F2)  and unit object e = u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='Luniv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For the torus H, we have HH = Sh0(H × X∗(H)) the dg derived category of locally  constant sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Convolution is simply the pushforward L1 ⋆ L2 = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (L1 ⊠ L2) along the multiplication  and addition map m : (H × X∗(H)) × (H × X∗(H)) → H × X∗(H), and the unit object is the universal  local system e = Luniv = exp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Dh supported on H × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' As with HG, we can also naturally regard HG as a monoidal category in HH-bimodule  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Whittaker objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Fix a maximal torus T ⊂ B ⊂ G, and let B− ⊂ G be the opposite Borel  subgroup, and U − ⊂ B− its unipotent radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Finite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the diagram  A1  U −  χ  �  r− � U\\G/U  where r− is induced by the inclusion U − ⊂ G, and χ : U → U/[U, U] → A1 is a non-degenerate character  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', nontrivial on each simple root group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Consider the natural factorization of r−  r− : U −  i−  � G/U  q  � U\\G/U  Note i− is a closed embedding transverse to the B-orbits in G/U, and q is smooth (a base change of  pt → BU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Thus for any F ∈ Sh(U\\G/U) that is left H-monodromic, so locally constant along the left  H-orbits in U\\G/U, we have canonical isomorphisms  r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  −F ≃ i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  −q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F ≃ i∗  −q∗F[2 dim r−] ≃ r∗  −F[2 dim r−]  Let ϕχ,1 : Sh(U −) → k-mod denote the vanishing cycles at the identity 1 ∈ U − with respect to the  non-degenerate character χ : U − → A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) The Whittaker functor is the composition  WG : HG  � k-mod  WG(F) = ϕχ,1r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  −F[− dim r−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  13  (2) The universal finite Whittaker sheaf WhG ∈ HG is the object corepresenting the Whittaker functor  WG(F) ≃ HomHG(WhG, F),  for all F ∈ HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The shift in the definition of WG is chosen to make WG exact for the perverse t-structure  on HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, for the unit object e ∈ HG, we have WG(e) ∼= k[X∗(H)] is concentrated in degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Coalgebra structure on Whittaker sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Here we define the natural coalgebra structure on WhG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  First, introduce the analogous Whittaker functor on U\\G ×U G/U:  WU\\G×U G/U : Sh(U\\G ×U G/U)  � k-mod  WU\\G×U G/U(F) = ϕχ+χ,1r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  −F[− dim r−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  where  A1  U − × U −  χ+χ  �  r− � U\\G ×U G/U  Now observe the Whittaker functor WG is naturally lax monoidal: for any F1, F2 ∈ HG, we have a  natural map:  WG(F1) ⊗ WG(F2) ≃ WG×G(F1 ⊠ F2) ≃ WU\\G×U G/U(δ∗(F1 ⊠ F2))  −→ WU\\G×U G/U(π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗(F1 ⊠ F2)) ≃ WG(F1 ⋆ F2)  In other words, we have a natural transformation of functors  WG ⊠ WG → WG ◦ m : HG ⊗ HG → k-mod  This induces a map between co-representing objects:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  α : mℓ(WhG) −→ WhG ⊗ WhG  By adjunction, we obtain a map:  β : WhG −→ WhG ⋆ WhG  which defines a comultiplication on WhG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The counit and coherences can be constructed analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In fact, the full additive ∞-subcategory  of HG generated by the monoidal unit e and Whittaker object WhG is closed under convolution and  in fact a classical category (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' equivalent as an ∞-category to a discrete category, since its Homs are  concentrated in degree 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The coalgebra structure on WhG comes from a coalgebra structure in this  classical subcategory, and thus its coherences are all strictly determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Microlocal description for Whittaker sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will not need the following microlocal interpretation  but mention it for conceptual clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will use an analogue for character sheaves discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Consider the cotangent bundle T ∗(G/U), and note its fiber at the identity coset U/U ∈ G/U is naturally  isomorphic to (g/u)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Thus the differential dχ : u− → A1 gives a covector  ξ : g/u  � g/b ≃ u−  dχ � A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Ξ : HG → k-mod denote the ∗-pullback along q : G/U → U\\G/U, followed by the microstalk at  ξ ∈ T ∗  U/U(G/U) (normalized so that the microstalks of perverse sheaves are concentrated in degree 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The following is a standard calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a general version in Betti Geometric Langlands, see [NT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The Whittaker functor is naturally isomorphic to the microstalk functor  WG ≃ Ξ : HG  � k-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  14  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Affine case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the natural closed embedding  i : U\\G/U  � Iu\\G/Iu  induced by the inclusion of constant loops G ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) The affine Whittaker functor is the composition  WG : HG  � k-mod  WG(F) = WG(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The universal affine Whittaker sheaf WhG ∈ HG is the object corepresenting the affine Whittaker  functor  WG(F) ≃ HomHG(WhG, F),  for all F ∈ HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The universal affine Whittaker sheaf is the pushforward of the finite Whittaker finite sheaf  WhG ≃ i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='WhG  In particular, for the unit object e ∈ HG, we have WG(e) ∼= k[X∗(H)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By adjunction,  WG(F) = WG(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F) ≃ HomHG(WhG, i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F) ≃ HomHG(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='WhG, F)  □  Note that i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' is monoidal, and therefore WhG has the induced coalgebra structure from WhG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Langlands duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G∨ be the dual group of G, viewed as a split group over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let B∨ ⊂ G∨ be  the distinguished Borel subgroup, U ∨ ⊂ B∨ its unipotent radical, and H∨ = B∨/U ∨ its universal Cartan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We may identify the Weyl group of G∨ with W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall the canonical identification B∨/B∨ ≃ �  G∨/G∨ where �  G∨/G∨ is the Grothendieck-Springer stack  of pairs (g, E) of an element g ∈ G∨, and a Borel subgroup E ⊂ G∨, such that g ∈ E, all up to conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall under the above identification, the map of adjoint quotients B∨/B∨ → G∨/G∨ corresponds to the  Grothendieck-Springer map �  G∨/G∨ → G∨/G∨ that forgets the Borel subgroup E ⊂ G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The projection B∨ → H∨ factors through B∨/B∨ → H∨, which corresponds to the projection  �  G∨/G∨ → H∨ that takes a pair (g, E) to the class [g] ∈ E/Eu ≃ H∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) The universal Steinberg stack is the derived fiber product of adjoint quotients  StG∨ = B∨/B∨ ×R  G∨/G∨ B∨/B∨  (2) The universal spectral affine Hecke category is the monoidal convolution category  IndCoh(StG∨) = IndCoh(B∨/B∨ ×R  G∨/G∨ B∨/B∨)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The smallness of the Grothendieck alteration �  G∨ → G∨ implies that the underived fiber  product B∨/B∨ ×G∨/G∨ B∨/B∨ has the expected dimension zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since G∨/G∨ and B∨/B∨ are both  smooth, we see that the derived structure on StG∨ is in fact trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note we can also naturally regard IndCoh(StG∨) as a monoidal category in QC(H∨)-  bimodule categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To deduce spectral consequences of results on HG, we will use a universal version of a result of  Bezrukavnikov [Bez16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since it is not yet in the literature, we state it here as an Ansatz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' we will provide  a proof in a sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is a monoidal equivalence of universal affine Hecke categories  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  Φ : IndCoh(StG∨)  ∼  � HG  with the following properties:  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  15  (1) Φ identifies the structure sheaf OStG∨ with the universal affine Whittaker object WhG as coalgebra  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Φ is naturally an equivalence of algebras in bimodules over QC(H∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (Here we regard the HH-  bimodule HG as a QC(H∨)-bimodule using the canonical monoidal equivalence HH ≃ Sh0(H) ≃  QC(H∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Ansatz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4 (2) is not used in this paper, we only record it for conceptual completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Coxeter presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To make calculations in the cocenter of HG, we will use a universal mon-  odromic version of a result of Tao-Travkin [TT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In Appendix C, we provide the necessary extensions to  apply their arguments to the universal monodromic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let G◦ be the neutral component of the loop group G, and let HG◦ ⊂ HG be the full subcategory of  objects supported on G◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then HG◦ is a monoidal subcategory of HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For each finite type subset J ⊂ft Ia, the finite universal Hecke category HLJ embeds into HG◦ as  a monoidal full subcategory of sheaves supported on Iu\\PJ/Iu, which is a pro-unipotent gerbe over  UJ\\LJ/UJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' These embeddings are compatible with inclusions J ⊂ J′ ⊂ft Ia in an obvious sense, and  induce a monoidal functor  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  colim⊗  J⊂ftIa HLJ  � HG◦  Here we regard all the Hecke categories as objects in stable presentable k-linear categories StL  k with  morphisms left adjoints, and we write colim⊗ to emphasize that the colimit is of monoidal categories in  HH-bimodues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' of algebra objects in HH-bimodues in StL  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem (Tao-Travkin [TT], see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The functor (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) is a monoidal equivalence of  HH-bimodule categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In [TT], the authors worked with the bi-I-equivariant version of the affine Hecke category and assumed  G was simply-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We sketch the necessary modifications to their argument in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Formulating a generalization of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1 for the whole category HG when G◦ ⫋ G is  more combinatorially complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will not need it since Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1 will suffice for our application  to the cocenter of any reductive G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Hochschild homology and cocenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We work in the setting of stable presentable k-linear cate-  gories StL  k with morphisms left adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' All higher algebra constructions will be following [Lur12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let A be an algebra object in StL  k , and Aop the opposite algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Let M be an A-bimodule, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' an A ⊗ Aop-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The Hochschild homology of A with values in M is the tensor  hh(A, M) = A ⊗A⊗Aop M  The trace map is the natural projection  tr : M  � A ⊗A⊗Aop M = hh(A, M)  induced by the unit of A in the first factor of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The cocenter of A is the Hocschild homology of A with values in the regular bimodule  hh(A) = hh(A, A) = A ⊗A⊗Aop A  The trace map is the natural projection  tr : A  � A ⊗A⊗Aop A = hh(A, A) = hh(A)  induced by the unit of A in the first factor of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  16  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Recall one typically calculates hh(A, M) as the geometric realization of the Hochschild complex (sim-  plicial object)  B•(A, M) = [M  M ⊗ A  ��  M ⊗ A ⊗ A · · · ]  ���  This results from tensoring the regular A-bimodule with the bar resolution  M  [M ⊗ A  �  M ⊗ A ⊗ A  ��  M ⊗ A ⊗ A ⊗ A · · · ]  ���  Given a monoidal subcategory A′ ⊂ A, there is a minor variation on the Hochschild complex that  equally well calculates hh(A, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note we can regard A as an algebra in A′-bimodules, and likewise,  regard M as an A-bimodule in A′-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then to calculate hh(A, M), we can form the relative  Hochschild complex  B•(A, M)A′ = [hh(A′, M)  hh(A′, M ⊗A′ A)  ��  hh(A′, M ⊗A′ A ⊗A′ A) · · · ]  ���  This results from tensoring the regular A-bimodule with the relative bar resolution inside of A′-bimodules  M  [M ⊗A′ A  �  M ⊗A′ A ⊗A′ A  ��  M ⊗A′ A ⊗A′ A ⊗A′ A · · · ]  ���  In particular, we can take A′ = ⟨1A⟩ ⊂ A to be the full subcategory generated by the unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Descended trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The following general construction will provide a useful way to characterize certain  objects of the cocenter of a monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let ∆ denote the simplex category of non-empty finite  ordered sets [n] = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' , n}, n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Suppose A is a monoidal category with product denoted by ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Given an algebra object a ∈ A, let L = LModa(A) and R = RModa(A) denote the category of left and  right a-modules in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let B = Bimoda(A) denote the category of a-bimodules in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note B is naturally  monoidal with product given by  m ⋆a n = colim∆op[m ⋆ n  m ⋆ a ⋆ n  ��  m ⋆ a ⋆ a ⋆ n · · · ]  ���  Similarly, L is a B ⊗ A-bimodule, and R is a A ⊗ B-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In fact, L and R are in duality with unit  and counit maps denoted by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  u : B  � L ⊗A R  c : R ⊗B L  � A  Here, u is the inverse of the natural functor L ⊗A R → B (sending x ⊗ y to x ⋆ y), which is an equivalence  by [BFN10, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall the dual bimodules L and R, with their unit and counit maps, provide a map on cocenters  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  h : hh(B) = B ⊗B⊗Bop B  u′  � (L ⊗A R) ⊗B⊗Bop B ≃ A ⊗A⊗Aop (R ⊗B L)  c′  � A ⊗A⊗Aop A = hh(A)  where u′ is induced by u, and c′ by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To avoid confusion, let us denote the usual trace maps by  trA : A  � hh(A)  trB : B  � hh(B)  Given m ∈ B, we can take its B-trace trB(m) ∈ hh(B) then its image h(trB(m)) ∈ hh(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Given m ∈ B, we call  tr(m) := h(trB(m)) ∈ hh(A)  the descended trace of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  17  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Functoriality of descended trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose f : A → A′ is a monoidal functor of monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then there is an evident commutative diagram  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  A  tr  �  f  � A′  tr  �  hh(A)  hh(f) � hh(A′)  Suppose in addition a ∈ A is an algebra object with image algebra object a′ = f(a) ∈ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then  applying f to bimodules provides a natural monoidal functor F : Bimoda(A) → Bimoda′(A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have  an evident commutative diagram  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  Bimoda(A)  tr  �  F  � Bimoda′(A′)  tr  �  hh(Bimoda(A))  h  �  hh(F )� hh(Bimoda′(A′))  h  �  hh(A)  hh(f)  � hh(A′)  where each h denotes the map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2) from the cocenter of bimodules to the cocenter induced by the left  and right module categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We immediately conclude:  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose f : A → A′ is a monoidal functor of monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Suppose a ∈ A is an algebra object with image algebra object a′ = f(a) ∈ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Under the induced functor hh(f) : hh(A) → hh(A′), we have an identification of descended traces  hh(f)(tr(m)) ≃ tr(F(m))  m ∈ Bimoda(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In particular, for the regular bimodule m = a with image F(m) = f(a) = a′, an identification of descended  traces  hh(f)(tr(a)) ≃ tr(a′)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Formula for descended trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We provide here a formula for the descended trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let ∆+ denote the augmented simplex category of (possibly empty) finite ordered sets [n] = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' , n},  n ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Given m ∈ B = Bimoda(A), consider its bar augmented simplicial object m• : ∆op  + → B given  by the assignments: mn = m ⋆ a⋆n, with each face map a contraction via the multiplication of a, and each  degeneracy map an insertion of the unit of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We can picture the diagram of face maps in the usual way  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='m• = [m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='m ⋆ a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='m ⋆ a ⋆ a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='m ⋆ a ⋆ a ⋆ a · · · ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='and the augmentation descends to an equivalence on the geometric realization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='colim∆op[m ⋆ a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='∼  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='m ⋆ a ⋆ a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='m ⋆ a ⋆ a ⋆ a · · · ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='Now let us take the descended trace of the bar augmented simplicial object to obtain a resolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='tr(m)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='colim∆op[tr(m ⋆ a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='∼  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='tr(m ⋆ a ⋆ a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='tr(m ⋆ a ⋆ a ⋆ a) · · · ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='using that we work in the setting of continuous functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Unwinding the definitions, for any a-bimodule of the form ℓ ⋆ r where ℓ ∈ L = LModa(A) and r ∈  R = RModa(A), we have a natural isomorphism tr(ℓ ⋆ r) ≃ trA(r ⋆a ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Thus the above resolution gives a  concrete formula for the descended trace  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  tr(m)  colim∆op[trA(m)  ∼  �  trA(m ⋆ a)  ��  trA(m ⋆ a ⋆ a) · · · ]  ���  18  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  where the face maps are given by contractions via the multiplication of a (and degeneracy maps given by  an insertion of the unit of a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, the new face map trA(m ⋆ a⋆(n+1)) → trA(m ⋆ a⋆n) is induced  by the left multiplication of the last factor on the first available thanks to the cyclic symmetry of the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Descended trace for coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' One can repeat the construction of the descended trace starting  with a coalgebra c ∈ A rather than an algebra a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We only prefer the algebra formulation due to our  lack of familiarity with “bi-comodules”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' But in any case, we will only work with the coalgebra c itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In  this case, we can take the following concrete formula as definition of the descended trace  tr(c) := lim∆[trA(c)  �� trA(c ⋆ c)  ��� trA(c ⋆ c ⋆ c) · · · ]  where the coface maps are given by expansions via the comultiplication of c (and degeneracy maps given  by an insertion of the counit of c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, the new coface map trA(c⋆n) → trA(c⋆n+1) is induced  by the left comultiplication of the last factor on the first available thanks to the cyclic symmetry of the  trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In our applications, where some additional hypotheses hold, we can relate the above definition for  coalgebras with the prior theory for algebras, and in particular take advantage of the prior recorded  functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Assume that A is compactly generated, and the monoidal product preserves the category of compact  objects Ac ⊂ A, so that A ≃ IndAc as monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the categories L, R, B, hh(B), hh(A) are  all compactly generated, and the functor tr preserves compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose our coalgebra is compact  c ∈ Ac, and denote by a ∈ Aop  c  the same object regarded as an algebra in the dual monoidal category  A∨ ≃ IndAop  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose in addition a ∈ Bimoda(A∨) is compact (for example, it is a summand of a ⋆ a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then chasing definitions, under the canonical equivalence hh(A∨)c ≃ hh(Aop  c ) ≃ hh(Ac)op ≃ hh(A)∨  c , the  colimit calculating the algebra descended trace tr(a) ∈ hh(Aop  c ) is equivalent to the limit calculating the  coalgebra descended trace tr(c) ∈ hh(Ac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Spectral realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this subsection, we consider the spectral realization IndCoh(StG∨) of the  universal affine Hecke category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, we compute the descended trace of the structure sheaf  OStG∨ as a coalgebra object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The arguments are quite general, so we will work with an abstract setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We stress that in this  subsection, all fiber products of stacks are derived fiber products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let X → Y be a proper morphism between smooth stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then IndCoh(X ×Y X) is naturally a  monoidal category via convolution: F ⋆ G = m∗δ∗  23(F ⊠ G) where δ23 is the diagonal map and m the  forgetful map in the correspondence  X ×Y X × X ×Y X  X ×Y X ×Y X  δ23  �  m  � X ×Y X  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In our application, we will take the induction map X = B∨/B∨ → G∨/G∨ = Y so that  X ×Y X = StG∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Alternatively, we could define the !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='-convolution: F⋆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='G = m∗δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  23(F⊠G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will denote by IndCoh(X×Y  X)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' the resulting monoidal category with the !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that δ23 is quasi-smooth, so δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  23 and δ∗  23,  and hence ⋆ and ⋆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' as well, only differ by an invertible twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Serre duality gives an equivalence of compact objects Coh(X ×Y X)op ≃ Coh(X ×Y X) and hence an  equivalence of their ind-completions IndCoh(X ×Y X)∨ ≃ IndCoh(X ×Y X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Note the dual IndCoh(X×Y X)∨ inherits a monoidal structure from IndCoh(X×Y X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The identification  IndCoh(X ×Y X)∨ ≃ IndCoh(X ×Y X) naturally lifts to an equivalence of monoidal categories  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  DSerre  X×Y X : IndCoh(X ×Y X)∨  ∼  � IndCoh(X ×Y X)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now denote by LY = Hom(S1, Y ) the derived loop space of Y , and ΛX/Y = π∗δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (T ∗−1  X×Y X) ⊂ T ∗(LY )  the Lagrangian defined via the correspondence  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  X ×Y X  (X ×Y X) ×X×X X ≃ LY ×Y X  δ  �  π  � LY  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  19  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Continuing in the setup of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, we find the commuting stack  LY = L(G∨/G∨) = Z2  G∨ = (G∨/G∨ × G∨/G∨) ×G∨/G∨ ({1}/G∨) ≃ ((G∨ × G∨) ×G∨ {1})/G∨  the derived fiber product of the commutator map c : G∨ × G∨ → G∨, and the inclusion of the identity  1 ∈ G∨, all up to conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We also find the Lagrangian of nilpotent codirections ΛX/Y = N as  calculated in [BNP17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Thanks to [BNP17], we have the following:  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) The functor π∗δ∗ lands in IndCohΛX/Y (LY ) and fits into a commutative diagram  with the trace map tr inducing a horizontal equivalence  IndCoh(X ×Y X)  tr  �  π∗δ∗  �❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  hh(IndCoh(X ×Y X))  ∼  � IndCohΛX/Y (LY )  (2) Similarly, the functor π∗δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' lands in IndCohΛX/Y (LY ) and fits into a commutative diagram with  the trace map tr inducing a horizontal equivalence  IndCoh(X ×Y X)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  tr  �  π∗δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  hh(IndCoh(X ×Y X)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=')  ∼  � IndCohΛX/Y (LY )  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It follows that the equivalence on trace categories induced by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) is also given by Serre  duality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e, the following diagram naturally commutes  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  hh(IndCoh(X ×Y X))∨  hh(DSerre  X×Y X) �  ∼  �  hh(IndCoh(X ×Y X)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=')  ∼  �  IndCohΛX/Y (LY )∨  DSerre  LY  � IndCohΛX/Y (LY )  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In the above situation, we compute tr(∆∗OX), where ∆ : X → X ×Y X is the diagonal  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have a commutative diagram with a derived Cartesian square on the left  X  ∆  �  LX  pX  �  ϕ  �  Lf  �❑  ❑  ❑  ❑  ❑  ❑  ❑  ❑  ❑  ❑  X ×Y X  LY ×Y X  δ  �  π  � LY  By base change we have  tr(∆∗OX) = π∗δ∗∆∗OX ≃ π∗ϕ∗p∗  XOX = (Lf)∗OLX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Decended trace of structure/dualizing sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We continue with the above general setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The structure sheaf OX×Y X is naturally a coalgebra object in IndCoh(X ×Y X) under convolution: its  comultiplication is given by the unit of adjunction  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  OX×Y X  � m∗m∗OX×Y X ≃ OX×Y X ⋆ OX×Y X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Similarly, the dualizing sheaf ωX×Y X is naturally an algebra object in in IndCoh(X ×Y X)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' under  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='-convolution: its multiplication is given by the counit of adjunction  ωX×Y X ⋆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ωX×Y X ≃ m∗m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='ωX×Y X  � ωX×Y X  20  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  In fact, the coalgebra OX×Y X, viewed as an algebra in IndCoh(X ×Y X)∨, and the algebra ωX×Y X are  identified under the monoidal equivalence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we have the following calculations of the descended traces of OX×Y X and ωX×Y X:  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assume further that X → Y is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Under the equivalence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3(1)  hh(IndCoh(X ×Y X))  ∼  � IndCohΛX/Y (LY )  there is a canonical identification of the descended trace of the coalgebra object OX×Y X with the  structure sheaf  tr(OX×Y X) ≃ OLY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Similarly, under the equivalence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3(2)  hh(IndCoh(X ×Y X)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=')  ∼  � IndCohΛX/Y (LY )  there is a canonical identification of the descended trace of the algebra object ωX×Y X with the  dualizing sheaf  tr(ωX×Y X) ≃ ωLY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We prove (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' then (1) follows from the Serre duality equivalences (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For the natural map π : LY ×Y X → LY , we have the adjunction  π∗ : IndCoh(LY ×Y X) ⇄ IndCoh(LY ) : π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Moreover, π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' is conservative (because π is proper and surjective) and preserves colimits (because π is  quasi-smooth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore by Berr-Beck, the canonical resolution associated to the comonad π∗π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' is an  equivalence:  ωLY  colim∆op[π∗π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='ωLY  ∼  �  π∗π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='π∗π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='ωLY  ��  π∗π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='π∗π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='π∗π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='ωLY · · · ]  ���  By standard identities including base-change, the canonical resolution can be identified with the simplicial  object  tr(ωX×Y X)  tr(ωX×Y X ⋆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ωX×Y X)  ��  tr(ωX×Y X ⋆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ωX×Y X ⋆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ωX×Y X) · · ·  ���  computing the descended trace of ωY ×XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Here is a more direct proof of (2) of the theorem that in fact motivates the definition of  the descended trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Set A = IndCoh(X ×Y X), B = BimodOX×Y X(IndCoh(X ×Y X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='-descent, we have a monoidal  equivalence B ≃ QCoh(Y ) where the monoidal structure on QCoh(Y ) is given by the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Moreover, under this equivalence the regular bimodule OX×Y X ∈ B corresponds to the structure sheaf  OY ∈ QCoh(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Thanks to [BFN10], we have a commutative diagram  QCoh(Y )  tr  �  p∗  �P  P  P  P  P  P  P  P  P  P  P  P  hh(QCoh(Y ))  ∼  � QCoh(LY )  where p : LY → Y is the natural base-point projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Hence we have an equivalence  QCoh(LY ) ≃ hh(QCoh(Y )) ≃ hh(B)  Under this equivalence, the natural map h : hh(B) → hh(A) is identified with the inclusion i : QCoh(LY ) ֒→  IndCohΛX/Y (LY ), where we view QCoh(LY ) ⊂ IndCoh(LY ) as ind-coherent sheaves with singular support  in the zero-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Thus we conclude  tr(OX×Y X) = h(trB(OX×Y X)) ≃ i(tr(OY )) ≃ p∗OY ≃ OLY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  21  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Application to Steinberg stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now we specialize our prior setup to X → Y the induction map  B∨/B∨ → G∨/G∨ so that X ×Y X = StG∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2, we have LY = Z2  G∨ and ΛX/Y = N the  nilpotent codirections in T ∗−1Z2  G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We immediately conclude the following:  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under the equivalence  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  hh(IndCoh(StG∨))  ∼  � IndCohN (Z2  G∨)  there is a canonical identification of the descended trace of the coalgebra object OStG∨ with the structure  sheaf  tr(OStG∨ ) ≃ OZ2  G∨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Automorphic realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this subsection, we give a presentation of the cocenter of the universal  Hecke category HG using “partial cocenters”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The partial cocenters are then interpretated geometrically  using parabolic character sheaves introduced by Lusztig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Character sheaves as cocenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We first review the interpretation of character sheaves on G as the  cocenter of HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Various versions of this statement appear in the literature, see [BNa], [BFO12] and [Lusb]  We will state here a universal monodromic version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let ShN (G/G) be the full subcategory of G-equivariant sheaves on G with nilpotent singular support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  This is a universally monodromic version of character sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the horocycle correspondence  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  U\\G/U  G  U  δG  �  πG � G  G  Then we have the horocycle functor  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  γ := πG!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗  G : HG = Shbimon(U\\G/U)  � ShN (G/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The following result is a special case of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 which we will prove in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is a canonical equivalence  hh(HG)  ∼  � ShN (G/G)  such that the composition HG  trG  −−→ hh(HG) ≃ ShN (G/G) is identified with γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Hochschild homology under HG◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall G◦ is the neutral component of the loop group G and  HG◦ ⊂ HG is the full subcategory of objects supported on G◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For any finite type J ⊂ft Ia, we have the (universal) finite Hecke category HLJ of the Levihoric  LJ, which lies inside HG◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Any HG◦-bimodule can be viewed as a HLJ-bimodule and we can form  the Hochschild homology hh(HLJ, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For J ⊂ J′ both of finite type, we have a natural functor  hh(HLJ, M) → hh(HLJ′, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary (of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any HG◦-bimodule M, the natural maps induce an equivalence  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  colimJ⊂ftIa hh(HLJ, M)  ∼  � hh(HG◦, M)  Moreover, for each J ⊂ft Ia, the equivalence naturally extends to a commutative diagram  M  trJ  �❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥  �  tr  �❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  hh(HLJ, M)  � colimJ′⊂ftIa hh(HLJ′ , M)  ∼  � hh(HG◦, M)  where the diagonal arrows are traces, and the left horizontal arrow is the natural map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  22  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, we have HG◦ ≃ colimJ⊂ftIa HLJ where the colimit is taken within monoidal  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a right HG◦-module M1 and a left HG◦-module M2, we then have  M1 ⊗HG◦ M2 ≃ colimJ⊂ftIa M1 ⊗HLJ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Applying this to the right HG◦-module HG◦ and the left HG◦-module M, we get an equivalence of HG◦-  bimodules  M ≃ HG◦ ⊗HG◦ M ≃ colimJ⊂ftIa HG◦ ⊗HLJ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Taking hh(HG◦, −) on both sides, we get  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  hh(HG◦, M) ≃ colimJ⊂ftIa hh(HG◦, HG◦ ⊗HLJ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Finally, observe that hh(HG◦, HG◦ ⊗HLJ M) ≃ HG◦ ⊗HLJ ⊗Hop  G◦ M ≃ hh(HLJ, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The rest of the corollary is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Hochschild homology under HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G be a connected reductive group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we would like to  calculate hh(HG, M) for a HG-bimodule M in terms of Hochschild homology under various finite Hecke  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  First we recall some constructions of Varshavsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let D◦ be the poset of finite type subsets J ⊂ft Ia  under inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the abelian group Ω = NG(I)/I with its action on Ia and induced action on  D◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let D be the category with objects J ⊂ft Ia, morphisms J → J′ given by ω ∈ Ω with ω(J) ⊂ J′, and  compositions induced by multiplication in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In other words, D is the groupoid D◦/Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note the natural  (faithful but not full) functor i : D◦ → D which is an equivalence if and only if G is simply-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For each ω ∈ Ω we define a monoidal auto-equivalence of HG as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Choose a lifting ˙ω of ω in  NG(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Using ˙ω as the base point, we identify I ˙ωI/Iu with H (via ˙ωh ↔ h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let C ˙ω ∈ HG be the  extension by zero of Luniv supported on I ˙ωI/Iu ≃ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that C ˙ω−1 is the inverse of C ˙ω under the  monoidal structure of HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We get a monoidal auto-equivalence  c ˙ω : HG  C ˙ω⋆(−)⋆C ˙ω−1 � HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We claim that c ˙ω is canonically independent of the choice of the lifting ˙ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, for a different lifting  ¨ω = ˙ωh for some h ∈ H, we have a canonical isomorphism  C ˙ω ≃ C¨ω ⊗R Luniv,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here R = k[X∗(H)] and Luniv,h is the stalk of Luniv at h ∈ H, an invertible R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' On the other  hand,  C ˙ω−1 ≃ C¨ω−1 ⊗R Luniv,h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Since Luniv,h and Luniv,h−1 are inverse to each other as invertible R-modules, the operations c ˙ω = C ˙ω ⋆  (−) ⋆ C ˙ω−1 and c¨ω = C¨ω ⋆ (−) ⋆ C¨ω−1 are canonically identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore we get a canonical monoidal  auto-equivalence  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  cω : HG → HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The canonicity of cω implies that they together give an action of Ω on HG as a monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The same construction shows that: for any HG-bimodule M, there is a canonical action of Ω on M  such that ω ∈ Ω acts by C ˙ω ⋆ (−) ⋆ C ˙ω−1, for any lifting ˙ω of ω in NG(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  If ω ∈ HomD(J, J′), cω sends HLJ to HLJ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore, the diagram of Hecke categories J �→ HLJ, for  J ⊂ft Ia, naturally extends along i to a functor from D to monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Using these functors, for  any HG-bimodule M, restriction to HLJ for J ⊂ft Ia, naturally extends to a functor from D to bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G be a reductive group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any HG-bimodule M, the natural maps induce an  equivalence  colimD hh(HLJ, M)  ∼  � hh(HG, M)  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  23  Moreover, for each J ⊂ft Ia, the equivalence naturally extends to a commutative diagram  M  tr  �❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦  trJ  �  tr  �❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  ❙  hh(HLJ, M)  � colimD hh(HLJ, M)  ∼  � hh(HG, M)  where the diagonal arrows are traces, and the left horizontal arrow is the natural map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any HG-bimodule M, there is a canonical map  cM : colimD hh(HLJ, M)  � hh(HG, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We need to show that this is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It suffices to set M = HG ⊗ HG (where HG acts on the first  factor of HG ⊗ HG on the right and the second on the left) and prove the natural map  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)  c : colimD hh(HLJ, HG ⊗ HG) ≃ colimD HG ⊗HLJ HG  � HG ≃ hh(HG, HG ⊗ HG)  is an equivalence of HG-bimodules (where the HG-action on both sides is induced by its action on the left  of the first factor of HG ⊗ HG and on the right of the second).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note c is induced by HG-bimodule maps,  so is naturally an HG-bimodule map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Thus it suffices to check c is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Consider the following commutative diagram  colimD◦ HG◦ ⊗HLJ HG  i  �  ∼ � colimD◦ hh(HLJ, HG◦ ⊗ HG)  ∼  � hh(HG◦, HG◦ ⊗ HG)  ∼  �  colimD HG ⊗HLJ HG  ∼ � colimD hh(HLJ, HG ⊗ HG)  c  � hh(HG, HG ⊗ HG)  Here the top middle arrow is an equivalence by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4, the right vertical map is the evident  induction equivalence (both are equivalent to HG), and i is the natural map induced by i : D◦ → D and  the inclusion HG◦ ֒→ HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Thus to show c is an equivalence, it suffices to show i is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Φ : D◦ → Cat∞ be the functor given by J �→ HG◦ ⊗HLJ HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The forgetful functor i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' :  Fun(D, Cat∞) → Fun(D◦, Cat∞) admits a left adjoint (left Kan extension along i) i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : Fun(D◦, Cat∞) →  Fun(D, Cat∞), so that colimD◦ Φ = colimD i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Using this we can rewrite i as  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7)  colimD(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='Φ)(J) → colimD HG ⊗HLJ HG  induced by the termwise functor φJ : (i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='Φ)(J) → HG ⊗HLJ HG for J ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We show that φJ is an  equivalence for each J ⊂ft Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed,  (i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='Φ)(J) =  �  ω∈Ω  Φ(ω(J)) =  �  ω∈Ω  HG◦ ⊗HLω(J) HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We have embeddings  iω : HG◦ ⊗HLω(J) HG → HG ⊗HLJ HG  given by x ⊗ y �→ (x ⋆ C ˙ω) ⊗ (C ˙ω−1 ⋆ y) (which is again canonically independent of the lifting ˙ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The  functor φJ is the direct sum of iω  ⊕iω :  �  ω∈Ω  HG◦ ⊗HLω(J) HG → HG ⊗HLJ HG,  which is easily seen to be an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since φJ is an equivalence for all J ∈ D, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Recall that Ω acts on any HG-bimodule M by conjugation, and it acts on HG◦ by monoidal auto-  equivalences, compatible with the bimodule structure on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' These actions induce an action of Ω on the  Hochshild homology hh(HG◦, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  24  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any G-bimodule M, there is a canonical equivalence between the Ω-coinvariants on  hh(HG◦, M) and hh(HG, M):  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8)  hh(HG◦, M)Ω  ∼  → hh(HG, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In particular,  hh(HG◦, HG)Ω ≃ hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let C : D◦ → Cat∞ be the functor given by CJ = hh(HLJ, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then this functor has a canonical  Ω-equivariant structure, hence colimJ∈D◦ CJ carries an action of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is a natural map  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9)  (colimJ∈D◦ CJ)Ω → colimJ∈D CJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6, the two sides above are equivalent to the two sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Thus it suffices to show that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that D = D◦/Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the projection  π : D → BΩ, which is a coCartesian fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The left Kan extension π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='C is colimJ∈D◦ CJ as a category  with Ω-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By [Lur09, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10], we have  colimD C ≃ colimBΩ π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='C ≃ (colimD◦ C)Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Geometry of trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For J ⊂ft Ia, define the ind-stack  YJ := Pu  J \\G/Pu  J  LJ  ,  where LJ denotes the quotient by the conjugation action of LJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the horocycle correspondence  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10)  Iu\\G/Iu  Pu  J \\G/Pu  J  UJ⊂ft  δJ  �  πJ  � Pu  J \\G/Pu  J  LJ  = YJ  To simplify the notation, we will set  HG,J = ShN (YJ) := colimw∈{WJ\\�  W/WJ } ShN  �Pu  J \\G≤w/Pu  J  LJ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here the colimit is taken over longest representatives in the WJ-double cosets of �  W (so that G≤w is a  union of PJ-double cosets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The notation ShN (−) means, viewed as sheaves on Pu  J \\G/Pu  J , the singular  support has nilpotent image under the moment map for the LJ × LJ-action by left and right translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Note that when J = ∅, HG,∅ imposes an Ad(H)-equivariance structure on sheaves on Iu\\G/Iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will see in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 that HG,J is closely related to the notion of parabolic character  sheaves for the loop group G defined by Lusztig in [Lusa].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Consider the functor  πJ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗  J : HG = Shbimon(Iu\\G/Iu)  � Sh(YJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  It is easy to check that the image of πJ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗  J lands in the full subcategory ShN (YJ) = HG,J (it suffices to  check on each PJ-double coset of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Hence we get a functor  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11)  πJ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗  J : HG  � HG,J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The following result gives a geometric interpretation of the partial cocenters hh(HLJ, HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is a special  case of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2, which we will state and prove in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For J ⊂ft Ia, the functor πJ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗  J fits into a commutative diagram with the trace map tr  inducing a horizontal equivalence  HG  tr  �  πJ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗  J  �▼  ▼  ▼  ▼  ▼  ▼  ▼  ▼  ▼  ▼  ▼  hh(HLJ, HG)  ∼  � HG,J  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  25  Substituting Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10 into Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6, we immediately obtain:  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have equivalences  colimJ∈D◦ HG,J ≃  hh(HG◦, HG),  colimJ∈D HG,J ≃  hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Moreover, for each J ⊂ft Ia, the equivalence above naturally extends to a commutative diagram  HG  πJ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗  J  �qqqqqqqqqqq  �  tr  �  HG,J  � colimD HG,J′  ∼  � hh(HG)  where the left horizontal arrow is the natural map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For ω ∈ Ω, let Hω  G be the full subcategory of HG consisting of sheaves  supported on the ω-component of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly, define Hω  G,J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note the action of Ω on HG preserves each  Hω  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore we have a decomposition by support  hh(HG) =  �  ω∈Ω  hh(HG)ω  where  hh(HG)ω ≃ colimJ∈D Hω  G,J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Descended trace of Whittaker object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall that WhG is naturally a coalgebra in HG, therefore  it makes sense to take its descended trace tr(WhG) ∈ hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let  WhG/G := tr(WhG) ∈ hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The goal of this subsection is to calculate the WhG/G in terms of character sheaves on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Reduction from G to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall the monoidal functor i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : HG → HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It induces a functor by passing  to cocenters  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  a : ShN (G/G)  ∼  � hh(HG)  hh(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' )� hh(HG)  where the first equivalence is given by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall I ⊂ Ia are the simple roots of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The corresponding maximal parahoric PI ⊂ G is the arc group  G0 = G[[t]] with pro-unipotent radical Pu  I ⊂ PI the arcs based at the identity Gu  0 = ker(G[[t]] → G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By  writing hh(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=') as the composition  hh(HG) → hh(HG, HG) → hh(HG),  and using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11 (applied to J = I), the functor a factors as the composition  a : ShN (G/G)� � iG/G � HG,I  � hh(HG)  where iG/G is the full embedding given by the direct image along the natural map  G  G → Gu  0 \\G/Gu  0  G  = YI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have the following relation between the descended traces of WhG and WhG:  tr(WhG) ≃ a(tr(WhG)) ∈ hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9, the universal affine Whittaker sheaf is the extension by zero of the finite Whittaker  finite sheaf WG ≃ i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='WhG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The statement then follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  26  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Whittaker character sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the diagram  A1  U −/U −  χ  �  r− � G/G  where r− is induced by the inclusion U − ⊂ G, and χ is induced by the same-named non-degenerate  character χ : U → U/[U, U] → Ga = A1 used in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let ϕχ,1 : Sh(U −/U −) → k-mod denote the vanishing cycles at the identity 1 ∈ U − with respect to the  function χ : U −/U − → A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) The Whittaker functor on character sheaves is the composition  WG/G : ShN (G/G)  � k-mod  WG/G(F) = ϕχ,1r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  −F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The Whittaker character sheaf WhG/G ∈ ShN (G/G) is the object corepresenting the Whittaker  functor  WG/G(F) ≃ HomShN (G/G)(WhG/G, F),  for all F ∈ ShN (G/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we arrive at the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A generalization in the context of nodal degenerations  of curves will appear in [NYb].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) For the trace map  trG = γ : HG = ShN (U\\G/U)  � ShN (G/G) ≃ hh(HG)  there is a canonical identification of the descended trace of the universal finite Whittaker sheaf  tr(WhG) ∈ hh(HG) with the Whittaker character sheaf WhG/G ∈ ShN (G/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) We have a canonical identification of the descended trace of WhG:  WhG/G = tr(WhG) ≃ a(WhG/G) ∈ hh(HG),  where a is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (2) follows immediately from (1) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To prove (1), it will be convenient to view HG as an algebra in HH = Sh0(H)-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Here Sh0  means sheaves with singular support within the zero-section, or equivalently local systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note HH ⊂ HG  is the full monoidal subcategory generated by the monoidal unit HH = ⟨1HG⟩ ⊂ HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall hh(HG) is canonically independent of whether we work absolutely or in HH-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To be  more precise, set H(n)  G  = HG ⊗ · · · ⊗ HG (n copies of HG) and H(n)H  G  = HG ⊗HH · · · ⊗HH HG (n copies of  HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' So in particular HG = H(1)  G = H(1)H  G  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Set also H(n)  G,H = hh(HH, H(n)H  G  ), so in particular  HG,H = H(1)  G,H = ShN  �U\\G/U  H  �  Here as usual ShN means sheaves with nilpotent singular support, or equivalently monodromic sheaves  with respect to the remaining H-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then the natural map from the absolute to HH-relative Hochschild complexes induces an equivalence  on colimits  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  hh(HG)  ≃  �  [HG  γ  �  q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='=q(1)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  H(2)  G  q(2)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  ��  H(3)  G · · · ]  q(3)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  ���  hh(HG)  [HG,H  γH  �  H(2)  G,H  ��  H(3)  G,H · · · ]  ���  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  27  Here the maps q(n)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  are the !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='-pushforwards along the natural quotient maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Moreover, the augmentations  are given by γ = πG!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗  G and γH = πG,H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗  G,H as appear in the diagram  G  G  G  U  �  πG  �  δG � U\\G/U  q  �  G  B  πG,H  �❁❁❁❁❁❁❁❁  δG,H � U\\G/U  H  with Cartesian square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note by base-change, γ = πG!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗  G indeed admits the natural factorization  γ : HG  q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  � HG,H  γH=πG,H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗  G,H� ShN ( G  G)  For 1 ≤ i ≤ n, let mn,i : H(n)  G,H → H(n−1)  G,H  denote the face map of the lower simplicial diagram (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  given by convolving the cyclically-ordered ith and (i + 1)st terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (By convention, we have m1,1 = γH,  and H(0)  G,H = ShN(G/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=')  Note by standard base-change identities, the natural base-change map is an isomorphism mℓ  n,imn,i ≃  mn+1,i+1mℓ  n+1,i, for all 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let γ(n)  H  : H(n)  G,H → hh(HG) denote the canonical map given by γH  applied to any cyclically-ordered total convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Set Wh(n)  G  = WhG ⊗ · · · ⊗ WhG (n copies of WhG) and Wh(n)  G,H = q(n)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Wh(n)  G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, we have  WhG = Wh(1)  G and write WhG,H = Wh(1)  G,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By convention, set Wh(0)  G,H = WhG/G ∈ H(0)  G,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The map α of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) gives  αH : mℓWhG,H → Wh(2)  G,H,  which yields for any n ≥ 1, and 1 ≤ i ≤ n a map:  αH  n,i : mℓ  n+1,iWh(n)  G,H → Wh(n+1)  G,H  Similarly, we have αH  0 : γℓ  HWhG/G → WhG,H (in fact, it is constructed in the next lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By adjunction, we obtain  βH  n,i : Wh(n)  G,H → mn+1,iWh(n+1)  G,H  Note that, by construction, this is exactly the natural map induced by the coalgebra structure of WhG  (after applying q(n)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly, by adjunction, we have βH  0 : WhG/G → γHWhG,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6 below, αH  0 and all αH  n,i are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Thus unwinding the identities, we  conclude the canonical resolution of WhG/G given by the monad T = γHγℓ  H is precisely the resolution  calculating the descended trace of WhG ∈ HG regarded as a coalgebra:  WhG/G  ∼  → [γHγℓ  HWhG/G  �� (γHγℓ  H)2WhG/G  ��� (γHγℓ  H)3WhG/G · · · ]  ∼  → [γHγℓ  HWhG/G  �� γ(2)  H γℓ  H  (2)WhG/G  ��� γ(3)  H γℓ  H  (3)WhG/G · · · ]  ≃ [γHWhG,H  �� γ(2)  H Wh(2)  G,H  ��� γ(3)  H Wh(3)  G,H · · · ]  ≃ [γWhG  �� γ(WhG ⋆ WhG)  ��� γ(WhG ⋆ WhG ⋆ WhG) · · · ]  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The maps αH  0 : γℓ  HWhG/G → WhG,H and αH  n,i : mℓ  n+1,iWh(n)  G,H → Wh(n+1)  G,H , for all  n ≥ 1 and 1 ≤ i ≤ n, are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  28  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By standard identities, it suffices to prove this for αH  0 and αH  1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will give an argument for αH  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  a similar but simpler argument works for αH  1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Set WG,H = Hom(WhG,H, −), WG/G = Hom(WhG/G, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will prove there is a canonical isomor-  phism of functors  WG,H(−)[−2ν] ≃ WG/G(γH(−)) : HG,H  � k-mod  where ν = dim N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, we will then obtain γℓ  H(WhG/G) ≃ WhG,H by adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Consider the commutative diagram whose right hand square is Cartesian  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  U −\\(G/U × G/U)/H  s  �  (U − × B)/U −  �δ  �  r  �  �π  � U −/U −  r  �  G\\(G/U × G/U)/H  (G × B)/G  δ  �  π  � G/G  Here B = G/B is the flag variety of G, and G acts on G×B by the adjoint action on G and the translation  action on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Thus (G × B)/G is isomorphic to the quotient of G by the adjoint action of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly,  U − acts on U − × B by the adjoint action on G and the translation action on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' From this perspective,  π(g, g1) = g, �π(u, g1) = u, and δ(g, g1) = (gg1, g1), �δ(u, g1) = (ug1, g1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Set �f = χ ◦ ˜π : (U − × B)/U − → A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By base change we have  WG/G(τ(F)) ≃ φ0f∗r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='π∗δ∗(F) ≃ φ0 �f∗r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Since δ is smooth of relative dimension ν = dim N, we have δ∗(F) ≃ δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F[−2ν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  WG/G(τ(F)) ≃ φ0 �f∗r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (F)[−2ν] ≃ φ0 �f∗�δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F[−2ν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now consider the leftward map �δ : (U − × B)/U − → U −\\(G/U × G/U)/H at the top of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Stratify B by U −-orbits B = ⊔w∈W Bw where B1 = U −B/B denotes the open U −-orbit, and in general  Bw = U − ˙w ≃ U −/U −  w where U −  w = U − ∩ wB where wB = ˙wB ˙w−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Stratify (U −×B)/U − by the pullback of the U −-orbits on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' So we have the strata (U −×U −/U −  w )/U −,  in particular the open stratum (U − × U −)/U − ≃ U −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Stratify U −\\(G/U × G/U)/H by the pullback of  the U − × U −-orbits on B × B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' So we have the strata U −\\(U −/U −  w × U −/U −  w′)/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The map �δ restricted to the w-stratum takes the form  �δw : (U − × Bw)/U −  � U −\\(Bw ×BH Bw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We claim that for w ̸= 1, for any Fw ∈ Sh(U −\\(Bw ×BH Bw)), we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  φ0 �f∗�δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  wFw ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To prove the claim, note that w ̸= 1 implies U −  w contains U−αi ≃ A1 for some simple root αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We are  studying the correspondence  U −  w \\U −/U −  w  U −/U −  w  �δw  �  �π  � U −/U −  f  � A1  where the second and third quotients are by the adjoint action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let U −  0 /U − ⊂ U −/U − denote the pre-  image of 0 ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the action provides an isomorphism U − ≃ U−αi × U −  0 with �f = f ◦ �π given simply  by the projection to U−αi ≃ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now U−αi ⊂ U −  w implies that �π∗�δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  wFw is constant along U−αi ≃ A1, and  hence φ0 �f∗π1∗�δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  wFw ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By the claim, we have  φ0 �f∗�δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F ≃ φ0 �f∗�δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1F1  where F1 is the restriction of s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F to the open stratum U −\\(B1 ×BH B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Observe that the composition  s ◦ �δ1 is nothing more than the composition  U −  r− � U\\G/U  q  � (U\\G/U)/H  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  29  used to define WG,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We conclude that  φ0 �f∗�δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F ≃ φ0�a∗�δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1F1 ≃ WG,H(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Combined with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) we get a canonical isomorphism  WG,H(F)[−2ν] ≃ WG/G(τ(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Finally, we combine the spectral and automorphic realizations of the descended Whittaker object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Assuming Ansatz (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5), taking cocenters of both sides of Φ we get an equivalence  hh(Φ) : hh(IndCoh(StG∨)) ≃ hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Combined with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5), we get an equivalence  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)  IndCohN (Z2  G∨) ≃ hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Combining Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5, we get:  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assume Ansatz (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Under the equivalence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6), the structure sheaf OZ2  G∨ ∈  IndCohN (Z2  G∨) corresponds to WhG/G ∈ hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Horocycle descent  This section contains a proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10, or more precisely its generalization to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Horocycle diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G be a connected reductive group, B ⊂ G a Borel subgroup, N = [B, B]  its unipotent radical, and H = B/N the universal Cartan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Consider the basic horocycle diagram  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  BG  BB  ǫ  �  δ  � BB ×BH BB  as a diagram over B(G × G) ≃ BG × BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note ǫ is smooth and proper, with fibers isomorphic to G/B,  and δ is smooth, with fibers isomorphic to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Z be an ind-stack with a G × G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We often turn the second G-action as a right action on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For subgroups G1 ⊂ G and G2 ⊂ G, we write G1\\Z/G2 instead of Z/(G1 × G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We can base-change diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) along Z/(G × G) → B(G × G) and get a Z-horocycle diagram  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  Z/∆G  Z/∆B  δ  �  ǫ  �  (N\\Z/N)/∆H  where we write ∆G, ∆B to emphasize the diagonal action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Horocycle adjunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let ν = dim N, and define functors  hc!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' = δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='ǫ∗[−2ν] : Sh(Z/∆G) → Sh((N\\Z/N)/∆H)  ch = ǫ∗δ∗ = ǫ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗ : Sh((N\\Z/N)/∆H) → Sh(Z/∆G)  hc∗ = δ∗ǫ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : Sh(Z/∆G) → Sh((N\\Z/N)/∆H)  Since ǫ is proper and δ is smooth of relative dimension ν, we have adjunctions (hc!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', ch) and (ch, hc∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Assume Z is smooth and let Λ ⊂ T ∗Z be a closed conic G × G-invariant subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any subgroup  G′ ⊂ G × G, consider the full subcategory ShΛ(Z/G′) ⊂ Sh(Z/G′) of G′-equivariant complexes F on Z  with singular support satisfying ss(F) ⊂ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The following statement is a special case of [MV88, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The functors hc!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' and hc∗ send ShΛ(Z/∆G) to ShΛ((N\\Z/N)/∆H), and the functor ch  sends ShΛ((N\\Z/N)/∆H) to ShΛ(Z/∆G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, if we let hcΛ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : ShΛ(Z/∆G) → ShΛ((N\\Z/N)/∆H)  be the restriction of hc!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', and similarly define chΛ and hcΛ,∗, then there are adjunctions (hcΛ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', chΛ) and  (chΛ, hcΛ,∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  30  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Unit diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the basic unit diagram  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  B\\G/B ≃ BB ×BG BB  BB  d  �  p  � BH  as a diagram over B(H × H) ≃ BH × BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Here d is the relative diagonal and p is the natural projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Note d is a closed embedding and p is smooth, with fibers isomorphic to BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Z be an ind-stack with a H × H-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We can base-change diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3) along H\\Z/H →  B(H × H) and get a Z-unit diagram  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  Z′ := Z ×H×H N\\G/N  Z ×H×H N\\B/N ≃ Z/∆B  p  �  d  �  Z/∆H  where we write ∆H, ∆B to emphasize the diagonal action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In forming the middle term Z/∆B, the action  of ∆B factors through ∆H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Unit adjunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note p∗ : Sh(Z/∆H) → Sh(Z/∆B) is an equivalence since the ∆B-action on Z  factors through ∆H and the kernel is the unipotent ∆N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall ν = dim N, so −ν = dim BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Define  functors  uℓ = p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='d∗[2ν] : Sh(Z′) → Sh(Z/∆H)  u = d∗p∗ = d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='p∗ : Sh(Z/∆H) → Sh(Z′)  ur = p∗d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : Sh(Z′) → Sh(Z/∆H)  Since d is proper and p is smooth of relative dimension −ν, we have adjunctions (uℓ, u) and (u, ur).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Assume Z is smooth and let Λ ⊂ T ∗Z be a closed conic H × H-invariant subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the full  subcategory ShΛ(Z/∆H) ⊂ Sh(Z/∆H) of ∆H-equivariant complexes F on Z with singular support  satisfying ss(F) ⊂ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider as well the full subcategory ShΛ(Z′) ⊂ Sh(Z′) of H × H-equivariant  complexes F on Z × N\\G/N with singular support satisfying ss(F) ⊂ Λ × N ′, where N ′ ⊂ T ∗(N\\G/N)  denotes the N × N-reduction of G × N ∗ ⊂ G × g∗ ≃ T ∗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The functors uℓ and ur send ShΛ(Z′) to ShΛ(Z/∆H), and the functor u sends ShΛ(Z/∆H)  to ShΛ(Z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, if we let uΛ,ℓ : ShΛ(Z′) → ShΛ(Z/∆H) be the restriction of uℓ, and similarly  define uΛ and uΛ,r, then there are adjunctions (uΛ,ℓ, uΛ) and (uΛ, uΛ,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' First, we show u respects the singular support conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Given F ∈ ShΛ(Z/∆H), viewed as a  ∆H-equivariant complex on Z, note p∗F ≃ F ⊠ F0 where F0 denotes the constant sheaf on N\\B/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let  d0 : BB → N\\G/B be the closed embedding so that d = id×d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then ss(d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (F⊠F0)) = ss(F)×ss(d0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F0) ⊂  Λ × N ′ since d0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F0 is H × H-bimonodromic hence has singular support in N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Finally, we show uℓ, ur respect the singular support conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Given F ∈ ShΛ(Z′), view F as an  H ×H-equivariant complex on Z ×N\\G/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Using the estimate of singular support for pullbacks d∗F and  d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F as in [KS90, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4, Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8], we see that ss(d∗F) and ss(d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F) are both contained in Λ  when viewed as sheaves on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore the same is true for p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='d∗F and p∗d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F since p is an N-gerbe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Descent for smooth stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Z be a smooth stack with a left G × G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will write the  first G-action as a left action and turn the second G-action as a right action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Λ ⊂ T ∗Z be a closed G × G-invariant subset such that under the moment map µ : T ∗Z → g∗ × g∗,  we have µ(Λ) ⊂ N ∗ × N ∗ where N ∗ ⊂ g∗ denotes the nilcone in the dual to the Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In many situations of interest, one has Z = Z− × Z+, Λ = Λ− × Λ+, where Z± are  smooth stacks with G-action, and Λ± ⊂ T ∗Z± is a closed G-invariant subset such that under the moment  map µ± : T ∗Z± → g∗, we have µ±(Λ±) ⊂ N ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this case, we view the action of G on Z− as a right  action and the one on Z+ as a left action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the Z-horocycle diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2) and Z-unit diagram  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) take the form  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  Z− ×G Z+  Z− ×B Z+  δ  �  ǫ  �  Z−/N ×H N\\Z+  Z′ = Z− ×H N\\G/N ×H Z+  Z− ×H N\\B/N ×H Z+ ≃ Z− ×B Z+  p  �  d  �  Z− ×H Z+  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  31  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A basic example is Z = G, with its natural G × G-action, and Λ = G × N ∗ ⊂ G × g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The category ShΛ(N\\Z/N) as a HG-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The assumptions on Λ imply that any object of  ShΛ(N\\Z/N) is H × H-monodromic, therefore we can also regard ShΛ(N\\Z/N) as a HG-bimodule in  HH = Sh0(H)-bimodules since Sh0(H) = Mod(End(e)) ⊂ HG is the full monoidal subcategory generated  by the monoidal unit e ∈ HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that  hh(HH, ShΛ(N\\Z/N))  ∼  � ShΛ((N\\Z/N)/∆H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here is the main technical theorem of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A generalization to “nilpotent categorical bimodules”  will appear in [NYb];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' in particular the assumption here that Λ ⊂ T ∗Z is Lagrangian is not necessary but  allows the proof to call on the generalities of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Z be a smooth stack with a G × G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Λ ⊂ T ∗Z be a closed G × G-invariant  conic Lagrangian such that under the moment map µ : T ∗Z → g∗ × g∗, we have µ(Λ) ⊂ N ∗ × N ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then there is a canonical equivalence  hh(HG, ShΛ(N\\Z/N))  ∼  � ShΛ(Z/∆G)  such that the functor ch defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2 factors as the composition  ch : ShΛ((N\\Z/N)/∆H) ≃ hh(HH, ShΛ(N\\Z/N))  � hh(HG, ShΛ(N\\Z/N)) ≃ ShΛ(Z/∆G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In the setting of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, the theorem gives an equivalence  ShΛ+(Z−/N) ⊗HG ShΛ−(N\\Z+)  ∼  � ShΛ(Z− ×G Z+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In the setting of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2, the theorem gives the equivalence  hh(HG)  ∼  � ShN (G/G)  that we stated in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In [NYb] we will prove an abstract version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 where ShΛ(Z) is replaced with  a Sh(G)-bimodule category satisfying a certain nilpotence condition reflecting that Λ maps to N ∗ × N ∗  under the moment map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, one can remove the assumption that Λ is a Lagrangian in Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, we will identify each term of the (relative) Hochschild complex computing  hh(HG, ShΛ(N\\Z/N)) as sheaves on a certain space, and augment the Hochschild complex by adding  the term ShΛ(Z/∆G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Finally we will apply Lurie’s criterion [Lur12, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3] to verify that the  augmented Hochschild complex is a colimit diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Nonlinear diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To start, we present some general patterns for constructing the (relative)  Hochschild complex for spaces, following [BNb].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In the subsequent Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9, we specialize to the case  we will use in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let K be an algebraic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let CorrK be the category of stacks with K-action with morphisms given  by K-equivariant correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let CorrK×K be the monoidal category of stacks with K × K-action with morphisms given by K × K-  equivariant correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The monoidal structure on CorrK×K is given by U ⋆ V := U ×K V , where  the quotient of K uses the second action of K on U and the first action of K on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note K with its  regular K × K-action is the monoidal unit in CorrK×K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Note that CorrK is naturally a module category for CorrK×K: for U ∈ CorrK×K and V ∈ CorrK, we  define the action of U on V to be U ⋆ V := U ×K V ∈ CorrK (quotient using the second action of K on  U and the action of K on V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' the first action of K on U induces a K-action on U ⋆ V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  32  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Consider a diagram of stacks with Cartesian square  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  X0  p0  �  b′  � X  f  �  p  �  Y  pt  b  � BK  In other words, we are given a stack X0 with K-action and a K-invariant map X0 → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Given (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2), observe that A = X0 ×Y X0 is naturally an algebra object in CorrK×K with product  given by the correspondence  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  A ⋆ A = (X0 ×Y X0) ×K (X0 ×Y X0)  X0 ×Y X ×Y X0  δ  �  πf  � X0 ×Y X0 = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here δ is induced by the natural map X = X0/K → X0 ×K X0 of the middle factors (which in turn is  induced by the diagonal map X0 → X0 × X0), and πf is induced by the projection of the middle factor  f : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The unit of A is given by the correspondence  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  K = pt ×BK pt  pt ×BK X ×BK pt ≃ X0 ×X X0  ǫp  �  δf  � X0 ×Y X0 = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here ǫp is induced by p : X → BK, and δf is induced by f : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note swapping the factors of A gives  an equivalence with its monoidal opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Given any map of stacks q : W → Y , observe that W0 := X0 ×Y W is naturally an A-module object in  CorrK with action given by the correspondence  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  A ⋆ W0 = (X0 ×Y X0) ×K (X0 ×Y W)  X0 ×Y X ×Y W  δ  �  πf  � X0 ×Y W = W0  where δ and πf are defined in the same as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Similarly, given any map of stacks q1 × q2 : W → Y × Y , W0 := X0 ×Y W ×Y X0 is naturally an  A-bimodule object in CorrK×K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The following is elementary to check;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' we leave further details to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We refer to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5 for  the terminology of relative Hochschild complex, which we borrow here for the monoidal category CorrK×K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Given any map of stacks q1 × q2 : W → Y × Y , we have an augmented simplicial object  B(A, W)• in CorrK×K such that:  (1) The underlying simplicial object of B(A, W)• is the relative Hochschild complex for the A-bimodule  W0 in CorrK×K (relative to the unit object K ∈ CorrK×K which is also an algebra object, and the  unit map K → A given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) is a map of algebras):  · ·  ��� W0 ×K×K A  ��  �� W0/∆K  �  (2) The augmentation map B(A, W)0 → B(A, W)−1 is given by the correspondence  W0/∆K = (X0 ×Y W ×Y X0)/∆K  W ×Y ×Y X  δ  �  πf  � W ×Y ×Y Y  where δ is induced by the natural map X = X0/K → X0 ×K X0, and πf is induced by f : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Our case of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We specialize the preceding constructions when the initial diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  takes the form  BN  p0  �  b′  � BB  f  �  p  �  BG  pt  b  � BH  where the maps are the embeddings U ⊂ B ⊂ G and projection B ։ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' So here K is simply the universal  Cartan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  33  Inside of CorrH×H, we have the algebra A = BN ×BG BN ≃ N\\G/N with the multiplication dia-  gram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3) given by usual convolution  A ⋆ A = (N\\G/N) ×H (N\\G/N)  N\\G ×B G/N  δ  �  πf  � N\\G/N = A  and the unit diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) taking the form  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)  H  N\\B/N  ǫp  �  δf  � N\\G/N = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Given a stack Z with G-action, applying the general discussion about A-modules to the induced map  q : W = G\\Z → BG, then we have W0 = N\\Z ∈ CorrH is naturally an A-module with action (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  given by usual convolution  A ⋆ W0 = (N\\G/N) ×H (N\\Z)  N\\G ×B Z  δ  �  πf  � N\\Z = W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Similarly, given a stack Z with G × G-action, applying the general discussion about A-bimodules to the  map q1 × q2 : W = G\\Z/G → BG × BG, then W0 = N\\Z/N ∈ CorrH×H is naturally an A-bimodule  object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8 takes the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Given a stack Z with G×G-action, we have an augmented simplicial object B(A, Z)• (in  the notation of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8 it should be called B(A, W)•, where W = G\\Z/G) in CorrH×H such that:  (1) The underlying simplicial object of B(A, Z)• is the relative Hochschild complex of the A-bimodule  N\\Z/N (relative to the algebra map H → A in CorrH×H given by the unit diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)):  · ·  ��� (N\\Z/N) ×H×H (N\\G/N)  ��  �� (N\\Z/N)/∆H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  (2) The augmentation map B(A, Z)0 → B(A, Z)−1 is given by the horocycle correspondence  (N\\Z/N)/∆H  Z/∆B  δ  �  πf  � Z/∆G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Augmented Hochschild complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For the next step towards the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, we shall  take the categories of sheaves termwise for the augmented simplicial object B(A, Z)• in CorrH×H provided  by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10, and impose singular support conditions to obtain the augmented Hochschild complex  for the HG-bimodule ShΛ(N\\Z/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Consider the monoidal category HH = Sh0(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any X ∈ CorrH×H, Sh(X) is a bimodule for HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Thanks to [GR19], passing to categories of sheaves gives a monoidal functor  CorrH×H → BimodHH(St)  where the target is the 2-category of stable presentable ∞-categories that are HH-bimodules (and the  monoidal structure is tensor product over the middle copy of HH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a morphism from X to Y in  CorrH×H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', a H × H-equivariant correspondence  X  C  p  �  q  � Y  the functor Sh(X) → Sh(Y ) is given by q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Passing to the categories of all sheaves termwise for B(A, Z)•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We obtain an augmented simplicial object Sh(B(A, Z))• in BimodHH(St).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we impose singular support conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Λ−1 ⊂ T ∗(Z/∆G) be the ∆G-reduction of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For  n ≥ 0, B(A, Z)n can be written as the quotient  B(A, Z)n ≃ (Z × Gn)/(B ×H B)n+1  where for n ≥ 1, the ith factor (0 ≤ i ≤ n) of B ×H B acts on Z × Gn by  (b, b′) · (z, g1, · · · , gn) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  (zb, b′−1g1, · · · , gn)  i = 0,  (z, · · · , gib, b′−1gi+1, · · · , gn)  1 ≤ i ≤ n − 1  (b′−1z, g1, · · · , gn−1, gnb)  i = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  34  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  For n = 0 the action of B ×H B acts on Z is by (b, b′) · z = b′−1zb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For n ≥ 0, set Λn ⊂ T ∗B(A, Z)n ≃ T ∗((Z × Gn)/(B ×H B)n+1) be the reduction of Λ × (G × N ∗)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The augmented simplicial category Sh(B(A, Z))• restricts to an augmented simplicial  object C• in BimodHH(StL  k ) with terms Cn = ShΛn(B(A, Z)n) for n ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Moreover:  (1) Let M = ShΛ(N\\Z/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the underlying simplicial object of of C• is equivalent to the relative  Hochschild complex B(HG, M)•:  · ·  ��� M ⊗HH⊗Hop  H HG  ��  �� M ⊗HH⊗Hop  H HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  (2) The augmentation map C0 → C−1 is given by the transform  ch : M ⊗HH⊗Hop  H HH ≃ ShΛ0((N\\Z/N)/∆H)  � ShΛ−1(Z/∆G)  associated to the horocycle transform construction (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2) applied to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The description of Cn in terms of M and HG follows from the categorical K¨unneth formula for  sheaves with prescribed singular support conditions on twisted products X1 ×H X2, see [NYa, Lemma  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It remains to check that the prescribed singular supports are respected by the given functors in  Sh(B(A, Z))•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For n ≥ 0, and the injection ϕ : [n − 1] → [n] whose image misses i, the corresponding face map of  B(A, Z)• is given by the functor ch resulting from the horocycle transform construction (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2) applied  to Zi  n = (Z × Gn)/Bi  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Here Bi  n ⊂ (B ×H B)n+1 is the subgroup where we replace the ith factor by the  trivial group and keep the other factors unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The G × G-action on Zi  n is the natural action along  where the ith factor of (B ×H B)n+1 originally acted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, the associated horocycle functors,  in particular the face map ch, respect the prescribed singular support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Similarly, for n ≥ 0, and the surjection ϕ : [n + 1] → [n] that identifies i and i + 1, the corresponding  degeneracy map of B(A, Z)• respects the singular support as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Observe the degeneracy map is given  by the unit functor u (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5) resulting from the unit transform construction (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) applied to  Zn,i = (Z × Gn)/Bn,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Here Bn,i ⊂ (B ×H B)n+1 is the subgroup where we replace the ith factor by the  group N × N and keep the other factors unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The H × H-action on Zn,i is the natural action along  where the ith factor of (B ×H B)n+1 originally acted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6, the associated unit functors, in  particular the degeneracy map u, respect the prescribed singular support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Finish of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, it remains to prove that the the  augmented simplicial object C• exhibits C−1 = ShΛ(Z/∆G) as the colimit of the underlying simplicial  object of C•, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', the Hochschild complex B(HG, M)•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Equivalently, letting C• be the augmented cosim-  plicial object obtained from C• by passing to right adjoints (note these are available by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6), it suffices to show that C• exhibits C−1 as the limit of the underlying cosimplicial object {Cn}n≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For this it suffices to check that C• satisfies the following strong form of the criteria of [Lur12, Corollary  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3]:  (1) The augmentation map d0  −1 = hc∗ : C−1 → C0 is: (a) conservative and (b) continuous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  preserves colimits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The following commutative squares are left adjointable for any order-preserving map α : [m] → [n]  (where m, n ≥ −1)  Cm  α  �  d0  m � Cm+1  α′  �  Cn  d0  n  � Cn+1  Here d0  m is the inclusion [m] → [m + 1] that whose image misses 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' d0  n is defined similarly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' and  α′ : [m + 1] → [n + 1] is the map defined by α′(0) = 0 and α′(i + 1) = α(i) + 1 for i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Left adjointability of the above square means the face maps d0  m, d0  n admit respective left adjoints  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  35  (d0  m)ℓ, (d0  n)ℓ (which in our case are already given by the construction of d0  m, d0  n as right adjoints),  and the associated base change map is an equivalence  (d0  n)ℓ ◦ α′  ∼  � α ◦ (d0  m)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1a) First, we check d0  −1 = hc∗ is conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It suffices to show that ch ◦ hc∗ contains the identity  functor as a direct summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This is essentially [MV88, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We use notation from the diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By definition ch ◦ hc∗ = ǫ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗δ∗ǫ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='. The fiber square along δ  can be identified with  Z/∆B  (Z × N)/∆B  p1  �  a1  �  Z/∆B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here the ∆B-action on N is by conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The map a1 is the action map of N on Z via N ×{1} → G×G,  and p1 is the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since δ is smooth of relative dimension ν = dim N, δ∗δ∗ ∼= p1∗a∗  1 ∼= p1∗a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1[−2ν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Hence  ch ◦ hc∗ ∼= ǫ∗p1∗a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1ǫ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' [−2ν] = p′  1∗a′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1[−2ν]  where p′  1 = ǫ ◦ p1, a′  1 = ǫ ◦ a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have a commutative diagram  (Z × N)/∆B  π  �  p′  1  �◆  ◆  ◆  ◆  ◆  ◆  ◆  ◆  ◆  ◆  a′  1  �qqqqqqqqqqq  Z/∆G  (Z × G)/∆G  π1  �  α1  �  Z/∆B  Here the ∆G-action (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ∆B-action) on G (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' N) is by conjugation, π1 is projection, and α1 is the  action map of G on Z via G × {1} → G × G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The map π is the base change of the Springer resolution  N  Ad(B) →  G  Ad(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Hence for F ∈ ShΛ(Z/∆G), we have  ch(hc∗(F)) ∼= p′  1∗a′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1F[−2ν] ∼= π1∗π∗π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1F[−2ν] ∼= π1∗Hom(π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='k, α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1F)[−2ν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The Springer sheaf contains the skyscraper sheaf at 1 ∈ G as a direct summand, hence π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='k contains  i∗k[−2ν] as a direct summand, where i : Z/∆G ֒→ (Z × G)/∆G corresponds to the inclusion of 1 into G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore ch(hc∗(F)) contains as a direct summand  π1∗Hom(i∗k[−2ν], α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F)[−2ν] ∼= π1∗i∗i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F ∼= F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We have shown that ch ◦ hc∗ contains the identity functor as a direct summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1b) follows from Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) We will check the required left adjointability for the augmentation map α = d0  −1 : [−1] → [0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' the  verification for other maps is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Consider the relevant categories and functors  ShΛ(Z/∆G)  hc∗  � ShΛ((N\\Z/N)/∆H)  ch  �  hc∗0 � ShΛ(N\\Z/N) ⊗HH⊗Hop  H HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  ch1  �  Here ch1 is the HG-action on ShΛ((N\\Z/N)/∆H) induced by the right G-action on Z, and hc∗0 is right  adjoint to the HG-action on ShΛ((N\\Z/N)/∆H) induced by the left G-action on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We seek to show the natural adjunction transformation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7)  τ : ch1 ◦ hc∗0  � hc∗ ◦ ch  is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, by proper base change, hc∗ ◦ ch = δ∗ǫ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='ǫ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗ can be identified with the functor  c∗p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' [−2ν] constructed from the diagram  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8)  (N\\Z/N)/∆H  ∆B\\(Z × G/B)  c  �  p  �  (N\\Z/N)/∆H  where in the middle term, b ∈ ∆B acts by b(z, gB) = (bzb−1, bgB), and the maps p and c are defined by  p(z, g) = z and c(z, g) = gzg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  36  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  On the other hand, ch1 ◦ hc∗0 can be identified with the functor m1∗m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  0[−2ν] constructed from the  diagram  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9)  (N\\Z/N)/∆H  Z×BG  ∆B  m1 �  m0  �  (N\\Z/N)/∆H  where in the middle term, b ∈ ∆B acts by b(z, g) = (bz, gb−1), and the maps m0 and m1 are defined by  m0(z, g) = gz and m1(z, g) = zg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We have an isomorphism between the two diagrams (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9) that is the identity on (N\\Z/N)/∆H  and on the middle terms it takes the form (z, gB) �→ (g−1z, g) ∈  Z×BG  ∆B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This isomorphism identifies  c∗p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' [−2ν] with m1∗m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  0[−2ν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' One checks that τ is the composition  ch1 ◦ hc∗0 ≃ m1∗m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  0[−2ν] ≃ c∗p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' [−2ν] ≃ hc∗ ◦ ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore τ is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This concludes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Singular and ind-version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will generalize Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 to the situation of stratified ind-stacks  (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Z be an ind-stack equipped with a partition into smooth locally closed substacks Z =  ⊔α∈P Z◦  α indexed by a poset P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For α ∈ P, assume {β ∈ P|β < α} is finite and Zα := ∪β≤αZ◦  β is closed  in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let i◦  α : Z◦  α → Z be the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Assume Z has a G × G-action preserving each Zα ⊂ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Λα ⊂ T ∗Z◦  α be a closed G × G-invariant  conic Lagrangian such that under the moment map µ : T ∗Z◦  α → g∗ × g∗, we have µ(Λα) ⊂ N ∗ × N ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Assume for β < α ∈ P, the composition (i◦  β)∗(i◦  α)∗ : Sh(Z◦  α) → Sh(Z◦  β) takes ShΛα(Z◦  α) to ShΛβ(Z◦  β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Define ShΛ(Z) to be the full subcategory of objects F such that (i◦  α)∗F ∈ ShΛα(Z◦  α), for all α ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For any Q ⊂ P that is locally down-closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', Q = Q1 \\ Q2 where Q1 and Q2 are down-closed,  let ZQ = ∪α∈QZ◦  α be the corresponding locally closed substack of Z, we can define the full subcategory  ShΛ(ZQ) ⊂ Sh(ZQ) in the same way by requiring objects to have image under (i◦  α)∗ lying in ShΛα(Z◦  α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now for a locally down-closed Q and a downclosed subset Q′ ⊂ Q with complement Q′′ = Q \\ Q′, the  assumptions guarantee that the usual functors in the recollement diagram of Sh(ZQ), Sh(ZQ′) and Sh(ZQ′′)  restrict to a recollement diagram  ShΛ(ZQ′′)  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  j∗  �  ShΛ(ZQ)  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='=j∗  �  i∗  �  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  ShΛ(ZQ′)  i∗=i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  where i : ZQ′ ֒→ ZQ and j : ZQ′′ ֒→ ZQ are the closed and open inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' From this we see that ShΛ(Z)  admits a stratification indexed by P in the sense of Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2, with strata categories ShΛα(Z◦  α) for  α ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For any subgroup G′ ⊂ G×G, let ShΛ(Z/G′) ⊂ Sh(Z/G′) denote the full subcategory of G′-equivariant  complexes F on Z whose underlying object lies in ShΛ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The assumptions on Λ imply that any object  of ShΛ(N\\Z/N) is H ×H-monodromic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The G×G actions on Z equip ShΛ(N\\Z/N) with a HG-bimodule  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' With the above setup, there is a canonical equivalence of stable ∞-categories  hh(HG, ShΛ(N\\Z/N))  ∼  � ShΛ(Z/∆G)  such that the functor ch defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2 factors as the composition  ch : ShΛ((N\\Z/N)/∆H) ≃ hh(HH, ShΛ(N\\Z/N))  � hh(HG, ShΛ(N\\Z/N)) ≃ ShΛ(Z/∆G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We explain that the steps in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 can be made to work for the ind-stack  Z with slight modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The augmented simplicial diagram in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10 makes sense for the  ind-stack Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now for each term B(A, Z)n = (Z × Gn)/(B ×H B)n+1 (where n ≥ 0), we assign the full  subcategory category Cn ⊂ Sh(B(A, Z)n) consisting of objects F whose pullback to Z◦  α × Gn has singular  support contained in Λα × (G × N ∗)n, for all i ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let C−1 = ShΛ(Z/∆G) as is already defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  37  With this definition of C• ∈ BimodHH(StL  k ), we claim that the statement of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We  need to check the maps in the augmented simplicial object Sh(B(A, Z)•) preserve the subcategories C•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For α ∈ P, applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12 to Z◦  α together with the Lagrangian Λα, we get an augmented  simplicial object Cα,• in BimodHH(StL  k ) whose terms are ShΛα,n(B(A, Z◦  α)n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let i◦  α,n : B(A, Z◦  α)n ֒→  B(A, Z)n be the locally closed embedding induced by i◦  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We claim that the functors i◦  α,n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : Cα,n →  Sh(B(A, Z)n) induce a functor of augmented simplicial objects  i◦  α,•!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : Cα,• → Sh(B(A, Z)•).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Indeed, for an injection ϕ : [n − 1] → [n], the corresponding face maps chα,ϕ : Cα,n → Cα,n−1 and  chϕ : Sh(B(A, Z)n) → Sh(B(A, Z)n−1) are given by special cases of the ch functor defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The isomorphism  chϕ ◦ i◦  α,n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ≃ i◦  α,n−1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ◦ chα,ϕ  follows from proper base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a surjection ϕ : [n + 1] → [n], the corresponding degeneracy maps  uα,ϕ : Cα,n → Cα,n+1 and uϕ : Sh(B(A, Z)n) → Sh(B(A, Z)n+1) are given by special cases of the unit  functor u defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The isomorphism  uϕ ◦ i◦  α,n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ≃ i◦  α,n+1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ◦ uα,ϕ  again follows from proper base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now observe that for n ≥ −1, Cn ⊂ Sh(B(A, Z)n) is generated by the images of i◦  α,n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (for α ∈ P) under  colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have checked above that the face and degeneracy maps for Sh(B(A, Z)•) preserve the images  of i◦  α,•!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' for fixed α ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since the face and degeneracy maps are continuous (they are left adjoints), they  preserve C•, therefore we get an augmented simplicial object C•, and the analog of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The last step is to check that the augmented cosimplicial object C•, obtained from C• by passing to  right adjoints, satisfies Lurie’s criterion [Lur12, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3] for a limit diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The same argument  as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='13 works for the current situation without change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Harder-Narasimhan subcategories  This main result of this section is Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7 giving a recollement structure on the cocenter hh(HG)  of the universal affine Hecke category HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The properties of this recollement mirror those of the recollement  structure on the Betti Langlands automorphic category ShN (BunG(E)) for a genus one curve E induced  by the Harder-Narasimhan stratification of BunG(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Most basically, the filtrations of both recollements  are indexed by Newton points NP ⊂ X∗(T )+  Q (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In a sequel, we will prove that hh(HG)  is equivalent to ShN (BunG(E)) so that the recollements match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this paper, we will focus on the specific  consequence stated in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2 that the associated graded for the minimum index 0 ∈ NP embeds  fully faithfully in hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Combinatorial pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Our starting point for the analysis of hh(HG) is the colimit description of  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11:  colimD HG,J  ∼  � hh(HG)  We will begin with the group theory of B´edard and Lusztig [Lusa] underlying the natural decomposition  of each HG,J into pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Combinatorial pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let W a be the affine Weyl group of G and �  W = X∗(T ) ⋊ W be the extended  affine Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For J ⊂ft Ia, let WJ ⊂ W a be the subgroup generated by J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' it is the Weyl group of  the Levi LJ of the parahoric subgroup PJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J�  W (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' �  W J) be the set of minimal length elements in  the cosets WJ\\�  W (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' �  W/WJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J �  W J′ be the set of minimal length elements in the double cosets  WJ\\�  W/WJ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For a group Γ and a subgroup Γ′ ⊂ Γ, let Γ  Γ′ denote the set of Γ′-conjugacy classes in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' More generally,  if δ is an automorphism of Γ, let  Γ  Adδ(Γ′) denote the set of orbits of Γ′ acting on Γ by twisted conjugation  γ′ · γ = γ′γδ(γ′−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' If γ ∈ Γ, we use  Γ  Adγ(Γ′) to mean the  Γ  AdAd(γ)(Γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  38  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Let us first review a combinatorial procedure of B´edard, as found in [Lusa].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J ⊂ft Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let SJ be  the set of sequences (Jn, J′  n, un)n≥0 such that  (1) J0 = J and Jn = Jn−1 ∩ Ad(u0 · · · un−1)Jn−1 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) J′  0 = J and J′  n = Jn−1 ∩ Ad(u0 · · · un−1)−1Jn−1 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (3) un ∈ J′  n(WJn−1)Jn for n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We call elements in SJ combinatorial J-pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that Jn and J′  n stabilize for n large hence un = 1 for  n large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' As explained in [Lusa, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5], the map (Jn, J′  n, un)n≥0 �→ u0u1 · · · um (for m ≫ 0) defines a  bijection  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  SJ  ∼  → J �  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For u ∈ J�  W, we denote the corresponding element in SJ by  u  J ∈ SJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The map σJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any J ⊂ft Ia, we define a map  σJ :  �  W  WJ  → SJ  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For c ∈  �  W  WJ , we set c0 = c, J0 = J′  0 = J and u0 ∈ J �  W J to be the WJ-double coset containing the  WJ-conjugacy class c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will inductively construct a sequence (cn, Jn, J′  n, un)n≥0 satisfying the following  conditions:  (1) (Jn, J′  n, un)n≥0 ∈ SJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) For n ≥ 1, cn ∈  WJn−1  Adu0···un−1 (WJ′n) is characterized by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  cn = {x ∈ WJn−1|u0 · · · un−1x ∈ c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (3) For n ≥ 1, un ∈ J′  n(WJn−1)Jn is the minimal length element in the (WJ′n, WJn)-double coset of cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  These conditions clearly characterize (cn, Jn, J′  n, un)n≥0 uniquely, given the initial terms (c0 = 0, J0 =  J, J′  0 = J, u0) as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To confirm this inductive procedure is well-defined, we need to show that, given  (ci, Ji, J′  i, ui)0≤i≤n−1 satisfying the above conditions (so that Jn and J′  n can already be defined using Jn−1  and u0 · · · un−1 as dictated by the requirement that (Jn, J′  n, un)n≥0 ∈ SJ), the set cn defined using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  is non-empty and consists of a single (u0 · · · un−1)-twisted conjugacy class under WJ′n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  First, to see cn is non-empty: from the inductive hypothesis, we know that for x ∈ WJn−2 (we understand  WJ−1 to be �  W), u0 · · · un−2x ∈ c if and only if x ∈ cn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' From the construction of un−1, we see any x ∈ cn−1  can be written as x = aun−1b for some a ∈ WJ′  n−1 and b ∈ WJn−1 = Ad(u0 · · · un−2)WJ′  n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Take any such  x = aun−1b and  u0 · · · un−2x = u0 · · · un−2aun−1b = a′u0 · · · un−1b  where a′ = Ad(u0 · · · un−2)a ∈ WJn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The above element then lies in the same WJn−1-conjugacy class as  u0 · · · un−1ba′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This implies u0 · · · un−1WJn−1 ∩ c ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Next, to see cn is a single (u0 · · · un−1)-twisted conjugacy class under WJ′  n: suppose y, y′ ∈ WJn−1 are  such that u0 · · · un−1y, u0 · · · un−1y′ ∈ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By the inductive hypothesis, un−1y, un−1y′ ∈ cn−1, hence there  exists z ∈ WJ′  n−1 such that un−1y = zun−1y′Ad(u0 · · · un−2)z−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let z′ = u−1  n−1zun−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  y = u−1  n−1zun−1y′Ad(u0 · · · un−2)z−1 = z′y′Ad(u0 · · · un−1)z′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now z′ = yAd(u0 · · · un−2)zy′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note z′ ∈ Ad(u−1  n−1)WJ′  n−1 and yAd(u0 · · · un−2)zy′−1 ∈ WJn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There-  fore z′ ∈ Ad(u−1  n−1)WJ′  n−1 ∩ WJn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since u−1  n−1 ∈ Jn−1(WJn−2)J′  n−1, we have Ad(u−1  n−1)WJ′  n−1 ∩ WJn−1 =  WAd(un−1)−1J′  n−1∩Jn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Note that Ad(un−1)−1J′  n−1 = Ad(u0 · · · un−1)−1Jn−1 hence Ad(u−1  n−1)J′  n−1 ∩  Jn−1 = Ad(u0 · · · un−1)−1Jn−1 ∩ Jn−1 = J′  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We conclude that z′ ∈ WJ′  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3) im-  plies y and y′ lie in the same (u0 · · · un−1)-twisted conjugacy class of WJ′n in WJn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  This completes the definition of the map σJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  39  Composing σJ with the bijection (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) we get a map  pJ :  �  W  WJ  σJ  −−→ SJ  ∼  → J�  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let (Jn, J′  n, un)n≥0 ∈ SJ and u = u0u1 · · · un ∈ J �  W (for large n) be its image under the  bijection (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let K be the stable value of Jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) (Compare [He07, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4]) K is the largest subset of J that is stable under Ad(u), and K is  also the stable value of J′  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We denote K by I(J, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The preimage of u  J under σJ consists of WJ-conjugacy classes of uy where y ∈ WK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Two elements  uy and uy′ (y, y′ ∈ WK) are in the same WJ-conjugacy class if and only if y, y′ lie in the same  u-twisted conjugacy class of WK, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', y �→ uy gives a canonical bijection  uWK  WK  ∼=  WK  Adu(WK)  ∼  → σ−1  J ( u  J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (1) For n large so that Jn−1 = K, we have Jn = Jn−1 ∩ Ad(u)Jn−1 and Jn−1 = Jn, hence Jn is  stable under Ad(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Also J′  n = Jn−1 ∩ Ad(u)−1Jn−1 = Jn−1 = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Conversely, if K1 ⊂ J is stable under Ad(u), we show inductively that K1 ⊂ Jn for all n ≥ 0, hence  K1 ⊂ K = I(J, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For n = 0, this is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assume K1 ⊂ Jn−1 for some n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let s ∈ K and let αs  be the corresponding simple root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For K′ ⊂ft Ia, let ΦK′ be the sub root system of the affine roots of  G spanned by K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since K1 is stable under Ad(u), αi = uαi′ for some αi ∈ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Write u = u0 · · · un−1x  where x ∈ WJn−1, then αi = u0 · · · un−1β where β = xαi′ ∈ ΦJn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since u0 · · · un−1 ∈ �  W Jn−1, u0 · · · un−1  sends Φ+  Jn−1 to positive roots, therefore β is also a simple root in ΦJn−1 for otherwise u0 · · · un−1β cannot  be a simple root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The equality αi = u0 · · · un−1β then implies αi ∈ Ad(u0 · · · un−1)Jn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore  αi ∈ Jn−1 ∩ Ad(u0 · · · un−1)Jn−1 = Jn for any αi ∈ K1, hence K1 ⊂ Jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) is immediate from the construction of σJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Relevant affine subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let A be the standard apartment with the action of �  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By a relevant  affine subspace of A, we will mean the intersection of a set of affine root hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let E be the set  of relevant affine subspaces of A, and let E be the set of W a-orbits on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Each subset K ⊂ft Ia gives  a relevant affine subspace A(K) = {x ∈ A|α(x) = 0, ∀α ∈ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Every relevant affine subspace is W a-  conjugate to one of the form A(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This induces a surjection {K ⊂ft Ia} ։ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This map may not be a  bijection in general: for K, K′ ⊂ft Ia, A(K) is in the W a-orbit of A(K′) if and only if there exists w ∈ W a  such that wKw−1 = K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For E ∈ E, we denote its image in E by [E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The set E is partially ordered such that [E] ≥ [E′] if and  only if E ⊃ wE′ for some w ∈ W a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For J ⊂ft Ia, let EJ be the subset of E consisting of relevant affine subspaces that contain A(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Clearly  WJ acts on EJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For K ⊂ J, we have A(K) ∈ EJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J ⊂ft Ia and u  J ∈ SJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let K = I(J, u) ⊂ J as specified in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) We define the J-type of u  J to be the element τJ( u  J ) ∈ WJ\\EJ given by the image of A(K), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', the  relevant affine space A(K) up to WJ-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) We define the coarse type of u  J to be the element τ( u  J ) ∈ E = W a\\E given by the image of A(K),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', the relevant affine space A(K) up to W a-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Taking the coarse type of a J-piece defines a map  τ : S :=  �  J⊂ftIa  SJ → E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Next, we give a way to compute the J-type starting from any WJ-conjugacy class of �  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For w ∈ �  W and  J ⊂ft Ia, the set Ew  J := {E ∈ EJ|w(E) = E} is non-empty (since A ∈ Ew  J ) and closed under intersection,  hence has a unique minimal element which we denote by EJ,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The next lemma explains the relation  between J-type and EJ,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  40  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J ⊂ft Ia, u ∈ J �  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Let K = I(J, u), then EJ,uy = A(K) for any y ∈ WK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Suppose w ∈ �  W is such that σJ(w) = u  J ∈ SJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the J-type of u  J is the WJ-orbit of EJ,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (3) Suppose w, w′ ∈ �  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then σJ(w) = σJ(w′) if and only if w′ is WJ-conjugate to an element w′′  with EJ,w = EJ,w′′ and w|EJ,w = w′′|EJ,w′′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let K = I(J, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the J-type of u  J is the WJ-orbit of A(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, w = xuyx−1  where y ∈ WK and x ∈ WJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have EJ,w = xEJ,uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore it suffices to consider the case w = uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Hence (2) follows from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We prove (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since Ad(u)K = K, u stabilizes A(K), hence uy stabilizes A(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By the minimality of  E := EJ,uy, we have E ⊂ A(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We claim that E = A(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let F be the fundamental alcove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider  the relevant affine subspace Span(E ∩ F) spanned by E ∩ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have Span(E ∩ F) = A(K′) for some  K′ ⊂ft Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since A(J) ⊂ E ⊂ A(K), we have K ⊂ K′ ⊂ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that A(K′) = Span(E ∩ CJ), where  CJ ⊂ A is the dominant WJ-chamber centered along A(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' On the other hand, since u ∈ J �  W, we have  uF ∈ CJ, and therefore Span(uyE ∩ uF) = Span(E ∩ uF) ⊂ Span(E ∩ CJ) = A(K′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The action of  u sends Span(E ∩ F) = Span(yE ∩ F) (note that y ∈ WK acts by identity on A(K)) isomorphically  to Span(uyE ∩ uF) = Span(E ∩ uF), hence we must have Span(E ∩ uF) = A(K′), and in particular  u stabilizes A(K′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now u stabilizes the affine roots ΦK′ spanned by K′ ⊂ J, and u ∈ J �  W implies  that u−1 sends positive roots Φ+  J ⊂ ΦJ to positive affine roots, therefore u preserves Φ+  K′, and hence  preserves K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By the maximality of K as a u-stable subset of J, we must have K′ = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore,  E ⊃ Span(E ∩ F) = A(K′) = A(K), forcing E = A(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We prove (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The statement is invariant under WJ-conjugation of w and w′ separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore  we may assume w = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let K = I(J, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose w′ is WJ-conjugate to w′′ with A(K) = EJ,w′′ and  w|A(K) = w′′|A(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then w′′ = uy for some y ∈ WK, hence σJ(w′) = σJ(w′′) = u  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Conversely, suppose  σJ(w′) = u  J , then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, w′ is WJ-conjugate to uy for some y ∈ WK, hence EJ,uy = A(K) by  (1) and uy|A(K) = u|A(K) since y ∈ WK acts by identity on A(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J ⊂ J′ ⊂ft Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let πJ′  J :  �  W  WJ →  �  W  WJ′ be the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then there is a unique  map δJ′  J : SJ → SJ′ making the following diagram commutative  �  W  WJ  σJ  �  πJ′  J  �  �  W  WJ′  σJ′  �  SJ  δJ′  J  � SJ′  In particular, for J ⊂ J′ ⊂ J′′ ⊂ft Ia, δJ′′  J′ ◦ δJ′  J = δJ′′  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since σJ is surjective, δJ′  J is unique if it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For the existence of δJ′  J , we need to show the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let w ∈ �  W and σJ(w) =  u  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We need to  show that σJ′(w) depends only on u and not on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, up to WJ-conjugacy we may assume  w = uy for some y ∈ WK, where K = I(J, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We need to show that σJ′(u) = σJ′(uy) for all y ∈ WK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For this, we use the criterion in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6(1), EJ,u = EJ,uy = A(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore  EJ′,u, EJ′,uy ⊂ A(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since y acts by identity on A(K), we have EJ′,u = EJ′,uy, and u|EJ′,u = uy|EJ′,uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore σJ′(u) = σJ′(uy) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J ⊂ J′ ⊂ft Ia, u ∈ J�  W and u′ ∈ J′�  W such that δJ′  J ( u  J ) = u′  J′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then  (1) ℓ(u) ≥ ℓ(u′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) A representative of the J-type of  u  J (as a relevant subspace containing A(J), up to WJ-action)  contains a representative of the J′-type of u′  J′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, τ( u  J ) ≥ τ( u′  J′ ) under the partial order  on E defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  41  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (1) By [HN14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5], u′ has minimal length among σ−1  J′ ( u′  J′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since u ∈ σ−1  J′ ( u′  J′ ) by con-  struction, we see that ℓ(u) ≥ ℓ(u′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6, the J-type of u  J is represented by EJ,u, and the J′-type of u′  J′ is represented by  EJ′,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Clearly EJ′,u ⊂ EJ,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J ⊂ J′ ⊂ft Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let u  J ∈ SJ and δJ′  J ( u  J ) = u′  J′ ∈ SJ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) We say that u  J is quasi-J′-reduced if ℓ(u) = ℓ(u′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) We say that u  J is J′-reduced if ℓ(u) = ℓ(u′), and τ( u  J ) = τ( u′  J′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Newton point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall the Newton point of w ∈ �  W is a point ν(w) ∈ X∗(T )+  Q (rational dominant  cone) characterized by the following property: for sufficiently divisible n, wn ∈ X∗(T ) is in the same  W-orbit as the translation element given by nν(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The Newton point is constant on each �  W-conjugacy  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore we have a map  ν :  �  W  �  W  → X∗(T )+  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let NP ⊂ X∗(T )+  Q be the image of this map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We also have the natural projection  κ : �  W → �  W/W a =: Ω  that factors through the conjugacy classes of �  W because Ω is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Combining ν and κ we get the  enhanced Newton map  �ν :  �  W  �  W  → NP × Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let �  NP ⊂ NP × Ω be the image of �ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For J ⊂ft Ia, there is a unique map �νJ : SJ → �  NP such that the following diagram is  commutative  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  �  W  WJ  �  σJ  � SJ  �νJ  �  �  W  �  W  �ν  � �  NP  where the left vertical map is the natural projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Moreover, for J ⊂ J′ ⊂ft Ia, we have νJ′ ◦ δJ′  J = �νJ :  SJ → �  NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since σJ is surjective, the uniqueness of �νJ is clear: it sends  u  J to �ν(u),where u ∈ J �  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To  show the diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) is commutative, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, it suffices to show that ν(uy) = ν(u) and  κ(uy) = κ(u) for all y ∈ WK, where K = I(J, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Since WK ⊂ W a, we have κ(uy) = κ(u) ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we show ν(uy) = ν(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any n ≥ 1 we have (uy)n = Ad(u)y · Ad(u2)y · · · Ad(un)y · un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Each  Ad(ui)y ∈ WK since K is stable under Ad(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let m be the order of Ad(u) on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then if n is divisible by  m|WK|, Ad(u)y ·Ad(u2)y · · · Ad(un)y = (Ad(u)y ·Ad(u2)y · · · Ad(um)y)n/m ∈ (WK)n/m = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore  (uy)n = un for n sufficiently divisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This implies ν(uy) = ν(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Straight elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Following Krammer [Kra09], one says w ∈ �  W is straight if ℓ(wn) = nℓ(w) for  all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A conjugacy class c ∈  �  W  Wa is called straight if it contains a straight element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let w ∈ �  W with Newton point ν ∈ NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then ℓ(w) ≥ ⟨2ρ, ν⟩, and the equality holds if  and only if w is a straight element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  42  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any translation element tλ ∈ �  W corresponding to a dominant λ ∈ X∗(T ), we have ℓ(tλ) =  ⟨2ρ, λ+⟩ for the unique dominant element λ+ in the W-orbit of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore, if w ∈ �  W has Newton  point ν ∈ NP, wn is conjugate to tnν for sufficiently divisible n, hence ℓ(wn) = ⟨2ρ, nν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore  ℓ(w) ≥ 1  nℓ(wn) = ⟨2ρ, ν⟩, and equality holds if and only if w is a straight element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  By [HN14, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3], there is a unique straight W a-conjugacy class for each enhanced Newton point  �ν ∈ �  NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The space B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will introduce a topological space B obtained by gluing copies of quotients of the  apartment in the building of G and passing to quotients, using the combinatorial pieces for the affine Weyl  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The space B will serve as an organizational tool of subquotient categories of HG,J that we study  in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  When G is almost simple, B will be a ∆-complex (a weaker notion than a simplicial complex), which is  a union of simplices where the intersection of two simplices is not necessarily a common face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In general,  B will be a union of poly-simplices (product of simplices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' D◦-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5 that D◦ denotes the poset of finite type subsets J ⊂ft Ia under  inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A D◦-set X is a functor  X : D◦ → Sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In other words, it is an assignment J �→ XJ ∈ Sets for each J ⊂ft Ia, and a map XJ → XJ′ whenever  J ⊂ J′, compatible with three-term inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For a D◦-set X, Tot(X) := �  J⊂ftIa XJ is a poset whose relations are of the form xJ ≤ yJ′, if J ⊂  J′ ⊂ft Ia and xJ ∈ XJ maps to yJ′ ∈ XJ′ under the map XJ → XJ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For a D◦-set X, we shall define a topological space |X| as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For J ∈ D◦, we let |∆J| ⊂ A be the  standard facet in the (reduced) apartment for T indexed by J (so that for J = ∅, |∆J| is the fundamental  alcove in A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then |∆J| is isomorphic to a product of simplices with codimension #J in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let |X| be the  space obtained as quotient of ⊔J⊂ftIaXJ × |∆J| by the relation (xJ, t) ∼ (yJ′, t) where J ⊂ J′, xJ �→ yJ′  under XJ → XJ′ and t ∈ |∆J| ⊂ |∆J′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The image of {xJ} × |∆J| in |X| (for xJ ∈ XJ) is called a J-facet  of |X|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' they are parametrized by XJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  When G is almost simple of rank r, each J-facet of |X| is a simplex of dimension r − #J, and |X| is a  ∆-complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In general, |X| is a union of poly-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The set Tot(X) is the set of all facets of |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The partial order on Tot(X) is the opposite of the closure  order of faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Let δ be the D◦-set given by the constant functor valued in the singleton set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The geometric realization |δ| can be identified with the fundamental alcove in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The standard apartment A of G gives rise to a D◦-set Fac(A): Fac(A)J is the set of J-facets in  A, and the transition maps Fac(A)J → Fac(A)J′ sends a J-facet F to the unique J′-facet in its  closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then we have a canonical homeomorphism |Fac(A)| ∼= A respecting the poly-simplicial  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (3) The same construction of (2) applies to any closed subset E ⊂ A that is a union of facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It gives  a D◦-subset Fac(E) ⊂ Fac(A), and |Fac(E)| is identified with E as a subspace of |Fac(A)| ∼= A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a D◦-set X, the poset Tot(X) also has a geometric realization |Tot(X)| (see Sec-  tion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then |Tot(X)| is always a simplicial complex, and it is a subdivision of |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  When G  is almost simple, |Tot(X)| is the barycentric subdivision of |X|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', |Tot(X)| ∼= sd(|X|) as simplicial  complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Construction of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We define a D◦-set S whose value at J ⊂ft Ia is SJ, and for J ⊂ J′ ⊂ft Ia,  the transition map is δJ′  J : BJ = SJ → SJ′ = BJ′ defined in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let B = |S| be the geometric realization of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For u  J ∈ SJ, we denote by B( u  J ) the corresponding  J-facet of B, with interior B( u  J )◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  43  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any finite or affine Weyl group W with an automorphism σ preserving the simple  reflections, we have the notion of σ-twisted J-pieces and one can similarly define B(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In our setting,  the normal slice to a facet B( u  J ) in B is isomorphic to B(WK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' u) for the finite Weyl group WK (where  K = I(J, u)) under the u-twisted conjugation action of WK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Functions on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have attached three invariants to each element in Tot(S) = �  J⊂ftIa SJ:  (1) The length function ℓ : Tot(S) → Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The coarse type τ : Tot(S) → E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (3) The enhanced Newton map �ν : Tot(S) → �  NP ⊂ NP × Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We may consider these functions as piece-wise continuous functions on the geometric realization B:  (1) ℓ : B → Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) τ : B → E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (3) �ν : B → �  NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  such that the value of ℓ, τ and ν on B( u  J )◦ is ℓ( u  J ), τ( u  J ) and νJ( u  J ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8, the functions ℓ and τ on B are both lower semicontinuous (non-increasing under  specialization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11, the function �ν is locally constant on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have decompositions of the  D◦-set S and the space B:  S =  �  �ν∈ �  NP  S�ν,  B =  �  �ν∈ �  NP  B�ν  where S�ν,J is the set of u  J such that �ν( u  J ) = �ν, and B�ν = |S�ν|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note each B�ν ⊂ B is open and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Essential part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Fix �ν = (ν, ω) ∈ �  NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='13, any piece u  J that appears in B�ν satisfies  ℓ(u) ≥ ⟨2ρ, ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let S♥  �ν,J ⊂ SJ be the subset consisting of  u  J with �ν(u) = �ν and ℓ(u) = ⟨2ρ, ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='13 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8, we see that the assignment J �→ S♥  �ν,J defines a D◦-subset S♥  �ν of S�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let  B♥  �ν = |S♥  �ν | and call it the essential part of B�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='13, B♥  �ν is the closure of the maximal facets  of B�ν indexed by straight elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For G = SL2, B is a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since G is simply-connected, Ω = 0 and �  NP = NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The  connected components of B are indexed by the possible Newton points ν ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that W a = ⟨s0, s1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Tn = (s1s0)n for n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For ν > 0, Bν is a cycle with two nodes and two edges:  T−n  {s1}  Tn  ∅  T−n  ∅  Tn  {s0}  There is only one conjugacy class with Newton point n > 0, namely that of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this case, we have  B♥  ν = Bν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For ν = 0, the graph Bν is an infinite linear tree:  1  {s1}  s1  ∅  1  ∅  s1  {s0}  s0s1s0  ∅  s0s1s0  {s1}  s1s0s1s0s1  ∅  · ·  1  {s0}  s0  ∅  s0  {s1}  s1s0s1  ∅  s1s0s1  {s0}  s0s1s0s1s0  ∅  · ·  We draw it in three segments to reflect that there are three conjugacy classes in W a with Newton point  0: the identity conjugacy class (corresponding to the vertical edge), the conjugacy class of s1 (upper ray)  and the conjugacy class of s0 (lower ray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this case, B♥  ν consists of the edge labelled 1  ∅ and its two end  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  44  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the case G = PGL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now �  W ∼= T Z  1/2 ⋊ ⟨s1⟩, where T 2  1/2 = T1 in the notation  of Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let ω = s1T1/2 be the length zero element in �  W − W a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then for n ≥ 1 odd, Tn/2 =  s1s0 · · · s1ω (length n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have NP = 1  2Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The set �  NP ⊂ NP × Ω ∼= 1  2Z≥0 × Z/2Z is  �  NP = {(n/2, n mod 2)|n ∈ Z≥0} ∪ {(0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For �ν = (n, 0) ∈ �  NP, B�ν is the same as Bn described in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For �ν = (n/2, 1) where n ≥ 1 odd, B�ν is of the form  T−n/2  {s1}  Tn/2  ∅  Tn/2  ∅  T−n/2  {s0}  There is only one conjugacy class with Newton point n/2 > 0, namely that of Tn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this case we have  B♥  �ν = B�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For �ν = (0, 1), B�ν is an infinite linear tree  · ·  s0ωs0  {s1}  s0ωs0  ∅  ω  {s0}  ω  ∅  ω  {s1}  s1ωs1  ∅  s1ωs1  {s0}  · ·  There is only one W a-conjugacy class in �  W with enhanced Newton point �ν = (0, 1), namely the conjugacy  class of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this case, B♥  �ν consists of the edge labelled ω  ∅ and its two end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Linear structure on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We shall describe B as glued from copies of quotients of the apartment A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For a relevant affine subspace E ∈ E, let W E ⊂ Aff(E) be the affine transformations of E of the form  w|E for some w ∈ Stab�  W (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let WE ⊂ W a be the subgroup that fixes E pointwise (this is a parabolic  subgroup of W a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that we have a short exact sequence  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  1 → WE → Stab�  W (E) → W E → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Consider the following category A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Its objects are pairs (E, w) where E ∈ E and w ∈ W E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' its morphisms  are defined by  MorA((E, w), (E′, w′)) = {g ∈ W a|gE ⊂ E′, w′(gE) = gE, w′|gE = g ◦ w ◦ g−1 ∈ Aff(gE)}  with the evident composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For each (E, w) ∈ A, we define a map of D◦-sets  ϕE,w : Fac(E) → S  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It sends a J-facet F = xFJ of E (where FJ is the standard J-facet, x ∈ W a/WJ) to σJ(x−1wx)  (note that x−1wx ∈  �  W  WJ is well-defined, it is the relative position between F and wF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose g : (E, w) → (E′, w′) is a morphism in A, then  ϕE,w = (ϕE′,w′|Fac(gE)) ◦ g : Fac(E) → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We lift w to an element of Stab�  W (E) and w′ to an element of Stab�  W (E′) and still denote them by  w and w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let F = xFJ ⊂ E be a J-facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' On the one hand, ϕE,w(F) = σJ(x−1wx) ∈ SJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' On the other  hand, gF = gxFJ ⊂ E′ is a J-facet of E′, and ϕE′,w′(g(F)) = σJ((gx)−1w′gx) = σJ(x−1(g−1w′g)x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Since g : (E, w) → (E′, w′) is a morphism in A, we have g−1w′g ∈ Stab�  W (E) and it has the same image  as w in W E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By the exact sequence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) we can write g−1w′g = wy for some y ∈ WE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We reduce to  showing  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  σJ(x−1wx) = σJ(x−1wyx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For this we use the criterion in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let E1 = EJ,x−1wx and E2 = EJ,x−1wyx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then E1 is the  minimal relevant affine subspace that contains A(J) and stable under x−1wx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now x−1E contains A(J)  and is stable under x−1wx, hence E1 ⊂ x−1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since y ∈ WE, x−1E is stable under x−1yx, hence also  stable under x−1wyx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This implies E2 ⊂ x−1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Moreover, because y fixes E pointwise, the actions of  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  45  x−1wx and x−1wyx on x−1E are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore both E1 and E2 are equal to the minimal relevant  affine subspace that contains A(J) and contained in x−1E, and stable under x−1wx|x−1E = x−1wyx|x−1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6(3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  By the above lemma, ϕE,w is invariant under AutA(E, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Using the transition maps g : E → E′ for  g ∈ MorA((E, w), (E′, w′)), we may form the colimit in the category of D◦-sets  colim(E,w)∈A Fac(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11 implies that the maps {ϕE,w} together induce a map of D◦-sets  ϕ : colim(E,w)∈A Fac(E) → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Taking geometric realizations, we get a map  |ϕ| : colim(E,w)∈A E → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The map ϕ is an isomorphism of D◦-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consequently, the map |ϕ| is a homeomorphism  respecting the facet structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We need to show for each J ⊂ft Ia, the map on the set of J-facets  ϕJ : colim(E,w)∈A FacJ(E) → SJ  is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To see ϕJ is surjective, note that for E = A and w ∈ �  W = W E, ϕA,w is the composition of �ϕA,w :  FacJ(A) = W a/WJ →  �  W  WJ (sending x �→ x−1wx) and σJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' As w runs over all of �  W, the images of �ϕA,w  cover all of  �  W  WJ , therefore the images of ϕA,w cover all of SJ since σJ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we show ϕJ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let (E1, w1), (E2, w2) ∈ A and Fi = xiFJ ∈ FacJ(Ei) for i = 1, 2 such  that ϕJ(F1) = ϕJ(F2) ∈ SJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This means σJ(x−1  1 w1x1) = σJ(x−1  2 w2x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now we invoke Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Upon right multiplying x2 by an element in WJ, we may assume EJ,x−1  1  w1x1 = EJ,x−1  2  w2x2, which we  denote by E, and that x−1  1 w1x1|E = x−1  2 w2x2|E, which we denote by w ∈ W E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then we have morphisms  x1 : (E, w) → (E1, w1) and x2 : (E, w) → (E2, w2) in A, and these morphisms send FJ (which is a facet of  E, because by definition E contains A(J)) to F1 and F2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This means F1 and F2 are already  identified in the colimit colim(E,w)∈A FacJ(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This proves ϕJ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12 tells us that B can be constructed as follows: for each W a-conjugacy  class in �  W, take a representative w and form the quotient space A/CW a(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then B is obtained from  the disjoint union of A/CW a(w) (w running over representatives of W a-conjugacy classes in �  W) by gluing  along relevant subspaces of dimension less that that of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' He-Nie function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Following the idea of He and Nie [HN14, ], we now define a function f : B → R≥0  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For (E, w) ∈ A, we have the function fE,w : E → R≥0 defined by x �→ ∥x − wx∥2, using a  fixed W-invariant positive definitive quadratic form ∥ ·∥2 on V = X∗(T )R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any morphism g : (E, w) →  (E′, w′) in A, one checks that  fE,w = fE′,w′ ◦ g : E → R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore {fE,w}(E,w)∈A gives a piecewise smooth function on colim(E,w)∈A E ∼= B, which we denote by  f : B → R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Moreover, using the quadratic form ∥ · ∥2 restricted to E, the differential dfE,w turns into a gradient  vector field ∇fE,w on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since fE,w is quadratic, ∇fE,w is a linear vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The next lemma shows  that for varying (E, w) ∈ A, the vector fields ∇fE,w assemble to a piecewise-linear continuous vector field  ∇f on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' If g : (E, w) → (E′, w′) is a morphism in A, then g∗ takes the vector field ∇fE,w on E  to the vector field ∇fE′,w′|gE on gE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  46  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let (E, w) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let VE ⊂ X∗(T )R be the vector space parallel to E, so that w − 1 is a map  E → VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let w ∈ End(VE) be the linear part of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Direct calculation shows  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  (∇fE,w)(x) = 2(w − 1)∗(w − 1)(x),  ∀x ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here (w − 1)∗ ∈ End(VE) is the adjoint of (w − 1) ∈ End(VE) under the quadratic form ∥ · ∥2 on VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Replacing (E, w) by (gE, gwg−1), we may assume that E ⊂ E′, w′(E) = E, w′|E = w and g is the  identity element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since w′ preserves E, the endomorphism w − 1 of VE′ preserves VE, and so is its adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Hence by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3), ∇fE′,w′|E = ∇fE,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Let Crit(f) ⊂ B be the vanishing locus of ∇f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Crit(f)�ν = Crit(f) ∩ B�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any �ν ∈ �  NP, the critical locus Crit(f)�ν is contained in the essential part B♥  �ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let x ∈ Crit(f)�ν, then there exists some (E, w) ∈ A (where w ∈ W E) such that x is the image  of a critical point y of fE,w under |ϕE,w| : E → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We choose such a (E, w) with E minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Choose  a connected component C of A − ∪E⊂HH (remove all affine root hyperplanes that contain E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let  �w ∈ Stab�  W (E) be the lifting of w such that �w(C) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then g = 1 gives a morphism (E, w) → (A, �w) in  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have y ∈ Crit(fE,w) = E ∩ Crit(fA, �  w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, ν = ν( �w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Apply [HN14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6] to the  subspace Crit(fE,w) of Crit(fA, �  w), and an alcove A ⊂ C that contains y in its closure, we conclude that  �wA := pos(A, �wA) (relative position, which is in the same W a-conjugacy class of �w) has length ⟨2ρ, ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The  image of A under |ϕA, �  w| : A → B is the maximal facet indexed by �wA ∈ S∅ = �  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since ℓ( �wA) = ⟨2ρ, ν⟩,  the closure of the maximal facet B( �  wA  ∅ ) is in the essential part B♥  �ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' On the other hand, y ∈ A implies x  is in the closure of B( �  wA  ∅ ), hence x ∈ B♥  �ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is likely that B♥  �ν is the smallest D◦-subset of B�ν whose geometric realization contains  Crit(f)�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In other words, it should be true that for any straight element w, the critical locus of fA,w :  A → R intersects the interior of the fundamental alcove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Below we prove a topological property of certain subsets of B�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is a key ingredient in the proof of  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Fix �ν = (ν, ω) ∈ �  NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let �ν ∈ �  NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A D◦-subset S′ ⊂ S�ν is a called downward if it satisfies:  Tot(S′) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  S′ contains S♥  �ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let u  J ∈ S�ν,J, u′  J′ ∈ S�ν,J′ satisfy ℓ(u) ≤ ℓ(u′) and τ( u  J ) ≤ τ( u′  J′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' If u′  J′ ∈ S′  J′, then u  J ∈ S′  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A subspace B′ ⊂ B�ν is called downward if it is of the form |S′| for a downward D◦-subset S′ of S�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) B♥  �ν is the smallest downward subspace of B�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Let n ≥ ⟨2ρ, ν⟩ and [E] ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let S�ν,≤(n,[E]) ⊂ S�ν be the D◦-subset consisting of u  J ∈ S�ν,J such  that ℓ(u) ≤ n and τ( u  J ) ≤ [E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The fact that this is a D◦-subset follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  When n > ⟨2ρ, ν⟩, S�ν,≤(n,[E]) is downward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  When n = ⟨2ρ, ν⟩ and [E] = [A], we have  S�ν,≤(⟨2ρ,ν⟩,[A]) = S♥  �ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We denote B�ν,≤(n,[E]) = |S�ν,≤(n,[E])|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (3) By definition, any downward subspace B′ ⊂ B�ν is a finite union of the form  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  B′ = B♥  �ν ∪  ��  i  B�ν,≤(ni,[Ei])  �  where ni > ⟨2ρ, ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='16, for any downward B′  �ν ⊂ B�ν, we have Crit(f)�ν ⊂ B♥  �ν ⊂ B′  �ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Fix �ν = (ν, ω) ∈ �  NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let B′ ⊂ B�ν be a downward subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the inclusion  Crit(f)�ν ֒→ B′ admits a deformation retract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, both inclusions Crit(f)�ν ⊂ B♥  �ν ⊂ B′ are  homotopy equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  47  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Abbreviate Crit(f)�ν by C�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='15, the gradient flow of f is well-defined as a flow Φt  on B�ν, for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The deformation retract will be constructed using the flow Φt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The downward subspace B′ is stable under the flow Φt, for t ≤ 0 non-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof of Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since B′ is a union of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4), it suffices to show that B♥  �ν and B�ν,≤(n,[E]) (where  n > ⟨2ρ, ν⟩ )are stable under the flow {Φt}t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since B♥  �ν = B�ν,≤(⟨2ρ,ν⟩,[A]), we suffices to show that  B�ν,≤(n,[E]) is stable under the flow {Φt}t≥0 whenever n ≥ ⟨2ρ, ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  First consider the case E = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since every point of B�ν lies in the image of |ϕA,w| : A → B for some  w ∈ �  W with �ν(w) = �ν, and the flow stays inside the image of |ϕA,w|, we only need to prove the same  statement for A with respect to the flow defined by ∇fA,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let A0 ⊂ A be the (closed) fundamental  alcove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any alcove A = yA0, let wA = y−1wy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Aw,≤n ⊂ A be the union of alcoves A such that  ℓ(wA) ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then Aw,≤n = |ϕA,w|−1(B�ν,≤(n,[A])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We only need to show that Aw,≤n is stable under the  flow Φt of ∇fA,w for t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Z ⊂ A be the union of all H ∩ H′ where H and H′ run over distinct affine  root hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let U ⊂ Aw,≤n be the subset of x ∈ Aw,≤n such that Φt(x) /∈ Z for all t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then U  is dense in Aw,≤n (using that Z has codimension two in A, so ∪t≥0Φt(Z) has codimension at least one in  A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore it is enough to show that Φt(x) ∈ Aw,≤n for x ∈ U and t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose this is not the case,  then for some x ∈ U and some t ≤ 0, Φt(x) lies in an alcove A with ℓ(wA) > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let t0 be the supremum  of such t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then Φt0+ǫ(x) ∈ A with ℓ(wA) ≤ n for small ǫ > 0 and Φt0−ǫ(x) ∈ A′ with ℓ(wA′) > n for small  ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Moreover, x0 := Φt0(x) is on the common face A ∩ A′ of A and A′, and does not lie on any other  affine root hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let n be a normal vector of H that points to A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  ⟨∇fA,w(x0), n⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We have wA′ = swAs where s is the simple reflection determined by hyperplane H = Span(A∩A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Hence  ℓ(wA′) = ℓ(wA) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Applying [HN14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1], we see that ⟨∇fA,w(x0), n⟩ > 0 (note that x0 = Φt0(x)  is a regular point of A ∩ A′ since x0 /∈ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This contradicts (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we consider the case of a general [E] ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any x ∈ B�ν,≤(n,[E]) we may find (E, w) ∈ A (with  E ∈ [E]) such that x lies in the image of |ϕE,w| : E → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Hence it is enough to prove the analogous  statement for E with respect to the gradient flow of fE,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let �w ∈ �  W be a lifting of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then |ϕE,w| is  the composition E ֒→ A  |ϕA, �  w|  −−−−→ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, E ∩ |ϕE,w|−1(B�ν,≤(n,[E])) = A ∩ |ϕA, �  w|−1(B�ν,≤(n,[E])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Moreover, the gradient flow of fE,w is the same as the gradient flow of fA, �  w restricted to E by Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore the case [E] = [A] proved above implies the case of a general [E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Now for any x ∈ B�ν, limt→−∞ Φt(x) ∈ C�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, we may assume x lies in the image of |ϕA,w| : A →  B�ν for some w ∈ �  W, and the corresponding statement is [HN14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Moreover, the calculation  in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' shows that the flow is contracting in a neighborhood of C�ν as t → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This implies that  the map H : [0, 1) × B′ → B′ given by H(s, x) = Φlog(1−s)(x) can be extended to a continuous function  H : [0, 1] × B′ → B′ by letting H(1, x) = limt→−∞ Φt(x) ∈ C�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then H gives a deformation retract from  B′ to C�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Geometric pieces and sheaves on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Here we turn to the geometry indexed by the prior  combinatorics, in particular sheaves on geometric pieces and the natural functors between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Cyclic reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the following general situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G be an algebraic group, and P, P ′ ⊂  G two parabolic subgroups with unipotent radicals P u and P ′u and Levi quotients L and L′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let δ : L′  ∼  → L be an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Q′ = Im(P ∩ Ad(x−1)P ′ → L) ⊂ L, Q =  δ(Im(P ′ ∩ Ad(x)P → L′)) ⊂ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then Q, Q′ are parabolic subgroups of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Qu and Q′u be the  unipotent radicals of Q and Q′ and M and M ′ be their Levi quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then M ′ = (P ∩ Ad(x−1)P ′)red  (where (−)red denotes Levi quotient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have a canonical isomorphism  δx : M ′ = (P ∩ Ad(x−1)P ′)red  Ad(x)  −−−−→ (P ′ ∩ Ad(x)P)red ∼= Im(P ′ ∩ Ad(x)P → L′)  δ−→ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  48  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma (Cyclic reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is a canonical map  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  P ′u\\(P ′xP)/P u  Adδ(L′)  → Q′u\\L/Qu  Adδx(M ′)  sending p′xp to pδ(p′) (where p is the image of p under P ։ L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' similar for p′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This map is a gerbe for  the unipotent group P ′u ∩ Ad(x)P u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We will refer to the above map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) as a cyclic reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The left P ′-translation and right P-translation on P ′xP gives a P ′ × P-equivariant isomorphism  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  P ′ ×P ′∩Ad(x)P P  ∼  → P ′xP,  (p′, p) �→ p′xp  where the action of P ′ ∩Ad(x)P on P is by Ad(x−1) : P ′ ∩Ad(x)P  ∼  → Ad(x−1)P ′ ∩P and left translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now quotienting both sides of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2) by left P ′u, right P u and Adδ(L′)-actions, we get isomorphisms  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  L′ ×P ′∩Ad(x)P L  Adδ(L′)  ∼  ←− P ′u\\P ′ ×P ′∩Ad(x)P P/P u  Adδ(L′)  ∼  → P ′u\\P ′xP/P u  Adδ(L′)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here the action on P ′∩Ad(x)P on L′ is via the projection P ′∩Ad(x)P → L′, whose image is δ−1(Q) ⊂ L′,  and right translation on L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly, the action on P ′ ∩Ad(x)P on L is via the projection P ′∩Ad(x)P  ∼  →  Ad(x−1)P ′ ∩ P → Q′ ⊂ L and left translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The normal subgroup P ′u ∩ Ad(x)P u acts trivially on  L′ × L, and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  (P ′ ∩ Ad(x)P)/(P ′u ∩ Ad(x)P u) ∼= δ−1(Q) ×M′ Q′  where the projection δ−1(Q) → M ′ is the composition δ−1(Q)  δ−→ Q ։ M  (δx)−1  −−−−→ M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) we get a P ′u ∩ Ad(x)P u-gerbe  L′ ×P ′∩Ad(x)P L  Adδ(L′)  → L′ ×δ−1(Q)×M′ Q′ L  Adδ(L′)  = (L′/δ−1(Qu)) ×M′ (Q′u\\L)  Adδ(L′)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Via the map (ℓ′, ℓ) �→ ℓδ(ℓ′) (for ℓ ∈ L, ℓ′ ∈ L′), we have an isomorphism  (L′/δ−1(Qu)) ×M′ (Q′u\\L)  Adδ(L′)  ∼  → Q′u\\L/Qu  Adδx(M ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Composing all these isomorphisms we get a P ′u ∩ Ad(x)P u-gerbe  P ′u\\(P ′xP)/P u  Adδ(L′)  → Q′u\\L/Qu  Adδx(M ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2 gives an equivalence of categories by pullback  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  Sh(Q′u\\L/Qu  Adδx(M ′) ) ≃ Sh(P ′u\\(P ′xP)/P u  Adδ(L′)  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Nilpotent sheaves under cyclic reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Next we show that sheaves with nilpotent singular support  correspond to each other under the above equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To simplify the statements, we introduce the  following terminology: For a group H acting on a smooth variety X, and F ∈ Sh(X), we say F is H-  nilpotent if µH(SS(F)) lies in the nilpotent cone of h∗ = (LieH)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' When there is ambiguity as to how H  acts on X, we will specify F is H-nilpotent for which H-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let  ShN (P ′u\\(P ′xP)/P u  Adδ(L′)  ) ⊂ Sh(P ′u\\(P ′xP)/P u  Adδ(L′)  )  be the full subcategory consisting of objects that are L′-nilpotent for left translation by L′ (equivalently,  L-nilpotent for the right translation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly define the full subcategory  ShN (Q′u\\L/Qu  Adδx(M ′) ) ⊂ Sh(Q′u\\L/Qu  Adδx(M ′) )  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  49  using either M ′-nilpotence for the left translation or M-nilpotenve for the right translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under the equivalence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5), the full subcategories ShN ( P ′u\\(P ′xP )/P u  Adδ(L′)  ) and ShN ( Q′u\\L/Qu  Adδx(M′) )  correspond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, pullback via the cyclic reduction map induces an equivalence  ShN (Q′u\\L/Qu  Adδx(M ′) ) ≃ ShN (P ′u\\(P ′xP)/P u  Adδ(L′)  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Choose a section to P ։ L and realize L as a Levi subgroup of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly realize L′ as a subgroup  of P ′, M as a subgroup of Q′ and M as a subgroup of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We have a commutative diagram  P ′ × P  πx  �  p′×p  � L′ × L  mδ  � L  π  �  P ′u\\(P ′xP )/P u  Adδ(L′)  c  � Q′u\\L/Qu  Adδx(M′) )  where the bottom arrow is the cyclic reduction map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1), p, p′ are the projections and mδ(ℓ′, ℓ) = ℓδ(ℓ′),  πx(g′, g) = g′xg, and π is the quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let F ∈ Sh( Q′u\\L/Qu  Adδx(M′) ), and �F = π∗F be its pullback to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let K = c∗F ∈ Sh( P ′u\\P ′xP/P u  Adδ(L′)  ), and  �K = π∗  xK be its pullback to P ′ × P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By definition, F ∈ ShN ( Q′u\\L/Qu  Adδx(M′) ) if and only if �F is M ′-nilpotent  for the left translation of M ′ on L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' K ∈ ShN ( P ′u\\P ′xP/P u  Adδ(L′)  ) if and only if �K is L-nilpotent for the right  translation actions on the P-factor of P ′ × P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore, we reduce to prove that the following are  equivalent:  (1) �F is M ′-nilpotent for the left translation on L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) �K = (p′ × p)∗m∗  δ �F is L-nilpotent for the right translation on the P-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Since p′ × p is smooth and equivariant for the right translation of L, (2) is equivalent to  (3) m∗  δ �F is L-nilpotent for the right translation on the L-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  It is easy to see that on L′×L, L-nilpotence with respect to the left translation is equivalent to L-nilpotence  with respect to the right translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Also mδ is equivariant with respect to the left translation action of  L (but not the the right translation), therefore (3) is equivalent to  (4) �F is L-nilpotent for the left translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  It remains to prove (1) ⇐⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let µL : T ∗L → l∗ be the moment map for the left transaltion of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall �  F is Q′u-equivariant for the left translation action, so µL(SS( �F)) ∈ n⊥  Q′ (here nQ′ = LieQ′u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then  (1) means the image of µL(SS( �F)) under the projection n⊥  Q′ → m′∗ is nilpotent, which is equivalent to  saying that µL(SS( �F)) lies in the nilpotent cone of n⊥  Q′, which is (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Geometric pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8 that  YJ = Pu  J \\G/Pu  J  LJ  regarded as an ind-stack over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For J ⊂ft Ia, Lusztig [Lusa, §3] defined a stratification of YJ indexed by  SJ: for u  J ∈ SJ, Y( u  J ) is the locally closed substack of YJ defined as the image of the projection  Y( u  J ) = Im(IuI ⊂ G → YJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We shall call Y( u  J ) geometric J-pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For each w ∈ �  W and any lifting ˙w ∈ NG(T ), ˙w ∈ Y( u  J ) if and only  if σJ( ˙w) = u  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Below we recall an inductive construction of Y( u  J ) that leads to a description of it in terms of a twisted  adjoint quotient, also due to Lusztig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  50  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Let (Jn, J′  n, un) ∈ SJ corresponding to u ∈ J �  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We describe the geometric piece Y( u  J ) using cyclic  reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Choose a lifting ˙un for each un in NLJn−1(T ) (here LJ−1 is understood to be G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For n ≥ 0, let  P ′  n+1 ⊂ LJn be the standard parabolic subgroup whose Levi is LJ′  n+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' let Pn+1 ⊂ LJn be the standard  parabolic subgroup whose Levi is LJn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For n ≥ 0 define  Zn(u0, · · · , un) = P ′u  n+1\\LJn/P u  n+1  Ad ˙u0··· ˙un(LJ′  n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  If we define LJ−1 = G, P ′  0 = P0 = PJ, then we can define Z−1 using the above formula, and have  Z−1 = Pu  J \\G/Pu  J  LJ  = YJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We have maps  Zn(u0, · · · , un)  Zn(u0, · · · , un+1) := P ′u  n \\P ′  nun+1Pn/P u  n  Ad ˙u0··· ˙un(LJ′  n+1)  �  � �  �  Zn+1(u0, · · · , un+1)  where the second map is the cyclic reduction in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The geometric piece Y( u  J ) is defined to be  the fiber product for n ≫ 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)  Y( u  J ) := Z−1(u0) ×Z0(u0) Z0(u0, u1) ×Z1(u0,u1) · · · ×Zn−1(u0,··· ,un−1) Zn−1(u0, · · · , un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The “shape” of the geometric piece Y( u  J ) is described by Lusztig:  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition (Lusztig [Lusa, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let (Jn, J′  n, un)n≥0 ∈ SJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Choose liftings ˙un ∈ NLJn−1(T ) for  n ≥ 0 such that ˙un = 1 whenever un = 1 (as usual, we understand LJ−1 = G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let ˙u = ˙u0 · · · ˙un for  n ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let K = I(J, u) ⊂ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then there is a canonical map  πJ,u : Y( u  J ) →  LK  Ad ˙u(LK)  which is an iterated gerbe for (pro-)unipotent groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  One can eliminate the choice of ˙u by writing the right side as uLK  LK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let n ≫ 0 such that Jn = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6), projection to the last factor followed by cyclic reduction  Y( u  J ) → Zn−1(u0, · · · , un) → Zn =  LK  Ad ˙u(LK)  gives the desired map, which is an iterated gerbe for (pro-)unipotent groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For each piece  u  J with K = I(J, u), pullback along πJ,u induces an equivalence of  categories  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7)  π∗  J,u : Sh(uLK  LK  ) ≃ Sh(  LK  Ad ˙u(LK))  ∼  → Sh(Y( u  J )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let ≥ be the Bruhat partial order on �  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In [He07, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='13], a partial order ≥J  on J �  W is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For u, u′ ∈ J �  W, define u ≥J u′ if there is a u′′ in the same WJ-conjugacy  class of u′ such that u ≥ u′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let w, w′ ∈ �  W be in the same WJ-conjugacy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall from loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' that w′ is said to be obtained  from w by a J-cyclic shift if ℓ(w′) = ℓ(w) and w′ = sws for some simple reflection s ∈ J such that either  ℓ(sw) = ℓ(w) − 1 or ℓ(ws) = ℓ(w) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Denote w ∼J w′ if w′ can be obtained from w by a sequence of  J-cyclic shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is shown in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' that u ≥J u′ if and only if there exists u′′ ∼J u′ such that u ≥ u′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem (He, loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For u ∈ J �  W, the closure of Y( u  J ) is the union of Y( u′  J ) for  u ≥J u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  51  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The functor chJ′  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J ⊂ J′ ⊂ft Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the image of PJ under the projection PJ′ → LJ′ is a  parabolic subgroup P J′  J ⊂ LJ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the diagram  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8)  YJ = Pu  J \\G/Pu  J  LJ  Pu  J′ \\G/Pu  J′  P J′  J  qJ′  J  �  pJ′  J  � Pu  J′ \\G/Pu  J′  LJ′  = YJ′  Define  chJ′  J = (pJ′  J )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (qJ′  J )∗ : Sh(YJ) → Sh(YJ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Finally, we recall the following key geometric input we will need to understand the natural transforms  between sheaves on different geometric pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We are grateful to Xuhua He for providing it in a recent  paper [He].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that the sheaves involved in the following theorem need not have nilpotent singular  support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem (He, [He]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J ⊂ J′ ⊂ft Ia, u ∈ J �  W and u′  J′ = δJ′  J ( u  J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let K = I(J, u) ⊂ J and  K′ = I(J′, u′) ⊂ J′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let iJ,u : Y( u  J ) ֒→ YJ and iJ′,u′ : Y( u′  J′ ) ֒→ YJ′ be the inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Suppose  u  J is not quasi-J′-reduced (namely ℓ(u) > ℓ(u′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then the image of chJ′  J ◦ iJ,u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' :  Sh(Y( u  J )) → Sh(YJ′) is supported on the closed substack YJ′,<ℓ(u) = ∪ℓ(v)<ℓ(u)Y( v  J′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Suppose  u  J is quasi-J′-reduced (recall this means ℓ(u) = ℓ(u′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then there is a unique x ∈  WJ′ ∩ �  W K′ such that x−1ux = u′ and K1 := x−1(K) ⊂ K′ (note that u′(K1) = K1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let  Ad(x−1) : Sh(uLK  LK  ) → Sh(u′LK1  LK1  )  be the functor induced by conjugation by ˙x−1, where ˙x ∈ NLJ′(T ) is any lifting of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' 3  Consider also the induction functor  u′IndK′  K1 = bK′  K1,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (aK′  K1)∗ : Sh(u′LK1  LK1  ) → Sh(u′LK′  LK′ )  given by the correspondence  u′LK1  LK1  u′P K′  K1  P K′  K1  aK′  K1  �  bK′  K1 � u′LK′  LK′  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then the outer square of the following diagram is commutative  Sh( uLK  LK )  Ad(x−1)�  π∗  J,u  ≀  �  Sh(  u′LK1  LK1 )  u′IndK′  K1� Sh( u′LK′  LK′ )  π∗  J′,u′  ≀  �  Sh(Y( u  J ))  γJ′,u′  J,u  �❴  ❴  ❴  ❴  ❴  ❴  ❴  ❴  ❴  ❴  iJ,u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  Sh(Y( u′  J′ ))  iJ′,u′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  Sh(YJ)  chJ′  J  � Sh(YJ′)  The same is true when iJ,u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' and iJ′,u′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' are replaced with iJ,u∗ and iJ′,u′∗ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We define the functor γJ′,u′  J,u  : Sh(Y( u  J )) → Sh(Y( u′  J′ )) as the composition  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9)  γJ′,u′  J,u  : Sh(Y( u  J ))  (π∗  J,u)−1  −−−−−→ Sh(uLK  LK  )  Ad(x−1)  −−−−−→ Sh(u′LK′  LK′ )  π∗  J′,u′  −−−−→ Sh(Y( u′  J′ )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (3) In particular, when u  J is J′-reduced (which implies K1 = K′ in the above notation), then γJ′,u′  J,u  is  an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  3Two liftings of x differ by multiplication by t ∈ T ⊂ LK, therefore the resulting functors Sh( uLK  LK ) → Sh(  u′LK1  LK1 ) are  canonically isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  52  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In the following we will quote results from [He] where the author primarily works with a finite  dimensional reductive group rather than a loop group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' However, [He, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5] it is explained that the results  there extend to loop groups in a straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) is proved in [He, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) From the diagram of the proof of [He, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3], we see that (using notation from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10)  pJ′  J (qJ′  J )−1(Y( u  J )) ⊂ Y( u′  J′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Using proper base change and the fact that pJ′  J is proper, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10) implies that for any F ∈ Sh(Y( u  J )),  chJ′  J iJ,u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F is a !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='-extension from Y( u′  J′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore, to prove the statement, it suffices to show that  i∗  J′,u′chJ′  J iJ,u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F is canonically isomorphic to γJ′,u′  J,u F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This is exactly the statement of [He, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For the ∗-version, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10) and the fact that qJ′  J  is smooth, we see that for any F ∈ Sh(Y( u  J )),  chJ′  J iJ,u∗F is a ∗-extension from Y( u′  J′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore, it suffices to show that i∗  J′,u′chJ′  J iJ,u∗F is canonically  isomorphic to γJ′,u′  J,u F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This is proved in the same way as the calculations towards the end of the proof of  [He, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3], using that qJ′  J is smooth to justify the base change steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (3) follows directly from (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Nilpotent sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For J ⊂ft Ia, recall that HG,J = ShN (YJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For J ⊂ J′ ⫋ Ia, it is standard to  check that chJ′  J sends HG,J to HG,J′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For each geometric piece u  J , we have the equivalence Sh(Y( u  J )) ≃ Sh( uLK  LK ) given in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Con-  sider the subcategory ShN ( uLK  LK ) ⊂ Sh( uLK  LK ) consisting of sheaves whose pullback to uLK is LK-nilpotent  for the left transaltion (equivalently, LK-nilpotent for the right translation), using the terminology intro-  duced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We define  ShN (Y( u  J )) ⊂ Sh(Y( u  J ))  to be the full category corresponding to ShN ( uLK  LK ) ∼= ShN (  LK  Ad ˙u(LK)) under the equivalence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For u  J ∈ SJ and iJ,u : Y( u  J ) ֒→ YJ be the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then:  (1) The category HG,J = ShN (YJ) consists of objects F ∈ Sh(YJ) such that i∗  J,uF ∈ ShN (Y( u  J )) for  all u  J ∈ SJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Alternatively, ShN (YJ) consists of objects F ∈ Sh(YJ) such that i∗  J,uF ∈ ShN (Y( u  J ))  for all u  J ∈ SJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The functors iJ,u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' and iJ,u∗ send ShN (Y( u  J )) (defined above) to ShN (YJ) = HG,J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We prove (1) and (2) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For (1), we give the argument for the ∗-pullback statement, and the  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='-version is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall the notation Zn(u0, · · · , un) and Zn(u0, · · · , un+1) introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5,  for n ≥ −1, and (u0, u1, · · · ) the sequence of elements in �  W that appear in any combinatorial piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Define  ShN (Zn(u0, · · · , un)) and ShN (Zn(u0, · · · , un+1)) using left LJ′  n+1-nilpotence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then HG,J = ShN (Z−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Fix n ≥ −1 and (u0, · · · , un) that appear as the first n + 1 terms of a combinatorial piece (so that  J0 = J, J1, J′  1, · · · , Jn+1, J′  n+1 are defined according the recipe in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have a stratification  Zn(u0, · · · , un) =  �  un+1∈  J′  n+1W Jn  Jn  Zn(u0, · · · , un, un+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Convention: W−1 := �  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We also have the cyclic reduction maps  cun+1 : Zn(u0, · · · , un, un+1) → Zn+1(u0, · · · , un, un+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any n ≥ −1 , F ∈ Sh(Zn(u0, · · · , un)) lies in ShN (Zn(u0, · · · , un)) if and only if for all  un+1 ∈ J′  n+1W Jn  Jn , the sheaf F♭  un+1 ∈ Sh(Zn+1(u0, · · · , un, un+1)) corresponding to F|Zn(u0,··· ,un,un+1) un-  der cyclic reduction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', c∗  un+1F♭  un+1 ∼= F|Zn(u0,··· ,un,un+1)) lies in ShN (Zn+1(u0, · · · , un+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  53  Proof of Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, let �  Zn(u0, · · · , un) = P ′u  n+1\\LJn/P u  n+1, and let �  Zn(u0, · · · , un+1) be the preimage  of Zn(u0, · · · , un+1) in �  Zn(u0, · · · , un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then �  Zn(u0, · · · , un+1) for various un+1 give a stratification of  �  Zn(u0, · · · , un) that is stable under the left translation by LJ′  n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By definition, F ∈ ShN (Zn(u0, · · · , un))  if and only if its pullback �F on �  Zn(u0, · · · , un) is left LJ′  n+1-nilpotent, which happens if and only if  �F| �  Zn(u0,··· ,un+1) is left LJ′  n+1-nilpotent for all un+1, if and only if F|Zn(u0,··· ,un+1) ∈ ShN (Zn(u0, · · · , un+1))  for all un+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4, F|Zn(u0,··· ,un+1) ∈ ShN (Zn(u0, · · · , un+1)) if and only if F♭  un+1 ∈ ShN (Zn+1(u0, · · · , un+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  We continue with the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that for each combinatorial piece (Jn, J′  n, un) ∈ SJ,  Jn will stabilizes for n ≥ r, where r is the semisimple rank of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Each u  J ∈ SJ gives an (r + 1)-tuple  (u0, · · · , ur), and by the above remark, u is determined by (u0, · · · , ur).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For each u ∈ SJ, we define  F♭  u ∈ ShN (Zr(u0, · · · , ur)) to be the result of restricting and apply cyclic reduction successively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In other  words, under the unipotent gerbe πJ,u : Y( u  J ) → Zr(u0, · · · , ur), π∗  J,uF♭  u ≃ F|Y( u  J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Using the Claim in the  previous paragraph repeatedly for n = −1, 0, · · · , r − 1, we conclude that for F ∈ Sh(YJ), F ∈ ShN (YJ)  if and only if F♭  u ∈ Sh(Zr(u0, · · · , ur)) lies in ShN (Zr(u0, · · · , ur)), for all J-pieces u  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By definition, the  latter is the same as saying F|Y( u  J ) = i∗  J,uF lies in ShN (Y( u  J )) for all u  J ∈ SJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  For any locally closed substack Z ⊂ YJ that is a union of geometric J-pieces, we define ShN (Z) ⊂ Sh(Z)  to be the full subcategory consisting of F ∈ Sh(Z) such that i∗  J,uF ∈ ShN (Y( u  J )) whenever u  J ⊂ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='13, for Z = YJ, this definition of ShN (YJ) coincides with the old one which is HG,J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The full subcategories ShN (Z) for closed unions of geometric pieces Z ⊂ YJ give a  stratification structure on HG,J indexed by the poset (SJ, ≤J) (in the sense recalled in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2), such  that the strata category corresponding to u  J ∈ SJ is ShN (Y( u  J )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Semi-orthogonal decomposition of the cocenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this section, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4 and  its generalization Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4 which describes the cocenter hh(HG) up to taking “associated graded”  indexed by enhanced Newton points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Colimit of character sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J ⊂ft Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7, we have a canonical equivalence  Sh(Y( 1  J )) ≃ Sh(LJ/LJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By definition, we have  ShN (Y( 1  J )) ≃ ShN (LJ/LJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We remark that ShN (LJ/LJ) can be viewed as a version of the category of character sheaves on LJ  allowing sheaves with infinite-dimensional stalks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='13, the closed embedding iJ,1 : Y( 1  J ) ֒→ YJ gives a fully faithful embedding  ιJ = iJ,1∗ : ShN (LJ/LJ) ≃ ShN (Y( 1  J )) ֒→ HG,J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For J ⊂ J′ ⊂ft Ia, there is the induction functor of character sheaves defined by Lusztig:  IndJ′  J : ShN (LJ/LJ) → ShN (LJ′/LJ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Via the embeddings ιJ and ιJ′, the induction functor is intertwined with chJ′  J : HG,J → HG,J′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore  they induce a functor on colimits  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  ι : colimD ShN (LJ/LJ) → colimD HG,J  where D is the groupoid D◦/Ω introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Combining the above functor with the equiv-  alence to hh(HG) given by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11, we get  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  colimJ∈D ShN (LJ/LJ) → hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The functor ι is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  54  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  This is the specific statement we seek for this paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' it is an easy consequence of a general theorem  describing all of hh(HG) up to “taking associated graded”, which we will state and prove next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Below we  will take the main step towards such a description of hh(HG) by considering hh(HG◦, HG) first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Semi-orthogonal decomposition of hh(HG◦, HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this section, we arrive at our first approximation  to the main goal, as encapsulated in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Instead of the cocenter hh(HG), we consider hh(HG◦, HG), which is equivalent to colimJ⊂ftIa HG,J by  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By the discussion in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12, HG decomposes into the direct sum of HG◦-bimodules  Hω  G according to the connected components of G indexed by ω ∈ Ω, we have a decomposition  hh(HG◦, HG) =  �  ω∈Ω  hh(HG◦, Hω  G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For �ν ∈ �  NP, recall the essential part B♥  �ν ⊂ B�ν, whose J-facets are indexed by the subset S♥  J,�ν ⊂ SJ  consisting of u  J such that �ν(u) = �ν and ℓ(u) = ⟨2ρ, ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall Tot(S♥  �ν ) = �  J⊂ftIa S♥  J,�ν is a poset as defined  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, and the order is opposite to the closure order of facets in B♥  �ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' If u  J ≤ u′  J′ in Tot(S♥  �ν ), we  have the functor  γJ′,u′  J,u  : ShN (Y( u  J )) → ShN (Y( u′  J′ )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  defined in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11 (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9), and here we restrict to sheaves with nilpotent singular support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Using the functors γJ′,u′  J,u  we may form the colimit  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  colim u  J ∈Tot(S♥  �ν ) ShN (Y( u  J )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For �ν = (0, 0), Tot(S♥  �ν ) consists of 1  J for J ⊂ft Ia, and can be identified with the poset D◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this case,  the above colimit is the same as colimJ⊂ftIa ShN (LJ/LJ) using the induction functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Fix ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then hh(HG◦, Hω  G) admits a semi-orthogonal decomposition indexed by  non-negative integers  hh(HG◦, Hω  G)0 ֒→ hh(HG◦, Hω  G)≤1 ֒→ · · · hh(HG◦, Hω  G)≤n ֒→ · · · ֒→ hh(HG◦, Hω  G) =  �  n≥0  hh(HG◦, Hω  G)≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In particular, each inclusion hh(HG◦, Hω  G)≤n ֒→ hh(HG◦, Hω  G) extends to a recollement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For each n ≥ 0, the n-th associated graded category hh(HG◦, Hω  G)n has the following description: it is  the direct sum  hh(HG◦, Hω  G)n ≃  �  �ν=(ν,ω)∈ �  NP,⟨2ρ,ν⟩=n  hh(HG◦, Hω  G)ν,  and for �ν = (ν, ω) ∈ �  NP,  hh(HG◦, Hω  G)ν = colim u  J ∈Tot(S♥  �ν ) ShN (Y( u  J )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In the argument we shall treat hh(HG◦, HG) as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is clear that the resulting semi-orthogonal  decomposition induces a semi-orthogonal decomposition for each summand hh(HG◦, Hω  G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Abbreviate HG,J by CJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let YJ,≤n be the union of geometric pieces Y( u  J ) with ℓ(u) ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9, YJ,≤n is closed in YJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let CJ,≤n = ShN (YJ,≤n) (the meaning of ShN is defined in the paragraph  preceding Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11(1)(2), for J ⊂ J′ ⊂ft Ia, the functors chJ′  J send CJ,≤n to  CJ′,≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore we may form the colimit  C≤n = colimJ⊂ftIa CJ,≤n  using the functors chJ′  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For �ν ∈ �  NP, define YJ,�ν,♥ to be the union of Y( u  J ) where �ν(u) = �ν and ℓ(u) = ⟨2ρ, ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let YJ,≤n,♥ be  the substack of YJ,≤n  YJ,≤n,♥ = YJ,≤n−1 ∪ (  �  �ν=(ν,ω)∈ �  NP,⟨2ρ,ν⟩=n  YJ,ν,♥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  55  Again by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9, YJ,≤n,♥ is closed in YJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Define CJ,≤n,♥ = ShN (YJ,≤n,♥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11(1)(2),  chJ′  J sends CJ,≤n,♥ to CJ′,≤n,♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore we may form the colimit  C≤n,♥ = colimJ⊂ftIa CJ,≤n,♥  using the functors chJ′  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For each J ⊂ft Ia, we have natural inclusions  CJ,≤0,♥ ֒→ CJ,≤0 ֒→ CJ,≤1,♥ ֒→ · · · ֒→ CJ,≤n−1 ֒→ CJ,≤n,♥ ֒→ CJ,≤n ֒→ · · ·  These inclusions are compatible with the functors chJ′  J , hence we get functors between the colimits over  J:  C≤0,♥  κ0  −→ C≤0  i0  −→ C≤1,♥  κ1  −→ · · · → C≤n−1  in  −→ C≤n,♥  κn  −−→ C≤n → · · ·  For each J, we have CJ ≃ colimn CJ,≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Commuting the order of taking colimits, we have  hh(HG) ≃ colimJ∈D◦ CJ ≃ colimn C≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The assertion of the theorem will follow from the two claims below:  (1) For n ≥ 0, the functor in : C≤n−1 → C≤n,♥ is fully faithful and extends to a recollement diagram  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  Cn,♥  jn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' �  jn∗ �  C≤n,♥  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  n=j∗  n  �  i∗  n  �  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  n  �  C≤n−1  in  �  where the category Cn,♥ is canonically equivalent to the direct sum of hh(HG◦, Hω  G)ν defined using  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3), for ω ∈ Ω and ⟨2ρ, ν⟩ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) For n ≥ 0, the functor κn : C≤n,♥ → C≤n is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We first prove Claim (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let �  NPn be the set of �ν = (ν, ω) ∈ �  NP such that ⟨2ρ, ν⟩ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For J ⊂ft Ia,  let  YJ,n,♥ =  �  �ν∈ �  NPn  YJ,�ν =  �  �ν∈ �  NPn  \uf8eb  \uf8ec  \uf8ed  �  u  J ∈S♥  J,�ν  Y( u  J )  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let CJ,n,♥ = ShN (YJ,n,♥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The decomposition of YJ,n,♥ above gives a decomposition  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  CJ,n,♥ =  �  �ν∈ �  NPn  CJ,�ν,♥ =  �  �ν∈ �  NPn  \uf8eb  \uf8ec  \uf8ed  �  u  J ∈S♥  J,�ν  ShN (Y( u  J ))  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then CJ,≤n,♥ carries a recollement structure  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)  CJ,n,♥  jJ,n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' �  jJ,n∗ �  CJ,≤n,♥  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  J,n=j∗  J,n  �  i∗  J,n �  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  J,n �  CJ,≤n−1  iJ,n  �  For J ⊂ J′ ⊂ft Ia,  u  J ∈ S♥  �ν,J,  u  J is quasi-J′-reduced since ℓ(u) = ℓ(u′) = ⟨2ρ, ν⟩ if  u′  J′ = σJ′  J ( u  J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11(2), the functor chJ′  J  respects the recollement structure on CJ,≤n,♥ and induces the  functor ⊕ u  J ∈S♥  J,�νγJ′,u′  J,u  : CJ,n,♥,�ν) → CJ′,n,♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1 of Appendix A, we conclude that that  the colimit C≤n,♥ = colimJ⊂ftIa CJ,≤n,♥ also has a recollement structure by taking termwise colimits of  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This gives the recollement diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Moreover, the category Cn,♥ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) is the direct  sum over �ν ∈ �  NPn of C�ν,♥ := colimJ⊂ftIa CJ,�ν,♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5) and the description of chJ′  J on CJ,n,♥ in terms  of γJ′,u′  J,u , we conclude that Cν,♥ is canonically equivalent to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we prove Claim (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We fix n ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Σ<n ⊂ E× �  NP be the set of pairs ([E], �ν) ∈ E× �  NP such  that ⟨2ρ, ν⟩ < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then Σ<n has a partial order ([E], �ν) ≤ ([E′], �ν) if [E] ≤ [E′] in E (no comparison if the  Newton points are different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We extend this partial order to a total order on Σ<n and denote it by ⪯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  56  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  For J ⊂ft Ia, let SJ,(n,[E],�ν) be the set of combinatorial pieces u  J ∈ SJ such that ℓ(u) = n, τ( u  J ) = [E]  and �ν(u) = �ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For ([E], �ν) ∈ Σ<n, define a finite subset of SJ by  SJ,⪯(n,[E],�ν) = SJ,≤n,♥ ∪  \uf8eb  \uf8ed  �  Σ<n∋([E′],�ν′)⪯([E],�ν)  SJ,(n,[E′],�ν′)  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8, for J ⊂ J′ ⊂ft Ia, the map δJ′  J  send SJ,⪯(n,[E],�ν) to SJ′,⪯(n,[E],�ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore the  assignment J �→ SJ,⪯(n,[E],�ν) gives a D◦-subset S⪯(n,[E],�ν) of S�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let  B⪯(n,[E],�ν) = |S⪯(n,[E],�ν)| ⊂ B�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let SJ,≺(n,[E],�ν) = SJ,⪯(n,[E],�ν) \\ SJ,(n,[E],�ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Thus we get a D◦-set S≺(n,[E],�ν) and its geometric realization  B≺(n,[E],�ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let YJ,⪯(n,[E],�ν) ⊂ YJ be the union of geometric pieces Y( u  J ) where u  J ∈ SJ,⪯(n,[E],�ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9, YJ,⪯(n,[E],�ν) is a closed substack of YJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly we define the closed substack YJ,≺(n,[E],�ν) ⊂ YJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The category of nilpotent sheaves  CJ,⪯(n,[E],�ν) := ShN (YJ,⪯(n,[E],�ν)),  CJ,≺(n,[E],�ν) := ShN (YJ,≺(n,[E],�ν))  are defined as in the paragraph preceding Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11 implies that chJ′  J  sends  CJ,⪯(n,[E],�ν) to CJ′,⪯(n,[E],�ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We form the colimit  C⪯(n,[E],�ν) := colimJ⊂ftIa CJ,⪯(n,[E],�ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Similarly define C≺(n,[E],�ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since ⪯ is a total order, C≺(n,[E],�ν) = C⪯(n,[E′],�ν′) if ([E′], �ν′) is a predecessor  of ([E], �ν), or if ([E], �ν) is the minimal element of Σ<n, C≺(n,[E],�ν) ≃ C≤n,♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Since CJ,≤n = colim([E],�ν)∈Σ<n CJ,⪯(n,[E],�ν), taking colimit over J we get  C≤n ≃ colim([E],�ν)∈Σ<n C⪯(n,[E],�ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To show that C≤n,♥ → C≤n is an equivalence, it suffices to show that for each ([E], �ν) ∈ Σ<n, the natural  functor  κ[E],�ν : C≺(n,[E],�ν) → C⪯(n,[E],�ν)  is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For this we apply the contraction principle for cosheaves on posets that we proved in  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, in the form of Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To set the stage for applying Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, we define a poset I whose underlying set is  I := D◦ ⊔ Tot(S(n,[E],�ν)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let f : I → D◦ be the projection that is the identity on D◦ and the natural projection on Tot(S(n,[E],�ν)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let s : D◦ ֒→ I be the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For the partial order, we include all order relations from D◦ and  Tot(S(n,[E],�ν)), and take the partial order generated by the following extra relations:  For any J ∈ D◦ and u  J ∈ SJ,(n,[E],�ν), we declare s(J) < u  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For any J ⊂ J′ in D◦, u  J ∈ SJ,(n,[E],�ν) such that δJ′  J ( u  J ) ∈ SJ′,≺(n,[E],�ν), we declare u  J < s(J′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By construction, f is a coCartesian map of posets, and every fiber f −1(J) has a unique minimal element  s(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We next define a cosheaf F on I in the sense of Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For J ∈ D◦, let Fs(J) = CJ,≺(n,[E],�ν) = ShN (YJ,≺(n,[E],�ν)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For u  J ∈ SJ,(n,[E],�ν), let F u  J = ShN (YJ,≺(n,[E],�ν) ∪ Y( u  J )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that Y≺(n,[E],�ν) ∪ Y( u  J ) is closed in  YJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The transition functors are defined as follows:  For J ⊂ J′ ∈ D◦, the functor Fs(J) → Fs(J′) is given by chJ′  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For J ∈ D◦ and u  J ∈ SJ,(n,[E],�ν), the functor Fs(J) → F u  J is ι∗ : ShN (YJ,≺(n,[E],�ν)) ֒→ ShN (YJ,≺(n,[E],�ν)∪  Y( u  J )) for the closed embedding ι : YJ,≺(n,[E],�ν) ֒→ YJ,≺(n,[E],�ν) ∪ Y( u  J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  57  For u  J ≤ u′  J′ in Tot(S(n,[E],�ν)) (which means δJ′  J ( u  J ) = u′  J′ ), the functor F u  J → F u′  J′ is given by chJ′  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here we use Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11(2) to verify that chJ′  J  indeed sends sheaves supported on Y( u  J ) to  sheaves supported on Y( u′  J′ ) since ℓ(u) = ℓ(u′) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For u  J < s(J′), where J ⊂ J′, u  J ∈ SJ,(n,[E],�ν) and δJ′  J ( u  J ) ∈ SJ′,≺(n,[E],�ν), the functor F u  J → Fs(J′)  is again given by chJ′  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Here we again use Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11(1)(2) to verify that chJ′  J indeed sends  sheaves supported on Y( u  J ) either to sheaves supported on YJ′,≤n−1 ⊂ YJ′,≺(n,[E],�ν) if ℓ(u′) <  ℓ(u) = n, or to sheaves supported on Y( u′  J′ ) ⊂ YJ′,≺(n,[E],�ν) if ℓ(u′) = ℓ(u) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By construction, s∗F is the cosheaf on D◦ whose value at J is CJ,≺(n,[E],�ν) and transition functors are  given by chJ′  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7)  C≺(n,[E],�ν) = colimD◦ s∗F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  On the other hand, we claim that for any J ∈ D◦, (  �  f F)J ≃ CJ,⪯(n,[E],�ν) and this assignment extends to  an equivalence of cosheaves on D◦ (for the definition of  �  f F for a coCartesian map f, see Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Indeed, since f −1(J) = {s(J)} ∪ SJ,(n,[E],�ν) as a poset has s(J) as the minimal element and no extra  relations besides those already in SJ,(n,[E],�ν), (  �  f F)J is the pushout of Fs(J) diagonally embedded into  ⊕F u  J (sum over all u  J ∈ SJ,(n,[E],�ν)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In other words, (  �  f F)J is the pushout of CJ,≺(n,[E],�ν) diagonally  embedded into ⊕ShN (YJ,≺(n,[E],�ν) ∪ Y( u  J )) for all  u  J ∈ SJ,(n,[E],�ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The latter is precisely CJ,⪯(n,[E],�ν)  because of the open-closed decomposition  YJ,⪯(n,[E],�ν) =  \uf8eb  \uf8ed  �  u  J ∈SJ,(n,[E],�ν)  Y( u  J )  \uf8f6  \uf8f8 ⊔ YJ,≺(n,[E],�ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The transition maps (  �  f F)J → (  �  f F)J′ are given by chJ′  J , therefore we get  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8)  colimD◦(  �  f  F)J ≃ colimD◦ CJ,⪯(n,[E],�ν) = C⪯(n,[E],�ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we would like to apply Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 to the situation of f : I → D◦ with section s, and the cosheaf  F on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The conclusion would be that the natural map colimD◦ s∗F → colimD◦ �  f F is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Combining with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8), we conclude that  κ[E],�ν : C≺(n,[E],�ν) = colimD◦ s∗F → colimD◦  �  f  F = C⪯(n,[E],�ν)  is an equivalence, proving Claim (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  It thus remains to check that the conditions for applying Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The first condition: we define L to be the cosheaf on Tot(S(n,[E],�ν)) = I \\ s(J ) whose value at u  J is  ShN (Y( u  J )), and the transition functors are given by chJ′  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have a natural map β : F|Tot(S(n,[E],�ν)) → L  termwise given by open restriction ShN (YJ,≺(n,[E],�ν) ∪ Y( u  J )) → ShN (Y( u  J )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  With this definition of L, we need to check that  L  F|Tot(S(n,[E],�ν))  β  �  (s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='s∗F)|Tot(S(n,[E],�ν))  α  �  is a recollement of cosheaves on Tot(S(n,[E],�ν)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2, s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='s∗F ≃ f ∗s∗F, therefore we can replace  the rightmost term above by (f ∗s∗F)|Tot(S(n,[E],�ν)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Termwise at u  J ∈ SJ,(n,[E],�ν), they fit into a recollement  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9)  ShN (Y( u  J ))  �  � ShN (YJ,≺(n,[E],�ν) ∪ Y( u  J ))  �  �  � ShN (YJ,≺(n,[E],�ν))  �  by the open-closed decomposition Y( u  J ) ∪ YJ,≺(n,[E],�ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' All functors above are continuous by Proposi-  tion B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, therefore (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9) is a recollement in StL  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For u  J ≤ u′  J′ in Tot(S(n,[E],�ν)), chJ′  J induces a morphism  of between the recollement (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9) and its counterpart for u′  J′ , which follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This  verifies the first condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  58  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  The second condition: by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11(3), L is locally constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The third condition: we need to check that |s(D◦)| ⊂ |I| is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let K :=  Tot(S⪯(n,[E],�ν));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' K′ := Tot(S≺(n,[E],�ν)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then I \\ D◦ = Tot(S(n,[E],�ν)) = K \\ K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any D◦-set X, let  D◦ ⊳ X be the poset whose underlying set is D◦ ⊔ X with the added relation J < x for any x ∈ XJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then  |D◦ ⊳ X| is homeomorphic to the mapping cylinder [0, 1]× |X| �  {0}×|X| |D◦| of the projection |X| → |D◦|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we can write I as a pushout of posets  I ∼= (D◦ ⊳ K)  �  D◦⊳K′  D◦  where the map D◦ ⊳K′ → D◦ is the identity on D◦ and the projection on K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under this presentation, the  inclusion s : D◦ ֒→ I corresponds to the inclusion of the second factor D◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Taking geometric realizations  we get  |I| ∼= ([0, 1] × |K|)  �  {0}×|K|∪[0,1]×|K′|  |D◦|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To show |D◦| ֒→ |I| is a homotopy equivalence, it suffice to show {0} × |K| ∪ [0, 1] × |K′| ֒→ [0, 1] × |K| is  a homotopy equivalence, or |K′| ֒→ |K| is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, |K| is a subdivision of B�ν,⪯(n,[E]), and |K′| is a subdivision of B�ν,≺(n,[E]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore it  suffices to check that the inclusion B�ν,≺(n,[E]) ֒→ B�ν,⪯(n,[E]) is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Both B�ν,≺(n,[E])  and B�ν,⪯(n,[E]) are downward subsets of B�ν in the sense of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='18 (see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There-  fore by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='20, the inclusions Crit(f)�ν ֒→ |B�ν,≺(n,[E])| and Crit(f)�ν ֒→ B�ν,⪯(n,[E]) are both  homotopy equivalences, hence the inclusion B�ν,≺(n,[E]) ֒→ B⪯(n,[E]) is also a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This  verifies the second condition and completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We expect that hh(HG) has a recollement structure indexed by the stronger poset ( �  NP, ≤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Under Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, this recollement structure should correspond to the Harder-Narasimhan stratifi-  cation on BunG(E), which is also indexed by �  NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Semi-orthogonal decomposition of hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now consider hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By the discussion in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12,  hh(HG) again decomposes into a direct sum hh(HG)ω indexed by ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Identify Ω with length zero elements in �  W, then Ω acts on �  W by conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The length function  ℓ : �  W → Z≥0 and the enhanced Newton point function �ν : �  W → �  NP are Ω-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore for  J ⊂ft Ia and ω ∈ Ω, we have a bijection  dω : SJ  ∼  → Sω(J)  sending u  J to ωuω−1  ω(J) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For �ν ∈ �  NP, the map dω induces a bijection  S♥  J,�ν  ∼  → S♥  ω(J),�ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  This induces an action of Ω on the D◦-set S♥  �ν , and on the total set Tot(S♥  �ν ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We can form the groupoid  [Tot(S♥  �ν )/Ω] whose objects are Tot(S♥  �ν ) and morphisms between u  J and u′  J′ ∈ Tot(S♥  �ν ) is the set of ω ∈ Ω  such that dω( u  J ) = u′  J′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The action of cω on HG (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5) induces an equivalence HG,J → HG,ω(J), which restricts to  an equivalence ShN (Y( u  J ))  ∼  → ShN (Y(dω( u  J ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore, the assignment u  J �→ ShN (Y( u  J )) also extends  to a functor  ShN (Y(−)) : [Tot(S♥  �ν )/Ω] → StL,  u  J �→ ShN (Y( u  J )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We can then form the colimit  colim u  J ∈[Tot(S♥  �ν )/Ω] ShN (Y( u  J )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For �ν = (0, 0), [Tot(S♥  �ν )/Ω] can be identified with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In this case, the above colimit is the same as  colimJ∈D ShN (LJ/LJ) under the induction functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  59  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For each ω ∈ Ω, the category hh(HG)ω admits a semi-orthogonal decomposition indexed  by non-negative integers  hh(HG)ω  0 ֒→ hh(HG)ω  ≤1 ֒→ · · · hh(HG)ω  ≤n ֒→ · · · ֒→ hh(HG)ω =  �  n≥0  hh(HG)ω  ≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In particular, each inclusion hh(HG)ω  ≤n ֒→ hh(HG)ω extends to a recollement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For n ≥ 0, the n-th associated graded category hh(HG)n has the following description: it is the direct  sum  hh(HG)ω  n ≃  �  �ν=(ν,ω)∈ �  NP,⟨2ρ,ν⟩=n  hh(HG)�ν  where, for �ν ∈ �  NP,  hh(HG)�ν = colim u  J ∈[Tot(S♥  �ν )/Ω] ShN (Y( u  J )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In particular, the functor ι from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) is identified with the embedding hh(HG)0 ≃ colimJ∈D ShN (LJ/LJ) ֒→  hh(HG)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, ι is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Again we will prove the statement for the whole hh(HG), which implies the statement for each sum-  mand hh(HG)ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7 we have hh(HG) ≃ hh(HG◦, HG)Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Each filtration step hh(HG◦, HG)≤n  in hh(HG◦, HG) constructed in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4 is stable under Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, taking Ω-coinvariants  (which is the same as taking colimits over BΩ) gives a semi-orthogonal decomposition of hh(HG) with fil-  tration pieces  hh(HG)≤n ≃ (hh(HG◦, HG)≤n)Ω  and associated graded category  hh(HG)n ≃ (hh(HG◦, HG)n)Ω  By the description of hh(HG◦, HG)n given in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4, we see hh(HG)n is a direct sum over those  �ν = (ν, ω) ∈ �  NP (such that ⟨2ρ, ν⟩ = n) of  (hh(HG◦, Hω  G)ν)Ω ≃  �  colim u  J ∈Tot(S♥  �ν ) ShN (Y( u  J ))  �  Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The same argument as in the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7 allows us to rewrite the right side as a colimit over  the groupoid [Tot(S♥  �ν )/Ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Endomorphisms of Whittaker functional  Throughout this section, we assume the coefficient field k is algebraically closed and char(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Fix τ a complex number with imaginary part Im(τ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G be a reductive group  over C with maximal torus T and t = LieT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this section, we shall adopt the following notations:  (1)  Φ the set of roots of (G, T );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �Φ = Z × Φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �Φ = Z × Z × Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Each α ∈ Φ, �Φ or �Φ defines an affine linear function on t as follows  α ∈ Φ, defines a linear functional α : t → C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  α = (n, α) ∈ �Φ defines an affine linear function α := α + n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  α = (n, m, α) ∈ �Φ defines an affine linear function α := α + n + mτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Under the above notations, let Hα = {x ∈ t : α(x) = 0}, and sα the reflection about Hα defined  by sα(x) = x − α(x)α∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For α ∈ Φ, put gα ⊂ g the root space, and for α = (n, α) ∈ �Φ or α = (n, m, α) ∈ �Φ, put  gα = gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (3) For J ⊂ Φ, �Φ or �Φ, put ǫJ = ∩α∈SHα, and define the following sets of affine subspaces of t  S = {ǫJ : J ⊂ Φ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  60  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  �S = {ǫJ : J ⊂ �Φ} 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �S = {ǫJ : J ⊂ �Φ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (4)  W the Weyl group of (G, T ), it naturally acts t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  W = X∗(T ) ⋊ W, it acts on t via (λ, w) · x = λ + w(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  W = (X∗(T ) × X∗(T )) ⋊ W (with the diagonal action of W), it acts on t via (λ1, λ2, w) · x =  λ1 + τλ2 + w(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (5)  For ǫ ∈ S, put Φǫ := {α ∈ Φ : α(ǫ) = 0}, and Wǫ = ⟨sα : α ∈ Φǫ⟩ ⊂ W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For ǫ ∈ �S, put Φǫ := {α ∈ �Φ : α(ǫ) = 0}, and Wǫ = ⟨sα : α ∈ Φǫ⟩ ⊂ �  W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For ǫ ∈ �S, put Φǫ := {α ∈ �Φ : α(ǫ) = 0}, and Wǫ = ⟨sα : α ∈ Φǫ⟩ ⊂ �  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For ǫ ∈ S, �S, or �S, put Lǫ the connected reductive subgroup of G containing T with roots Φǫ,  so that it has Weyl group Wǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (6)  For ǫ ∈ S, put W ǫ = NW (Wǫ)/Wǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For ǫ ∈ �S, put �  W ǫ = N�  W (Wǫ)/Wǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For ǫ ∈ �S, put �  W ǫ = N�  W (Wǫ)/Wǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (7) Put CG the set of isomorphism classes of cuspidal sheaves on NG/G, where NG ⊂ g is the nilpotent  cone of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For definition, see [Lus87, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  C = {(ǫ, B, F) : ǫ ∈ S, F ∈ CLǫ, B ⊃ T a Borel subgroup of Lǫ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �C = {(ǫ, B, F) : ǫ ∈ �S, F ∈ CLǫ, B ⊃ T a Borel subgroup of Lǫ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �C = {(ǫ, B, F) : ǫ ∈ �S, F ∈ CLǫ, B ⊃ T a Borel subgroup of Lǫ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (8) We also use the notation WG, SG, CG, etc, to emphasis the dependence on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (9) For J ⊂ Φ, �Φ or �Φ, put LJ = LǫJ, WJ = WǫJ, W J = W ǫJ, zJ = X∗(Z(LJ)◦) ⊗Z k, CJ = CLJ,  �CJ = �CLJ, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Combinatorial descriptions of character sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this section, we shall recall the combina-  torial description of the dg category of character sheaves in [Lia] (for simply-connected group), and [Lib]  (for reductive group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Using the same idea, we give a combinatorial description of the colimit category  colimJ∈D ShN (LJ/LJ), the latter may be thought of as an elliptic version of character sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The main  results and the relationship between various versions of character sheaves is summarized in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Sheaves of categories on affine spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Λ be a free Z-module of finite rank, and ΛC be com-  plexified affine space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let F be a locally finite set of complex affine subspaces of ΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any ǫ ∈ F,  denote ǫ the linear space parallel to ǫ, and zǫ = (ǫ ∩ Λ) ⊗Z k the finite dimensional affine space over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Put QCohǫ the constant sheaf of categories on ǫ with value QCoh(z∗  ǫ[−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a map of sets f : G → F,  denote QCohG = �  c∈G QCohf(c), viewed as a sheaf of categories on ΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let K be a discrete group of affine  linear transformations acting properly discontinuously on ΛC, assume that the linear part of K preserves  Λ, and F is stable under the K-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assume also that K acts on G, such that f is K-equivariant, then  QCohG is naturally a K-equivariant sheaf of categories on ΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore, for any K-invariant open subset  U ⊂ ΛC, we have natural K-action on Γ(U, QCohG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Denote the invariant category by Γ(U, QCohG)K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let A be an dg-algebra over k, and K be a discrete group acting strictly on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The  smash product k[K]#A is by definition a dg algebra whose underlying dg vector space is k[K] ⊗ A, and  the multiplication is given by (w ⊗ a) · (w′ ⊗ a′) = (ww′ ⊗ w′−1(a)a′), for all w ∈ K and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is  natural equivalence of dg categories (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' [Lia, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2]):  (k[K]#A)-mod ≃ (A-mod)K  Using smash product, we can write Γ(ΛC, QCohG)K more concretely as follows: let G � K be the set of  K-orbits on G, and let [G � K] ⊂ G be a set of representatives of each orbit, and denote Kc the stabilizer  4Note that by definition �S is in natural bijection with the set E of relevant affine subspaces in the standard apartment  A = tR, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  61  of K at c ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then we have equivalence of categories:  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  Γ(ΛC, QCohG)K ≃  �  c∈[G�K]  (Sym(zf(c)[1])-mod)Kc ≃  �  c∈[G�K]  k[Kc]#Sym(zf(c)[1])-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let K′ ⊂ K be a subgroup, and G′ ⊂ G be a K′-stable subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have a pair of adjoint functors:  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  IndG/K  G′/K′ : Γ(ΛC, QCohG′)K′ ⇄ Γ(ΛC, QCohG)K : ResG/K  G′/K′  And when the context is clear, we shall simply denote them by Ind, Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we take Λ = X∗(T ), and hence ΛC = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In Notation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, the groups W, �  W and �  W naturally act  on S, �S and �S respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any w ∈ W, �  W or �  W, denote its image in W by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore Ad(w)  gives idenfications Lǫ ≃ Lw(ǫ), and CLǫ ≃ CLw(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This gives an action of W, �  W and �  W on C, �C and �C  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let f : C → S, �f : �C → �S and �f : �C → �S be the projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore the categories  Γ(t, QCohC)W , Γ(t, QCoh�C)  �  W , and Γ(t, QCoh�C)  �  W  are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We have the following combinatorial description of character sheaves on G and g:  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G be a connected reductive group over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There are equivalences of categories:  (1) Lg : ShN (g/G) ≃ Γ(t, QCohC)W ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) LG : ShN (G/G) ≃ Γ(t, QCoh�C)�  W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (1) This is essentially the generalized Springer decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We use notations from Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  in particular, I denotes the set of simple roots of G with respect to T and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  D := {(J, F) : J ⊂ I, F ∈ CJ = CLJ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By [Lia, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2], we have an equivalence of categories:  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  ShN (g/G) ≃  �  (J,F )∈D  k[W J]#Sym(zJ[1])-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let �f : D → C be the map sending (J, F) to (ǫJ, B∩LJ, F), and let f be the composition D  �  f−→ C → C�W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then f is bijective by [Lia, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1] (the content here is: if LJ carries a cuspidal local system, then  any two parabolics containing LJ as a Levi subgroup are conjugate in G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' One can further identify W J  with Wc, the stabilizer of c = f(J, F) under the W-action on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore, (1) holds by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) is proved in [Lib, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  In view of Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 implies the following explicit description of the category of  character sheaves on the group G, analogous to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any connected reductive group G over C, there is a canonical equivalence  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  ShN(G/G) ≃  �  {(J,F ):J⊂ftIa,F ∈CJ}/Ω  k[�  W J]#Sym(zJ[1])-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let F ∈ ShN (G/G) be an object whose image on the right side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5) lies in a single  summand indexed by (J, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the support of F is contained in the closed subset of G consisting of  elements whose semisimple parts lie in Ad(G)(exp(ǫJ)), where the exponential map exp : t → T normalized  to have kernel X∗(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we explain the relationship between character sheaves on the group and on the Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There  is a natural functor β1 : ShN(G/G) → ShN (g/G) defined as follows: let D be a Ad(G)-invariant star-  shaped open subset of g such that exp : D → G is an isomorphism onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let j : D/G ֒→ g/G be  the open inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then j∗ : ShN (g/G) → ShN (D/G) is an equivalence of categories, and we put  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)  β1 = (j∗)−1 ◦ exp |∗  D/G : ShN (G/G) → ShN (g/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The functor β1 perserves limit, and therefore admits a left adjoint α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  62  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under the equivalence in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, the diagrams naturally commute:  ShN (g/G)  α1  �  ≃  Lg  � Γ(t, QCohC)W  Ind �  ShN(G/G)  ≃  LG  �  β1  �  Γ(t, QCoh�C)�  W  Res  �  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This follows from the proof of [Lib, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1]: under the notation loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', the functor  ShN (G/G) → ShN (g/G) is identified with the natural functor limI∈FG ChG → ChG(I = 0), which is  identified with the functor limI∈FG QCoh  �  W  �C → QCoh  �  W  �C (I = 0), which is then further identified with Res :  Γ(t, QCoh�C)�  W → Γ(t, QCohC)W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The other commutative square follows by adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Now let P ⊃ T be a parabolic subgroup of G, with Levi subgroup L ⊃ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then we have a pair of  adjoint functors by parabolic induction and restriction:  IndG  L⊂P : ShN (L/L) ⇄ ShN (G/G) : ResG  L⊂P  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition ([Lib],Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under the equivalence in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3(2), the diagrams nat-  urally commute:  ShN (L/L)  IndG  L⊂P  �  ≃  LL  � Γ(t, QCoh�CL)�  WL  Ind  �  ShN (G/G)  ≃  LG  �  ResG  L⊂P  �  Γ(t, QCoh�CG)�  WG  Res  �  Next we will prove a double affine version of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Instead of considering character sheaves  on the Lie algebra or the group, we consider the colimit colimJ∈D ShN (LJ/LJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6  we have identified the initial filtered piece hh(HG)0 of hh(HG) with this colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' From the point of view of  Betti geometric Langlands, this colimit can be thought of as an elliptic version of character sheaves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=',  nilpotent sheaves on the semistable locus of BunG(E) for an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is an equivalence of categories  LG : colimJ∈D ShN (LJ/LJ) ≃ Γ(t, QCoh�C)  �  W  where the colimit is formed using the parabolic induction functors as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We may rewrite the colimit on the left side as a limit  lim  J∈Dopp ShN(LJ/LJ)  using the parabolic restriction functors that are right adjoint to the induction functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We view �  W as a subgroup of �  W in two ways: �  W = (X∗(T ) × 0) ⋊ W, and �  W τ = (0 × X∗(T )) ⋊ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Denote W a ⊂ �  W, W a,τ ⊂ �  W τ the corresponding affine Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let A := X∗(T )R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall we have  fixed an imaginary τ so that t = AC = A × Aτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a subset X ⊂ A, Xτ denotes the corresponding subset  of Aτ, which in turn is a subset of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The affine space A is stratified by facets given by connected components of intersections of affine  roots hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any J ⊂ft Ia, let AJ be the corresponding facet in the fundamental alcove, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=',  AJ = {x ∈ A|α(x) = 0, β(x) > 0, for all α ∈ J, β ∈ Ia \\ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let star(AJ) ⊂ A be the star of AJ, namely,  star(AJ) is the union of facets whose closures contain AJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Denote the open subset:  VJ := A × star(AJ)τ ⊂ A × Aτ = AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We have an isomorphism of topological groupoids  colimJ∈D◦ star(AJ)τ/WJ = Aτ/W a,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  63  Dividing Ω on both side, we get an isomorphism of topological groupoids  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7)  colimJ∈D star(AJ)τ/WJ ≃ Aτ/�  W τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let X∗(T ) ⋊ W τ  J ⊂ �  W be the subgroup generated by X∗(T ) × 0 and W τ  J ⊂ W a,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Abstractly, it is  isomorphic to the semi-direct product X∗(T ) ⋊ WJ ≃ �  WJ, the extended affine Weyl group of LJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7) induces an isomorphism of topological groupoids  colimJ∈D VJ/(X∗(T ) ⋊ W τ  J ) ≃ t/�  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Since all arrows in the above colimit are open embeddings, it further induces an equivalence  Γ(t, QCoh�C)  �  W ≃ Γ(t/�  W, QCoh�C)  ≃  lim  J∈Dopp Γ(VJ/(X∗(T ) ⋊ W τ  J ), QCoh�C)  ≃  lim  J∈Dopp Γ(VJ, QCoh�C)X∗(T )⋊W τ  J  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8)  On the other hand, choose any x ∈ AJτ, and put tx : t → t the translation by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then there is an  canonical isomorphism of sheaves on VJ  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9)  tx,∗(QCoh�CJ)|VJ ≃ QCoh�C|VJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  This is because affine spaces in �C having nonempty intersections with VJ are precisely those of the form  tx(ǫ), for ǫ in �CJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The isomorphism (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9) is naturally X∗(T ) ⋊ W τ  J ≃ �  WJ equivariant, and induces an  equivalence on global sections:  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10)  Γ(VJ, QCoh�C)X∗(T )⋊W τ  J ≃ Γ(t−1  x (VJ), QCoh�CJ)  �  WJ ≃ Γ(t, QCoh�CJ)  �  WJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here, the second equivalence is because t−1  x (VJ) is convex and has nonempty intersection with each affine  space in �CJ (and hence the intersection is contractible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Moreover for J ⊂ J′ ⊂ft Ia, under (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10) the  restriction map:  Γ(VJ, QCoh�C)X∗(T )⋊W τ  J → Γ(VJ′, QCoh�C)X∗(T )⋊W τ  J′  is identified with the restriction map defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2):  Res  �CJ′/�  WJ′  �CJ/�  WJ  : Γ(t, QCoh�CJ)  �  WJ → Γ(t, QCoh�CJ′ )  �  WJ′  It is also clear that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10) is equivariant under the action of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore, the functor Dopp → StR  k  defined by J �→ Γ(VJ, QCoh�C)X∗(T )⋊W τ  J can be identified with the functor J �→ Γ(t, QCoh�CJ )�  WJ (using the  restriction functors and the Ω-action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In summary, we have  Γ(t, QCoh�C)  �  W ≃  lim  J∈Dopp Γ(VJ, QCoh�C)X∗(T )⋊W τ  J  by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8)  ≃  lim  J∈Dopp Γ(t, QCoh�CJ)  �  WJ  by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10)  ≃  lim  J∈Dopp ShN(LJ/LJ)  by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 (2) and Prop 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11)  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8, the diagrams naturally commutes:  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12)  ShN (g/G)  α1  �  ≃  Lg  �  ShN (G/G)  α2  �  ≃  LG �  colimJ∈D ShN(LJ/LJ)  ≃  LG �  Γ(t, QCohC)W  Ind1 � Γ(t, QCoh�C)�  W  Ind2  � Γ(t, QCoh�C)�  W  where α2 is the natural map into the colimit corresponding to J = I, and Ind1, Ind2 are the induction  functors defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  64  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The first square is commutative by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For the second square, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11), the right  adjoint of α2 corresponds to the restriction map Γ(t, QCoh�C)�  W → Γ(VI, QCoh�C)�  W , and the latter can be  identified with Γ(t, QCoh�C)�  W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Passing to left adjoints gives the commutativity of the second square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Fourier-Sato transform and Whittaker sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this section, we study the Lie algebra version  of the Whittaker sheaf, and identify its image under the equivalence in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The main tool  here is the Fourier-Sato transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Consider the diagram  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  Ga  u−/U −  f  �  r− � g/G  where r− is induced by the inclusion u− ֒→ g, and f = dχ : u− → Ga is the differential of the generic  character χ : U − → Ga used to define the Whittaker functor on the group, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) The Whittaker functor on ShN (g/G) is the functor  Wg/G : ShN (g/G) → k-mod,  Wg/G(F) = ϕf,0 ◦ r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  −F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The Whittaker sheaf on the Lie algebra is the object Whg/G ∈ ShN (g/G) corepresenting the  Whittaker functor Wg/G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=',  Wg/G(F) ≃ HomShN (g/G)(Whg/G, F),  for all F ∈ ShN (g/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let  T : ShN (g/G) ≃ Sh(N/G)  be the Fourier-Sato transform, normalized to be t-exact with respect to the perverse t-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let e ∈ u be the regular nilpotent element corresponding to f ∈ (u−)∗ under the  invariant form on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let ie : {e} → N/G be the natural map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then there is a natural commutative  diagram:  ShN (g/G) �  T  Wg/G  �◆  ◆  ◆  ◆  ◆  ◆  ◆  ◆  ◆  ◆  ◆  Sh(N/G)  i∗  e[r]  �  k-mod  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since f is linear, we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  ϕf,0(K) = ϕid,0(f∗(K))  for any Gm-monodromic sheaf K on u− (where Gm acts by dialition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Moreover, for any Gm-monodromic  sheaf E on Ga, we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  ϕid,0(E) ≃ i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1(T(E))[1]  where i1 : {1} ֒→ Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We have the dual maps to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  Ga  i  � u  g  π  �  ,  where π is the quotient map to g/b− ∼= u, and i(c) = ce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any F ∈ Sh(N/G), we have  Wg ◦ T(F) ≃ ϕid,0 ◦ f∗ ◦ r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  − ◦ T(F) ≃ ϕid,0 ◦ TGa ◦ i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ◦ π∗(F)[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=']  ≃ i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  1 ◦ i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ◦ π∗(F)[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='] ≃ RΓ(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  π−1(e)F)[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='] = RΓ(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  π−1(e)∩N F)[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='] ≃ i∗  e[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=']  where the first isomorphism uses (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2), the second uses the standard property of the Fourier-Sato trans-  form under linear maps (here TGa is the Fourier-Sato transform for Sh(Ga)), and the third uses (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The last isomorphism is because π−1(e)∩N = (e+b−)∩N is the free orbit of e under the adjoint action of  U −, which is contractible and along which F is contant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Although we ignored all the cohomological shifts  in the above calculation, we still conclude that Wg/G ◦ T ≃ i∗  e[r], because both functors are t-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  65  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The diagram naturally commutes:  ShN (G/G)  β1  �  WG/G  �❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ShN (g/G)  Wg/G  �  k-mod  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let D ⊂ g as in the definition of β1 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='f (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Put D′ := exp(D) ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let f : U − → Ga and  f ′ : u− → Ga be the function induced by a non-degenerate character of u−/[u−, u−], then the diagram  commutes:  Ga  =  �  u− ∩ D  exp  ≃  �  f  �  r0  � D/G  exp  ≃  �  j  � g/G  Ga  U − ∩ D′  f ′  �  r′  0  � D′/G  j′  � G/G  The Whittaker functor Wg/G on the Lie algebra g can be identified with ϕf,− ◦ r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  0 ◦ j∗, therefore  Wg/G ◦ β1 = ϕf,0 ◦ r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  0 ◦ j∗ ◦ (j∗)−1 ◦ exp |∗  D/G ◦ j′∗ = ϕf ′,1 ◦ r′  0  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ◦ j′∗ = WG/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Generalized Springer correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We recall the generalized Springer correspondence due to  Lusztig [Lus84, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a nilpotent orbit O, let AG(O) = CG(e)/CG(e)◦ for e ∈ O, which  is a finite group well-defined up to inner automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a finite group Γ, let Irr(Γ) be the set of  irreducible representations of Γ over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the generalized Springer correspondence is a bijection  P := {(O, χ) : O a nilpotent orbit of G, χ ∈ Irr(AG(O))}  GSpr  1:1  �  �D := {(J, F, θ) : J ⊂ I, F ∈ CJ, θ ∈ Irr(W J)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Preg ⊂ P be the subset of pairs (O, χ) where O = Oreg is the regular nilpotent orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since AG(Oreg) =  Z(G)/Z(G)◦, we can identify Preg with Irr(Z(G)/Z(G)◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  On the other hand, recall the set D from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Dreg ⊂ D be the subset of those (J, F) ∈ D such  that Supp(F) = NLJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under the map  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  P  GSpr  −−−→ �D → D  the subset Preg maps bijectively to Dreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, it induces a bijection  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  Irr(Z(G)/Z(G)◦)  1:1  −−→ Dreg  sending χ to the unique pair (Jχ, Fχ) ∈ Dreg such that GSpr(Oreg, χ) = (Jχ, Fχ, θχ) for some θχ ∈  Irr(W Jχ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let (J, F, θ) = GSpr(Oreg, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We claim that Supp(F) = NLJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By the construction of the general-  ized Springer correspondence, IC(Oreg, χ) is a direct summand of IndG  LJ(F), where IndG  LJ is the pull-push  functor along the diagram (and PJ is a parabolic subgroup of G containing LJ as a Levi subgroup)  NG/G  NPJ/PJ  qJ  �  pJ  �  NLJ/LJ  If Supp(F) was properly contained in NLJ, pJ(q−1  J Supp(F)) would not meet Oreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore we must  have Supp(F) = NLJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This shows that the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) sends Preg to Dreg, and the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5) is bijective because for each (J, F) ∈ Dreg, IndG  LJ(F) has full support on NG, and its  restriction to Oreg has rank one, hence it has a unique direct summand of the form IC(Oreg, χ) for some  χ ∈ Irr(Z(G)/Z(G)◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  66  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under the equivalence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4), the Lie algebra Whittaker sheaf  Whg/G ∈ ShN (g/G)  is identified with  �  (Jχ,Fχ)∈Dreg  Sym(zJχ[1])[−r] ∈  �  (J,F )∈D  k[W J]#Sym(zJ[1])-mod ≃ ShN (g/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here we use the bijection in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5 to index elements in Dreg by χ ∈ Irr(Z(G)/Z(G)◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let AJ = k[W J]#Sym(z∗  J[−2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let ktriv be the 1-dimensional AJ-module in degree 0 where W J  acts trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then W J-equvariant Koszul duality gives an equivalence  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)  HomSym(z∗  J[−2])(ktriv, −) : AJ-perf ≃ k[W J]#Sym(zJ[1])-modfd  where we identify Sym(zJ[1]) with EndSym(z∗  J[−2])(ktriv)opp, and -modfd stands for the category of finite  dimensional modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under this equivalence, ktriv corresponds to Sym(zJ[1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Shc,N(g/G) ⊂ ShN (g/G) be the full subcategory of constructible sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6),  we have equivalences  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7)  Shc,N (g/G) ≃ �  (J,F )∈D k[W J]#Sym(zJ[1])-modfd  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)  ∼  � �  (J,F )∈D AJ-perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Moreover, the composition  Shc(N/G)  T−→ Shc,N(g/G) ≃  �  (J,F )∈D  AJ-perf  is identified with the functor HomN/G(⊕(J,F )∈DIndG  LJ⊂PJ F, −), where we use  End(IndG  LJ⊂PJF) ≃ Aopp  J  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Prop 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2, T−1Whg/G is isomorphic to (ie)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' [r] where ie : {e} → N/G is the inclusion of a regular  nilpotent element e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, T−1Whg/G ∈ Shc(N/G), hence Whg/G corresponds to a collection of  perfect AJ-modules MJ,F, one for each (J, F) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For any (J, F) ∈ D, put eJ ⊂ NLJ a regular nilpotent element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then by Prop 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2  HomNG/G(T−1Whg/G, IndG  LJ⊂PJ(F)) ≃ i∗  e(IndG  LJ⊂PJ(F))[r] = i∗  eJ(F)[r] =  �  ksign,  (J, F) ∈ Dreg  0,  (J, F) /∈ Dreg  where ksign is the 1-dimensional End(IndG  LJ⊂PJ(F))-module where W J acts by the sign representation  (this is because we identify the Weyl group actions on Springer fibers via Fourier transform, therefore the  regular Springer fiber corresponds to the sign representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore we have an isomorphism of right  AJ-modules (the right AJ-action comes from post-composing with the right AJ-action on itself)  HomAJ(MJ,F , AJ) ≃  �  ksign,  (J, F) ∈ Dreg  0,  (J, F) /∈ Dreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Note also that there is an equivalence of categories between left and right AJ-modules:  HomAJ(−, AJ) : AJ-perf  ∼  � Aopp  J  perf  under which the 1-dimensional trivial module ktriv corresponds to ksign[r] by a Koszul resolution calcula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' From this we conclude that  MJ,F ≃  �  ktriv[−r],  (J, F) ∈ Dreg  0,  (J, F) /∈ Dreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore, under the equivalence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7), Whg/G corresponds to  ⊕(J,F )∈Dregktriv[−r] ∈  �  (J,F )∈D  k[W J]#Sym(z∗  J[−2])-perf,  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  67  which is further identified under Koszul duality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)with  ⊕(J,F )∈DregSym(zJ[1])[−r] ∈  �  (J,F )∈D  k[W J]#Sym(zJ[1])-modfd ⊂  �  (J,F )∈D  k[W J]#Sym(zJ[1])-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Calculation of endormorphisms of the Whittaker sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8 the descended  trace WhG/G ∈ hh(HG) of the universal affine Whittaker sheaf WhG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We can now calculate its endomor-  phism algebra explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For χ ∈ Irr(Z(G)/Z(G)◦), denote Tχ = Z(Lχ)◦, Wχ = W Jχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let T ∨  χ the torus over k dual to Tχ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=',  X∗(T ∨  χ ) = X∗(Tχ)), and t∨  χ = Lie(T ∨  χ ) = z∗  Jχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall the functor α2 : ShN (G/G) → colimJ∈D ShN(LJ/LJ) corresponding to J = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Define  WhG,I := α2(WhG/G) ∈ colimJ∈D ShN (LJ/LJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is a canonical isomorphism of dg algebras:  EndcolimJ∈D ShN (LJ/LJ)(WhG,I) ≃  �  χ∈Irr(Z(G)/Z(G)◦)  O(T ∨  χ × T ∨  χ × t∨  χ[−1])Wχ  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, we have  WhG,I = α2(WhG/G) ≃ α2 ◦ α1(Whg/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9, the image of WhG,I in Γ(t, QCoh�C)�  W is  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  Ind2 ◦ Ind1(Lg(Whg/G)) ∈ Γ(t, QCoh�C)  �  W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For any ǫ ∈ S, put Λǫ = X∗(T ) ∩ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For c = (ǫ, B, F) ∈ C, the summand corresponds to c  in Γ(t, QCohC)W is equivalent to the category k[W ǫ]#Sym(zǫ[1])-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Similarly the summand corre-  sponds to c in Γ(t, QCoh�C)�  W (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  in Γ(t, QCoh�C)�  W ) is equivalent to k[�  W ǫ]#Sym(zǫ[1])-mod (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  k[�  W ǫ]#Sym(zǫ[1])-mod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We also have �  W ǫ = Λǫ ⋊ W ǫ, and �  W ǫ = (Λǫ × Λǫ) ⋊ W ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore, Ind2 ◦ Ind1  can be identified with the direct sum of the following induction functors over (J, F) ∈ D:  Ind  �  W ǫ  W ǫ : k[W J]#Sym(zJ[1])-mod → k[�  W J]#Sym(zJ[1])-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1), the image of WhG,I in Γ(t, QCoh�C)�  W is  Ind2 ◦ Ind1  \uf8eb  \uf8ed  �  χ∈Irr(Z(G)/Z(G)◦)  Sym(zJχ[1])  \uf8f6  \uf8f8 =  �  χ∈Irr(Z(G)/Z(G)◦)  Ind  �  W Jχ  W Jχ Sym(zJχ[1]) ∈ Γ(t, QCoh�C)  �  W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore (all direct sums are over χ ∈ Irr(Z(G)/Z(G)◦))  End(WhG,I)  ≃  �  χ  Endk[�  W Jχ ]#Sym(zJχ[1])  �  Ind  �  W Jχ  W Jχ Sym(zJχ[1])  �  ≃  �  χ  Homk[W Jχ ]#Sym(zJχ[1])  �  Sym(zJχ[1]), Res  �  W Jχ  W Jχ Ind  �  W Jχ  W Jχ Sym(zJχ[1])  �  ≃  �  χ  �  k[ΛJχ × ΛJχ] ⊗ Sym(zJχ[1])  �W Jχ  =  �  χ  O(T ∨  χ × T ∨  χ × t∨  χ[−1])Wχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In the last identity we use ΛJχ = X∗(Tχ) = X∗(T ∨  χ ), t∨  χ ∼= z∗  Jχ and Wχ = W Jχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is a canonical isomorphism of dg algebras:  Endhh(HG)(WhG/G) ≃  �  χ∈Irr(Z(G)/Z(G)◦)  O(T ∨  χ × T ∨  χ × t∨  χ[−1])Wχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  68  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5, we have WhG/G ≃ a(WhG/G) where a is the natural map ShN(G/G) ֒→ HG,I →  hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2, we have a fully-faithful embedding  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  colimJ∈D ShN(LJ/LJ) ֒→ hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Under this embedding, we can identify α2 : ShN (G/G) → colimJ∈D ShN (LJ/LJ) (corresponding to the  term J = I) with a : ShN (G/G) → hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Hence WhG/G can be identified with the image of WhG,I  under the embedding (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore  Endhh(HG)(WhG/G) ≃ EndcolimJ∈D ShN (LJ/LJ)(WhG,I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The desired statement now follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assume Ansatz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then there is a canonical isomorphism of dg algebras:  O(Z2  G∨) ≃  �  χ∈Irr(Z(G)/Z(G)◦)  O(T ∨  χ × T ∨  χ × t∨  χ[−1])Wχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In particular, H0(O(Z2  G∨)) ≃ �  χ∈Irr(Z(G)/Z(G)◦) O(T ∨  χ × T ∨  χ )Wχ is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Although we primarily focus on the derived structure and reduceness of the commuting  stack, our theorem also gives new perspectives even for coarse moduli spaces, which we will elaborate in  Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Additional applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this section, we use similar method to compute two additional endo-  morphism algebras of interest: (i) the derived spherical Hecke algebra and (ii) endomorphisms of parabolic  inductions (for simplicity, we will restrict to inductions from a torus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' First, we will make the automorphic  calculations, then explain their spectral implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Automorphic calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For each of the categories Γ(t, QCohC)W , Γ(t, QCoh�C)�  W and Γ(t, QCoh�C)�  W ,  there is a “principal block” (direct summand) corresponding to the affine space ǫ = t and the skyscraper  sheaf on T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' They are described as follows:  k[W]#Sym((t∨)∗[1])-mod ≃ k[W]#O(t∨[−1])-mod ⊂ Γ(t, QCohC)W ,  k[�  W]#Sym((t∨)∗[1])-mod ≃ k[W]#O(T ∨ × t∨[−1])-mod ⊂ Γ(t, QCoh�C)  �  W ,  k[�  W]#Sym((t∨)∗[1])-mod ≃ k[W]#O(T ∨ × T ∨ × t∨[−1])-mod ⊂ Γ(t, QCoh�C)  �  W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A simple perverse sheaf F ∈ ShN (G/G) lies in the principal block if and only if the  following two conditions hold:  (1) The support of F (which is closed by definition) contains 1 ∈ G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Every simple constituent of the Fourier transform T−1(β1F) ∈ Sh(N/G) is a summand of the  Springer sheaf Spr ∈ Sh(N/G) (normalized to be perverse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', stalks along regular nilpotent  elements lie in degree r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Moreover, when F ∈ ShN (G/G) is in the principal block, Lg(β1F) ∈ k[W]#Sym((t∨)∗[1])-mod is Koszul  dual to  HomN/G(Spr, T−1(β1F)) ∈ End(Spr)opp-mod ≃ k[W]#Sym(t∨[−2])-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' First, assume F lies in the principal block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6, β1F lies in the principal block and  is nonzero because the restriction functor Res : k[�  W]#Sym((t∨)∗[1])-mod → k[W]#Sym((t∨)∗[1])-mod is  faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Being in the principal block, all simple constituents of β1F are summands of the Grothendieck-  Springer sheaf (direct image of constant sheaf under b/B → g/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  This implies that all simple con-  stituents of T−1(β1F) are summands of Spr, which verifies condition (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since all simple summands of  the Grothendieck-Springer sheaf have full support, and β1F ̸= 0, its support contains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, the  support of F contains 1, which verifies condition (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Conversely, assume the conditions (1) and (2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Using Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5 and condition (1), we see that  LG(F) lies in a summand in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5) indexed by (J, F) such that 1 ∈ exp(ǫJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This implies that, up to  changing J by the action of Ω, we may arrange J ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under the restriction functor Res : Γ(t, QCoh�C)�  W →  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  69  Γ(t, QCohC)W , the block indexed by (J, F) with J ⊂ I on the source maps into the block indexed by the  same (J, F) on the target, under the decomposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6, LG(F) lies  in the principal block if and only if Lg(β1F) lies in the principal block, which is guaranteed by condition  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Recall the natural monoidal embedding i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : HG → HG, and let a : ShN (G/G) ≃ hh(HG) → hh(HG)  denote the induced functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose F ∈ ShN (G/G) lies in the principal block, with the corresponding k[W]#O(T ∨ ×  t∨[−1])-module LG(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then there is a canonical equivalence of dg algebras  Endhh(HG)(a(F)) ≃  �  O(T ∨) ⊗ EndO(T ∨×t∨[−1])(LG(F))  �W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that a(F) can be identified with α2(F) ∈ colimJ∈D ShN (LJ/LJ), the latter identified with  the full subcategory hh(HG)0 by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9, we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  LG(α2(F)) ≃ Ind  �  W  �  W (LG(F)) ∼= O(T ∨) ⊗ LG(F) ∈ k[W]#O(T ∨ × T ∨ × t∨[−1])-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  From this we see  Endhh(HG)(a(F))  ≃  Endk[W]#O(T ∨×T ∨×t∨[−1])(LG(α2(F)))  ≃  Endk[W]#O(T ∨×T ∨×t∨[−1])(O(T ∨) ⊗ LG(F))  ≃  �  O(T ∨) ⊗ EndO(T ∨×t∨[−1])(LG(F))  �W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Write trG : HG → hh(HG), trG : HG → hh(HG) for the trace maps, and recall the natural isomorphism  trG|HG ≃ a ◦ trG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Let kG/G denote the constant sheaf on G/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is a canonical equivalence  of dg algebras  Endhh(HG)(a(kG/G)) ≃ O(T ∨ × (t∨)∗[1] × (t∨)∗[2])W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Let eG be the monoidal unit of the universal affine Hecke category HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  There is a canonical  equivalence of dg algebras  Endhh(HG)(trG(eG)) ≃ k[W]#O(T ∨ × T ∨ × t∨[−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In both calculations, we are computing End(a(F)) for some F ∈ ShN (G/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We shall show in both  cases F lies in the principal block of ShN(G/G), and identify the corresponding k[W]#O(T ∨ × t∨[−1])-  module LG(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then we conclude the calculation by invoking Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Note that T−1(β1kG/G) = T−1(kg/G) ≃ k0/G[− dim G], which is a direct summand of Spr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By  the criterion in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2, LG(kG/G) lies in the principal direct summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Under Koszul duality,  Lg(β1kG/G) = Lg(kg/G) ∈ k[W]#Sym(t[1])-mod corresponds to  HomN/G(Spr, k0/G[− dim G]) ∈ k[W]#Sym(t∨[−2])-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Using the Cartesian diagram  {0}/B  ν0  �  � �  i′  0  � u/B  ν  �  {0}/G� �  i0  � N/G  70  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  we have by adjunction and proper base change  HomN/G(Spr, i0∗k0/G[− dim G])  =  Hom(ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='ku/B[−r], k0/G[− dim G])  ≃  Hom(ku/B[−r], ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='i0∗k0/G[− dim G])  ≃  Hom(ku/B[−r], i′  0∗ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  0k0/G[− dim G])  =  Hom(ku/B, i′  0∗k0/B)  ≃  H∗  B(pt) ≃ Sym(t∨[−2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  This shows that the Koszul dual of Lg(β1kG/G) is Sym(t∨[−2]) as a natural k[W]#Sym(t∨[−2])-module,  hence Lg(β1kG/G) ≃ ktriv as a k[W]#Sym((t∨)∗[1])-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In other words, LG(kG/G) ∈ k[�  W]#Sym((t∨)∗[1]) is one-dimensional over k with the trivial action of  k[W]#Sym((t∨)∗[1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Using the fact that kG/G appears as a direct summand of IndG  T ⊂B(kT/T ), and the  compatibility of LG with parabolic induction in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7, we see that the action of the lattice  part X∗(T ) ⊂ �  W on LG(kG/G) is also trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We conclude that LG(kG/G) ≃ ktriv as an object in  k[�  W]#Sym((t∨)∗[1])-mod = k[W]#O(T ∨ × t∨[−1])-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore we have a W-equivariant equivalence  of dg algebras  EndO(T ∨×t∨[−1])(LG(kG/G)) ≃ O((t∨)∗[1] × (t∨)∗[2])  By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, we get  Endhh(HG)(a(kG/G)) ≃ (O(T ∨) ⊗ EndO(T ∨×t∨[−1])(LG(kG/G)))W ≃ O(T ∨×(t∨)∗[1] × (t∨)∗[2])W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Let eG be the monoidal unit of HG, then trG(eG) ≃ a(trG(eG)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We claim that trG(eG) is in fact  the “universal Grothendieck-Springer sheaf”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Indeed, consider the following commutative diagram with  a Cartesian square on the left  H  pH  �  U\\B/U  qH  �  B  U  δ  �  �  π  � G  G  H  H  B  B  q′  H  �  π′  � G  G  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2, trG = γ is the horocycle functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore  trG(eG) = π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗(eG) = π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='δ∗q∗  H(exp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' kh)[r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By proper base change, we can identify it with  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  trG(eG) ≃ π′  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='q′∗  H(pH!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' exp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' kh)[r] ≃ IndG  T ⊂B(pH!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' exp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' kt[r]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Under the equivalence ShN (T/Ad(T )) = Sh0(T/Ad(T )) ≃ O(T ∨×t∨[−1])-mod, pH!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' exp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' kt[r] corresponds  to the free rank one module O(T ∨ ×t∨[−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7, LG(trG(eG)) ∼= k[W]#O(T ∨ ×t∨[−1])  is also the free rank one k[W]#O(T ∨ × t∨[−1])-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1), LG(a(trG(eG))) is again the free rank  one k[W]#O(T ∨ × t∨[−1])-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore End(trG(eG)) ≃ End(a(trG(eG))) ≃ End(LG(a(trG(eG))))  is the dg algebra k[W]#O(T ∨ × T ∨ × t∨[−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The functor a is analogous to the compact induction functor for p-adic representations  c-ind : G(OK)-rep → G(K)-rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here K is a local non-archimedean field with valuation ring OK, and G  is a split reductive group over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, Endhh(HG)(a(kG/G)) is analogous to the spherical Hecke  algebra EndG(K)(c-ind(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Spectral consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now let us introduce the objects corresponding to a(kG/G), trG(eG) ∈ hh(HG)  on the spectral side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Introduce the natural closed embedding i2 : G∨/G∨ → Z2  G∨ induced by g �→ (g, 1), and the coherent  sheaf i2∗OG∨/G∨ on Z2  G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Introduce as well the natural projection p : Z2  B∨ → Z2  G∨, and the coherent sheaf  p∗OZ2  B∨ on Z2  G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  71  Consider the closed embedding of derived stacks  σ : B∨\\G∨/B∨ ≃ BB∨ ×BG∨ BB∨� �  � StG∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Taking direct image of ind-coherent sheaves gives a monoidal functor  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  σ∗ : IndCoh(B∨\\G∨/B∨)  � IndCoh(StG∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let 2ρ∨ ∈ X∗(T ∨) be the sum of positive roots of G∨ with respect to B∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let N be the number of  positive roots of G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' When ρ∨ ∈ X∗(T ∨), we consider the sheaf  A := OB∨\\G∨/B∨(−ρ∨, −ρ∨)[N] ∈ IndCoh(B∨\\G∨/B∨)  obtained as the tensor product of pullbacks of O(−ρ∨) from both factors BB∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' When ρ∨ is not a character  of T ∨, let ν : G∨  1 → G∨ be the double cover for which ρ∨ is a character of T ∨  1 = ν−1(T ∨), then we have  B∨\\G∨/B∨ = G∨  1 /(B∨  1 × B∨  1 /∆(ker(ν))) (where B∨  1 = ν−1(B∨), two factors of B∨  1 act on G∨  1 by left and  right translations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since (−ρ∨, ρ∨) is always character of B∨  1 × B∨  1 /∆(ker(ν)), it defines a line bundle  on G∨  1 /(B∨  1 × B∨  1 /∆(ker(ν))) and hence on B∨\\G∨/B∨, which we still denote by OB∨\\G∨/B∨(−ρ∨, −ρ∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In any case, we have the object A := OB∨\\G∨/B∨(−ρ∨, −ρ∨)[N] ∈ IndCoh(B∨\\G∨/B∨), and the object  σ∗A ∈ IndCoh(StG∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The object σ∗A ∈ IndCoh(StG∨) carries a canonical algebra structure for which its de-  scended trace is  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  tr(σ∗A) ≃ i2∗OG∨/G∨ ∈ IndCoh(Z2  G∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We prove the lemma in the case ρ∨ ∈ X∗(T ∨);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' the general case can be deduced by passing to double  cover G∨  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We construct an algebra structure on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let VG∨ = B∨/B∨×G∨/G∨ BG∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the correspondence  StG∨ = B∨/B∨ ×G∨/G∨ B∨/B∨  B∨/B∨ ×G∨/G∨ BB∨  c  �  d  � B∨/B∨ ×G∨/G∨ BG∨ = VG∨  where the maps are induced by the natural maps B∨/B∨ ←֓ BB∨ → BG∨ on the second factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider  the adjoint functors  ΠL = d∗(c∗ ⊗ p∗  2O(−ρ∨)[N]) : IndCoh(StG∨)  � IndCoh(VG∨) : Π = c∗(d∗ ⊗ p∗  2O(−ρ∨))  �  From these we get a natural isomorphism of endo-functors of IndCoh(StG∨):  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  Π ◦ ΠL ≃ (−) ⋆ σ∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The monad structure on Π ◦ ΠL then gives an algebra structure on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5) and the Barr-Beck-Lurie theorem, we conclude that the category RModσ∗A(IndCoh(StG∨))  of right σ∗A-modules in IndCoh(StG∨) can be identified with IndCoh(VG∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly, BimodA(IndCoh(StG∨))  can be identified with IndCoh(WG∨), where  WG∨ = BG∨ ×R  G∨/G∨ BG∨  is the spectral stack hosting the derived spherical Hecke category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under this equivalence, the regular  bimodule A itself corresponds to the monoidal unit ∆∗OBG∨ ∈ IndCoh(WG∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3, we  have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)  tr(σ∗A) ≃ tr(∆∗OBG∨) ∈ IndCoh(Z2  G∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here the right side a priori lies in hh(IndCoh(WG∨)), which is a full subcategory of IndCoh(Z2  G∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  To compute tr(∆∗OBG∨), we apply Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5 to the map f : X = BG∨ → G∨/G∨ = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The  induced map on the loop spaces Lf : LX = G∨/G∨ → Z2  G∨ = LY can be identified with i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5 gives  tr(∆∗OBG∨) ≃ i2∗OG∨/G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6) we get (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  72  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  To state the next result, we need an variant of Ansatz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The monoidal equivalence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) in Ansatz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4 holds:  Φ : IndCoh(StG∨)  ∼  � HG,  Moreover, Φ takes σ∗A to i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='kU\\G/U ∈ HG (where kU\\G/U ∈ HG is the constant sheaf and i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : HG ֒→ HG),  compatibly with their respective algebra structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall assuming equivalence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1), we have the equivalence (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6))  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7)  IndCohN (Z2  G∨)  ∼  � hh(HG) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Assume Ansatz 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then under the equivalence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7), i2∗OG∨/G∨  corresponds to a(kG/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Assume the monoidal equivalence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under the equivalence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7), p∗OZ2  B∨ corresponds  to tr(eG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (1) To see the first identification, we first have an analogous (but simpler) version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In place of the universal finite Whittaker sheaf WhG ∈ HG, we consider the constant sheaf kU\\G/U ∈ HG  as an algebra object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then it is straightforward to check its descended trace trG(kU\\G/U) ∈ hh(HG) ≃  ShN (G/G) (as a bimodule over itself) is the constant sheaf kG/G ∈ ShN (G/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By the functoriality of descended trace in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5 applied to i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : HG → HG, we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8)  trG(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='kU\\G/U) ≃ a(kG/G) ∈ hh(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Comparing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4), we conclude that i2∗OG∨/G∨ matches with a(kG/G) under (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Under the monoidal equivalence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1), the monoidal unit ∆∗OB∨/B∨ ∈ IndCoh(StG∨) corresponds  to the monoidal unit eG ∈ HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore tr(eG) corresponds to tr(∆∗OB∨/B∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5 applied  to the map f : X = B∨/B∨ → G∨/G∨ = Y , we conclude that tr(∆∗OB∨/B∨) ≃ p∗OZ2  B∨ , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Assume Ansatz 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then there is a canonical equivalence of dg alge-  bras  EndZ2  G∨ (i2∗OG∨/G∨) ≃ O(T ∨ × (t∨)∗[1] × (t∨)∗[2])W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Assume the monoidal equivalence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then there is a canonical equivalence of dg algebras  EndZ2  G∨ (p∗OZ2  B∨ ) ≃ k[W]#O(T ∨ × T ∨ × t∨[−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Functions on the commuting stack  The goal of this section is to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Almost commuting pairs of semisimple elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G∨ be a connected reductive group over  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We first recall some results of Borel, Friedman and Morgan [BFM, §4] concerning almost commuting  pairs with compact simple and simply-connected Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will state an extension of their results  to almost commuting pairs of semisimple elements in complex reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is easy to adapt their  proof to this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let G∨,sc be the simply-connected cover of G∨,der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let �  G∨ = G∨,sc × (ZG∨)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the natural map  π : �  G∨ → G∨ is a finite central isogeny whose kernel contains ker(G∨,sc → G∨,der) = π1(G∨,der).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any  c ∈ π1(G∨,der), let Z2  G∨(c) ⊂ Z2  G∨ be the open and closed substack of pairs (x, y) in G∨ such that, for  some (any) liftings �x, �y of x and y to �  G∨, [�x, �y] = c, all up to G∨-conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' An almost commuting pair  in �  G∨ refers to a pair (�x, �y) ∈ �  G∨2 as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have a decomposition  Z2  G∨ =  �  c∈π1(G∨,der)  Z2  G∨(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let (x, y) ∈ Z2  G∨(c) with x, y semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the simulteneous centralizer CG∨(x, y), which is a  reductive (possibly disconnected) subgroup of G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let S ⊂ CG∨(x, y) be a maximal torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is shown in  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  73  [BFM, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1] that the G∨-conjugacy class of S is independent of the choice of (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let us  fix such a torus S and denote it by Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let L∨  c = CG∨(Sc), a Levi subgroup of G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is known that Sc = Z(L∨  c )◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let T ∨  c = L∨  c /L∨,der  c  be  the quotient torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have T ∨  c = Sc/(Sc ∩ L∨,der  c  ) (T ∨  c is denoted Sc in [BFM]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Wc = W(Sc, G∨) =  NG∨(Sc)/L∨  c be the relative Weyl group of G∨ with respect to Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then Wc also acts on T ∨  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Fix a pair  (xc, yc) ∈ (L∨,der  c  )2 such that (xc, yc) ∈ Z2  G∨(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have a map  �ιc : Z2  Sc → Z2  G∨(c)  sending (t1, t2) ∈ Z2  T ∨  c to (t1xc, t2yc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we state the generalized form of results in [BFM] replacing elements in compact groups to semisim-  ple elements in reductive groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' the proofs in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' works without change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition ([BFM, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Any other choice of (x′  c, y′  c) ∈  (L∨  c )2 such that (x′  c, y′  c) ∈ Z2  G∨(c) is L∨  c -conjugate to (xc, yc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Moreover, CL∨,der  c  (xc, yc) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The map �ιc factors through Z2  T ∨  c and is Wc-invariant for the diagonal action, inducing a map of  derived stacks  ιc : Z2  T ∨  c /Wc → Z2  G∨(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (3) The map ιc restricts to a bijection on the set of semisimple closed points:  (T ∨  c (C) × T ∨  c (C))/Wc ↔ |Z2  G∨(c)(C)ss|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here |Z2  G∨(c)(C)ss| denotes the set of G∨-conjugacy classes of semisimple pairs (x, y) ∈ Z2  G∨(c)(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Chevalley restriction theorem for the commuting stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We finish the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2,  which is restated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let c ∈ π1(G∨,der).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Z2  G∨(c) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' 5  (2) The map ιc defined in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1(2) induces an isomorphism on classical functions:  ι∗  c : H0O(Z2  G∨(c))  ∼  → H0O(Z2  T ∨  c )Wc = O(T ∨  c × T ∨  c )Wc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (1) Let Xc = Spec H0O(Z2  G∨(c)), which is understood to be empty if Z2  G∨(c) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' First, Xc is  connected if non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, assuming Z2  G∨(c) ̸= ∅, and suppose for the contrary that Xc decomposes  into a disjoint union of open and closed non-empty subschemes X′  c  � X′′  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let C2  G∨(c) ⊂ C2  G∨ be the  preimage of Z2  G∨(c) in the commuting scheme C2  G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Z2  G∨(c) = Z′  c  � Z′′  c and C2  G∨(c) = C′  c  � C′′  c be the  corresponding decompositions by taking preimages of X′  c and X′′  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now C′  c contains a closed G∨-orbit,  which then consists of semisimple pairs (x, y) ∈ C2  G∨(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This implies Z′  c contains a semisimple pair (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1(2)(3), Z′  c meets the image of ιc : (T ∨  c ×T ∨  c )/W → Z2  G∨(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly, Z′′  c also contains  a semisimple pair, hence it also meets the image of ιc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' But T ∨  c × T ∨  c is irreducible, we get a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  This proves that Xc is connected if non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 shows that Spec H0O(Z2  G∨) = �  c∈π1(G∨,der) Xc has exactly #Irr(Z(G)/Z(G)◦) con-  nected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since #Irr(Z(G)/Z(G)◦) = #π1(G∨,der), and each Xc is connected if non-empty, all  of them must be non-empty and connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) We claim that Xc is irreducible, reduced and normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Above we have shown that Xc is a connected  component of Spec H0O(Z2  G∨), hence isomorphic to Spec O(T ∨  χ ×T ∨  χ )Wχ for a unique χ ∈ Irr(Z(G)/Z(G)◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In particular, Xc is irreducible, reduced and normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The map ιc induces a map on coarse moduli spaces  ι′  c : (T ∨  c × T ∨  c ) � Wc → Xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We next claim that ι′  c induces a bijection on closed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, by a result of Richardson [Ric88],  the closed points in Xc are in bijection with G∨-orbits of (x, y) ∈ Z2  G∨(c) such that the Zariski closure  A(x, y) ⊂ G∨ of the subgroup generated by x and y is reductive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since x commutes with y, A(x, y) is a  5Although implicit in [BFM] but we did not find an explicit statement of the non-emptiness of Z2  G∨(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  74  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  commutative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' it is reductive if and only if x and y are both semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1(3),  ι′  c induces a bijection on closed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Since (T ∨  c × T ∨  c ) � W and Xc are reduced and irreducible of finite type over C, and ι′  c is a bijection on  closed points, to show ι′  c is an isomorphism if suffices to show that ι′  c is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We have a map �s = (�s1, �s2) : Z2  G∨(c) → (G∨ � G∨)2 = (T ∨ � W)2 by recording the G∨-invariants of x  and y in a commuting pair (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This induces a map s = (s1, s2) : Xc → (T ∨ � W)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have maps of  affine schemes  Sc × Sc  p−→ (T ∨  c × T ∨  c ) � Wc  ι′  c  −→ Xc  s−→ (T ∨ � W)2  where the composition is the square of the natural projection Sc → T ∨ � W, hence finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since p is  surjective, s ◦ ι′  c : (T ∨  c × T ∨  c ) � Wc → (T ∨ � W)2 is also finite, therefore ι′  c is finite as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This finishes  the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G∨  Q be the split form of G∨ over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then Z2  G∨ has a Q-form Z2  G∨,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is easy to  deduce from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2 a description of global functions on Z2  G∨,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, we have a decomposition  of Z2  G∨,Q into open and closed substacks Z2  G∨,Q([c]) indexed by Galois orbits [c] ∈ π1(G∨,der)/Gal(Q/Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  If Q[c] is the fixed field of the stabilizer of any c ∈ [c] under Gal(Q/Q) (so Q[c] is the smallest cyclotomic  extension of Q over which Z2  G∨(c) is defined), we have an isomorphism Z2  G∨,Q([c]) ∼= Z2  G∨,Q[c](c), the latter  being the descent of Z2  G∨(c) to Q[c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any c ∈ [c], T ∨  c has a Q[c]-form T ∨  c,Q[c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We conclude  H0O(Z2  G∨,Q)  ∼  →  �  [c]∈π1(G∨,der)/Gal(Q/Q)  O(T ∨  c,Q[c] × T ∨  c,Q[c])Wc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Identifying χ and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' There is a canonical isomorphism  δ : Irr(Z(G)/Z(G)◦) ≃ π1(G∨,der)  because both sides are canonically dual to the torsion part of X∗(T )/root lattice of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  On the other hand, by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, we have two expressions of the ring H0(O(Z2  G∨)):  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  �  χ∈Irr(Z(G)/Z(G)◦)  O(T ∨  χ × T ∨  χ )Wχ ≃ H0(O(Z2  G∨)) ≃  �  c∈π1(G∨,der)  O(T ∨  c × T ∨  c )Wc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Fratila [Fra16] and Bonnaf´e [Bon04] observe a mysterious coincidence that the pairs (T ∨  c , Wc) that  appear on the right side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) are exactly those coming from Levi subgroups of G that support  cuspidal local systems on the regular nilpotent orbit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', those pairs (T ∨  χ , Wχ) that appear on the left  side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' They verified this fact by a case-by-case explicit computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Below we give a uniform  conceptual proof of this combinatorial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let χ ∈ Irr(Z(G)/Z(G)◦) and c = δ(χ) ∈ π1(G∨,der).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the following hold:  (1) Put Lχ := LJχ as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4, then L∨  c (defined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) is the dual Levi of Lχ (up to  G∨-conjugacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) There is an isomorphism of pairs (T ∨  χ , Wχ) ≃ (T ∨  c , Wc) compatible with their respective actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Moreover, such an isomorphism is canonical up to conjugation by Wχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (1) Let L∨  χ ⊂ G∨ be the dual Levi to Lχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that the canonical map π1(L∨,der  χ  ) → π1(G∨,der) is  always injective, and c = δ(χ) lies in its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore it makes sense to form Z2  L∨  χ(c) ⊂ Z2  L∨  χ, which is  non-empty by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Viewing c as an element in π1(G∨,der), we can form the Levi L∨  c of G∨  by taking a maximal torus Sc ⊂ CG∨(x, y) and take its centralizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now Z(L∨  χ)◦ is a torus in CG∨(x, y),  hence we may take Sc to contain Z(L∨  χ)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This implies L∨  c = ZG∨(Sc) ⊂ ZG∨(Z(L∨  χ)◦) = L∨  χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since T ∨  c  and T ∨  χ are the abelianizations of L∨  c and L∨  χ respectively, we have a surjection T ∨  c ։ T ∨  χ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular,  we have the inequality  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  dim T ∨  c ≥ dim T ∨  χ , for all χ ∈ Irr(Z(G)/Z(G)◦) and c = δ(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We remark that, since L∨  c is a Levi subgroup of L∨  χ up to conjugation, L∨  c is conjugate to L∨  χ if and only  if dim T ∨  c = dim T ∨  χ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  75  On the other hand, by taking the Krull dimensions of all direct summands on both sides of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1), we  get  �  χ∈Irr(Z(G)/Z(G)◦)  2 dim T ∨  χ =  �  c∈π1(G∨,der)  2 dim T ∨  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Combined with the inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2), we are forced to have an equality dim T ∨  χ = dim T ∨  δ(χ) for every χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  As we remarked earlier, this implies that L∨  δ(χ) is conjugate to L∨  χ in G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Let c = δ(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that Wχ = NG∨(L∨  χ)/L∨  χ, T ∨  χ is the abelianization of L∨  χ, and the same is  true if χ is replaced with c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Any g ∈ G∨ that conjugates L∨  χ to L∨  c induces an isomorphism of pairs  ιg : (T ∨  χ , Wχ) ≃ (T ∨  c , Wc) compatible with the actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Different choices of g differ by right multiplication  by NG∨(L∨  χ), therefore the resulting ιg differ by Wχ-conjuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Combining Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4, we get the following description of the dg ring of  derived functions on Z2  G∨ promised in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assume Ansatz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then there is a canonical isomorphism of dg algebras:  O(Z2  G∨) ≃  �  c∈π1(G∨,der)  O(T ∨  c × T ∨  c × t∨  c [−1])Wc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' From groups to Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will deduce the Lie algebra analogue of the Chevalley restriction  theorem from the group case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let G∨ be a connected reductive group with Lie algebra g∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let C2  g∨ be the commuting scheme for g∨,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', it is the (classical) fiber over 0 of the Lie bracket map [·, ·] : g × g → g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then G∨ acts on C2  g∨ by  diagonal adjoint action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We are interested in calculating the C-algebra O(C2  g∨)G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let T ∨ ⊂ G∨ be a maximal torus and t∨ = LieT ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The embedding t∨ × t∨ ֒→ C2  g∨ induces a map of  stacks  ιg∨ : (t∨ × t∨)/W → C2  g∨/G∨  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The map ιg∨ induces an isomorphism on global (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' invariant) functions  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  ι∗  g∨ : O(C2  g∨)G∨  ∼  → O(t∨ × t∨)W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For the statement of the theorem we may replace G∨ by another group isogenous to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular,  we may assume the derived group G∨,der is simply-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We have the natural map  s : C2  g∨ → (t∨ � W)2  by taking the invariants of both elements in the commuting pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let �C2  g∨ be the formal completion of C2  g∨  along the fiber s−1(0, 0) (which classifies commuting pairs of nilpotent elements ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Analogously, let C2  G∨ ⊂ G∨ × G∨ be the commuting scheme for G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have the natural map  s : C2  G∨ → (T ∨ � W)2  by taking the invariants of both elements in the commuting pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let �C2  G∨ be the formal completion of  C2  G∨ along the fiber s−1(1, 1) (which classifies commuting pairs of unipotent elements ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2 below, the exponential map gives a G∨-equivariant isomorphism  exp : �C2  g∨  ∼  → �C2  G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Moreover, letting �  t∨ × t∨ and  �  T ∨ × T ∨ be the formal completions at (0, 0) and (1, 1) respectively, we have  a commutative diagram of formal schemes  �  t∨ × t∨  ιg∨  �  exp  �  �C2  g∨  exp  �  �  T ∨ × T ∨  ιG∨ � �C2  T ∨  76  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  where ιG∨ is the obvious analogue of ιg∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The vertical maps are isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This gives a commutative  diagram of algebras  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  O(�C2  g∨)G∨  �ι∗  g∨  � O( �  t∨ × t∨)W  O(�C2  G∨)G∨  �ι∗  G∨ �  ∼  =  �  O( �  T ∨ × T ∨)W  ∼  =  �  Now the top arrow is obtained from the map of S = O(t∨)W ⊗ O(t∨)W -algebras (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) by completing  along the maximal ideal of O(t∨)W ⊗O(t∨)W corresponding to (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly, the bottom row is obtained  from the map of S = O(T ∨)W ⊗ O(T ∨)W -algebras  ι∗  G∨ : O(C2  G∨)G∨ → O(T ∨ × T ∨)W  by completing along the maximal ideal of O(T ∨)W ⊗ O(T ∨)W corresponding to (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, ι∗  G∨ is an isomorphism (using that G∨,der is simply-connected this follows directly,  but in fact this is true without assuming G∨,der is simply-connected), hence so is its completed version  �ι∗  G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By the commutative diagram (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2) where both vertical arrows are isomorphisms, we conclude that  �ι∗  g∨ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We would like to deduce that ι∗  g∨ is an isomorphism from the fact that �ι∗  g∨ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note  that we have a Gm-action on C2  g∨ by simultaneously scaling the elements in the commuting pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This  gives a grading R := O(C2  g∨)G∨ = ⊕n≥0Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly, T = O(t∨ × t∨)W has a grading T = ⊕n≥0Tn as  well as S = ⊕n≥0Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The map ι∗  g∨ is a map of graded S-algebras:  ι∗  g∨ : R = ⊕n≥0Rn → T = ⊕n≥0Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The map �ι∗  g∨ is obtained from ι∗  g∨ by S+-adic completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is clear that the S+-adic completion of T is  given by �T = �  n≥0 Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Note that R/S+R = O(C2  N ∨)G∨ where C2  N ∨ = s−1(0, 0) ⊂ N ∨,2 is the scheme of commuting nilpotent  elements in g∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly, for R = O(C2  G∨)G∨ and S+ the augmentation ideal of S, R/S+R = O(C2  U ∨)G∨  where C2  U ∨ = s−1(1, 1) ⊂ U∨,2 is the the scheme of commuting unipotent elements in G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Lemma  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2 below (or rather a simpler version of the same argument) shows that R/S+R ∼= R/S+R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, R/S+R is the coordinate ring of the central fiber of (T ∨ × T ∨) � W → (T ∨/ � W)2, hence  finite dimemsional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We conclude that R/S+R is finite-dimensional over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This implies that S+R and  R+ = ⊕n>0Rn define the same topology on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore the S+-adic completion �R is �  n≥0 Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The  fact that  ι∗  g∨ : �R =  �  n≥0  Rn → �T =  �  n≥0  Tn  is an isomorphism implies its restriction to each degree Rn → Tn is bijective, hence ι∗  g∨ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The exponential map (x, y) �→ (exp(x), exp(y)) induces a G∨-equivariant isomorphism of  formal schemes  exp2 : �C2  g∨  ∼  → �C2  G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Nil be the category of pairs (R, I) where R is a C-algebra with a nilpotent ideal I (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' IN = 0  for some N > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We write R = R/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the category of formal schemes over C full faithfully embeds  into the category of functors Fun(Nil, Sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The formal completion �  g∨ of g∨ along the nilpotent cone N ∨ is the functor that sends (R, I) to the set  of x ∈ g∨ ⊗ R such that x (the image of x in g∨ ⊗ R) lies in N ∨(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly, the formal completion �  G∨  of G∨ along the unipotent variety U∨ sends (R, I) to the set of g ∈ G∨(R) such that g ∈ U∨(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  77  Similarly, the completion �C2  g∨ sends (R, I) to the set of commuting pairs (x, y) ∈ C2  g∨(R) such that  x, y ∈ N ∨(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The completion �C2  G∨ sends (R, I) to the set of commuting pairs (g, h) ∈ C2  G∨(R) such that  g, h ∈ U∨(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We first check that the exponenital map and logarithm map give inverse isomorphisms between the  functors �  g∨ and �  G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To construct these maps, we choose a faithful representation V of G∨ and embed G∨  into GL(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We define �gl(V ) and �  GL(V ) as the formal completions of gl(V ) and GL(V ) along the closed  subsets of nilpotent and unipotent matrices, viewed as functors on Nil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note that any x ∈ �gl(V )(R, I)  is nilpotent (because xn has entries in I, hence nilpotent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly, any g ∈ �  GL(V )(R, I) is unipotent  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', g − 1 is nilpotent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The exponential map exp : �gl(V )(R, I) → �  GL(V )(R, I) is defined using the  usual formula exp(x) = �  n≥0  xn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' which is a finite sum since x is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The logarithm map log :  �  GL(V )(R, I) → �gl(V )(R, I) is defined using the usual formula log(g) = − �  n≥1  (1−g)n  n  which is a finite  sum since g − 1 is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Clearly exp and log give inverse isomorphisms between the functors �gl(V )  and �  GL(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We claim that exp(�  g∨(R, I)) ⊂ �  G∨(R, I) and log( �  G∨(R, I)) ⊂ �  g∨(R, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Once this is shown, exp and  log then restrict to give inverse bijections between �  g∨(R, I) and �  G∨(R, I) functorially for (R, I) ∈ Nil,  hence giving inverse isomorphisms (clearly G∨-equivariant) between the formal schemes �  g∨ and �  G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To  check exp(�  g∨(R, I)) ⊂ �  G∨(R, I), let {vi} be a collection of tensors on V such that G∨ is defined as  the simultaneous stabilizer of {vi} in GL(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then g∨ is the simultaneous annihilator of {vi} in gl(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  If x ∈ �  g∨(R, I), then xvi = vi, hence exp(x)vi = vi and therefore exp(x) ∈ �  G∨(R, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  This proves  exp(�  g∨(R, I)) ⊂ �  G∨(R, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The other inclusion log( �  G∨(R, I)) ⊂ �  g∨(R, I) is proved in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now we have a G∨ × G∨-equivariant isomorphism exp2 : �  g∨ × �  g∨  ∼  → �  G∨ × �  G∨ with inverse log2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We  claim that for any (R, I) ∈ Nil, exp2 sends �C2  g∨(R, I) to �C2  G∨(R, I), and log2 sends �C2  G∨(R, I) to �C2  g∨(R, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Once this is checked, exp2 and log2 then restrict to give inverse bijections between �C2  g∨(R, I) and �C2  G∨(R, I)  functorially for (R, I) ∈ Nil, hence giving inverse isomorphisms between the formal schemes �C2  g∨ and �C2  G∨,  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now suppose (x, y) ∈ �C2  g∨(R, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since exp(x) and exp(y) are polynomials in x and y, and  [x, y] = 0, we see that exp(x) commutes with exp(y) as elements in G∨(R), hence a fortiori in �  G∨(R, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  This verifies that exp2 sends �C2  g∨(R, I) to �C2  G∨(R, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The statement about log2 is proved similarly, using  that log(g) and log(h) are polynomials in g and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Same argument as in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1 proves also the Chevalley restriction theorem for  the Lie algebra-group commuting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Cg∨,G∨ be the subscheme of (x, g) ∈ g∨ × G∨ such that  Adg(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have the obvious map of stacks  ιg∨,G∨ : (t∨ × T ∨)/W → Cg∨,G∨/G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G∨ be a reductive group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The map ιg∨,G∨ induces an isomorphism on global (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' in-  variant) functions  ι∗  g∨,G∨ : O(Cg∨,G∨)G∨  ∼  → O(t∨ × T ∨)W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Sketch of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We let S = O(t∨)W ⊗ O(T ∨)W equipped with the grading coming from the scaling  action on t∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then S/S+ ∼= O(T ∨)W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then R = O(Cg∨,G∨)G∨ = ⊕n≥0Rn and T = O(t∨ × T ∨)W =  ⊕n≥0Tn are both graded S-algebras with gradings coming from scaling actions on g∨ and t∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The same  argument as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1 shows that the map induced by ι∗  g∨,G∨ on the S+-adic completions �R → �T  is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is easy to see that �T = �  n≥0 Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We claim that �R = �  n≥0 Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, we  have R/S+R = O(CN ∨,G∨)G∨ where, CN ∨,G∨ is the scheme of commuting pairs in N ∨ × G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The  exponenital map gives a G∨-equivariant isomorphism CN ∨,G∨  ∼  → CU ∨,G∨ where the nilpotent cone is  replaced with the unipotent variety U∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1, O(CU ∨,G∨)G∨ ∼= O(T ∨)W is free of rank one  over S/S+ = O(T ∨)W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore R/S+R ∼= O(CN ∨,G∨)G∨ ∼= O(T ∨)W ∼= R0 = O(G∨)G∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This implies  78  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  S+R = R+ = ⊕n>0Rn, which implies �R = �  n≥0 Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore the isomorphism �R  ∼  → �T implies the  isomorphism before completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Calculating some colimits  This appendix collects some results about colimits of categories called upon in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='We will  consider colimits of stable presentable ∞-categories viewed as objects of the ∞-category StL of stable  presentable ∞-categories with morphisms given by left adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (The discussion extends verbatim if we  additionally work in the k-linear context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=')  The main new result here is a categorical analog of the contraction principle for sheaf cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It  allows us to reduce the calculation of a colimit indexed by a complicated diagram to a simpler one, assuming  a certain contractibility condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This is a key categorical ingredient in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recollements and stratifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recollements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For recollement in the setting of ∞-categories, we recommend the reference [Lur12,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In our setting of stable presentable ∞-categories, we will largely follow the reference [AMR, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall a recollement [AMR, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1] of stable presentable ∞-categories is a diagram of adjoint  triples  C0  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  j∗  �  C  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='=j∗  �  i∗  �  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  C1  i∗=i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  where j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', j∗, and i∗ = i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' are fully faithful with Im(j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=') = Ker(i∗), Im(j∗) = Ker(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ), and Im(i∗ = i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=') =  Ker(j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' = j∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note the recollement is determined by the subdiagram  C0  C  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='=j∗  �  C1  i∗=i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  with the rest of the notion comprising properties to be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In our topological setting, we call C0 the open subcategory and C1 the closed subcategory of the rec-  ollement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6 We refer to C0 and C1 as the recollement complements of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Suppose we have a commutative diagram in StL  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  C0  f0  �  C  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='=j∗  �  f  �  C1  i∗=i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  f1  �  D0  D  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='=j∗  �  D1  i∗=i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  such that both rows extend to recollements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We say the above diagram (or the functors (f0, f, f1)) is a  morphism of recollements if the square on the left is both left and right adjointable with respect to the  horizontal arrows j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' = j∗ (equivalently, the square on the right is both left and right adjointable with  respect to the horizontal arrows i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' = i∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  If in a morphism of recollement as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1), both f0 and f1 are equivalences, then f is also an  equivalence (this follows from [Lur12, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Stratifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will use the more general notion of a recollement of stable presentable ∞-  categories indexed by a poset P as introduced in [AMR, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3] under the name stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since we  work in the topological rather than algebraic setting, we will adapt the notation and terminology in the  definitions to reflect this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a finite totally ordered poset, a stratification also goes by the name semi-  orthogonal decomposition [BK90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' When we apply the below definitions, our poset will be the opposite of  the non-negative natural numbers Z≥0 with their usual total order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  6The terminology is intended to model sheaves on a space with an open-closed decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note this is opposite to the  convention of [AMR] where C0 is called a closed subcategory since it models quasi-coherent sheaves on a scheme supported  along a closed subscheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  79  Given a stable presentable ∞-category, an open subcategory (called a closed subcategory in [AMR,  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1]) is an adjoint triple  C0  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  j∗  �  C  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='=j∗  �  where j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', j∗ are fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We write Copen for the poset of open subcategories of C with respect to  inclusion under j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='.  Given a poset P, a P-stratification (following [AMR, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2]) of a stable presentable ∞-category  C is a functor  Z• : P  � Copen  such that C = �  p∈P Zp and for any p, q ∈ P, there exists a factorization  �  r≤p,q Zr  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  � Zp  Zq  �✤  ✤  ✤  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  � Z  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='=j∗  �  Given a P-stratification Z•, its pth stratum [AMR, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3] is the quotient  Cp = Zp/  �  q<p  Zq  By construction, there is a recollement  �  q<p Zq  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  j∗  �  Zp  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='=j∗  �  i∗  �  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  Cp  i∗=i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose X = ∪p∈Z≥0Xp is a topological space written as the increasing union of closed  subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose C = colimp∈Z≥0 Sh(Xp) where the transition maps are given by extension by zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  This fits into the above formalism as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For p ∈ Z≥0, set Up = X \\ Xp−1 (convention: X−1 = ∅)  viewed as the increasing union of closed subspaces Up = ∪q∈Z≥0Up ∩Xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Set Zp = colimq∈Z≥0 Sh(Up ∩Xq)  where the transition maps are given by extension by zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then we have a stratification C = Z0 ←֓ Z1 ←֓ Z2 · · · with respect to the poset P = (Z≥0)op with open  subcategories  Zp  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  j∗  �  C  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='=j∗  �  with inclusions given by !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='-extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The strata are given by Cp = Zp/Zp+1 = Sh(Xp \\ Xp−1) noting that  Up \\ Up+1 = Xp \\ Xp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By construction, there are recollements  Zp+1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  j∗  �  Zp  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='=j∗  �  i∗  �  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  Cp  i∗=i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  The structure is equivalent to giving the inclusions Y0 ֒→ Y1 ֒→ Y2 · · · of the complementary closed  categories Yp = Sh(Xp) with their recollements  Cp  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  j∗  �  Yp  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='=j∗  �  i∗ �  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' �  Yp−1  i∗=i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  80  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Interaction of colimits and recollement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The following gives natural hypotheses for when a  colimit of stable presentable ∞-categories with recollement inherits a recollement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let I be a small category, C′  , C• : I → StL functors to stable presentable ∞-categories  (with morphisms given by left adjoints), and i• : C′  → C• a map of functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Assume the following:  (a) For each J ∈ I, the functor iJ extends to a recollement in StL:  C′′  J  jJ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' �  jJ∗ �  CJ  jJ  �  i∗  J  �  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  J  �  C′  J  iJ  �  In particular, iJ is fully faithful and C′′  J = CJ/iJ(C′  J) (hence the assignment J �→ C′′  J extends  naturally to a functor C′′  : I → StL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (b) For all maps I → J in I, the natural commutative diagram  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  C′′  I  �  CI  jI  �  �  C′  I  iI  �  �  C′′  J  CJ  jJ  �  C′  J  iJ  �  is a morphism of recollements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Here the vertical morphisms are given as part of the data of  C′′  , C• and C′  as functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then the map  i = colim i• : colimI C′ � colimI C•  extends to a recollement  colim C′′ j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  j∗  �  colim C•  j  �  i∗  �  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  colim C′ i  �  with i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ≃ colimI i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  J, i∗ ≃ colimI i∗  J, j = colimI jJ, j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' = colimI jJ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' and j∗ = colimI jJ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Assume further we are given a functor f : I0 → I, denote by C′  0• = C′  ◦ f, C0• = C• ◦ f and  C′′  0• = C′′  ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the natural commutative diagram is a morphism of recollements:  colim C′′  0•  �  colim C0•  �  �  colim C′  0•  �  �  colim C′′ colim C•  �  colim C′ �  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (1) By assumption (a), we have adjoint triples (i∗  J, iJ, i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  J) in StL, so that the unit u : id → i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  JiJ  and counit ǫ : i∗  JiJ → id are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Assumption (b) implies (i∗ := colim i∗  J, i = colim iJ, i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' :=  colim i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  J) is also an adjoint triple, with counit i∗i → id an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore i is fully faithful,  and colim CJ is a recollement of colim C′  J and colim CJ/ colim C′  J ≃ colim C′′  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Moreover, the natural map  j : colim C• → colim C′′  is naturally isomorphic to colimI jJ, by the functoriality of cokernel (cokernel of  a functor f : A → B is by definition the pushout of 0 ← A  f−→ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since taking colimit preserves adjoint  pairs, so we also have j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' = colimI jJ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' and j∗ = colimI jJ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Follows directly from the functoriality of colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose we have a morphism of recollement as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) in which f0 is an equivalence,  then the functor  f  �  i∗ : C  �  C1  D1 → D  is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  81  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Applying Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1 to the pushout of the recollements (C0 ← C ← C1) and (0 ← D1 ← D1) along  (0 ← C1 ← C1), we get a recollement (C0 ← C �  C1 D1 ← D1) that fit into a commutative diagram  C0  f0  �  C �  C1 D1  �  f � i∗  �  D1  �  D0  D  �  D1  �  It is easy to check the left square is both left and right adjointable, hence the above diagram is a morphism  of recollements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since both of the vertical arrows at the two ends are equivalences, so is the middle vertical  arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Posets and cosheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Generalities on posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a poset (partially ordered set) I, let N(I) denote its nerve, viewed as  a simplicial set and hence a quasi-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We say a subset J ⊂ I is up-closed if i ∈ J , i ≤ i′ implies  i′ ∈ J ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' similarly, we say a subset J ⊂ I is down-closed if i ∈ J , i′ ≤ i implies i′ ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For a poset I, let |I| be the geometric realization of I, defined as the usual geometric realization of  the simplicial set N(I) (hence a simplicial complex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, the r-dimensional faces of |I| are  in bijection with non-degenerate r-dimensional simplices in N(I), which are parametrized by strictly  increasing chains i0 < i1 < · · · < ir in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A poset I is called acyclic if |I| is weakly contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  An order-preserving map of posets f : I → J is called coCartesian, if for any pair j ≤ j′ in J , and any  i ∈ f −1(j), the set {i′ ∈ I|i ≤ i′, j′ ≤ f(i′)} has a unique minimal element ∂j→j′(i) that maps to j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  An order-preserving map of posets f : I → J is called a left covering map, if for each pair j ≤ j′  in J , and any i ∈ f −1(j), there is a unique i′ ∈ f −1(j′) such that i ≤ i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A left covering map is in  particular coCartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' If f is a left covering map, then each fiber f −1(j) is a discrete poset (any pair  of distinct elements are incomparable), and ∂j→j′ gives a map f −1(j) → f −1(j′) for each pair j ≤ j′  in J , compatible with compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Such a structure is simply a functor J → Sets (called a J -set in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Conversely, given a functor X : J → Sets, if we let I = �  j∈J Xj and define the partial  order on I to be generated by i ≤ Xj→j′(i) for all j ≤ j′ ∈ J and i ∈ Xj, then the natural projection  f : I → J is a left covering map of posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We conclude that a left covering map of posets f : I → J is  the same datum as a J -set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', a functor J → Sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Generalities on cosheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let I be a poset with nerve N(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We call a functor F : I → StL a  cosheaf on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In other words, for each i ∈ I, F assigns a stable presentable ∞-category Fi, and for each  arrow ϕ : i → j in I, there is a functor Fϕ : Fi → Fj, together with higher compatibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We call  colimI F ∈ StL the (category of) cosections of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let f : I → J be an order-preserving map of posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G be a cosheaf on J , then we define f ∗G to  be the cosheaf on I given by the composition I  f−→ J  G−→ StL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' If f is injective, then we write f ∗F as F|I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In the following, we gather some useful tools for calculating colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Most of them can be found in  Lurie [Lur09, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Pushforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let f : I → J be an order-preserving map of posets and F be a cosheaf on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We  denote by f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F : J → StL a left Kan extension of F along the map f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then colimI F  ∼  → colimJ f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F by  the general properties of left Kan extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Suppose f : I → J is coCartesian, then for j ≤ j′ in J , there is a natural functor  colimi∈f −1(j) Fi → colimi′∈f −1(j′) Fi′  induced by the functors (for i ∈ f −1(j))  Fi → F∂j→j′ (i) → colimi′∈f −1(j′) Fi′  where the second map uses that ∂j→j′(i) ∈ f −1(j′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' These maps give a cosheaf  �  f F on J whose value at  j is colimi∈f −1(j) Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  82  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition (Lurie, special case of [Lur09, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose f : I → J is a co-  Cartesian map of posets and F is a cosheaf on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then  �  f F is a left Kan extension of F along f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In  particular, there is an equivalence colimI F ≃ colimJ  �  f F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Another instance of pushforward is extension by zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let ı : I1 ⊂ I be the inclusion of an up-closed  subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then for a cosheaf F1 on I1, the cosheaf ı!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F1 on I is obtained by extension by zero: it assigns to  each i ∈ I \\ I1 the zero category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Dually, let \uf6be : I0 ⊂ I be the inclusion of a down-closed subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let F0 be a cosheaf on I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We define  j∗F0 to be extension by zero of F0 as a cosheaf on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then j∗ : Fun(I0, StL) → Fun(I, StL) is a right  adjoint to the restriction j∗ : Fun(I, StL) → Fun(I0, StL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let sd(I) be the set of non-degenerate simplices in the nerve N(I): they are strictly  increasing non-empty chains i = (i0 < · · · < ir) in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Equip sd(I) with a partial order by defining i ≤ i′  if and only if i′ is part of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have a map of posets π : sd(I) → I sending i = (i0 < · · · < ir) to i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For a cosheaf F on I, let sd(F) = π∗F be the pullback cosheaf on sd(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The map π : sd(I) → I is cofinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In particular, for any cosheaf F on I, there is a  canonical equivalence  colimsd(I) sd(F) ≃ colimI F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To check π is cofinal, we apply Quillen’s Theorem A as in [Lur09, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For i ∈ I,  the category M = sd(I) ×I Ii/ consists of strictly increasing non-empty chains i0 < · · · < ir in I such  that i ≤ i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We need to show that the geometric realization |M| of M is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let L ⊂ M be  the subposet where i0 = i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' let K ⊂ M be the subposet where i < i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then M = L ⊔ K, and we can  identify L ∼= K⊲ (the right cone of K) because L has a maximal element that is i itself as a chain, and its  complement is isomorphic to K via i = (i0 < · · · < ir) → i′ = (i1 < · · · < ir).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Also we have i ≤ i′ (where  i′ ∈ K and i ∈ L is obtained from i′ by adding i), and all the other order relations are consequences of  it and the fact that L and K are subposets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We conclude that M ∼= (K × [1]) �  K×{0} K⊲ (where [1]  is the poset {0, 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The geometric realization |M| is thus homeomorphic to the cone over |K|, hence  contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Double colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In some situations we want to rewrite colimI F as a colimit of colimits, where the  inner colimit is taken over a subposet of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lurie’s theorem [Lur09, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10] gives a general  framework for doing this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We recall the statement of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' in the simplified situation where the colimits  are parametrized by posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose K is a poset and F : K → StL is a cosheaf on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J be another  poset with an assignment J ∋ j �→ Kj ⊂ K such that j ≤ j′ in J implies Kj ⊂ Kj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition (Lurie, special case of [Lur09, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under the above notation, assume:  Kj is up-closed for each j ∈ J , and for each k ∈ K, the subposet Jk := {j ∈ J |k ∈ Kj} is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then  the natural maps colimKj F → colimK F give an equivalence  colimj∈J (colimKj F) ≃ colimK F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  As an immediate application, we get a Mayer-Vietoris type theorem for colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let I be a poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Suppose I = I0 ∪ I1, where I0 and I1 are both up-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let F be a  cosheaf on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let I01 = I0 ∩ I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the natural maps give an equivalence  colimI0 F|I0  �  colimI01 F|I01  colimI1 F|I1  ∼  → colimI F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let D1 be the poset of proper subsets of {0, 1} under inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Apply Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8 to the  assignment ∅ �→ I01, {0} �→ I0 and {1} �→ I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  83  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Locally constant cosheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A locally constant cosheaf on I is one such that Fϕ is an equivalence  for all arrows ϕ in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition (Lurie, [Lur09, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let f : I ֒→ J be an order-preserving map of  posets and F a locally constant cosheaf on J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' If |I| ֒→ |J | is a weak homotopy equivalence, then the  natural functor  colimI f ∗F → colimJ F  is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In particular, if F is a cosheaf on an acyclic poset I, then for any i ∈ I, the natural map Fi → colimI F  is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recollement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Given maps of cosheaves on a poset I  F′′  F  β  �  F′  α  �  we say it is a recollement of cosheaves, if  (1) For each i ∈ I, evaluating α and β at i gives a recollement in StL:  F′′  i  βi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  �  βi∗ �  Fi  βi  �  α∗  i  �  α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  i  �  F′  i  αi  �  (2) For i ≤ i′ in I, the commutative diagram where the vertical maps are the transition functors for  F, F′ and F′′  F′′  i  �  Fi  βi  �  �  F′  i  αi  �  �  F′′  i′  Fi′  βi′  �  F′  i′  αi′  �  is a morphism of recollements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In this situation, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1 is applicable to conclude that  colimI F′′  colimI F  colim β  �  colimI F′  colim α  �  extends to a recollement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A caution to those who bring intuition of cosheaves with values in a stable category  (the ∞-category StL is itself not stable): one needs the hypothesis of a recollement of cosheaves in the  statement of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Else a “triangle of cosheaves” need not lead to a “triangle of cosections”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For a toy example of what could go wrong, consider the diagram of cosheaves  Q  F  �  K  �  on the poset ∆opp  1  = {0 ← ∅ → 1}, where each cosheaf is given by a horizontal row of the diagram  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  0  �  0  �  �  � Vect ⊕ 0  i  �  0  �  Vect  �  =  �  d  � Vect ⊕ Vect  p  �  0  Vect  �  ∼  � 0 ⊕ Vect  Here d is the diagonal embedding, i is inclusion to the first factor, and p is projection to the second factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Note at each element of ∆opp  1  , the corresponding vertical sequence in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) is a recollement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' But note  the right squares are not adjointable with respect to the vertical maps, so we do not have a recollement of  cosheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  84  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Then we have  colim∆opp  1  Q  colim∆opp  1  F  �  colim∆opp  1  K  �  0  0  �  Vect  �  So while colim∆opp  1  Q is the cokernel of colim∆opp  1  F ← colim∆opp  1  K, we see K is not the kernel of  colim∆opp  1  Q ← colim∆opp  1  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Thus we do not obtain a recollement on cosections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Expansion of cosheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this section, we prove the invariance of colimits of cosheaves under a  certain modification called expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Expansion of posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let I be a poset and I′ ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) We call I a weak elementary expansion of I′ if:  I \\ I′ contains a unique minimal element, which we denote by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let I′  >0 = {i ∈ I′|0 < i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then I′  >0 is up-closed in I and acyclic (in particular non-empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We denote the above situation by I′ ր I or I′ ր0 I if want to indicate the minimal element in  I \\ I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) We call I an expansion of I′ if there is a finite sequence of up-closed subsets In ⊂ I (1 ≤ n ≤ N)  such that I0 = I′, IN = I and In is a weak elementary expansion of In−1:  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  I′ = I0 ր I1 ր I2 ր · · · ր IN = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Note that a weak elementary expansion may not be an expansion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' it is an expansion if I′ is up-closed  in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We give an example of a weak elementary expansion that we use in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Fix n ≥ 0, and set  [n] = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let Dn be the partially ordered set of proper subsets J ⫋ [n] with respect to inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Fix a nonempty J ⊂ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Set DJ  n ⊂ Dn to be the subsets of [n] not contained in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) The inclusion DJ  n ⊂ Dn is a weak elementary expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) More generally, for any poset I with an embedding Dn ֒→ I whose image is up-closed, I is a weak  elementary expansion of I′ := I \\ (Dn \\ DJ  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (1) Clearly DJ  n is up-closed in Dn and Dn \\ DJ  n has the minimal element ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We need to show that  DJ  n is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We can identify the geometric realization of the nerve N(Dn) with the standard n-simplex  ∆n with vertices �0, · · · , �n in such a way that J ∈ Dn corresponds to the barycenter of the face σJ whose  vertices are {�i|i ∈ [n]\\J} (in particular, σ∅ is the maximal face of ∆n, and J1 ⊂ J2 if and only if σJ1 ⊃ σJ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Under this identification, the geometric realization of N(DJ  n) is ∆J  n := ∪J′∈DJ  nσJ′ = ∪J′⫋[n],J′̸⊂JσJ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We  need to show that ∆J  n is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Without loss of generality, we may assume J = {0, 1, · · · , j}, for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let ∆j ⊂ ∆n be the face  with vertices �0, · · · ,�j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then ∆J  n is the union of faces whose vertices do not contain all vertices �i with  j + 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Define a deformation retraction ∆n → ∆j by linearly extending the map on vertices given  by �i �→ �i, if i ≤ j, and �i �→ �j, if i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Restriction of this deformation retraction to ∆J  n ⊂ ∆n gives a  deformation retraction ∆J  n → ∆j, proving that ∆J  n is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) Let ∅ ∈ Dn be the unique minimal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then I′  >∅ = DJ  n is up-closed in Dn hence also in I by  assumption, and is acyclic by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Expansion of cosheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let I be a poset and I′ ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let ϕ : E → F be a map of cosheaves on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) If I′ ր0 I, we say that ϕ is a weak elementary expansion along I′ ր0 I if  ϕ|I′ : E|I′ → F|I′ is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  85  For i ∈ Inew = I \\ I′, the following square is a pushout  E0  �  ϕ0 � F0  �  Ei  ϕi  � Fi  (2) Suppose I is an expansion of I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We call ϕ an expansion along I′ ⊂ I if there exists a sequence  of elementary expansions as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1), where In are up-closed, such that:  ϕ|I′ : E|I′ → F|I′ is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For each 1 ≤ n ≤ N, let 0n ∈ In \\ In−1 be the unique minimal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then for any  i ∈ In \\ In−1, the following square is a pushout  E0n  �  ϕ0n � F0n  �  Ei  ϕi  � Fi  The key property of an expansion of cosheaves is the invariance of colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Let I′ ր0 I be a weak elementary expansion of posets, and let ϕ : E → F  be a weak elementary expansion of cosheaves on I along I′ ր0 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then ϕ induces an equivalence  on colimits  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  colim ϕ : colimI E  ∼  → colimI F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The same holds if I′ ⊂ I is an expansion and ϕ is an expansion of cosheaves on I along I′ ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (1) Let Iold = I′ and Inew = I \\ Iold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let I+ = I ⊔ {0′}, and extend the partial order from I to  I+ by adding the relations 0 < 0′ < i for all i ∈ Iold  >0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We first claim that the map of posets I → I+ is cofinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, we apply Quillen’s Theorem A to the  map i, and note that for any i ∈ I+, I ×I+ I+  ≥i either has a unique minimal element i if i ∈ I, or when  i = 0′, I ×I+ I+  ≥0′ = {i ∈ Iold|0 ≤ i} = Iold  >0 which is acyclic by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore the assumption  for applying Quillen’s Theorem A is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Extend E to a cosheaf E+ on I+ by assigning E+  0′ := F0 and the functors E+  0′ → E+  i  for i ∈ Iold are  given by the composition F0 → Fi  ϕ−1  i  −−→ Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The cofinality of I in I+ implies  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  colimI E  ∼  → colimI+ E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Next we would like to apply Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8 to write colimI+ E+ as a double colimit where the  outer colimit is parametrized by I again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We apply it to K = I+, J = I and the following assignment  I ∋ i �→ I+  i : if 0 ≤ i, let I+  i = {j ∈ I|j ≤ i} ∪ {0′};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' otherwise let I+  i = {j ∈ I|j ≤ i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We check that the  acyclicity condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let k ∈ I+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' If k ∈ I, then {i ∈ I|k ∈ I+  i } contains k as the unique minimal  element hence is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' If k = 0′, then {i ∈ I|0′ ∈ I+  i } contains 0′ as the unique minimal element, hence  again is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8 gives a natural equivalence  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  colimi∈I(colimI+  i E+) ≃ colimI+ E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For each i ∈ Iold, I+  i  has a unique maximal element i, hence colimI+  i E+ ≃ E+  i  = Ei ≃ Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For i ∈  Inew, I+  i  does not intersect Iold  >0 since Iold  >0 is up-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This implies {0, 0′, i} is cofinal in I+  i , hence  colimI+  i E+ ≃ E0′ �  E0 Ei ≃ Fi by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Combining these calculations we see that the right side of  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) is canonically equivalent to colimI F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Combined with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3) we conclude that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) We would like to reduce to the situation of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let I′ = I0 ր01 I1 ր02 I2 ր · · · ր0n IN = I be  a sequence of elementary expansions, with each In up-closed in I such that ϕ satisfies the conditions in  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For 1 ≤ n ≤ N, let En be the cosheaf on I that is F|In on In and is E|I\\In on I \\ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For I \\ In ∋ i ≤ i′ ∈ In, the transition functor (En)i = Ei → (En)i′ = Fi′ is defined to be the composition  86  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  Ei  ϕi  −→ Fi → Fi′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then E0 = E and EN = F and there is a natural map ϕn : En−1 → En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It suffices to  show that ϕn induces an equivalence on colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Inew  n  = In \\In−1 and Iold = I \\Inew  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We claim that I is an elementary expansion of Iold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed,  Inew  n  has a unique minimal element 0n by assumption, and Iold  >0n = (In−1)>0n because In−1 is up-closed  in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By construction, ϕn : En−1 → En is a weak elementary expansion along Iold ր I in the sense of  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore we reduce to the case of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A categorical contraction principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' If Y is a manifold with a flow {Φt}t>0 that contracts Y  to a subspace Z ⊂ Y as t → 0, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', the flow {Φt}t>0, viewed as an R>0-action, extends to an action  {Φt}t≥0 of the monoid R≥0 with Z = Φ0(Y )) and if F is a constructible sheaf on Y locally-constant (hence  constant) on flow lines, then the restriction map gives an isomorphism RΓ(Y, F) ∼= RΓ(Z, F|Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This is  usually called the contraction principle for sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Below we formulate and prove a categorical analog of  this principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let J ⊂ I be a subposet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Denote s : J → I the inclusion and I0 = I\\J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let  F : I → StL be a functor, denote by L : I0 → StL the cokernel of the natural map (s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='s∗F)|I0 → F|I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Assume that  (1) The maps of cosheaves on I0  L  F|I0  β  �  (s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='s∗F)|I0  α  �  is a recollement of cosheaves in the sense of Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) L is a locally constant cosheaf on I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (3) The inclusion |s| : |J | ֒→ |I| is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then the natural map is an equivalence:  colimJ s∗F ≃ colimI F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Before proving the theorem, we give a situation where it is easy to compute the left Kan extension  s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='s∗F of s∗F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let f : I → J be a coCartesian map of finite posets such that each fiber f −1(j) contains a  unique minimal element s(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then s : J → I is a section to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let I0 = I \\ s(J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let F be a cosheaf  on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The pullback f ∗s∗F is the left Kan extension s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (s∗F) of s∗F along s : J → I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For each i ∈ I, {j ∈ J |s(j) ≤ i} has a unique maximal element f(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore the value of s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (s∗F)  at i is (s∗F)f(i) = Fs(f(i)) = (f ∗s∗F)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Since the map f is coCartesian, f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='F is equivalent to  �  f F by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We get the following  corollary from Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Under the above notations, assume the conditions in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then the  natural maps are equivalences  colimJ s∗F ≃ colimI F ≃ colimI  �  f  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1 will be given after recalling some facts about simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Simplicial complexes and exit poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a poset I we have a simplicial complex |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a simplicial  complex Y we define its exit poset P(Y ) to be the set of (closed) faces σ of Y with the partial order given  by inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' From the definition, faces in |I| are parametrized by non-degenerate simplices in N(I),  therefore we get a canonical isomorphism of posets  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  sd(I) ∼= P(|I|)opp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For any subset I′ ⊂ I with the inherited poset structure, |I′| ⊂ |I| is a closed subcomplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The  complement |I| \\ |I′| is the open star star ◦(|I \\ I′|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  87  For a simplicial complex Y , let sd(Y ) be its barycentric subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then sd(|I|) can be identified  with |sd(I)| as simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  For a simplicial complex X, by a cosheaf F on X we mean a cosheaf F on P(X)opp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', a functor  F : P(X)opp → StL (equivalently, by passing to right adjoints, it is the same datum as a functor P(X) →  StR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will write  �  X F for colimP(X)opp F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We write FZ = F|P(Z)opp for the restriction of a cosheaf to  a subcomplex Z ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let F be a cosheaf on a poset I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1), sd(F) is a cosheaf on P(|I|)opp, hence a cosheaf on |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6, there is a canonical equivalence  �  |I|  sd(F) ≃ colimI F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Simplicial collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Recall a simplex τ ⊂ K in a simplicial complex K is called a free face if there  exists a unique maximal simplex σ ⊂ K such that τ ⫋ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In this case, the simplicial collapse of K along  the free face τ ⊂ K is defined to be the subcomplex K′ = K \\ (∪τ⊂γ⊂σγ◦) ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note the pair K′ ⊂ K  implicitly knows the free face τ as the unique minimal face of K removed to obtain K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Given general  simplicial complexes K0 ⊂ K, by a simplicial collapse of K to K0, we will mean a finite sequence of  subcomplexes K0 ⊂ · · · ⊂ Kn−1 ⊂ Kn ⊂ · · · ⊂ KN = K such that Kn−1 results from Kn via a simplicial  collapse along a free face τn ⊂ Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma (Whitehead [Whi39, Corollary on page 252]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For an inclusion of finite simplicial complexes  Z0 ⊂ Y0, after the second barycentric subdivision Z = sd2(Z0) ⊂ sd2(Y0) = Y , there is always a simplicial  collapse of the closed star U = star(Z) ⊂ Y to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We give a sketch here and refer to [Whi39] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  After the first barycentric subdivision, Z1 = sd(Y0) will be a full subcomplex of Y1 = sd(Y0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' if the  vertices of a simplex of Y1 lie in Z1, then the simplex itself lies in Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To see this, recall the k-simplices  σ ⊂ Y1 are given by k-flags of simplices σ0 ⊂ σ1 ⊂ · · · ⊂ σk ⊂ Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The vertices of σ are the 0-flags  σ0, σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' , σk ⊂ Y0 that occur in the flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' If these 0-flags satisfy σ0, σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' , σk ⊂ Z1, then the simplices  satisfy σ0, σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' , σk ⊂ Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Thus the k-flag representing σ indeed satisfies σ0 ⊂ σ1 ⊂ · · · ⊂ σk ⊂ Z0, and  hence σ ⊂ Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Now starting from the full subcomplex Z1 ⊂ Y1, after a barycentric subdivision Z = sd(Z1) ⊂ sd(Y1) =  Y , we can always find a simplicial collapse of the closed star U = star(Z) ⊂ Y to Z as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Such a  simplicial collapse will be determined by a total order on the simplices of U \\ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We can take any total  order compatible with the following partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  First, consider the prior open star U ◦  1 = star ◦(Z1) ⊂ Y1, and the complement U ◦  1 \\ Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Note every  relatively open simplex σ◦ ⊂ U \\ Z lies in a unique relatively open simplex ˆσ◦ ⊂ U ◦  1 \\ Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Partially order the simplices of U ◦  1 \\Z1 opposite to their natural order by closure inclusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' so the partial  order begins with the maximal simplices of U ◦  1 \\ Z1 and proceeds to the minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (Note U ◦  1 \\ Z1 has no  vertices – its simplex closures must contain vertices from both Z1 and its complement – so its minimal  simplices will at least be edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Partially order the simplices σ◦ ⊂ U \\ Z by the above partial order on the simplices ˆσ◦ ⊂ U ◦  1 \\ Z1  containing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' From here, it suffices to independently define for each τ ◦ ⊂ U ◦  1 \\ Z1 with partial closure  τ = τ ◦ ⊂ U ◦  1 , a simplicial collapse of the subcomplex τ ∩ U to the subcomplex ∂τ ∩ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Thus it remains to solve the following general local problem: given a closed simplex τ1 and a proper  closed facet ρ1 ⊂ τ1, consider the barycentric subdivision τ = sd(τ1) and its subcomplex ρ = sd(ρ1) ⊂ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let U = star(ρ) ⊂ τ be the closed star of ρ ⊂ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then we need a simplicial collapse of U to U ∩ ∂τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To  achieve this, we can take any total order on the simplices of U \\ (U ∩ ∂τ) compatible with the following  partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' First, partially order the simplices of U \\ (U ∩ ∂τ) opposite to their natural order by closure  inclusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' so the partial order begins with the maximal simplices of U \\ (U ∩ ∂τ) and proceeds to the  minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then refine this partial ordering, by further ordering the simplices of U \\ (U ∩ ∂τ) starting with  those with the maximal number of vertices not in ρ and ending with those with the minimal number of  vertices not in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is elementary to check this indeed gives a simplicial collapse of U to U ∩ ∂τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  88  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let E = s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='s∗F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We need to show that the adjunction map α : E → F  induces an equivalence colimI E ≃ colimI F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let Y0 = |I| and Z0 = |I1| = |s(J )|, which are finite simplicial complexes by assumption, and Z0 ⊂ Y0  is a closed subcomplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We denote Y = sd2(Y0) and Z = sd2(Z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then sd(F) and sd(E) define cosheaves  on Y0 and Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We abuse the notation to denote the pullback of sd(F), sd(E) to Y and Z still by F and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6, it suffices to show the natural map  �  Y E →  �  Y F is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The locally constant cosheaf L on I0 defines a locally constant cosheaf sd(L) on sd(I) \\ sd(s(J )) via  pullback by the first vertex map sd(I) \\ sd(s(J )) → I \\ s(J ) = I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In other words, L pullbacks to a  locally constant cosheaf sd(L) on P(Y0)opp \\ P(Z0)opp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Iterating this procedure, we get a locally constant  cosheaf sd3(L) on P(Y )opp \\ P(Z)opp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For simplicity we still denote it by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For a cosheaf F′ on Y , we  denote its restriction to P(Y )opp \\ P(Z)opp by F′  Y \\Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The assumptions on the original F imply the maps  α and β restricted to Y \\ Z extend to a recollement of cosheaves on P(Y )opp \\ P(Z)opp:  L  �  � FY \\Z  β  �  �  � EY \\Z  α  �  Let U ◦ = star ◦(Z) ⊂ Y be the open star of Z, and U = star(Z) ⊂ Y the closed star of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Set  V = Y \\ U ◦ and W = U ∩ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='9, we have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2)  �  U  FU  �  �  W FW  �  V  FV  ∼  →  �  Y  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  The same applies to E in place of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We will prove the following claims:  (1) The diagram  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3)  �  W EW  �  �  �  W FW  �  �  V EV  � �  V FV  is a pushout square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) The natural map  �  U EU →  �  U FU is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We first show that these two claims imply the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let D1 be the poset of proper subsets of {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We define a cosheaf A on D1 by  A∅ =  �  W  EW ,  A{1} =  �  U  EU,  A{0} =  �  V  EV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Define another cosheaf A′ on D1 by  A′  ∅ =  �  W  FW ,  A′  {1} =  �  U  FU,  A′  {0} =  �  V  FV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then α induces a map of cosheaves A → A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The inclusion {{0}} ⊂ D1 is a weak elementary expansion  in the sense of Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2(1), and A → A′ is a weak elementary expansion along {{0}} ր D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6(1) applies to the situation to conclude that colimD1 A  ∼  → colimD1 A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2) and  its analog for E, we conclude that  �  Y E  ∼  →  �  Y F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We prove (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By the second assumption, over V , the termwise recollement  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4)  LV  FV  βV  �  EV  αV  �  extends to a recollement of cosheaves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', all six-term diagrams relating the values of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4) at two faces  σ ⊂ τ in V are morphisms of recollements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1(1), we conclude that the colimits also fit  into recollements  �  V LV  �  �  �  V FV  βV  �  �  �  �  V EV  αV  �  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  89  The same holds for W in place of W, so we have a diagram where the two rows extend to recollements  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5)  �  W LW  �  �  W FW  βW  �  �  �  W EW  αW  �  �  �  V LV  �  V FV  βV  �  �  V EV  αV  �  Moreover, by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1(2) this is a morphism of recollements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now consider the map  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6)  �  W  LW →  �  V  LV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We claim that W ֒→ V is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, since the inclusions Z ⊂ Y , Z ⊂ U are  homotopy equivalences, the inclusion U ⊂ Y is as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore there is a deformation retraction of  Y to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (See for example [Hat02, Chapter 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=') Removing U ◦ then gives a deformation retraction of  V = Y \\ U ◦ to W = U \\ U ◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' in particular, the inclusion W ֒→ V is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since LV is  locally constant, we can then apply Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11 to conclude that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Then  by Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2 applied to the morphism of recollements (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5), we conclude that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3) is a pushout  diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We prove (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6, we have a sequence of subcomplexes Z = K0 ⊂ K1 ⊂ · · · KN = U  such that Kn−1 is a simplicial collapse of Kn along a free face τn, 1 ≤ n ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We claim P(Kn)opp  is a weak elementary expansion of P(Kn−1)opp with minimal element in P(Kn)opp \\ P(Kn−1)opp given  by the unique maximal face σn in Kn containing τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, we use notations from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let  d = dim σn, we can identify P(σn) with Dd after choosing an identification of the vertices of σn with  [d], and then P(Kn−1 ∩ σn)opp is identified with DJ  d for some proper subset J ⊂ [d] corresponding to τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then the partition P(Kn)opp = P(Kn−1)opp ⊔(Dd \\DJ  d ) satisfies the conditions in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3(2), which  verifies that P(Kn−1)opp րσn P(Kn)opp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore P(Z)opp ⊂ P(U)opp is an expansion in the sense of  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We claim that αU : EU → FU is an expansion of cosheaves along P(Z)opp ⊂ P(U)opp in the sense of  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Indeed, by construction, αU is an equivalence when restricted on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any two faces  τ ⊂ σ of U not contained in Z, we have a morphism of recollements  Lσ  �  Fσ  βσ  �  �  Eσ  ασ  �  �  Lτ  Fτ  βτ  �  Eτ  ατ  �  where the left vertical map is an equivalence since L is locally constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2, the right  square above is a pushout square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This verifies the second condition for an expansion of cosheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Claim  (2) now follows from Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Functoriality of sheaves with Lagrangian singular support  Let k be a field of characteristic 0, and let X be a smooth algebraic stack over C of finite type, L ⊂ T ∗X  be a closed (C×-)conic Lagrangian substack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Denote Sh(X) the ∞-category of sheaves of k-modules on  X, and ShL(X) ⊂ Sh(X) the full subcategory of sheaves with singular support contained in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' It is  known that ShL(X) is compactly generated [AGK+, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='8], Sh(X) is presentable, and the inclusion  ShL(X) → Sh(X) preserves both limits and colimits [AGK+, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='5] The goal of this section is to  prove the following:  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let f : X → Y be a representable map between finite type smooth algebraic stacks  over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (1) Let LY ⊂ T ∗Y be a closed conic Lagrangian substack, then the functors f ∗, f !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : ShLY (Y ) → Sh(X)  preserve both limits and colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  90  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  (2) Let LX ⊂ T ∗X be a closed conic Lagrangian substack, then the functors f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', f∗ : ShLX(X) → Sh(Y )  preserve both limits and colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (1) We show the statement for f ∗, the one for f !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' follows from similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Since f ∗ : Sh(Y ) →  Sh(X) is a left adjoint, it preserves colimits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' hence f ∗|ShLY (Y ) also preserves colimits since the inclusion  ιLY : ShLY (Y ) ֒→ Sh(Y ) does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Let F : I → ShLY (Y ) be a diagram of sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have the natural map  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  φ : f ∗(lim  i∈I Fi) → lim  i∈I f ∗Fi  in Sh(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To show it is an equivalence, it suffices to show the same after pulling back to a smooth cover  pX : �  X → X, since p∗  X is conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let pY : �Y → Y be a smooth surjective map from a scheme �Y ,  and form the Cartesian diagram  �  X  pX  �  �  f  � �Y  pY  �  X  f  � Y  Then p∗  Xφ can be factored as  p∗  Xf ∗(lim Fi) = �f ∗p∗  Y (lim Fi) ≃ �f ∗ lim(p∗  Y Fi) → lim �f ∗p∗  Y Fi = lim p∗  Xf ∗Fi ≃ p∗  X lim f ∗Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Here we use that p∗  X and p∗  Y preserves limits (being smooth, they agree with p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  X and p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Y up to a shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' To  show the above composition is an equivalence, it suffices to show the middle arrow is an equivalence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=',  �f ∗ : ShL �  Y (�Y ) → Sh( �  X) preserves limits (here L�Y ⊂ T ∗ �Y is the transport of LY under the Lagrangian  correspondence between T ∗ �Y and T ∗Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Henceforth we assume both X, Y are smooth schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any closed conic Lagrangian LY ⊂ T ∗Y ,  there is a Whitney stratification {Yα}α∈S of Y , so that LY ⊂ ∪αT ∗  XαX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' So suffices to prove the claim for  LY is conormal to a Whitney stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now pick a Whitney stratification {Xβ}β∈T of X, so that the  images of f ∗ land in ShM(X) ⊂ Sh(X), for M = ∪βT ∗  XβX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now let ix : x → X be the inclusion of a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  By [NYa, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='11], the functor i∗  x : ShM(X) → k-mod preserves limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now let F : I → ShLY (Y )  be a diagram of sheaves in ShLY (Y ), we need to show the natural map φ in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Equivalently, we need to check that φ induces an equivalence on each stalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Taking stalks at x ∈ X, φx  can be identified with the composition of equivalences  φx : (f ∗(lim Fi))x ≃ (lim Fi)f(x) ≃ lim(Fi)f(x) ≃ lim f ∗(Fi)x ≃ (lim f ∗(Fi))x  Here the second (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' last) equivalence uses the fact that i∗  f(x) : ShLY (Y ) → k-mod (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ix : ShM(X) →  k-mod) preserves limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This show φ is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  (2) We only prove that f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' preserves limits, and the case of f∗ follows from a similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' By the  same reduction using smooth covers as in (1), we can assume X, Y are smooth schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We can further  assume LX is conormal to a Whitney stratification X = ∪α∈SXα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now pick a relative compactification  X of f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='e, f factors as X  j−→ X  p−→ Y , where j is an open embedding, and p is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now suffices to  show that j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : ShXα(X) → Sh(X) preserves limits, since p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' = p∗ preserves limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let X = ∪α′∈S′Xα′ be  a Whitney stratification on X that refines the partition X = (∪Xα) ∪ (X\\X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Define L′  X and L′  X to be  the conormals to the new stratifications of X and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now suffices to show that j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' : ShL′  X(X) → ShL′  X(X)  preserves limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let F : I → ShL′  X(X) be a diagram of sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Consider the natural map ϕ : j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (lim Fi) →  lim j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='(Fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We only need to check that ϕ is an equivalence on each stalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For x ∈ X, it is clear that ϕx is  an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For x ∈ X \\ X, we have (j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (lim Fi))x = 0 and (lim j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (Fi))x = lim(j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' (Fi))x = 0 since i∗  x  preserves limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Therefore ϕx is also isomorphism, and hence ϕ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  91  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Universal version of Tao-Travkin Theorem  The method of [TT] can be applied to deduce the following theorem:  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let G◦ be the neutral component of the loop group G = G((t)) of a connected reductive  group G over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let HG◦ ⊂ HG be the full monoidal subcategory of the universal affine Hecke category  consisting of sheaves supported on Iu\\G◦/Iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then the natural maps induce an equivalence of stable  presentable monoidal categories:  colim⊗  J⊂ftIa HLJ  ∼  � HG◦  where the colimit is of stable presentable monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  In this section, we shall adopt the notations in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='. Denote by I the convolution Schubert 1-category  for the affine Weyl group W a, which is denoted Word1  fr and defined in [TT, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Objects in  I are sequences w = (w1, · · · , wn) where each wi ∈ W a lies in a finite standard parabolic subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Recall G≤w = IwI ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For w = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', wn) ∈ I, put Xw = Iu\\G≤w1 ×Iu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ×Iu G≤wn/Iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Let  X◦  w ⊂ Xw be the open stratum where G≤wi are replaced with Gwi = IwiI, and let ∂Xw = Xw \\ X◦  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Then Xw is stratified into the union of Xw′, for w′ = (w′  1, · · · , w′  n) ∈ I such that w′  i ≤ wi for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Define Sh′(X◦  w) ⊂ Sh(X◦  w) to be the full subcategory of locally constant sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Denote by Sh′(Xw) ⊂  Sh(Xw) the full subcategory consisting of sheaves that are locally constant on each stratum Xw′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Similarly  define Sh′(∂Xw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  We have a functor H : I → StL  k , which assigns to an object w ∈ I the category Sh′(Xw), and to  a morphism w1 → w2 in I, the functor H(w1) → H(w2) given by convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' This is the universal  monodromic analogue of [TT, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Sketch of proof of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The proof of [TT, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='3] goes through, except we need to  check the analogous statement for [TT, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2]: for every birational map w1 → w2 in I, the  diagram  colimw′  1→w1 strict emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' H(w′  1)  �  �  H(w1)  �  colimw′  2→w2 strict emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' H(w′  2)  � H(w2)  is coCartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Now by same proof as in [TT, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2], we have a natural equivalence for every  w ∈ I:  colimw′→w strict emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' H(w′) ≃ Sh′(∂Xw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Therefore we are left to check the following Lemma, analogous to [TT, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' For any birational map w1 → w2 in I, the natural diagram is coCartesian:  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  Sh′(∂Xw1)  �  �  Sh′(Xw1)  �  Sh′(∂Xw2)  � Sh′(Xw2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' We have a morphism of recollements (see Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1)  Sh′(X◦  w1)  �  �  � Sh′(Xw1)  �  �  �  � Sh′(∂Xwl)  �  �  Sh′(X◦  w2)  �  � Sh′(Xw2)  �  �  � Sh′(∂Xw2)  �  Since w1 → w2 is birational, X◦  w1 ∼= X◦  w2, the left vertical arrow above is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The desired  statement follows from Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  □  92  PENGHUI LI, DAVID NADLER, AND ZHIWEI YUN  References  [AGK+] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Arinkin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Gaitsgory, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Kazhdan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Raskin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Rozenblyum, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Varshavsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The stack of local systems  with restricted variation and geometric Langlands theory with nilpotent singular support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='01906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [AMR]  David Ayala, Aaron Mazel-Gee, and Nick Rozenblyum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Stratified noncommutative geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='14602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Bez16]  Roman Bezrukavnikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' On two geometric realizations of an affine Hecke algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Publications math´ematiques de  l’IH ´ES, 123(1):1–67, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [BFM]  Armand Borel, Robert Friedman, and John W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Morgan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Almost commuting elements in compact Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  arXiv:math/9907007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [BFN10] David Ben-Zvi, John Francis, and David Nadler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Integral transforms and Drinfeld centers in derived algebraic  geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Journal of the American Mathematical Society, 23(4):909–966, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [BFO12] Roman Bezrukavnikov, Michael Finkelberg, and Victor Ostrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Character D-modules via Drinfeld center of Harish-  Chandra bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Inventiones mathematicae, 188(3):589–620, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [BK90]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Bondal and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Kapranov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Representable functors, Serre functors, and mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Mathematics of the USSR-  Izvestiya, 35(3):519–541, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [BNa]  David Ben-Zvi and David Nadler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The character theory of a complex group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' arXiv:0904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [BNb]  David Ben-Zvi and David Nadler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Nonlinear Traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' arXiv:1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='7175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [BNP17] David Ben-Zvi, David Nadler, and Anatoly Preygel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A spectral incarnation of affine character sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Compositio  Mathematica, 153(9):1908–1944, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Bon04]  C´edric Bonnaf´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' ´El´ements unipotents r´eguliers des sous-groupes de Levi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Canadian Journal of Mathematics,  56(2):246–276, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [BRY22] Yuri Berest, Ajay C Ramadoss, and Wai-Kit Yeung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Representation Homology of Topological Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' International  Mathematics Research Notices, 2022(6):4093–4180, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Fra16]  Dragos Fratila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' On the stack of semistable G-bundles over an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Mathematische Annalen, 365(1-2):401–  421, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [GR19]  Dennis Gaitsgory and Nick Rozenblyum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A Study in Derived Algebraic Geometry Volume I: Correspondences and  Duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' American Mathematical Society, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Hat02]  Allen Hatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Algebraic Topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Cambridge University Press, Cambridge ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' New York, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [He]  Xuhua He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' A generalization of cyclic shift classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [He07]  Xuhua He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The G-stable pieces of the wonderful compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Transactions of the American Mathematical  Society, 359(7):3005–3024, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [He18]  Xuhua He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Cocenter of p-adic groups, I: Newton decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Forum of Mathematics, Pi, 6:e2, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [HN14]  Xuhua He and Sian Nie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Minimal length elements of extended affine Weyl groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Compositio Mathematica,  150(11):1903–1927, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Kra09]  Daan Krammer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The conjugacy problem for Coxeter groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Groups, Geometry, and Dynamics, pages 71–171,  2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [KS90]  Masaki Kashiwara and Pierre Schapira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Sheaves on Manifolds, Volume 292 of Grundlehren Der Mathematischen  Wissenschaften [Fundamental Principles of Mathematical Sciences].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Springer-Verlag, Berlin, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Lia]  Penghui Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Derived categories of character sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='04289 [math].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Lib]  Penghui Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Derived categories of character sheaves II: Canonical induction/restriction functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [LN21]  Penghui Li and David Nadler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Uniformization of semistable bundles on elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Advances in Mathematics,  380:107572, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [LNY]  Penghui Li, David Nadler, and Zhiwei Yun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' On the cocenter of the affine Hecke category and Betti geometric  Langlands in genus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Lur09]  Jacob Lurie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Higher Topos Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Number 170 in Annals of Mathematics Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Princeton University Press,  Princeton, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='J, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Lur12]  Jacob Lurie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Higher Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Lusa]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lusztig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Parabolic character sheaves, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' arXiv:math/0302151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Lusb]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lusztig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Truncated convolution of character sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' arXiv:1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='1082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Lus84]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Lusztig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Intersection cohomology complexes on a reductive group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Inventiones Mathematicae, 75(2):205–272,  1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Lus87]  George Lusztig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Fourier transforms on a semisimple Lie algebra over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' In Arjeh M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Cohen, Wim H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Hesselink,  Wilberd L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' van der Kallen, and Jan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Strooker, editors, Algebraic Groups Utrecht 1986, volume 1271, pages  177–188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Springer Berlin Heidelberg, Berlin, Heidelberg, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [MV88]  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Mirkovi´c and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Vilonen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Characteristic varieties of character sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Inventiones Mathematicae, 93(2):405–418,  1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [NT]  David Nadler and Jeremy Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The Whittaker functional is a shifted microstalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='09216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [NYa]  David Nadler and Zhiwei Yun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Automorphic gluing functor in Betti Geometric Langlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='12318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [NYb]  David Nadler and Zhiwei Yun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Compatibilities of automorphic gluing functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [Ric88]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Richardson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Conjugacy classes of n-tuples in Lie algebras and algebraic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Duke Mathematical Journal,  57(1), 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  [TT]  James Tao and Roman Travkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' The affine Hecke category is a monoidal colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='10998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  FUNCTIONS ON THE COMMUTING STACK VIA LANGLANDS DUALITY  93  [Whi39]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Whitehead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Simplicial Spaces, Nuclei and m-Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=' Proceedings of the London Mathematical Society,  s2-45(1):243–327, 1939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='  Yau Mathematical Sciences Center, Tsinghua University, Beijing, China  Email address: lipenghui@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='cn  Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720-3840  Email address: nadler@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='edu  Department of Mathematics, MIT, 77 Massachusetts Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content=', Cambridge, MA 02139  Email address: zyun@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfvgE1/content/2301.02618v1.pdf'}
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+Applications of elastic instability and elastic turbulence: Review,
+limitations, and future directions
+C. Sasmal*1
+Soft Matter Engineering and Microfluidics Lab, Department of Chemical Engineering, Indian Institute of Technology Ropar,
+Punjab, India-140001.
+(*Electronic mail: csasmal@iitrpr.ac.in)
+(Dated: 9 January 2023)
+Viscoelastic fluids are a subclass of complex fluids used in widespread applications ranging from biological to large-
+scale industrial settings. These fluids are often associated with various complex flow phenomena due to the presence
+of non-linear elastic stresses, originating due to the stretching and relaxation phenomena of microstructure (such as
+polymer molecules in the case of a viscoelastic polymer solution) in a deformed flow field. One such phenomenon is
+elastic instability (EI) which emerges due to the interaction between elastic stresses and streamline curvature present
+in a flow system at small values of the Renolds number (ratio of the inertial to that of the viscous forces) when the
+Weissenberg number (ratio of the microstructure relaxation time to that of the rate of flow deformation) exceeds a
+critical value. On further increasing this Weissenberg number to higher values, the unstable flow field caused due
+to this elastic instability transits to a more chaotic and turbulent-like flow structure called elastic turbulence (ET). The
+fluctuating hydrodynamics arising due to this ET flow exhibit many statistical resembles seen for the regular Newtonian
+turbulence occurring at large values of the Reynolds number. Over the last two decades or so, an extensive investigation
+has been performed on this particular topic in the complex fluids research community. Some excellent articles recently
+present this ET phenomenon’s development, understanding, and progress in the literature. This article focuses on the
+application perspectives of this phenomenon. In particular, this article aims to provide a comprehensive review of the
+investigations conducted so far in the literature to demonstrate the potential of these EI and ET phenomena in three main
+application areas, namely, microfluidic mixing, microscale heat transfer, and chemically enhanced oil recovery (EOR)
+process. Additionally, a detailed discussion of the limitations and future directions of these EI and ET phenomena from
+an application point of view is also presented in this article.
+I.
+INTRODUCTION
+Adding a minute amount of solid polymers or surfactants,
+even in parts per million (ppm) amount, into a simple fluid
+like water dramatically changes the flow behaviour of the re-
+sulting solution. This is due to the induction of fluid elasticity
+within the solution, resulting from the molecular conforma-
+tion changes of the polymer or surfactant molecules due to
+their stretching and relaxation phenomena in a deformed flow
+field1,2. It could also be induced naturally in a solution due
+to the presence of microstructure that stretches and relaxes
+in a deformed flow field, for instance, blood3,4, emulsions5,6,
+foams7,8, particle suspensions9,10, etc. The extent of this fluid
+elasticity in a solution is generally expressed in terms of a non-
+dimensional time, named Weissenberg number (Wi), defined
+as the product of the microstructure relaxation time (λ) and
+the flow deformation rate ( ˙γ), i.e., Wi = λ ˙γ11. The mecha-
+nisms through which this fluid elasticity could influence the
+coherent flow structures (and the associated transport phe-
+nomena such as the transfers of momentum, heat, and mass) of
+a flow field also depend on the Reynolds number (Re), defined
+as the ratio of the inertial (∼ O(ρU2
+ch)) to that of the viscous
+forces (∼ O(η Uch
+Lch )), i.e., Re = LchUchρ
+η
+where Lch is the char-
+acteristic length scale, Uch is the characteristic velocity scale,
+ρ and η are the fluid density and viscosity, respectively. The
+effect of fluid elasticity on the flow dynamics at sufficiently
+high values of the Reynolds number (where the flow field is
+in the turbulent regime) has been studied extensively over the
+past several decades, which is popularly known as the elasto-
+inertial turbulent (EIT) flow12. This particular EIT flow has
+gained extensive attention from researchers due to its associ-
+ation with the turbulent drag reduction (DR) phenomenon13.
+It was first reported by Tom in 1948 when he found a signif-
+icant reduction in the friction drag in a high Reynolds num-
+ber turbulent pipe flow of monochlorobenzene solvent due to
+the addition of a minute amount of poly(methyl methacrylate)
+(PMMA) solid polymers into it14. Subsequently, many other
+experimental studies were conducted with different combina-
+tions of polymers and solvents. Also, they observed the same
+Tom phenomenon (named after Tom) with the reduction in the
+drag to various extents15–17. With the advancement of com-
+putational techniques and hardware, later on, extensive direct
+numerical simulations (DNS) were also performed to investi-
+gate the underlying physics behind this phenomenon in more
+detail18–20. Many sophisticated experimental techniques, such
+as Schlieren photographs or particle image velocimetry (PIV),
+were also employed to investigate this phenomenon. All these
+experimental and numerical studies provided detailed infor-
+mation on the flow fields and polymer conformations at dif-
+ferent lengths and time scales, which facilitated a deeper and
+better understanding of this phenomenon. Some excellent re-
+view articles are present on the development and progress of
+the investigations and understandings of this DR phenomenon
+in the literature. However, even after almost 75 years of its
+findings, the DR phenomenon due to polymer additives ( and
+so the EIT flow ) still attracts a lot of attention in the complex
+fluids research community12.
+On the other hand, when the Reynolds number becomes
+vanishingly small, the effect of the fluid elasticity on the flow
+arXiv:2301.02395v1  [physics.flu-dyn]  6 Jan 2023
+
+2
+dynamics gives rise to a new flow regime called elastic turbu-
+lence (ET)21. A recent numerical study has shown that there
+is a continuous pathway (a single linearly unstable modal
+branch) present that connects these ET and EIT flow states22.
+Before the transition of this ET regime, elastic instability (EI)
+first originates within the system, resulting from the interac-
+tion between the streamline curvature and the normal elas-
+tic stresses present within the system. Therefore, elastic in-
+stability is the precursor of this elastic turbulent flow state.
+McKinley and co-workers23,24 proposed criteria for curvilin-
+ear geometries for the onset of this purely elastic instability,
+as written below
+M =
+�
+τ11
+η ˙γ
+λU
+R ≥ Mcrit
+(1)
+where τ11 is the normal elastic stress in the flow direction
+along a curved streamline, ˙γ is the characteristic value of the
+local flow deformation rate, R is the characteristic radius of
+the streamline curvature. When this M parameter exceeds a
+critical value, Mcrit, a purely elastic instability is originated in
+a flow system. McKinley et al.24 presented a detailed investi-
+gation wherein they calculated the values of this Mcrit param-
+eter for various geometries such as Taylor-Couette, lid-driven
+cavity, flow past a confined cylinder, etc. Furthermore, they
+also showed how this critical M parameter should be modified
+to account for the solvent viscosity ratio (defined as the ratio
+of the solvent to that of the zero-shear viscosity of the poly-
+mer solution) and the shear-thinning behaviour of the fluid. A
+very good agreement was found between this scaling predic-
+tion and the corresponding experimental observations for the
+onset of elastic instability, for instance, in a lid-driven cav-
+ity25,26.
+The EI phenomenon in a flow system creates an unstable
+flow field, which transits to a further chaotic and turbulent-like
+flow state or the ET state as the Weissenberg number further
+increases. The term "elastic turbulence" was first coined in
+1965 by Vinogradov and Manin27 when they studied the flow
+of polymer melts through a micro-scale contraction/expansion
+geometry. However, Groisman and Steinberg28 were the ones
+who investigated this ET flow thoroughly for a system com-
+prising a cylindrical cell with a rotating circular plate filled
+with polyacrylamide polymer solutions. For the first time,
+they observed that elastic turbulence could generate a fluc-
+tuating flow field with a broad range of spatial and temporal
+scales; likewise, it is seen in regular hydrodynamic Newto-
+nian turbulence. In particular, they showed that the turbulence
+generated due to these polymer additives at a very low value
+of the Reynolds number would be comparable to the corre-
+sponding Newtonian turbulence generated in a pipe flow at
+a Reynolds number as high as 105. Subsequently, Steinberg
+and co-workers carried out further experimental studies on
+this ET flow in other different geometries such as curvilin-
+ear channel29,30, rotating disks31, flow past an obstacle32,33,
+etc. Further extensive studies on the statistical analysis of the
+velocity and pressure fluctuations and the establishment of the
+scaling relations of the exponents of the power-law decay of
+elastic energy and pressure fluctuations in the ET regime were
+also conducted by the same research group34,35. Many ex-
+perimental studies on the ET flow in a microchannel having
+single or multiple pore constrictions have also been presented
+in the literature36,37. This particular geometry is considered
+one of the simplified model porous systems for understanding
+the micro-scale flow dynamics in an actual porous media38.
+Studies on a more complicated model porous system, such as
+a microchannel having many cylindrical pillars present in it,
+have also been recently carried out in the elastic turbulent flow
+regime39–44. A comprehensive review on the flows of com-
+plex viscoelastic fluids in a porous media has recently pre-
+sented by Kumar et al.45. The corresponding numerical evi-
+dence on the existence of elastic turbulence is also presented
+in the literature. However, this number is relatively less in
+comparison to that of experimental ones. This is mainly be-
+cause of the existence of the well-known "High Wessenberg
+Number Problem (HWNP)" that occurs for viscoelastic fluid
+simulations, particularly for simulations of constant shear vis-
+cosity viscoelastic fluids46. This problem leads to the loss
+of numerical stability at sufficiently high Weissenberg num-
+bers where the ET flow is expected to exist. Despite that,
+some studies have been carried out by employing various nu-
+merical stabilization techniques such as the log-conformation
+tensor approach47. Among a few studies, for instance, Berti
+et al.48 performed an investigation in a two-dimensional pe-
+riodic Kolmogorov flow utilizing the Oldroyd-B viscoelastic
+constitutive model and found a disordered and turbulent-like
+flow state with increased drag and Lyapunov exponent once
+the Weissenberg number exceeded a critical value. Their en-
+ergy spectrum of the velocity fluctuations showed a power-
+law decay with an exponent value close to that obtained both
+in the experiments and theoretical predictions. Ardekani and
+co-workers performed full-scale two-dimensional numerical
+simulations for various geometries such as microchannel with
+pore constrictions49 and flow past obstacles50 to delineate the
+mechanisms for originating the elastic instability and turbu-
+lence in such micro-geometries. Sasmal and co-workers also
+conducted extensive numerical simulations to investigate the
+EI and ET phenomena in the flows of viscoelastic micellar so-
+lutions51–53 as well as polymer solutions in complex porous
+media54. A comprehensive review of the investigations and
+understandings of the elastic instability and elastic turbulence
+phenomena has been presented recently in some excellent re-
+view articles21,55,56. Therefore, the readers are requested to
+go through them to gain further insights and to be aware of
+the latest development and new directions in the EI and ET
+phenomena. The main aim of this article is to present the ap-
+plication perspectives of these two phenomena.
+Therefore, it is readily evident that the EI and ET phenom-
+ena originate in a chaotic and turbulent-like flow state, like-
+wise the regular Newtonian turbulence, even at negligible val-
+ues of the Reynolds number. This low Reynolds number cri-
+terion of these two phenomena naturally leads to their appli-
+cations in microfluidic geometries where a laminar flow con-
+dition persists due to small-scale dimensions. As a result, the
+rate of any transport process, such as heat transfer or mixing in
+these micro-scale geometries, is dominated mainly by molec-
+ular diffusion, which in turn necessitates a much longer time
+
+3
+to carry out these process57. Over the years, a voluminous
+amount of studies, in terms of theory, simulations and exper-
+iments, have been carried out in the development of various
+methods to increase the rate of transport process in these mi-
+crofluidic geometries. These methods are broadly classified
+into two categories, namely, active and passive methods58–60.
+In active methods, the micro-scale geometries are fabricated
+in such a way that a secondary flow pattern could be generated
+inside the flow system (for instance, the Dean flow61), thereby
+facilitating a greater rate of either the heat transfer62,63 or the
+mixing process64,65. On the other hand, in passive methods,
+the rate of transport processes is increased with the help of ex-
+ternal driving forces such as mechanical rotation, electric and
+magnetic fields, or an acoustic field66–68. All these externally
+applied fields would help generate chaotic convection inside
+a microfluidic geometry, thereby increasing the rate of trans-
+port processes. In this perspective, the EI and ET phenomena
+have a considerable potential in microfluidic applications for
+enhancing the rate of various transport processes, such as heat
+transfer or mixing, as they can inherently generate chaotic and
+turbulent-like flow structures inside a microfluidic geometry.
+This has already been demonstrated by a large number of ex-
+perimental as well as numerical studies carried out for various
+micro-scale geometries. Apart from heat transfer and mixing
+applications in micro-scale geometries, the EI and ET phe-
+nomena can also significantly influence the polymer flooding
+used in the enhanced oil recovery (EOR) process69,70. In this
+process, a polymer solution is injected into an oil reservoir to
+displace the oil present in the reservoir’s porous rock struc-
+ture. This rock is made of billions of interconnected tortu-
+ous micropores through which polymer solution and oil flow.
+Therefore, there is a high possibility of inducing these EI
+and ET phenomena inside a porous matrix during the poly-
+mer flooding, which may significantly influence the oil dis-
+placement efficiency. It requires a well understanding of these
+two phenomena during the flows of viscoelastic fluids through
+a micro-scale porous geometry to increase the effectiveness
+of the polymer flooding process.
+In particular, this article
+presents a comprehensive review of these potential applica-
+tions utilizing the EI and ET phenomena demonstrated so far
+in the literature. In particular, we focus on three application
+domains where these two phenomena could contribute signif-
+icantly: microfluidic mixing, micro-scale heat transfer, and
+chemically enhanced oil recovery (EOR) process. In addition,
+a detailed discussion of the limitations and future directions
+is also presented, which would help in the development and
+progress of these two phenomena from an application point of
+view in the future.
+II.
+APPLICATIONS IN MICROFLUIDIC MIXING
+Mixing of fluids in microscale geometries was the first
+potential application of the EI and ET phenomena that was
+demonstrated in the literature. In doing so, Groisman and
+Steinberg29 conducted an experimental study in a curvilinear
+microchannel wherein they showed the mixing efficiency of
+two fluids entering the microchannel through two inlets, as
+schematically shown in sub-Fig. 1(a). In their experiments,
+they used a pure solvent consisting of 65% saccharose and
+1% NaCl in water as simple Newtonian fluid and a viscoelas-
+tic polymer solution by adding 80 ppm polyacrylamide in the
+solvent to show the difference in the mixing behaviour arising
+due to the elastic turbulence phenomenon. A fluorescent dye
+was mixed into the fluid entering through one of the inlets, and
+the image was taken using a charge-coupled device (CCD) in
+the presence of fluorescent light at a position downstream of
+the inlet, as marked in Fig. 1(a). The surface plots of the dye
+concentration in pure solvent and polymer solutions are de-
+picted in sub-Figs. 1(b) and (c), respectively. It can be seen
+that the two fluids (with finite and zero fluorescent dye con-
+centrations) were flowing side by side, and they did not mix
+with each other in the case of the pure solvent. On the other
+hand, efficient mixing of the two fluids occurred in the case of
+the polymer solution under the same operating conditions. It
+was possible due to the presence of the EI and ET phenomena
+in the latter solution, which increased the chaotic convection
+inside the microchannel and hence the mixing efficiency. It
+was confirmed by Groisman and Steinberg29 through the sta-
+tistical analysis of the temporal fluctuations of the velocity and
+dye concentration at a point inside the microchannel. They
+observed several statistical characteristics which ensured the
+existence of elastic turbulence in the flow system, such as a
+power-law decay with an exponent value of around 2.3 in the
+power spectrum of the velocity fluctuations and exponential
+tails of the probability distributions of the dye concentration
+profile fluctuations. In another subsequent study, Groisman
+and Steinberg71 demonstrated the mixing capability of the ET
+phenomenon in a Couette-Taylor (CT) geometry consisting of
+a cylindrical cell with a flat bottom and a concentric rotat-
+ing upper plate inside it. Figure 2 depicts the mixing patterns
+of an ink drop placed at the center of the CT cell filled with
+either viscoelastic polymer solutions (sub-Fig. 2(a)) or with
+a Newtonian solvent (sub-Fig. 2(b)). It can be seen that the
+ink started to spread over the surface after a certain time by
+the toroidal vortex resulting from the ET phenomenon when
+it was placed in a viscoelastic polymer solution. As time pro-
+gressed further, those toroidal structures were split into more
+fine structures either due to the excitation of the fluid mo-
+tion at small spatial scales or due to a significant stretching of
+the fluid elements along their Lagrangian trajectories by ran-
+domly fluctuating large-scale eddies. Furthermore, the con-
+trast in the fine structures gradually decreased, suggesting the
+progress of mixing with time. Finally, at time t = 8 min, the
+dye concentration became completely homogeneous in the so-
+lution, and a complete mixing was achieved. This time was
+almost four orders of magnitude smaller than the time re-
+quired for the mixing due to molecular diffusion. Hence, it
+was concluded that this mixing was definitely achieved due
+to the chaotic and random flow structures resulting from the
+elastic turbulence phenomenon. On the other hand, in Newto-
+nian solvent (sub-Fig. 2), the ink concentration was inhomo-
+geneous even after 9 hr of the experiments due to the absence
+of inertial effects and chaotic convection under the same op-
+erating conditions. Poole et al.73 carried out an experimental
+study in a similar kind of geometry and showed that the emul-
+
+4
+FIG. 1. Experimental study of the mixing performance in a curvilinear microchannel utilizing the EI and ET phenomena29. (a) The experi-
+mental setup consisted of a curvilinear microchannel with two inlets and smoothly connected sixty half-rings (numbered 1, 2, .., 30) with outer
+and inner radii of R1 = 6mm and R2 = 3mm, respectively. The depth of the channel was 3mm. Fluids with a fixed concentration of fluorescent
+dye (lower inlet) and zero concentration (upper inlet) entered the microchannel through the two inlets. Dye concentration profiles in (b) pure
+solvent and (c) viscoelastic polymer solutions. Here the bright white color corresponds to a finite dye concentration.
+sification rate of two immiscible liquids was substantially in-
+creased due to a greater mixing caused by the ET phenomenon
+when the dispersed medium was viscoelastic Boger fluids. In
+contrast, they remained separated when the dispersed medium
+was Newtonian.
+A study in the same curvilinear microchannel geometry as
+used by Groismann and Steinberg29 (however, it was 30 times
+smaller than used by Groismann and Steinberg29) was also
+carried out by Burghelea et al.74. This study also found a
+chaotic mixing of two fluids when a minute amount of poly-
+mers were added to them. They also reported that the mix-
+ing time in the chaotic elastic turbulence regime was three
+to four orders of magnitude shorter than due to the molecu-
+lar diffusion. Furthermore, the mixing time was found to be
+almost independent of the diffusion coefficient of the macro-
+molecules that were present in the solution. Therefore, they
+suggested that this ET phenomenon could efficiently mix flu-
+ids with additives of low diffusivities, such as large DNA
+molecules, viruses, particles, living cells, etc. Li et al.72 also
+performed an experimental study to demonstrate the poten-
+tial of elastic turbulence phenomenon in enhancing the mix-
+ing phenomenon in a curvilinear microchannel. They used
+glycerol water and CTAC/NaSal (cetyltrimethyl ammonium
+chloride/sodium salicylate) surfactant solutions in their exper-
+iments as Newtonian and viscoelastic fluids, respectively. Fig-
+ure 3 shows the details of the geometry used in their study
+and the corresponding surface plot of the dye distribution pro-
+files both in glycerol water and surfactant solutions. The ge-
+ometry had two inlets through which fluids entered into the
+microchannel, and the fluid entering through the upper in-
+let was colored with a fluorescent dye, as shown in Fig. 3.
+An almost uniform distribution of dye, particularly down-
+stream of the microchannel, was seen at any cross-sectional
+area of the microchannel in the case of flows of surfactant
+solutions (sub-Fig. 3(c)), resulting from the elastic instability-
+induced chaotic mixing of two fluids. In contrast, a highly
+non-uniform distribution of dye concentration profile was ob-
+served due to the absence of span-wise flow fluctuations when
+glycerol water was used as the working fluid. Tatsumi et al.75
+did an investigation in a serpentine microchannel with New-
+tonian sucrose and viscoelastic polyacrylamide (dissolved in
+sucrose) polymer solutions and also found an enhancement in
+the mixing phenomenon in the latter solution.
+The corresponding three-dimensional numerical simula-
+
+Field of view
+Lasersheet5
+FIG. 2. Mixing patterns of an ink drop placed at the center of a Couette-Taylor (CT) cell utilizing the ET phenomenon at different times71 in
+viscoelastic polymer solutions (a) and a Newtonian solvent (b).
+tions for showing the mixing performance of both Newtonian
+and viscoelastic fluids in a curvilinear microchannel were per-
+formed by Li et al.77. They used the Giesekus model to realize
+the fluid viscoelasticity and covered a wide range of the Weis-
+senberg number at a fixed value of the polymer viscosity ratio
+(ratio of the solvent viscosity to that of the zero shear-rate
+viscosity of the polymer solution) of 0.25. Likewise the ex-
+periments, they also found a significant enhancement in the
+mixing performance in the case of viscoelastic fluids com-
+pared to the case of Newtonian fluids. Apart from the mixing
+phenomenon, they also discussed the flow dynamics results
+in detail in terms of evaluating the microstructure extension,
+secondary flow structures, root mean square velocity fluctu-
+ations, etc. This enhancement in the mixing phenomenon of
+viscoelastic fluids was even observed in the flow through a
+straight microchannel, which was evident in another subse-
+quent numerical study from the same research group76. Fig-
+ure 4 displays the dye concentration profile and the mixing in-
+dex (defined in the caption of the figure) inside the microchan-
+nel both for Newtonian and viscoelastic fluids. In the case of
+Newtonian fluids (sub-Fig. 4(a)), the dyed and undyed fluids
+moved side by side without mixing in the span-wise direction.
+In contrast, a chaotic mixing of the two fluids was evident in
+the whole domain in the case of viscoelastic fluids. It was
+demonstrated more quantitatively in sub-Fig. 4(c), wherein
+the temporal variation of the mixing index was presented. For
+Newtonian fluids, the value of the mixing index remained con-
+stant at around 0.5, and there was almost no mixing in these
+simple fluids. However, it gradually decreased with time for
+viscoelastic fluids, thereby suggesting the presence of chaotic
+mixing in these fluids. Grilli et al.78 presented a numerical
+study on the mixing performance of viscoelastic fluids in a
+microchannel with a periodic array of cylindrical obstacles
+present in it. They simulated a range of the Weissenberg num-
+ber between 0 (Newtonian) and 1.6 using the Oldroyd-B vis-
+coelastic fluid model at a constant fixed value of the polymer
+viscosity ratio of 0.59. Furthermore, a stability analysis based
+on the dynamic mode decomposition (DMD) technique was
+utilized in their study to identify the most energetic mode re-
+sponsible for the unsteady chaotic flow behaviour. They also
+observed a power-law scaling behaviour both in the flow and
+dye concentration fluctuations; likewise, it was seen in the ex-
+periments in the elastic turbulence regime. As proof of the
+presence of ET flow and to show its potential in microfluidic
+applications such as in the mixing process, a numerical ex-
+periment was conducted wherein they observed the mixing of
+
+t=0
+30sec
+=0
+4min
+90sec
+15min
+2 min
+4min
+8min
+2hr
+(a)
+(b)6
+FIG. 3. Experimental study of mixing phenomenon of two viscoelastic fluids in a curvilinear microchannel72. The details of the geometry used
+in the study are shown in (a), whereas (b) and (c) depict the dye concentration profile inside the microchannel when the working fluids were
+glycerol water and viscoelastic CTAC/NaSal surfactant solutions, respectively. In (a), p1, p2.. are the probe lines where the dye concentration
+was measured to obtain the mixing efficiency inside the microchannel. The flow direction is shown by the green arrow.
+FIG. 4. Numerical study of mixing phenomenon in a straight three-dimensional microchannel76. A dye was mixed with the fluid placed
+at the lower half of the channel, and its profile inside the microchannel at t∗ = 160 (where t∗ is the non-dimensional time ) is shown for
+(a) Newtonian and (b) viscoelastic fluids. Here crms is the mixing index defined as the root mean square of the dye concentration c, i.e.,
+crms =
+�
+∑N
+i=1(ci −cm)2/N, where ci, cm, and N are the dye concentration at a grid point, mean dye concentration, and the total number
+of grid points in the whole computational domain, respectively. A zero value of crms corresponds to perfect mixing, whereas a value of 0.5
+corresponds to no mixing. (c) Variation of crms with the non-dimensional time both for viscoelastic and Newtonian fluids. The values of the
+Reynolds and Weissenberg numbers were fixed at 1 and 30, respectively, in these simulations.
+a dye placed on the upper wall of the microchannel. As the
+Weissenberg number gradually increased in their simulations,
+the spreading of the dye also progressively increased inside
+the microchannel due to the increase in the chaotic convection
+resulting from the elastic turbulence phenomenon.
+Gan et al.79 performed an experimental study to inves-
+tigate the mixing phenomenon of two fluids in a conver-
+gent/divergent microfluidic geometry, as schematically shown
+in Fig.79(a). The geometry had three inlets, namely, two side
+inlets (orange arrow) and one main inlet (green arrow). The
+
+Total flowrate:1.6uL/min(Re=0.l)
+Width:100μum,depth:50μm
+(a)
+Coloredwithdye
+Coloredwithdye
+Iniet
+Uncolored
+Inlet
+Uncolored
+555
+Outlet
+Outlet
+(c)
+(b)0
+0.4
+0.8
+0.50
+(a)
+0.45
+viscoelasticfluid
+Newtonian fluid
+0.40
+0
+2
+4
+6
+8
+10
+Crms
+a
+0.35
+(b)
+0.30
+(c)
+0.25
+0
+2
+4
+6
+8
+10
+0
+30
+60
+90
+120
+150
+180
+t*7
+FIG. 5. Experimental study of the mixing performance in a convergent/divergent microfluidic geometry79. (a) The geometry consisted of two
+side inlets (orange arrow) and one main inlet (green arrow), wherein the flow was happening from left to right. The fluid entering through
+the main inlet was mixed with a fluorescent dye. The dye concentration profile at the upstream and downstream sections of the geometry
+in (b) viscous Newtonian (glycerol water) and (c) viscoelastic fluids (polyethylene oxide (PEO) dissolved in glycerol water) under the same
+conditions.
+mainstream fluid (1 wt% polyethylene oxide (PEO) dissolved
+in 55 wt% glycerol water) entering the geometry had higher
+viscoelastic properties. It was mixed with a finite concentra-
+tion of fluorescent dye. In contrast, sidestream fluids having
+less viscoelastic properties (0.1 wt% PEO dissolved in wa-
+ter) or only viscous properties (glycerol water) entered into
+the geometry without any dye in them. The dye concentration
+profiles inside the geometry both for viscous Newtonian and
+viscoelastic polymer solutions are depicted in sub-Figs. 5(b)
+and (c), respectively. It can be seen that the dye spread very
+little when the fluids were viscous Newtonian both upstream
+and downstream sections of the geometry; however, it became
+prominent in the case of viscoelastic polymer solutions, par-
+ticularly at the downstream section of the geometry. There-
+fore, it suggested a greater mixing of the fluids in the latter
+case. In particular, they observed a mixing efficiency (whose
+values of 0 and 100 correspond to no mixing and perfect mix-
+ing conditions) of as high as 68% downstream of the geom-
+etry due to the emergence of viscoelastic instability. In an-
+other study80, they demonstrated the use of this viscoelastic
+instability for efficient mixing of fluids in an abruptly con-
+tracted microchip (8:1 contraction ratio) made out of poly-
+methyl methacrylate (PMMA) polymer. According to them,
+the device was cheap, disposable, and straightforward to fab-
+ricate yet effective for mixing over a short length with a rela-
+tively high flow rate.
+Hong et al.81 recently carried out an experimental inves-
+tigation to demonstrate the use of EI and ET phenomena in
+the enhancement of fluids mixing in a novel gear-shaped mi-
+crochannel setup, as schematically shown in Fig. 6(a) and (b).
+In particular, they made this setup by combining two geome-
+tries (which were already proven effective in inducing elastic
+instability and turbulence in a system), namely, a serpentine
+microchannel and a microchannel with step expansion and
+contraction or with a side-well structure. They found that the
+gear-shaped microchannel provides a more effective mixing
+of the two fluids than the serpentine microchannel under iden-
+tical conditions. This is depicted in Fig. 6(c)-(d), wherein both
+the surface plot of dye concentration (left) and its fluorescent
+intensity along a plane (right) are presented. From both plots,
+it can be seen that the dye was more uniformly distributed
+in the case of a gear-shaped microchannel than in a serpen-
+tine microchannel, whereas the pressure drop was almost the
+same in both cases. It led to an increased mixing efficiency
+of the two fluids in the former microchannel with a smaller
+mixing length than needed for the latter one. Their study fur-
+ther demonstrated that this enhancement in the mixing of vis-
+coelastic fluids ultimately facilitated the production of silica
+nanoparticles with more uniform size and shape distributions
+than that obtained with Newtonian fluids. Yang et al.82 per-
+formed an experimental study on a serpentine microchannel
+and showed that the orthogonal injection of the fluid into the
+primary flow in the microchannel enhanced the mixing effi-
+ciency in a short mixing length.
+All the above-mentioned studies have used pressure-driven
+flows for originating the elastic instability and, subsequently,
+the elastic turbulence. However, very few corresponding stud-
+ies are present for the electrokinetic-driven flows to induce
+these EI and ET phenomena inside a micro-scale system and
+their influence on the mixing process. Among a few studies,
+Bryce and Freeman83 performed an experimental study us-
+ing a long microchannel with many constrictions (2:1 ratio)
+present in it. They showed that the mixing efficiency was de-
+creased in viscoelastic fluids than in Newtonian fluids under
+the same operating conditions. This starkly contrasts with that
+observed in pressure-driven flows, where two to three orders
+of magnitude enhancement was observed despite the develop-
+ment of large-scale elastic instabilities inside the system in the
+EK-driven flows. According to them, it was attributed to the
+absence of shear-driven or any other deformations in this par-
+ticular geometry wherein mostly extensional-dominated elas-
+tic instabilities were present. However, a very recent numeri-
+cal study by Khan and Sasmal84 found an increase in the mix-
+ing efficiency in the same kind of geometry as that used by
+
+Downstream
+Upstream
+(a)
+0001
+1000
+1000
+(b)
+25
+C
+All dimensions in
+Jim
+3000
+Channel depth 1508
+FIG. 6. (a) Schematic of the gear-shaped microchannel used in the study of Hong et al. 81 for efficient mixing of fluids, which is a combination
+of serpentine and expansion and contraction (or side-well structure) microchannels (b) Schematic of the flow arrangement wherein dyed and
+dye-free solutions were simultaneously injected through two inlets with the same flow rates. The mixing zone consisted of 22 connected half
+rings. The mixing pattern inside the system at two flow rates for only serpentine (c and d) and gear-shaped (e and f) microchannels. Here the
+right-hand sub-figures represent the fluorescent intensity profiles (at a plane marked by the green arrow in (c)) normalized by the maximum
+intensity after background subtraction.
+Bryce and Freeman83. They used the Oldroyd-B viscoelas-
+tic constitutive equation to mimic the rheological behaviour
+of a Boger fluid and the Poisson–Boltzmann (PB) equation to
+calculate the ion distribution in the system for a wide range
+of electric field strength and viscosity ratio. Once the Weis-
+senberg number exceeded a critical value, an electro-elastic
+instability was developed, resulting in a chaotic and fluctu-
+ating convective flow field inside the system. This eventually
+led to the mixing of two fluids present in two halves of the sys-
+tem. It is shown in Fig. 7(i)-(ii) wherein the fluid present in the
+upper half of the channel was mixed with a dye, and the lower
+half was dye-free. When the fluids were in Newtonian nature
+(sub-Fig. 7i(a)), they moved side by side without any mixing.
+However, as fluid viscoelasticity was introduced into the flu-
+ids, the mixing of the two fluids started (sub-Fig. 7i(b)), which
+was further incremented as the Weissenberg number further
+increased (sub-Fig. 7i(c)). It is further evident in Fig. 7(ii),
+where the variation of the mixing efficiency parameter η (val-
+ues of 0 and 100 corresponded to no-mixing and perfectly
+mixing conditions) with the Weissenberg number was pre-
+sented, and a drastic increase in its value was observed once
+the Weissenberg number exceeded a critical value. This dif-
+ference in the observations between the studies of Bryce and
+Freeman83 and Khan and Sasmal84 may be attributed to the
+difference in the rheological behaviour of the working fluids.
+Khan and Sasmal84 used a perfectly Boger fluid in their sim-
+ulations, whereas Bryce and Freeman83 did not provide the
+rheological details of their working fluids. In another study by
+Sasmal85, it has been shown that the mixing of two viscoelas-
+tic fluids could even be achieved in a straight microchannel
+utilizing the electro-elastic instability and turbulence phenom-
+ena. These were locally generated inside the system by plac-
+ing patches of constant wall zeta potential on both the top
+and bottom walls of the microchannel. Figure 7(iii) shows
+the distribution of dye concentration inside the microchannel
+(the upper half was filled with a dyed fluid, whereas the lower
+half was filled with the same fluid with no dye) at different
+values of the Weissenberg number. The fluids traveled side by
+side without cross-convective mixing in the cases of Newto-
+nian (sub-Fig. 7iii(a)) and viscoelastic fluids with low values
+of the Weissenberg number (sub-Fig. 7iii(b)). As the Weis-
+senberg number increased to higher values, mixing between
+the two fluids was observed, which was seen both in the dye
+distribution presented in sub-Figs. 7iii(c) and (d) at two differ-
+ent values of the Weissenberg number and in the variation of
+the mixing efficiency with the Weissenberg number depicted
+in Fig. 7(iv). Therefore, this study proposed a very simple
+and effective approach for mixing two viscoelastic fluids in a
+straight microchannel under the influence of an electric field.
+However, it should be experimentally verified so that the po-
+tential of this proposed approach and the phenomena of EI and
+ET can be further established for mixing fluids in micro-scale
+systems.
+III.
+APPLICATIONS IN MICROSCALE HEAT TRANSFER
+The corresponding potential in applying EI and ET phe-
+nomena for the microscale heat transfer process was investi-
+gated much later than the mixing process. Whalley et al.86 was
+probably the first who showed this potential in 2015, almost
+24 years later than the experiment carried out for the mixing
+process by Groisman and Steinberg29. They used a square ser-
+pentine microchannel and Boger polymer solutions comprised
+
+Q=0.5ml/h
+Q=12ml/h
+(a)
+Serpentine Microchannel
+(Re=1.7, Wi=0.8)
+00.51 (e)
+(Re=41.7,Wi=18.2)
+"Gear-Shaped"Microchannel
+(c)
+Serpentine
+200μm/250μm
+Serpentine
++SideWellStructure
+H50μm
+(b)
+MixingZone
+100 μm
+Dye-Free
+(d)
+f
+Solution
+Outlet
+Gear
+Gear
+Junction
+Dyed
+Solution
+Stable
+Unstable9
+FIG. 7. Electrokinetically driven flows in a step expansion and contraction microchannel84 wherein the fluid present in the upper half was
+mixed with a dye, and that present in the lower half was dye free. (i) Dye distribution profile for Newtonian (a) and viscoelastic fluids at
+two values of the Weissenberg number, namely, 6 (b) and 15 (c). (ii) Variation of the mixing efficiency parameter (η) with the Weissenberg
+number in the same geometry. Mixing two viscoelastic fluids in a straight microchannel with patches of constant wall zeta potential on both
+top and bottom walls of the microchannel85 in electrokinetic-driven flows utilizing the electro-elastic instability and turbulence phenomena.
+(iii) Dye distribution profile for Newtonian (a) and viscoelastic fluids with values of the Weissenberg number, namely, 1 (b), 2(c), and (3). (iv)
+The corresponding variation of the mixing efficiency parameter with the Weissenberg number in this geometry.
+of high molecular-weight polyacrylamide polymers dissolved
+in a Newtonian solvent consisting of 65% sucrose, 1% NaCl,
+and 34% water in their investigation.
+Based on the varia-
+tions of the non-dimensional heat transfer rate and/or Nus-
+selt number and the non-dimensional pressure drop and/or
+friction factor with the Weissenberg number, they identified
+three regimes, namely, regime I (Wi < 5) wherein the heat
+transfer rate and pressure drop in viscoelastic polymer solu-
+tions were almost the same as that obtained with a Newto-
+nian solvent, regime II (5 < Wi < 25) wherein the pressure
+drop showed a significant enhancement in polymer solutions
+compared to that in a Newtonian solvent; however, the Nus-
+selt number was hardly influenced, and regime III (Wi > 25)
+wherein the pressure drop reached a plateau value, whereas
+
+(a)
+(b)
+60
+40
+-
+Steady
+(c)
+20
+Unsteadyregime
+(Electro-elastic instability)
+0
+5
+10
+15
+Wi
+(i)
+m(i)
+60
+0.00
+0.50
+1.00
+(a)
+Nomixing
+Mixing
+40
+(b)
+20
+Steady and
+stableregime
+[(c)
+Unstableand
+chaoticregime
+(d)
+0.5
+1.5
+2
+2.5
+3
+Wi
+(ili)
+(iv)10
+FIG. 8. Variation of (a) the normalized friction factor ( f Re) and (b) the surface averaged Nusselt number (Nuavg) with the Weissenberg
+number in a square serpentine microchannel86. Here ’Vis’ and ’New’ stand for the results of viscoelastic polymer solutions and Newtonian
+solvent, respectively. f and Re are the friction factor and Reynolds number, respectively.
+the Nusselt number showed a dramatic augmentation in its
+value, Fig. 8. At the highest Weissenberg number consid-
+ered in their study, an enhancement of up to 300% in the
+heat transfer rate was achieved in viscoelastic polymer solu-
+tions compared to that in a Newtonian solvent. For the same
+square serpentine microchannel geometry, Abed et al.87 also
+performed a further detailed study. In this study, in addition
+to constant viscosity Boger viscoelastic polymer solutions (as
+used by Whalley et al.86 in their study), they also consid-
+ered shear-thinning viscoelastic polymer solutions comprised
+of the same polyacrylamide polymers but dissolved in a dif-
+ferent solvent, namely, a mixture of water and glycerine to
+investigate the effect of the polymer solution type on the heat
+transfer rate and pressure drop. They also proposed the def-
+inition of a modified Weissenberg number (Wi∗) to collapse
+the Nusselt number data obtained at various polymer concen-
+trations onto a master plot. It basically incorporates the ef-
+fects of geometric dimensions, Prandtl number (Pr) and Weis-
+senberg number (Wi) defined as Wi∗ = W
+L PrWi, where W and
+L are the depth and length of the square serpentine channel,
+respectively. The physical significance of this modified num-
+ber is that it is the ratio of elastic stress to thermal diffusion
+stress. Figure 9 depicts the variation of the normalized sur-
+face averaged Nusselt number with the modified Weissenberg
+number both for Boger and shear-thinning polymer solutions
+along with the Newtonian limit under otherwise identical con-
+ditions. It can be seen from this figure that one needs to span
+more range of the modified Weissenberg number for shear-
+thinning polymer solutions than that for Boger polymer so-
+lutions for the same relative increase in the normalized sur-
+face averaged Nusselt number. Therefore, it suggests that the
+surface-averaged Nusselt number is a strong function of not
+only the modified Weissenberg number but also the degree of
+shear-thinning behaviours. Furthermore, they also observed a
+substantial increase in the heat transfer rate due to the pres-
+ence of EI and ET phenomena in viscoelastic polymer so-
+lutions, for instance, approximately 200% and 380% at low
+and high polymer concentrations, respectively, than that ob-
+tained in Newtonian solvents alone. Li et al.92 also conducted
+an experimental study for this square serpentine microchan-
+nel with viscoelastic polyacrylamide solutions (at two differ-
+ent concentrations, namely, 100 and 200 ppm) and Newtonian
+sucrose solutions (50 wt%) flowing into it at different Weis-
+senberg and Reynolds numbers. Once again, the heat transfer
+rate was greater in viscoelastic solutions than in Newtonian
+fluids. Furthermore, it was increased with the polymer con-
+centrations at any Reynolds number. However, detailed anal-
+ysis and explanation of the results were absent in this study as
+the main motivation was to show the potential of a Titanium-
+Platinum (Ti-Pt) film that they developed to measure the tem-
+perature for investigating heat transfer in microfluidic applica-
+tions. Later, they performed another detailed investigation us-
+ing this Ti-Pt film for the same square serpentine microchan-
+nel geometry93.
+Copeland et al.88 conducted an experimental investigation
+on how the elastic turbulence phenomenon could influence the
+convective heat transfer phenomena inside a miniature viscous
+disk pump (VDP)94. It is easy to fabricate, and it has sim-
+ple maintenance and good flow control capability compared
+to other mechanical pumps available for transporting fluids in
+various microfluidic applications. This pump consists of a ro-
+tating disk, C shaped microchannel, and two ports, one for the
+fluid inlet and the other for its outlet. They performed both
+the heat transfer and mixing experiments on this microdevice
+
+3.5
+3.5
+80ppm PA in sucrosesolution
+3.0
+3.0
+120 ppmAA in sucrose solution
+00
+f Re (Vis) / fRe (New)
+2.5
+Regime III
+C
+Regime II
+Regime II
+8
+2.0
+2.0
+RegimeI
+Regimei
+Regime II
+1.5
+00
+00
+1.0
+1.0
+lewt.limit
+Newt limit
+0.5
+0.5
+[a)
+OS0 ppmPAA insuxrose solution
+(b)
+O120 ppmPAA in sucrose solution
+0
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+100
+Wi
+Wi11
+FIG. 9. Variation of the surface averaged normalized Nusselt num-
+ber with the modified Weissenberg number in a square serpentine
+microchannel87. Here the results presented as blue symbols are for
+constant viscosity Boger fluids comprised of polyacrylamide poly-
+mers dissolved in a mixture of water (W) and sucrose (SUC) sol-
+vents at different concentrations (in ppm), whereas red symbols are
+for shear-thinning polymer solutions obtained by dissolving the same
+polymers in a mixture of water and glycerine (GLY) solvents.
+FIG. 10. Variation of the normalized average Nusselt number with
+the modified Reynolds number (ReETC)88 in a microfluidic viscous
+disk pump. The definition of ReETC is provided in the texts. Here
+Nu and Nu0 are the average Nusselt numbers obtained in viscoelastic
+polymer solutions and Newtonian solvent, respectively, and ˙γ is the
+shear rate originating due to the rotation of the disk.
+over a wide range of shear rates originating due to the rotation
+of the disk. Figure 10 shows the variation of the normalized
+Nusselt number with the modified Reynolds number defined
+as ReETC = ˙γ ρ
+µ L2
+c
+�
+ρc
+ρco
+�m
+. This definition of the Reynolds
+number included the effects of the local shear rate ( ˙γ), stream-
+wise development length scale (Lc), flow static density (ρco),
+FIG. 11.
+Variation of the normalized average Nusselt number
+(Nu/Nus) with the Weissenberg number (Wi)89.
+Here ’HPAM’
+stands for hydrolyzed polyacrylamide polymer solutions.
+FIG. 12. Variation of the average Nusselt number with the Weis-
+senberg number in viscoelastic polymer solutions comprised of hy-
+drolyzed polyacrylamide polymers (200 ppm) dissolved in 65% su-
+crose and 1% NaCl solutions90.
+absolute viscosity (µ), and polymer concentration (ρc). The
+values of m and ρco were used as 2.38 and 336.1 ppm in this
+relation. All of these parameters could greatly influence the
+transition and development of the ET phenomenon, and hence
+this definition of the Reynolds number could explain the re-
+sults better, as suggested by them. The figure clearly shows
+that the normalized Nusselt number increases with the mod-
+ified Reynolds number due to the presence of the elastic tur-
+bulence phenomenon. In particular, they observed an aug-
+mentation of around 240% in viscoelastic polymer solutions
+
+ReETC
+3000
+146.0
+△
+292.1
+x
+2000
+△438.1
+△
+口
+X 584.2
+8
+1000
+Nu
+0
+Nuo
+0
+1
+2
+3300ppmHPAM65%sucrose
+200ppmHPAM_65%sucrose
+6
+100ppmHPAM_65%sucrose
+LinearFitofConcatenatedData
+5
+4
+n
+Nu oc 1.2 Wi
+2 
+Newt. liquid
+0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+WiPower-law dependence:
+NucWio
+名
+10
+M
+Exponential dependence:
+Nu oc ewi
+Elasticturbulence
+regime
+2
+Elastic instability
+regime
+Wi
+105
+4.5
+4
+口口
+口
+3.5
+3
+Shear-thinning solutions
+O50-W/GLY
+0
+100-W/GLY
+2.5
+200-W/GLY
+2
+口
+O80-W/SUC
+口
+1.5
+120-W/SUC
+Newt.limit
+500-W/SUC
+1
+0.5
+0
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+Wi*12
+FIG. 13. (a) Variations of the average Nusselt number and (b) pressure drop gradient with the Weissenberg number for geometries with
+different dimensions91.
+(polyacrylamide + sucrose + NaCl) than in Newtonian sucrose
+solutions.
+Traore et al.95 conducted an experimental study for a von-
+Karman swirling flow geometry consisting of a cylindrical cup
+(of radius 40 mm) with two disks (of radius 39 mm) placed
+at the center. The top disk was allowed to rotate at a higher
+temperature, whereas the bottom was kept fixed and main-
+tained at a lower temperature. The distance between them
+was 60 mm. They used polyacrylamide polymers dissolved in
+an aqueous solution of sucrose as the working fluids for their
+experiments. Their analysis of the temperature fluctuations
+revealed characteristics similar to that observed for a passive
+scalar in the case of the mixing process in many earlier exper-
+iments carried out in the elastic turbulent regime29,71,96. For
+instance, the probability distribution functions of the tempera-
+ture fluctuations showed the presence of exponential tails and
+exponential decay of the second-order moment. Furthermore,
+the power spectrum of the temperature fluctuations obeyed a
+power-law decay with an exponent value of 1.1. They ob-
+served an enhancement in the heat transfer rate of up to four
+times in polymer solutions as compared to that obtained in
+solvent alone (without polymers) under the same conditions
+due to the presence of elastic turbulence in the former flu-
+ids. Furthermore, they calculated that a comparable increase
+in the heat transfer rate could be obtained by inertial turbu-
+lence at a Reynolds number of 1600. However, they noticed
+that the relative increase in the efficiency of the heat trans-
+fer rate was significantly lower than that obtained in the mix-
+ing process utilizing this ET phenomenon. Yao et al.89 pre-
+sented a further detailed investigation for the same geometry
+and polymer solutions by varying the polymer concentration,
+sucrose proportion in the solvent, and degree of salinity in the
+solution. They found the occurrence of elastic instability at
+earlier values of the swirling velocity and Weissenberg num-
+ber as the polymer concentration increases and the salinity de-
+creases. The heat transfer rate was higher in viscoelastic poly-
+mer solutions than in Newtonian sucrose solutions, and it in-
+creased with the Weissenberg number. They found a linear re-
+lationship between the normalized Nusselt number and Weis-
+senberg number as Nu/Nus ∝ 1.2Wi in the elastic turbulence
+regime, where Nu and Nus are the average Nusselt numbers
+obtained in viscoelastic polymer solutions and Newtonian sol-
+vents, respectively, Fig. 11. The normalized average Nusselt
+number was seen to be almost independent of the polymer
+concentration in the ET regime. Furthermore, at low rotation
+speeds, the extent of heat transfer enhancement increased with
+the reduction in the salinity of the polymer solution. Once
+the rotation speed exceeded a critical value, it became inde-
+pendent of the salinity of the polymer solution. In another
+study by the same authors90 for the same geometry and poly-
+mer solutions, once again, they found an enhancement in the
+heat transfer rate in viscoelastic polymer solutions compared
+to Newtonian solvent. This study also found the critical value
+of the Weissenberg number (∼ 1.14) for the onset of the elas-
+tic instability and a power-law decay in the injected power
+spectrum with an exponent of 3.9. Moreover, the variation
+of the Nusselt number with the Weissenberg number showed
+an exponential dependence in the elastic instability regime,
+whereas a power-law dependence in the fully-developed elas-
+tic turbulent regime, as schematically shown in Fig. 12.
+The influence of the geometry dimension on the elastic tur-
+bulence and subsequent heat transfer phenomena was recently
+studied by Yang et al.91. They used three different geometries,
+namely, straight microchannel, serpentine microchannel, and
+helically coiled microchannel, to realize one (1D), two (2D),
+and three-dimensional (3D) effects, respectively, of the flow
+field on these phenomena. The variations of the pressure drop
+gradient and average Nusselt number with the Weissenberg
+number in geometries with different dimensions are depicted
+in Fig. 13. Both the average Nusselt number and pressure drop
+
+(a) 3.0
+1D,200ppm
+(b)
+400
+2D,200ppm
+1D.200ppm
+2.5
+3D,200 ppm
+2D,200ppm
+nN
+dropgradient
+Wi2.2
+300
+3D,200ppm
+(Pa/mm)
+W2.0
+1.5
+Nusselt r
+200
+Wi1.9
+AP/AL
+1.0
+Wil.s
+0.5
+100
+Wi1.s
+Wil.0
+0.0
+0
+0
+5
+10
+15
+20
+25
+30
+0
+5
+10
+15
+20
+Weissenbergnumber,Wi
+Weissenbergnumber,Wi13
+FIG. 14. Surface plots of the non-dimensional temperature distri-
+bution along with the velocity vector plots at three different cross-
+sectional areas (C1, C2, and C3) of the microchannel97. Here the
+first row represents the results for a Newtonian fluid, whereas the
+second and third rows depict the results for viscoelastic fluids with
+Wi = 5 and 20, respectively.
+gradient increased with the Weissenberg number irrespective
+of the geometry dimension. At a fixed Weissenberg number,
+the average Nusselt number increased as the geometry dimen-
+sion increased from 1D to 3D. In contrast, the pressure drop
+gradient first increased as the geometry dimension increased
+from 1D to 2D and then decreased upon further increasing to
+3D. Therefore, they suggested that 3D geometry is best suit-
+able for increasing the heat transfer performance by utilizing
+the elastic turbulence phenomenon as it showed reduced pres-
+sure drop gradient and higher heat transfer performance than
+2D geometry under identical flow conditions. Furthermore,
+a non-linear dependence of both the average Nusselt number
+and pressure drop gradient with the Weissenberg number was
+observed, as can be seen from Fig. 13.
+Although a consid-
+erable number of experimental studies have been carried out
+on how the EI and ET phenomena influence the heat transfer
+aspects in various geometries; however, the number of cor-
+responding numerical studies is very limited. This is mainly
+due to the ’High Weissenberg Number Problem (HWNP)’ en-
+countered in viscoelastic fluid simulations, as mentioned ear-
+lier. Among very few studies, Li et al.97 was probably the first
+who numerically simulated the heat transfer performance in
+a three-dimensional square serpentine microchannel (whose
+wall is maintained at a higher temperature than the fluid inlet
+temperature) in the elastic turbulence regime. They used the
+Oldroyd-B fluid model to realize the fluid viscoelasticity and
+the log-conformation approach to stabilize the numerical sim-
+ulations. Figure 14 shows the non-dimensional temperature
+distribution along with the velocity vectors at three different
+cross-sectional areas of the microchannel both for Newtonian
+(first row) and viscoelastic fluids with Wi = 5 (second row)
+and 20 (third row). It can be evident that for Newtonian flu-
+ids, the temperature distribution showed a perfect symmetry
+around the center of the channel, and the isotherms adopted a
+circular structure. All these suggest that heat transfer in New-
+tonian fluids primarily occurred by the conduction mode in
+this microchannel geometry due to the absence of fluid ad-
+vection at these low Reynolds number flows. However, as
+viscoelasticity was gradually introduced into the Newtonian
+fluid, a dramatic change happened both in the temperature dis-
+tribution and velocity vectors. First of all, the symmetry that
+was seen for Newtonian fluids was completely lost, and the
+temperature distribution became more uniform. These ten-
+dencies became more prominent as the fluid viscoelasticity
+further increased. This happened due to the increased chaotic
+convection inside the microchannel resulting from the elas-
+tic turbulence phenomenon. As expected, the heat transfer
+rate also increased inside the microchannel, as evident from
+Fig. 15(b), wherein the variation of the surface-averaged Nus-
+selt number with the Weissenberg number is shown.
+Fig-
+ure 15(a) shows the temporal variation of the non-dimensional
+temperature at various values of the Weissenberg number. For
+Newtonian fluids, the temperature did not show any fluctua-
+tion and remained at a steady value. In contrast, it became in-
+creasingly fluctuating as the fluid viscoelasticity gradually in-
+creased due to the increased intensity of the ET phenomenon.
+A similar observation was also found in their other study99,
+which mainly focused on developing the numerical algorithm
+for simulating high Weissenberg number problems.
+Recently, Gupta et al.98 performed a numerical study on the
+mixed convective heat transfer phenomena inside a lid-driven
+cavity filled with viscoelastic fluids. This geometry is con-
+sidered to be one of the widely studied benchmark problems
+in the domain of flow and heat transfer phenomena. They
+used a large range of pertinent non-dimensional numbers like
+the Weissenberg and Reynolds numbers and presented exten-
+sive results and discussion on both the flow dynamics and
+heat transfer phenomena inside the cavity. In their simula-
+tions, two viscoelastic fluid models, namely, Oldroyd-B and
+FENE-P were used to show the competitive effect of the fluid
+elasticity and shear-thinning behaviours on the generation of
+the elastic turbulence phenomenon and subsequent influence
+on the heat transfer enhancement inside the cavity. Figure 16
+presents the variation of the time and surface-averaged Nus-
+selt number with the Weissenberg number for both fluids. It
+can be observed that the flow inside the cavity transited from a
+steady to unsteady chaotic regime after a critical value of the
+Weissenberg number due to the establishment of the EI and
+ET phenomena. The corresponding heat transfer rate was also
+drastically increased for Oldroyd-B viscoelastic fluids. They
+suggested that the heat transfer rate inside the cavity could
+be increased by more than 100% using the ET phenomenon.
+However, it was not observed to that extent for FENE-P vis-
+coelastic fluids, which show shear-thinning behaviours. It was
+because the shear-thinning behaviours tend to suppress the
+elastic instability100. A similar kind of observation was also
+seen in the experiments of Abed et al.87 for a square serpen-
+tine microchannel, Fig. 9.
+
+Temperature
+0.8
+0.8
+0.7
+0.65
+0.6
+0.55
+0.5
+0.45
+0.4
+0.35
+0.8
+0.3
+0.25
+0.2
+02
+9.4
+0.6
+0.15
+0.1
+0.05
+C1
+C2
+C314
+FIG. 15. (a) Temporal variation of the non-dimensional temperature at a probe location inside the microchannel97. Note that here Wi = 0
+stands for Newtonian fluids. (b) Variation of the surface-averaged Nusselt number with the Weissenberg number. The inset figure shows the
+variation of the RMS value of non-dimensional temperature, and ’NF’ represents Newtonian fluid.
+FIG. 16. Variation of the time and surface-averaged Nusselt number
+with the Weissenberg number in a lid-driven cavity.98. The results
+are presented for two viscoelastic fluid models, namely, Oldroyd-B
+and FENE-P.
+IV.
+APPLICATIONS IN ENHANCED OIL RECOVERY
+(EOR) PROCESS
+In the chemical EOR process, particularly in the polymer
+flooding EOR process, the flow dynamics occurring within
+the micron-sized rock pores could significantly influence the
+macroscopic performance of this process due to the induc-
+tion of elastic instability and elastic turbulence phenomena.
+Therefore, the study of the flow dynamics of single-phase vis-
+coelastic fluids in a microfluidic porous geometry has recently
+received immense attention38,42,44,54. However, in an actual
+EOR process, multi-phases are always present, for instance,
+oil and a polymer solution in the case of polymer flood-
+ing EOR process. Therefore, several studies were also con-
+ducted with multi-phases to investigate the oil displacement
+efficiency utilizing the EI and ET phenomena. For instance,
+Clarke et al.101 performed a detailed experimental study on
+the capability of a viscoelastic polymer solution (partially hy-
+drolyzed polyacrylamide (HPAM)) to displace a synthetic oil
+in a model fabricated porous media. Some other solutions,
+such as glycerol and xanthan, were also used to compare.
+They provided both microscopic flow details (utilizing streak
+photography and particle image velocimetry techniques) and
+macroscopic flow behaviours such as pressure drop and appar-
+ent viscosity. Figure 17(a1) shows the temporal variation of
+the average velocity at a sampled region inside the porous ma-
+trix. It can be seen that viscoelastic HPAM solution showed
+much larger flow fluctuations than glycerol under the same
+conditions. Also, the velocity speed in the sampled region in-
+creased gradually for the HPAM solution after a critical value
+of the flow rate (filled squares). In contrast, it remained al-
+most at the same value for the glycerol solution, Fig. 17(a2).
+The apparent viscosity (which is proportional to the pressure
+drop) also increased abruptly in HPAM solution after a criti-
+cal value of the flow rate. Both these signatures suggest the
+presence of elastic turbulence in the HPAM solution. Fig-
+ure 17(a3) depicts the distribution of the oil (red colour re-
+gion) and displacing fluid phases for both water-wet and oil-
+wet conditions. In both cases, it has been observed that the
+oil droplet had a moving meniscus (the bright halos) when the
+HPAM solution was used as the displacing fluid. Therefore,
+the oil droplets were in a moving condition due to the pres-
+ence of higher velocity fluctuations resulting from the elastic
+turbulence phenomenon in HPAM solutions, resulting in the
+higher oil displacement using these viscoelastic polymer so-
+lutions. Similar results and observations were also presented
+in another study by the same group102. The direct evidence of
+the oil droplet fluctuations and the movement of its meniscus
+caused due to this elastic turbulence in HPAM polymer solu-
+tions was presented in their another subsequent study103. It
+was obtained with the help of the nuclear magnetic resonance
+(NMR) pulsed field gradient (PFG) diffusion measurements
+
+(a) 0.15
+(b)
+9
+Wi=0
+Wi=5
+Wi=10
+Wi=20
+0.02
+0.01
+0.10
+8
+0.00
+nN
+0.01
+0.1
+10
+Wi
+0.05
+7
+Viscoelasticfluid
+NF2.Nu=6.20
+0.00
+0
+100200300400
+50060070080090010
+0.01
+0.1
+1
+10
+t
+IM80
+Oldroyd-B
+FENE-P
+60
+Unstable region
+(Elastic instabilities and
+Steady region
+elasticturbulence)
+40
+O
+.0..0....0.0..0
+20
+8
+0
+10-1
+100
+101
+102
+Wi15
+FIG. 17. (a1) Temporal variation of the averaged velocity measured within a sampled region of 100 µm square at the center of a pore inside
+the porous geometry (a2) Variation of the apparent viscosity (left side) and fractional velocity spread (right side) with the flow rate at the same
+sampled region (a3) Multiphase flows through the model fabricated porous media. Here the red regions represent the oil phase and the presence
+of bright halos on each oil droplet indicates the moving menisci101.
+FIG. 18. Two-phase flow of synthetic oil and dispensing (a) PEO
+and (b) HPAM polymer solutions under the same flow conditions at
+a particular probe area inside a porous matrix104.
+conducted in a three-dimensional (3D) opaque porous struc-
+ture (sandstone). Therefore, this in-situ experimental study,
+for the first time, established the presence of elastic turbulence
+in flows of viscoelastic polymer solutions once the flow rate
+exceeds a critical value and its subsequent influence on the
+enhancement in the breakup and mobilization of trapped oil
+droplets inside the porous matrix. This ultimately leads to a
+higher displacement efficiency of oil in viscoelastic polymer
+solutions.
+Hincapie et al.104 presented an experimental study on de-
+tailed pore-scale flow visualization of both single and two-
+phase flow dynamics inside a micromodel (composed of three
+layers wherein the middle layer was made of silicon and had
+the porous structure. It was then sandwiched with top and bot-
+tom layers made of glass for easy visualization purpose) that
+mimics a real porous matrix. Their flow visualization exper-
+iments revealed different micro-scale phenomena originating
+due to the viscoelastic instability during the flooding of vis-
+coelastic HPAM polymer solutions into this porous matrix,
+namely, streamline crossing, changing flow direction, flow
+penetration into small corners, and formation of local vor-
+tices. All these micro-scale phenomena collectively resulted
+in a larger displacement of synthetic oil saturated initially in
+the porous matrix when an HPAM polymer solution was used.
+This is also evident in Fig. 18 wherein the distribution of oil
+and dispensing phases is depicted under the same flow con-
+ditions at a particular probe area inside the porous matrix. It
+can be easily seen that the HPAM polymer solution displaced
+more oils from the porous matrix due to the establishment of
+elastic turbulence inside it. They also observed other probe
+areas and found the same trend104. Furthermore, in their ex-
+periments, an enhancement in the apparent viscosity was also
+seen after a critical value of the flow rate likewise Clarke et
+al.101,102. In another study from the same research group, they
+provided more analysis on how polymer concentration, salin-
+ity, pre-shearing of polymer solutions, molecular weight, etc.,
+would tend to influence the shear-thickening and elastic tur-
+bulence phenomena inside the porous matrix105. Liu et al.106
+
+2000
+Waterwet (hydrophilic)
+Oil wet (hydrophobic)
+1800
+1600
+1400
+0.24wt%Xanthan
+1200
+1000
+800
+600
+400
+Glycerol 84%
+(al)
+200
+0.12% HPAM (3630S)
+b)
+0.00
+0.20
+0.40
+0.60
+0.80
+1.00
+Time,(s)
+0.24wt%HPAM3630S
+0.15
+3.00
+(a2)
+2.50
+spread
+0.1
+2.00
+1.50
+1.00
+0.50
+(a3)
+0.05
+0.00
+1
+Flow rate, q (ul s-1)
+10
+100(a)
+(b)
+Oil
+Oil
+Oil16
+developed a novel polymer, named star-like amphiphilic poly-
+acrylamide (SHPAM), consisting of nano-SiO2 as the core
+and a layer of amphiphilic chains as the shell using a facile
+free radical polymerization method, to demonstrate its higher
+displacement efficiency than HPAM polymer solutions from a
+geological rock core (sandstone) initially saturated with crude
+oil. Their nuclear magnetic resonance (NMR) spectroscopy
+results revealed that SHPAM polymers have higher displace-
+ment efficiency than HPAM polymers under the same oper-
+ating conditions (and even at a lower concentration) due to
+the higher shear-thickening and viscoelastic properties in the
+former polymers owing to the presence of cross-linked mi-
+crostructure.
+De et al.107 presented a detailed and systematic experi-
+mental study on the mechanism of residue oil displacement
+in a model porous media consisting of a microchannel hav-
+ing several cylindrical micropillars placed in it using dis-
+placing fluids of different rheological characteristics.
+Fig-
+ure 19 represents the steady-state snapshots of the distribu-
+tion of oil and different displacing fluid phases (namely, wa-
+ter, xanthan, HPAM, and viscoelastic surfactant (VES) so-
+lution comprised of cationic surfactant cetyltrimethylammo-
+nium bromide (CTAB), sodium salicylate (NaSal) and sodium
+chloride (NaCl) dissolved in de-mineralized water) almost at
+the same values of the capillary number. When the capillary
+number was small, large oil blobs were seen to be present ir-
+respective of the displacing fluid type. However, as this num-
+ber was gradually incremented, oil ganglia of large sizes were
+still present when the displacing fluids were water and xan-
+than, sub-Figs. 19(a) and (b). On the other hand, in the case
+of viscoelastic HPAM and VES solutions (sub-Figs. 19(c) and
+(d)), those became considerably small in size compared to that
+seen in water and xanthan. This was due to the presence of the
+viscoelastic instability effect in these two displacing fluids,
+which disrupted large oil blobs into small ones and facilitated
+a larger displacement of oils from the porous media. This can
+be seen in sub-Fig. 19(e) wherein the remaining oil saturation
+was presented against the value of the capillary number for
+different displacing fluids. It decreased as the capillary num-
+ber increased for all displacing fluids; however, the extent of
+this decrease was more for HPAM and VES. Due to the ab-
+sence of elastic instability in water and xanthan (inelastic and
+weakly elastic shear-thinning fluids, respectively), the remain-
+ing oil saturation was comparatively high in these two fluids.
+Zhong et al.110 performed both experiments and numeri-
+cal simulations (based on the volume of fluid (VOF) method)
+using a quartz sand epoxy resin as the model porous me-
+dia and hydrophobically associating water-soluble polymers
+(HAWP). Their simulation results revealed that the fluid vis-
+coelasticity in the case of polymer flooding leads to a larger
+sweep area and stable front than water flooding, resulting in
+a decrease of the residual oil saturation for polymer flooding.
+However, an additional pressure drop was also observed in the
+case of polymer flooding in their simulations, as was seen in
+the corresponding experiments101,102. These observations of
+a larger sweep area and stable front in the case of viscoelas-
+tic polymer flooding were also seen in the experiments per-
+formed by Vik et al.111 and the dynamic pore network mod-
+eling by Salmo et al.112. Molecular-scale simulations were
+also performed to understand the mechanism of the trapped
+oil displacement from a dead micro-pore zone in porous me-
+dia by Fan et al.108.
+They conducted molecular dynamics
+(MD) simulations with various values of the polymer chain
+length (N) and injected pore volume (PV). Figure 20(a-b)
+shows the snapshots of the distribution of oil (black-coloured
+molecules) and displacing fluid molecules (red and green-
+coloured molecules represent water and polymers, respec-
+tively) inside the nanopore. It can be seen that in the case
+of water flooding (sub-Fig. 20(a)), the oil molecules remained
+at the dead end of the nanopore even at higher values of PV,
+whereas they came out from the dead zone in the case of poly-
+mer flooding (sub-Figs. 20(b)). The latter tendency further
+incremented as the polymer chain length increased. They pro-
+posed a mechanism for this enhanced displacement efficiency
+of the oil during the polymer flooding based on the pulling ef-
+fect of elastic polymer molecules. Similar findings were also
+seen in the experiments of Wang et al.109 in a microscopic
+pore of a porous media, sub-Figs. 20(c) and (d). Furthermore,
+they observed the formation of "oil thread" during the poly-
+mer flooding (sub-Fig. 20(f)), resulting in the origin of a new
+mechanism for the high displacement efficiency of this flood-
+ing process. A detailed theoretical analysis was presented to
+explain this phenomenon, and the presence of elastic stresses
+in polymer solutions was found to be responsible for the for-
+mation and stabilization of this oil thread.
+To understand the transport mechanism of oil blobs in an
+actual porous media during the polymer flooding process, fur-
+ther studies were conducted with a simple model porous sys-
+tem consisting of an expansion/contraction microchannel and
+placing one oil droplet inside it. For instance, Xie et al.113
+conducted an extensive two-dimensional Lattice-Boltzmann
+method (LBM) based numerical investigation using the sim-
+ple Maxwell model to account for the fluid viscoelasticity of
+the displacing fluid. In the case of simple Newtonian dis-
+placing fluid, the droplet was seen to pass through the con-
+stricted (or the pore-throat) region easily as time progressed,
+sub-Fig. 21(a). However, in the case of viscoelastic displac-
+ing fluid, the droplet started to oscillate in front of the en-
+trance of the constricted region. It also did not pass through
+this region, sub-Fig. 21(b). It was due to the presence of elas-
+tic instability in the polymer flooding case, which generated
+large and fluctuating vortices around the entrance of the con-
+stricted region. This, in turn, blocked the movement of the dis-
+persed droplet into this constricted region. No such vortices
+were formed for Newtonian fluid flooding, and as a result, the
+droplet passed smoothly through the constricted region. In a
+subsequent experimental study, Xie et al.115 also observed this
+oscillating trap of the droplet likewise seen in their numerical
+simulations. Once again, the droplet passed easily through the
+constricted region when the displacing fluids were Newtonian
+(sub-Fig. 21(c)) and inelastic shear-thinning (sub-Fig. 21(d));
+however, it was inhibited when the displacing fluids were vis-
+coelastic (sub-Figs. 21(e) and (f)). They also proposed scal-
+ing relationships for the amplitude of droplet oscillation and
+droplet length and found a good agreement with the corre-
+sponding experimental results. Both these were found to in-
+
+17
+FIG. 19. Snapshots of the distribution of oil and different displacing fluids, namely, (a) water, (b) xanthan, (c) HPAM, (d) VES, inside the
+porous matrix at different capillary numbers (Ca). The latter was defined as the ratio of the viscous to that of the surface tension forces, i.e.,
+Ca = µu
+σ where µ, u and σ are the displacing fluid viscosity, Darcy’s velocity, and the interfacial tension between the displacing and displaced
+fluids, respectively. (e) Variation of the remaining percentage oil saturation with the capillary number for various displacing fluids107.
+crease with fluid viscoelasticity.
+Xie et al.113 also performed simulations for a system com-
+prising a straight microchannel with a side dead zone where
+the droplet was present.
+The droplet was displaced from
+the dead zone and merged with the main flow during the
+viscoelastic fluid flooding due to elastic instability-induced
+chaotic convection, whereas it remained trapped inside the
+dead end during the Newtonian fluid flooding, as was also
+seen in earlier experiments109 and molecular-scale simula-
+tions108. This further establishes the role of elastic instability
+and elastic turbulence phenomena in displacing the trapped
+oil ganglia in a porous media. A very recent study by Mo-
+hamed et al.114 also proved it by examining the morphologies
+of oil globules in a three-dimensional porous structure with
+the help of an in-situ high-resolution microcomputed tomog-
+raphy (µ-CT) technique. The snapshots of oil globules inside
+the three-dimensional micro-pore structure are presented in
+Figure 22 both for water and polymer solution flooding. It
+can be observed that a big oil blob that was present in the cir-
+cled area of the pore structure during the water flooding (sub-
+Fig. 22) was not present during the polymer flooding (sub-
+Fig. 22(b)) under the same conditions. They proposed that the
+elastic turbulence phenomenon in the latter case fragmented
+and mobilized the oil globule, and hence a higher displace-
+ment of oil was obtained. This was also reflected in their cal-
+culation of the residual oil saturation, which decreased with
+the increased fluid viscoelasticity. A correlation between the
+residual oil saturation and the Weissenberg number was also
+
+Ca:210-5
+Ca:4-10
+110-3
+Ca:8-104
+Water
+Hpam
+Oil phase
+Oil phase
+(al)
+(a3)
+(e1)
+(a)
+(c)
+Xanthari10
+Ca:1-10
+Ca:3-105
+Ca:1-10
+Ca:1103
+VES
+Oil phase
+Oil phase
+(d2)
+(b)
+(d)
+50
+HPAM
+45
+VES
+Water
+40
+Xanthan
+saturation
+35
+30-
+25
+20-
+15-
+10-
+(e)
+5
+10-5
+104
+10-3
+Ca18
+FIG. 20. Molecular dynamics (MD) simulations of (a) water and (b) polymer flooding through a nanopore with a dead zone where an oil
+droplet (black-coloured molecules) is trapped108. Here N and PV are the polymer chain length and injected pore volume, respectively. The
+corresponding experimental results of (c) water and (d) polymer flooding in a microscopic pore of a porous media by Wang et al.109. The
+distribution of (e) oil and water and (f) oil and polymer solutions inside the porous media109. Here the red-dyed regions represent the oil phase.
+proposed in their study. A similar observation was also seen
+in earlier experiments of Qi et al.116, and they also proposed a
+correlation between the residual oil saturation and the Debo-
+rah number, as provided by Mohamed et al.114. Irfan et al.117
+also conducted a recent experimental investigation on a three-
+dimensional porous structure made of Berea sandstone and
+found an enhancement in the residual oil displacement from it
+by the use of HPAM polymer solution as the displacing fluid.
+They also concluded that elastic turbulence was responsible
+for induced pressure and velocity fluctuations at small pores
+of the porous media, increasing the oil displacement.
+Druetta and Picchioni119 performed a numerical study us-
+ing the upper convected Maxwell (UCM) and Oldroyd-B
+viscoelastic fluid models on a two-dimensional porous rock
+structure at relatively low values of the Weissenberg num-
+ber where the elastic turbulence phenomenon was not present.
+However, still, they observed an increase in the oil displace-
+ment efficiency of around 15.4% during the viscoelastic fluid
+flooding compared to the traditional water flooding. This was
+attributed to a larger sweep area (without local channels for
+flow) in viscoelastic fluid flooding than in water flooding.
+A larger sweep area was caused due to more penetration of
+polymer solutions into small pores of the porous structure.
+This was evident both in their numerical solutions and experi-
+ments. However, they also suggested that apart from the fluid
+viscoelasticity, the interfacial tension (IFT) between the dis-
+placing and displaced fluids also plays an essential role in the
+sweeping process. Parsa et al.118 conducted an experimental
+study to investigate the pore-scale interaction between an oil
+blob and displacing fluid in a three-dimensional micromodel
+porous media consisting of a square quartz capillary filled
+with randomly and loosely packed monodisperse borosilicate
+glass beads. They used confocal microscopy to configure the
+displacement of oil within the porous media and also to ob-
+tain a detailed velocity field within the displacing fluid. Fig-
+ure 23 represents the snapshot of an oil ganglion trapped (pur-
+
+(a)
+(b)
+waterflooding(7.25PV)
+N=250(0.85PV)
+(d)
+(e)
+(f)
+Oilthread19
+FIG. 21. Numerical simulations of different states of a non-wetting oil droplet during its transport through an expansion/contraction mi-
+crochannel when the displacing fluids were (a) Newtonian and (b) viscoelastic polymer solutions113. The corresponding experimental results
+when the displacing fluids were (c) Newtonian, (d) inelastic shear-thinning, (e) viscoelastic with lower relaxation time, and (f) viscoelastic
+with higher relaxation time. Here Ca and De are the capillary and Deborah numbers, respectively. The arrows show the direction of droplet
+motion.
+FIG. 22.
+Visualization of trapped oil globules (red) in a three-
+dimensional porous structure during (a) water flooding and (b) vis-
+coelastic polymer solution flooding114.
+ple) inside the micromodel porous media along with the veloc-
+ity vector fields (blue arrows) in the displacing fluid at three
+different cases, namely, after initial water flooding, polymer
+solution flooding, and the chase water flooding.
+They ob-
+served that the oil globule was present inside the porous media
+even after polymer solution flooding (sub-Fig. 23(b)), which
+was only completely removed after flooding with the chase
+FIG. 23.
+Snapshot of an oil ganglion (purple) trapped in three-
+dimensional micromodel porous media along with the velocity vec-
+tor field (blue arrows) immediately after the (a) initial water flooding,
+(b) polymer flooding, and (c) chase water flooding118. Here the size
+of the arrows dictates the velocity field strength.
+water (sub-Fig. 23(c)). Therefore, they showed that polymer
+solution flooding will not always facilitate oil displacement.
+However, it should be mentioned here that there was no elas-
+tic turbulence present in the system, as was confirmed by their
+study. However, they noticed significant local changes in the
+velocity field due to polymer solution flooding, leading to the
+origin of sufficiently large viscous forces at the interface of the
+immiscible fluids. They proposed that these large and hetero-
+geneous local changes in the flow field resulted in increased
+
+(a)
+(b)
+0.08s
+0.259s
+0.133s
+0.309s
+0.357s
+0.315s
+0.387s
+0.320s
+0.392s
+0.328s
+0.400s
+0.3325
+(a)
+(b)
+Flow direction of displacing fluids
+L=0
+t=0
+=23
+t=0
+t = 0.6s
+t=1.5s
+=1.53
+(=1.95
+=2.5s
+(c)
+(d)
+(e)
+(f)(a)
+(b)(a)
+(b)
+(c)
+500um20
+oil displacement.
+V.
+LIMITATIONS
+As mentioned earlier and discussed, the elastic instability
+and turbulence phenomena are generated in a system where
+the effect of inertial forces is negligible compared to that of
+viscous and elastic forces. In other words, these unstable and
+chaotic flow regimes are generated in a system when the elas-
+ticity number (El), defined as the ratio of the Weissenberg
+(Wi) to that of the Reynolds number (Re), i.e., El = Wi
+Re, be-
+comes much larger than one. Therefore, it is only possible
+to generate in a micro-scale system whose dimensions are of
+micron or millimeter sizes if we take the realistic values of
+the physical properties of a fluid, such as density, viscosity,
+relaxation time of polymer molecules, or any other micro-
+scopic structure, etc. This is the reason why all the studies
+regarding the potential applications of these two phenomena
+were so far carried out for microfluidic applications. How-
+ever, this requirement of small-scale dimensions for generat-
+ing these two phenomena may limit their use in many prac-
+tical applications. To understand this, we can take the ex-
+ample of flow through a straight pipe for which the pressure
+drop (∆p) varies inversely with the fourth power of the ra-
+dius of the pipe (R), i.e., ∆p ∼
+1
+R4 . We know this from the
+well-known Hagen-Poiseuille equation120. It suggests that as
+the radius of the pipe gradually decreases, the pressure drop
+increases non-linearly. Hence, the mechanical power needed
+to pump the fluid inside the system also increases abruptly if
+all other parameters in the Hagen-Poiseuille equation remain
+fixed. On top of that, an additional abrupt pressure drop is
+created in a system once the elastic instability and turbulence
+phenomena set in. In fact, this behaviour is considered one
+of the characteristic features of these phenomena, which has
+been found in many earlier experimental as well as numeri-
+cal studies74,78,86,87,92,95,96. The requirement of this extra sub-
+stantial mechanical power for pumping the viscoelastic fluids
+in the elastic turbulence regime may preclude its application
+(either in the enhancement of the rate of heat transfer or mix-
+ing process) from the viewpoint of the operational limit of
+an instrument. Furthermore, one may need to specially de-
+sign the whole system to sustain such a huge pressure drop in
+such small-scale microsystems, which in turn, may increase
+the operational cost. One needs to be more cautious when ap-
+plying these EI and ET phenomena to an application system
+consisting of sophisticated and flexible microfluidic compo-
+nents, which may be damaged due to the presence of this high-
+pressure gradient. In the case of electrokinetic-driven flows,
+one needs to apply a high-voltage difference across a system
+to pump the fluid and generate the electro-elastic turbulence,
+which may also become problematic in many applications.
+The next limitation in applying the EI and ET phenomena
+to any practical application is the rheological property of the
+fluid. Most of the studies performed so far on the potential of
+the application of these phenomena used the Boger fluid121.
+It shows a constant shear viscosity but exhibits high exten-
+sional viscosity122,123. This is a special type of fluid that is
+made manually in the lab to investigate the explicit effect of
+fluid elasticity on various flow phenomena124–130. One has
+to be careful in choosing the polymers and its concentration
+as well as the solvent to make such type of fluid121. There-
+fore, the working fluids in any application should be of Boger
+fluid type so that an unstable and chaotic flow field could be
+created inside the system to enhance the rate of any transport
+phenomena. Although many fluids that are routinely encoun-
+tered in many practical microfluidic applications such as poly-
+mer solutions, emulsions, suspensions, many biofluids includ-
+ing blood, saliva, DNA and protein suspensions, cerebrospinal
+fluid, suspensions of cells and bioparticles, etc.,4,131–136 ex-
+hibit non-Newtonian behaviours; however, they hardly show
+the Boger fluid type behaviour. All these fluids exhibit elas-
+tic behaviours along with other non-linear behaviours such as
+shear-thinning, shear-thickening, viscoplasticity, thixotropic,
+etc137. These other rheological behaviours significantly influ-
+ence the EI and ET phenomena in a system. For instance,
+the shear-thinning behaviour of a viscoelastic fluid has been
+shown to suppress the onset of the elastic instability in a sys-
+tem100. This, in turn, may inhibit the generation (or the in-
+tensity) of elastic turbulence in a system. It has been, in fact,
+observed both in experimental and numerical studies wherein
+a reduction in the rate of the heat transfer process occurred due
+to the suppression of the elastic turbulence phenomenon ow-
+ing to the shear-thinning properties of a viscoelastic fluid87,98.
+Therefore, one has to opt for a Boger fluid to utilize the full
+potential of elastic turbulence in any practical application.
+This may be easy for heat transfer applications for which a
+coolant could be made in such a way that it should exhibit the
+same rheological behaviours as that of a Boger fluid. How-
+ever, problems may arise in the case of mixing applications
+wherein the making and rheological behaviours of working
+fluids are not on our hands. Of course, one can add a minute
+amount of solid polymers or surfactants into the working fluid
+to make it viscoelastic. However, it does not guarantee that
+the resulting solution would behave like a constant viscosity
+Boger fluid, as the making of such fluid depends on many fac-
+tors. Moreover, a particular application may not allow such
+addition of polymers or surfactants into the main working flu-
+ids (although they are present in parts per million (ppm) quan-
+tities) as it may create problems for their further downstream
+applications or processing.
+Therefore, one has to perform
+a rigorous investigation before applying the EI and ET phe-
+nomena to any particular application, particularly related to
+enhancing the mixing process.
+The requirement of geometrical configuration may also
+sometimes limit the application of these two phenomena to
+any practical application. As already mentioned earlier and
+also seen in most of the studies, the onset of elastic instabil-
+ity happens due to the interaction between the normal elas-
+tic stresses and the streamline curvature present in a sys-
+tem23,24. The latter requires a curved geometry, which some-
+times may become difficult to fabricate for microfluidic ap-
+plications. The type and number of curved surfaces and their
+arrangement can significantly influence the onset and gener-
+ation of elastic turbulence phenomenon. Hence, one has to
+perform a thorough optimization study to select a particular
+
+21
+geometry. All these may become expensive from a practical
+perspective as compared to other options that are available to
+perform the same duty.
+VI.
+CONCLUSIONS AND FUTURE DIRECTIONS
+The elastic instability and elastic turbulence phenomena,
+indeed, have the potential to increase the rate of transport pro-
+cesses such as heat transfer or mixing processes. However,
+the application of these phenomena is limited to only micro-
+scale systems wherein the effect of inertial forces is negligi-
+ble compared to viscous and elastic forces. Furthermore, the
+working fluid has to be non-Newtonian viscoelastic in nature,
+which in turn, precludes the applicability of these phenom-
+ena for an application wherein a simple Newtonian fluid is
+handled. Among various non-linear rheological characteris-
+tics that a fluid can show, elasticity should be the dominant
+one, which promotes the generation of these phenomena in
+a microfluidic system. The constant shear viscosity and high
+extensional viscosity Boger fluid121 should be the ideal choice
+for this purpose. Based on the discussion presented in the
+preceding section, it can be readily acknowledged that these
+phenomena have a higher potential in microfluidic heat trans-
+fer applications than micro mixing applications due to lesser
+restrictions in the former applications for applying these phe-
+nomena. Although a considerable number of studies have al-
+ready been conducted to show the potential of these phenom-
+ena in increasing the rate of either the heat transfer or the mix-
+ing process or in enhancing the oil displacement efficiency in
+the EOR process; however, still a large scope is present as
+far as the application point of view is concerned of these two
+phenomena.
+• Most studies on the applications of the EI and ET phe-
+nomena were carried out for curvilinear or serpentine
+microchannels. The presence of a highly curved sur-
+face in this geometry produces high streamline curva-
+ture in the flow field, which in turn, facilitates the gen-
+eration of elastic turbulence in this geometry.
+How-
+ever, a detailed study on how the number and angle of
+curving could influence these phenomena and, subse-
+quently, the rate of transport processes is still missing
+in the literature. A direct relationship between the pres-
+sure drop and the rate of transport processes should be
+established; likewise, it was done for the regular hy-
+drodynamic turbulence138. More investigations should
+be carried out for other micro-scale geometries, for in-
+stance, a microchannel with in-built obstacles present
+in it. Although this geometry has already been used
+to induce elastic instability in a number of experimen-
+tal and numerical studies32,50,53,78,139,140; however, the
+corresponding study on the potential in enhancing the
+rate of transport processes in this geometry is not in-
+vestigated yet. Several factors, such as the shape of the
+obstacle, the number of obstacles, and the gap between
+two consecutive obstacles, could influence the rate of
+transport processes. Hence, a detailed investigation is
+needed on the same. A microchannel with either step
+expansion and contraction or micro constrictions could
+also be investigated to see its potential in enhancing the
+rate of transport processes. This particular geometry
+has also shown the potential to create elastic instability
+and turbulence36,49,53,141. Moreover, the fabrication of
+this latter geometry would be relatively easier than that
+of the curvilinear or serpentine microchannel.
+• Research efforts should be spent on establishing the cor-
+relations for the average Nusselt number (in the case of
+heat transfer applications) as a function of the relevant
+dimensionless numbers such as Weissenberg, Reynolds,
+Prandtl, and Richardson numbers. Although some stud-
+ies have already attempted to establish such correla-
+tions89,90; however, physical insights and/or scaling ar-
+guments behind the selection of a particular form of the
+correlation and the power-law exponents of different
+dimensionless numbers (particularly, the Weissenberg
+number) was somehow missing in those studies. More
+detailed and rigorous investigations are needed to estab-
+lish such correlations, including the effect of geomet-
+ric parameters and polymer concentration along with
+other thermo-physical properties (such as specific heat,
+thermal conductivity, etc.) and flow conditions prop-
+erly and systematically.
+Furthermore, the studies on
+heat transfer applications of the EI and ET phenom-
+ena are mostly limited to forced convection. In con-
+trast, almost no study (either experimental or numeri-
+cal) is present (except one recent numerical study by
+Gupta et al.98) on how these phenomena could influ-
+ence the other two modes of heat transfer, namely, nat-
+ural or free convection and mixed convection. These
+two modes of heat transfer are also used in many mi-
+crofluidic applications142. Furthermore, the phenomena
+of boiling and condensation happening in micro-scale
+geometries are also widely used in many microfluidic
+applications143,144. Many studies have been conducted
+on how adding polymer or surfactant molecules into a
+solvent like water could influence these phenomena145.
+However, those studies did not look into the problem
+from a perspective of elastic instability and turbulence
+phenomena, which could be generated inside a drop
+and could influence these processes significantly. All
+these previous studies investigated how adding either
+polymer or surfactant molecules influenced the surface
+tension and dynamic viscosity and, subsequently, the
+boiling and condensation heat transfer phenomena in a
+micro-scale geometry. Therefore, a large scope for fu-
+ture investigations is present in this particular area of
+heat transfer phenomena utilizing the EI and ET phe-
+nomena.
+• Droplet-based microfluidics has become a promising
+technique in past decades in many cutting-edge tech-
+nological applications, such as fluid mixing146,147, cell
+encapsulation and delivery148,149, cell sorting150, drug
+discovery and genetic applications151, sensing152, and
+many others153. A great potential is present in this par-
+ticular area of microfluidics wherein the introduction
+
+22
+of elastic instability and turbulence could dramatically
+influence the transport phenomena inside a drop, such
+as mixing, which in turn, could significantly influence
+further downstream processes such as chemical reac-
+tions147.
+• In recent days, lots of investigations in terms of both ex-
+periments and simulations have been conducted on the
+heat transfer enhancement capability of nanofluids154.
+These fluids are formed by adding a small amount of
+nanoparticles (made of metals, oxides, carbides, carbon
+nanotubes, etc.) into a base fluid like water, glycerol,
+or oil155. Using such nanofluids, many experimental,
+as well as numerical studies, have found a significant
+enhancement in the heat transfer rate as compared to
+that achieved in base fluids only in a microfluidic sys-
+tem such as microchannel or heat sink156–158. This en-
+hancement in the heat transfer rate in nanofluids is ba-
+sically due to an increase in the effective thermal con-
+ductivity of these fluids owing to the higher values of
+the thermal conductivity of the nanoparticles. A gen-
+eration of elastic turbulence in these nanofluids flowing
+in a microfluidic system could increase the heat transfer
+rate by many-fold than that achieved either only using
+nanofluids or the elastic turbulence phenomenon alone.
+Although some studies are present in the literature on
+viscoelastic nanofluids159–162; however, no study has
+so far attempted to investigate the elastic turbulence
+phenomenon and subsequently the heat transfer rate in
+these fluids.
+• Although a substantial number of studies have been
+conducted to show the potential of the ET phenomenon
+in increasing the oil displacement efficiency during the
+polymer flooding process; however, still, a complete un-
+derstanding of the mechanism behind this enhancement
+is missing in the literature.
+All those earlier studies
+have used a microporous model structure to carry out
+the investigation, which may not mimic the actual situ-
+ation under an oil reservoir. For instance, the materials
+used for making these micromodels are often silicon,
+glass, polymers, etc., which may not exhibit the same
+surface characteristics as the minerals that are present
+in the rock. It can significantly influence the contact
+angle and hence the corresponding multiphase flow dy-
+namics inside a porous media163. Therefore, the micro-
+models prepared for this kind of multiphase flow dy-
+namics study (in particular, the influence of the ET phe-
+nomenon) should mimic the surface wettability, miner-
+alogy, and roughness parameters of natural rocks. The
+Rock-on-a-Chip (ROC) approach164,165 should be con-
+sidered in future studies, which will facilitate a better
+understanding of the influence of the ET phenomenon
+on the oil displacement mechanism in a porous media.
+A three-dimensional ROC instead of a two-dimensional
+one should be employed in understanding the trans-
+port mechanism, along with sophisticated experimental
+techniques (such as confocal microscopy) to visualize
+and analyze the flow fields. Furthermore, almost all
+previous studies were carried out at room temperature
+and pressure, whereas the oil reservoirs are primarily
+present at elevated temperatures and pressure. These
+two parameters could significantly impact the perme-
+ability and the interfacial tension between two immis-
+cible fluids and, subsequently, the multiphase flow dy-
+namics inside the porous media166,167. Although sev-
+eral studies have emphasized that the ET phenomenon
+is responsible for higher oil displacement efficiency
+during polymer flooding; however, no detailed statis-
+tical analysis on the temporal and spatial fluctuations of
+either the velocity or the pressure (at different probe lo-
+cations) was presented, which could firmly establish the
+claim further.
+• Polymeric surfactants are believed to be promising in
+the chemically enhanced oil recovery process168,169 due
+to their ability to reduce the interfacial tension and in-
+crease the solution viscosity. Both these tend to facili-
+tate more oil displacement in a porous media. However,
+most studies on these polymeric surfactants related to
+the EOR process focused on their synthesis and char-
+acterization. There is almost no study present in the
+literature which focuses on the oil displacement mech-
+anism (as well as the ET phenomenon) at the pore level
+in these displacing fluids. This is particularly impor-
+tant to investigate as this is still a debatable subject
+whether these polymeric surfactants should be used in
+the EOR process due to their high synthesis and han-
+dling costs168.
+Therefore, the study of the ET phe-
+nomenon in the presence of these polymeric surfactants
+and other phases, such as oil, deserves significant atten-
+tion in the near future.
+• All the previous studies on the ET phenomenon during
+the EOR process considered the rheological properties
+of the displacing fluid but not the displaced fluid, i.e.,
+the crude oil. However, it can be readily acknowledged
+that crude oil exhibits various non-Newtonian charac-
+teristics, such as shear-thinning, yield stress, thixotropy,
+or even viscoelastic170–175. The rheological properties
+of the displaced fluid could also significantly regulate
+the ET phenomenon and the subsequent oil displace-
+ment efficiency.
+Therefore, careful pore-scale inves-
+tigations (comprising both numerical simulations and
+experiments) should be conducted in this regard. Fur-
+thermore, thorough and systematic studies of the effect
+of polymer type, polymer concentration, and molecular
+weight on the ET phenomenon should also be carried
+out so that it can be appropriately utilized during the
+EOR process.
+• Most studies related to either the generation of EI
+and ET phenomena or to demonstrate their potential
+in heat transfer rate or mixing enhancement applica-
+tions have been conducted in pressure-driven flows.
+In comparison, very few studies were conducted for
+electrokinetically-driven generation and applications of
+the EI and ET phenomena83–85,176,177.
+However, it
+
+23
+can be readily acknowledged that electrokinetic-driven
+flows are often used to transport fluids in micro-scale
+geometries for the following reasons i) the EK-based
+microdevices do not have any moving mechanical parts
+as they rely on the application of an electric field, and
+hence, they are easy to handle ii) the electrokinetic
+flows offer less resistance to the flow than pressure-
+driven flows due to almost plug-like velocity profile in
+the former flows178. Therefore, a huge scope is present
+for further future studies in these areas of electro-elastic
+instability and electro-elastic turbulence both from the
+application and fundamental understanding point of
+view.
+From the discussions presented herein, it is clear that a great
+potential for the applications of elastic instability and elas-
+tic turbulence phenomena is present in micro-scale systems
+to increase the rate of various transport processes. However,
+in applying so, we should also keep in mind the limitations
+that are discussed in the preceding section of this article. So
+far, the studies carried out to show the application potential of
+these two phenomena were limited to lab-scale experiments.
+Furthermore, the problem setup used either in experiments
+or simulations was not related to any direct application; in-
+stead, it was a prototype. Therefore, in the future, experiments
+should be conducted for a direct application to show the real
+potential of these two phenomena. For example, we could uti-
+lize these phenomena in the micro heat sink applications for
+cooling electronic chips179,180 or enhancing the reaction rate
+(and ultimately increasing the percentage of desired products
+in a chemical reaction) in a microsystem by enhancing the cor-
+responding fluid mixing181. Therefore, such practical studies
+are definitely needed in the future to establish the potential of
+the EI and ET phenomena for real-world applications.
+1R. B. Bird, C. F. Curtiss, R. C. Armstrong, and O. Hassager, Dynamics of
+Polymeric Liquids, volume 2: Kinetic Theory (Wiley, 1987).
+2C. M. Schroeder, “Single polymer dynamics for molecular rheology,” Jour-
+nal of Rheology 62, 371–403 (2018).
+3G. B. Thurston, “Viscoelasticity of human blood,” Biophysical Journal 12,
+1205–1217 (1972).
+4A. N. Beris, J. S. Horner, S. Jariwala, M. Armstrong, and N. J. Wagner,
+“Recent advances in blood rheology: A review,” Soft Matter (2021).
+5H.-r. Zhao, J. Wu, M.-x. Xu, and K.-m. Zhang, “Advances in the rheology
+of emulsion explosive,” Journal of Molecular Liquids 336, 116854 (2021).
+6P. L. Fuhrmann, S. Breunig, G. Sala, L. Sagis, M. Stieger, and E. Scholten,
+“Rheological behaviour of attractive emulsions differing in droplet-droplet
+interaction strength,” Journal of Colloid and Interface Science 607, 389–
+400 (2022).
+7Z. Briceño-Ahumada, A. Mikhailovskaya, and J. A. Staton, “The role of
+continuous phase rheology on the stabilization of edible foams: A review,”
+Physics of Fluids 34, 031302 (2022).
+8D. Kim, Y. Seol,
+and Y. Kim, “Numerical study on rheology of two-
+dimensional dry foam,” Physics of Fluids 33, 052111 (2021).
+9A. Jain and E. S. Shaqfeh, “Transient and steady shear rheology of
+particle-laden viscoelastic suspensions,” Journal of Rheology 65, 1269–
+1295 (2021).
+10H. M. Shewan, G. E. Yakubov, M. R. Bonilla, and J. R. Stokes, “Vis-
+coelasticity of non-colloidal hydrogel particle suspensions at the liquid–
+solid transition,” Soft Matter 17, 5073–5083 (2021).
+11R. B. Bird, R. C. Armstrong, and O. Hassager, “Dynamics of polymeric
+liquids, volume 1: Fluid mechanics,” (1987).
+12Y. Dubief, V. E. Terrapon, and B. Hof, “Elasto-inertial turbulence,” An-
+nual Review of Fluid Mechanics 55, 2023 (2022).
+13C. M. White and M. G. Mungal, “Mechanics and prediction of turbulent
+drag reduction with polymer additives,” Annual Review of Fluid Mechan-
+ics 40, 235–256 (2008).
+14B. A. Toms, “Some observations on the flow of linear polymer solutions
+through straight tubes at large reynolds numbers,” in Proceedings of the 1st
+International Congress on Rheology, North-Holland, Amsterdam 2, 135–
+141 (1948).
+15C. S. Wells Jr and J. G. Spangler, “Injection of a drag-reducing fluid into
+turbulent pipe flow of a newtonian fluid,” Physics of Fluids 10, 1890–1894
+(1967).
+16P. S. Virk, E. Merrill, H. Mickley, K. Smith, and E. Mollo-Christensen,
+“The Toms phenomenon: turbulent pipe flow of dilute polymer solutions,”
+Journal of Fluid Mechanics 30, 305–328 (1967).
+17H. C. Hershey and J. L. Zakin, “Existence of two types of drag reduction in
+pipe flow of dilute polymer solutions,” Industrial & Engineering Chemistry
+Fundamentals 6, 381–387 (1967).
+18T. Min, J. Y. Yoo, H. Choi, and D. D. Joseph, “Drag reduction by polymer
+additives in a turbulent channel flow,” Journal of Fluid Mechanics 486,
+213–238 (2003).
+19B. E. Owolabi, D. J. Dennis, and R. J. Poole, “Turbulent drag reduction
+by polymer additives in parallel-shear flows,” Journal of Fluid Mechanics
+827 (2017).
+20C.-F. Li, R. Sureshkumar,
+and B. Khomami, “Influence of rheological
+parameters on polymer induced turbulent drag reduction,” Journal of Non-
+Newtonian Fluid Mechanics 140, 23–40 (2006).
+21S. S. Datta, A. M. Ardekani, P. E. Arratia, A. N. Beris, I. Bischofberger,
+G. H. McKinley, J. G. Eggers, J. E. López-Aguilar, S. M. Fielding, A. Fr-
+ishman, et al., “Perspectives on viscoelastic flow instabilities and elastic
+turbulence,” Physical Review Fluids 7, 080701 (2022).
+22M. Khalid, V. Shankar, and G. Subramanian, “Continuous pathway be-
+tween the elasto-inertial and elastic turbulent states in viscoelastic channel
+flow,” Physical Review Letters 127, 134502 (2021).
+23P. Pakdel and G. H. McKinley, “Elastic instability and curved streamlines,”
+Physical Review Letters 77, 2459 (1996).
+24G. H. McKinley, P. Pakdel, and A. Öztekin, “Rheological and geomet-
+ric scaling of purely elastic flow instabilities,” Journal of Non-Newtonian
+Fluid Mechanics 67, 19–47 (1996).
+25P. Pakdel and G. H. McKinley, “Cavity flows of elastic liquids: purely
+elastic instabilities,” Physics of Fluids 10, 1058–1070 (1998).
+26P. Pakdel, S. H. Spiegelberg, and G. H. McKinley, “Cavity flows of elastic
+liquids: two-dimensional flows,” Physics of Fluids 9, 3123–3140 (1997).
+27G. Vinogradov and V. Manin, “An experimental study of elastic tur-
+bulence,” Kolloid-Zeitschrift und Zeitschrift für Polymere 201, 93–98
+(1965).
+28A. Groisman and V. Steinberg, “Elastic turbulence in a polymer solution
+flow,” Nature 405, 53–55 (2000).
+29A. Groisman and V. Steinberg, “Efficient mixing at low reynolds numbers
+using polymer additives,” Nature 410, 905–908 (2001).
+30Y. Jun and V. Steinberg, “Elastic turbulence in a curvilinear channel flow,”
+Physical Review E 84, 056325 (2011).
+31T. Burghelea, E. Segre,
+and V. Steinberg, “Elastic turbulence in von-
+Karman swirling flow between two disks,” Physics of fluids 19, 053104
+(2007).
+32A. Varshney and V. Steinberg, “Drag enhancement and drag reduction in
+viscoelastic flow,” Physical Review Fluids 3, 103302 (2018).
+33A. Varshney and V. Steinberg, “Elastic wake instabilities in a creeping flow
+between two obstacles,” Physical Review Fluids 2, 051301 (2017).
+34Y. Jun and V. Steinberg, “Power and pressure fluctuations in elastic turbu-
+lence over a wide range of polymer concentrations,” Physical review letters
+102, 124503 (2009).
+35S. Yamani, B. Keshavarz, Y. Raj, T. A. Zaki, G. H. McKinley,
+and
+I. Bischofberger, “Spectral universality of elastoinertial turbulence,” Phys-
+ical Review Letters 127, 074501 (2021).
+36C. A. Browne, A. Shih, and S. S. Datta, “Bistability in the unstable flow
+of polymer solutions through pore constriction arrays,” Journal of Fluid
+Mechanics 890 (2020).
+37E. M. Ekanem, S. Berg, S. De, A. Fadili, T. Bultreys, M. Rücker, J. South-
+wick, J. Crawshaw, and P. F. Luckham, “Signature of elastic turbulence
+of viscoelastic fluid flow in a single pore throat,” Physical Review E 101,
+042605 (2020).
+
+24
+38C. A. Browne, A. Shih, and S. S. Datta, “Pore-scale flow characterization
+of polymer solutions in microfluidic porous media,” Small 16, 1903944
+(2020).
+39C. A. Browne and S. S. Datta, “Elastic turbulence generates anomalous
+flow resistance in porous media,” Science Advances 7, eabj2619 (2021).
+40D. W. Carlson, K. Toda-Peters, A. Q. Shen, and S. J. Haward, “Volu-
+metric evolution of elastic turbulence in porous media,” Journal of Fluid
+Mechanics 950, A36 (2022).
+41D. Kawale, E. Marques, P. L. Zitha, M. T. Kreutzer, W. R. Rossen, and
+P. E. Boukany, “Elastic instabilities during the flow of hydrolyzed poly-
+acrylamide solution in porous media: effect of pore-shape and salt,” Soft
+Matter 13, 765–775 (2017).
+42S. J. Haward, C. C. Hopkins, and A. Q. Shen, “Stagnation points control
+chaotic fluctuations in viscoelastic porous media flow,” Proceedings of the
+National Academy of Sciences 118, e2111651118 (2021).
+43S. S. Datta, “Patches of patches: Spatial fluctuations of elastic turbulence
+in porous media,” Science Talks 3, 100044 (2022).
+44D. M. Walkama, N. Waisbord, and J. S. Guasto, “Disorder suppresses
+chaos in viscoelastic flows,” Physical Review Letters 124, 164501 (2020).
+45M. Kumar, J. S. Guasto, and A. M. Ardekani, “Transport of complex and
+active fluids in porous media,” Journal of Rheology 66, 375–397 (2022).
+46R. Keunings, “On the high weissenberg number problem,” Journal of Non-
+Newtonian Fluid Mechanics 20, 209–226 (1986).
+47A. Afonso, P. J. Oliveira, F. Pinho, and M. Alves, “The log-conformation
+tensor approach in the finite-volume method framework,” Journal of Non-
+Newtonian Fluid Mechanics 157, 55–65 (2009).
+48S. Berti, A. Bistagnino, G. Boffetta, A. Celani, and S. Musacchio, “Two-
+dimensional elastic turbulence,” Physical Review E 77, 055306 (2008).
+49M. Kumar, S. Aramideh, C. A. Browne, S. S. Datta, and A. M. Ardekani,
+“Numerical investigation of multistability in the unstable flow of a polymer
+solution through porous media,” Physical Review Fluids 6, 033304 (2021).
+50M. Kumar and A. M. Ardekani, “Elastic instabilities between two cylin-
+ders confined in a channel,” Physics of Fluids 33, 074107 (2021).
+51M. B. Khan and C. Sasmal, “Effect of micelle breakage rate on flows
+of wormlike micellar solutions through pore throats,” Journal of Non-
+Newtonian Fluid Mechanics 307, 104853 (2022).
+52M. B. Khan and C. Sasmal, “Elastic instabilities and bifurcations in flows
+of wormlike micellar solutions past single and two vertically aligned mi-
+crocylinders: Effect of blockage and gap ratios,” Physics of Fluids 33,
+033109 (2021).
+53M. B. Khan and C. Sasmal, “Effect of chain scission on flow characteristics
+of wormlike micellar solutions past a confined microfluidic cylinder: A
+numerical analysis,” Soft Matter 16, 5261–5272 (2020).
+54A. Chauan, S. Gupta, and C. Sasmal, “Effect of geometric disorder on
+chaotic viscoelastic porous media flows,” Physics of Fluids 34, 093105
+(2022).
+55V. Steinberg, “Elastic turbulence: an experimental view on inertialess ran-
+dom flow,” Annual Review of Fluid Mechanics 53, 27–58 (2021).
+56V. Steinberg, “New direction and perspectives in elastic instability and tur-
+bulence in various viscoelastic flow geometries without inertia,” Low Tem-
+perature Physics 48, 492–507 (2022).
+57H. Bruus, Theoretical Microfluidics, Vol. 18 (Oxford University Press,
+2007).
+58N. Kockmann, Transport phenomena in micro process engineering
+(Springer Science & Business Media, 2007).
+59P. K. Panigrahi, Transport Phenomena in Microfluidic Systems (John Wiley
+& Sons, 2016).
+60H. E. Meijer, M. K. Singh, T. G. Kang, J. M. Den Toonder, and P. D.
+Anderson, “Passive and active mixing in microfluidic devices,” in Macro-
+molecular Symposia, Vol. 279 (Wiley Online Library, 2009) pp. 201–209.
+61D. Di Carlo, “Inertial microfluidics,” Lab on a Chip 9, 3038–3046 (2009).
+62M. E. Steinke and S. G. Kandlikar, “Single-phase heat transfer enhance-
+ment techniques in microchannel and minichannel flows,” in Interna-
+tional Conference on Nanochannels, Microchannels, and Minichannels,
+Vol. 41642 (2004) pp. 141–148.
+63L. Léal, M. Miscevic, P. Lavieille, M. Amokrane, F. Pigache, F. Topin,
+B. Nogarède, and L. Tadrist, “An overview of heat transfer enhancement
+methods and new perspectives: Focus on active methods using electroac-
+tive materials,” International Journal of heat and mass transfer 61, 505–524
+(2013).
+64K. Ward and Z. H. Fan, “Mixing in microfluidic devices and enhancement
+methods,” Journal of Micromechanics and Microengineering 25, 094001
+(2015).
+65L. Capretto, W. Cheng, M. Hill, and X. Zhang, “Micromixing within mi-
+crofluidic devices,” Microfluidics , 27–68 (2011).
+66V. Hessel, H. Löwe, and F. Schönfeld, “Micromixers—a review on passive
+and active mixing principles,” Chemical Engineering Science 60, 2479–
+2501 (2005).
+67M. Sheikholeslami, M. Gorji-Bandpy, and D. D. Ganji, “Review of heat
+transfer enhancement methods: Focus on passive methods using swirl flow
+devices,” Renewable and Sustainable Energy Reviews 49, 444–469 (2015).
+68S. Li, H. Zhang, J. Cheng, X. Li, W. Cai, Z. Li, and F. Li, “A state-of-
+the-art overview on the developing trend of heat transfer enhancement by
+single-phase flow at micro scale,” International Journal of Heat and Mass
+Transfer 143, 118476 (2019).
+69D. O. Shah, Improved oil recovery by surfactant and polymer flooding (El-
+sevier, 2012).
+70A. M. Firozjaii and H. R. Saghafi, “Review on chemical enhanced oil re-
+covery using polymer flooding: Fundamentals, experimental and numeri-
+cal simulation,” Petroleum 6, 115–122 (2020).
+71A. Groisman and V. Steinberg, “Elastic turbulence in curvilinear flows of
+polymer solutions,” New Journal of Physics 6, 29 (2004).
+72F.-C. Li, H. Kinoshita, X.-B. Li, M. Oishi, T. Fujii, and M. Oshima, “Cre-
+ation of very-low-reynolds-number chaotic fluid motions in microchannels
+using viscoelastic surfactant solution,” Experimental Thermal and Fluid
+Science 34, 20–27 (2010).
+73R. Poole, B. Budhiraja, A. Cain, and P. Scott, “Emulsification using elas-
+tic turbulence,” Journal of Non-Newtonian Fluid Mechanics 177, 15–18
+(2012).
+74T. Burghelea, E. Segre, I. Bar-Joseph, A. Groisman, and V. Steinberg,
+“Chaotic flow and efficient mixing in a microchannel with a polymer so-
+lution,” Physical Review E 69, 066305 (2004).
+75K. Tatsumi, Y. Takeda, K. Suga, and K. Nakabe, “Turbulence character-
+istics and mixing performances of viscoelastic fluid flow in a serpentine
+microchannel,” in Journal of Physics: Conference Series, Vol. 318 (IOP
+Publishing, 2011) p. 092020.
+76H.-N. Zhang, F.-C. Li, Y. Cao, K. Tomoaki, and B. Yu, “Direct numerical
+simulation of elastic turbulence and its mixing-enhancement effect in a
+straight channel flow,” Chinese Physics B 22, 024703 (2013).
+77F.-C. Li, H.-N. Zhang, Y. Cao, K. Tomoaki, K. Haruyuki, and O. Marie,
+“A purely elastic instability and mixing enhancement in a 3d curvilinear
+channel flow,” Chinese Physics Letters 29, 094704 (2012).
+78M. Grilli, A. Vázquez-Quesada, and M. Ellero, “Transition to turbulence
+and mixing in a viscoelastic fluid flowing inside a channel with a periodic
+array of cylindrical obstacles,” Physical review letters 110, 174501 (2013).
+79H. Y. Gan, Y. C. Lam, N. T. Nguyen, K. C. Tam, and C. Yang, “Efficient
+mixing of viscoelastic fluids in a microchannel at low reynolds number,”
+Microfluidics and Nanofluidics 3, 101–108 (2007).
+80H. Y. Gan, Y. C. Lam,
+and N.-T. Nguyen, “Polymer-based device for
+efficient mixing of viscoelastic fluids,” Applied Physics Letters 88, 224103
+(2006).
+81S. O. Hong, K.-S. Park, D.-Y. Kim, S. S. Lee, C.-S. Lee, and J. M. Kim,
+“Gear-shaped micromixer for synthesis of silica particles utilizing inertio-
+elastic flow instability,” Lab on a Chip 21, 513–520 (2021).
+82H. Yang, G. Yao, and D. Wen, “Efficient mixing enhancement by orthog-
+onal injection of shear-thinning fluids in a micro serpentine channel at low
+reynolds numbers,” Chemical Engineering Science 235, 116368 (2021).
+83R. Bryce and M. Freeman, “Abatement of mixing in shear-free elonga-
+tionally unstable viscoelastic microflows,” Lab on a Chip 10, 1436–1441
+(2010).
+84M. B. Khan and C. Sasmal, “Electro-elastic instability in electroosmotic
+flows of viscoelastic fluids through a model porous system,” European
+Journal of Mechanics-B/Fluids 97, 173–186 (2023).
+85C. Sasmal, “A simple yet efficient approach for electrokinetic mixing of
+viscoelastic fluids in a straight microchannel,” Scientific Reports 12, 1–13
+(2022).
+86R. Whalley, W. Abed, D. Dennis, and R. Poole, “Enhancing heat trans-
+fer at the micro-scale using elastic turbulence,” Theoretical and Applied
+Mechanics Letters 5, 103–106 (2015).
+87W. M. Abed, R. D. Whalley, D. J. Dennis, and R. J. Poole, “Experimental
+
+25
+investigation of the impact of elastic turbulence on heat transfer in a ser-
+pentine channel,” Journal of Non-Newtonian Fluid Mechanics 231, 68–78
+(2016).
+88D. Copeland, C. Ren, M. Su, and P. Ligrani, “Elastic turbulence influ-
+ences and convective heat transfer within a miniature viscous disk pump,”
+International Journal of Heat and Mass Transfer 108, 1764–1774 (2017).
+89G. Yao, J. Zhao, X. Shen, H. Yang, and D. Wen, “Effects of rheologi-
+cal properties on heat transfer enhancements by elastic instability in von-
+karman swirling flow,” International Journal of Heat and Mass Transfer
+152, 119535 (2020).
+90G. Yao, H. Yang, J. Zhao, and D. Wen, “Experimental study on flow and
+heat transfer enhancement by elastic instability in swirling flow,” Interna-
+tional Journal of Thermal Sciences 157, 106504 (2020).
+91H. Yang, G. Yao, and D. Wen, “Flow resistance and convective heat trans-
+fer by elastic turbulence in 1d/2d/3d geometries,” International Journal of
+Thermal Sciences 176, 107512 (2022).
+92D.-Y. Li, X.-B. Li, H.-N. Zhang, F.-C. Li, S.-Z. Qian,
+and S. W. Joo,
+“Measuring heat transfer performance of viscoelastic fluid flow in curved
+microchannel using Ti–Pt film temperature sensor,” Experimental Thermal
+and Fluid Science 77, 226–233 (2016).
+93D. Li, X. Li, H. Zhang, F. Li, S. Qian, and S. W. Joo, “Efficient heat trans-
+fer enhancement by elastic turbulence with polymer solution in a curved
+microchannel,” Microfluidics and Nanofluidics 21, 10 (2017).
+94D. Blanchard, P. Ligrani,
+and B. Gale, “Miniature single-disk viscous
+pump (single-dvp), performance characterization,” (2006).
+95B. Traore, C. Castelain, and T. Burghelea, “Efficient heat transfer in a
+regime of elastic turbulence,” Journal of Non-Newtonian Fluid Mechanics
+223, 62–76 (2015).
+96T. Burghelea, E. Segre, and V. Steinberg, “Mixing by polymers: Experi-
+mental test of decay regime of mixing,” Physical review letters 92, 164501
+(2004).
+97D.-Y. Li, H. Zhang, J.-P. Cheng, X.-B. Li, F.-C. Li, S. Qian, and S. W. Joo,
+“Numerical simulation of heat transfer enhancement by elastic turbulence
+in a curvy channel,” Microfluidics and Nanofluidics 21, 1–16 (2017).
+98S. Gupta, A. Chauhan,
+and C. Sasmal, “Influence of elastic instabil-
+ity and elastic turbulence on mixed convection of viscoelastic fluids in
+a lid-driven cavity,” International Journal of Heat and Mass Transfer 186,
+122469 (2022).
+99H.-N. Zhang, D.-Y. Li, X.-B. Li, W.-H. Cai, and F.-C. Li, “Numerical
+simulation of heat transfer process of viscoelastic fluid flow at high weis-
+senberg number by log-conformation reformulation,” Journal of Fluids En-
+gineering 139 (2017).
+100L. Casanellas, M. A. Alves, R. J. Poole, S. Lerouge, and A. Lindner, “The
+stabilizing effect of shear thinning on the onset of purely elastic instabili-
+ties in serpentine microflows,” Soft Matter 12, 6167–6175 (2016).
+101A. Clarke, A. M. Howe, J. Mitchell, J. Staniland,
+and L. A. Hawkes,
+“How viscoelastic-polymer flooding enhances displacement efficiency,”
+SPE Journal 21, 0675–0687 (2016).
+102A. Clarke, A. M. Howe, J. Mitchell, J. Staniland, L. Hawkes,
+and
+K. Leeper, “Mechanism of anomalously increased oil displacement with
+aqueous viscoelastic polymer solutions,” Soft Matter 11, 3536–3541
+(2015).
+103J. Mitchell, K. Lyons, A. M. Howe, and A. Clarke, “Viscoelastic polymer
+flows and elastic turbulence in three-dimensional porous structures,” Soft
+Matter 12, 460–468 (2016).
+104R. Hincapie, A. Rock, J. Wegner, and L. Ganzer, “Oil mobilization by
+viscoelastic flow instabilities effects during polymer eor: A pore-scale vi-
+sualization approach,” in SPE Latin America and Caribbean Petroleum
+Engineering Conference (OnePetro, 2017).
+105A. Rock, R. Hincapie, J. Wegner, and L. Ganzer, “Advanced flow behav-
+ior characterization of enhanced oil recovery polymers using glass-silicon-
+glass micromodels that resemble porous media,” in SPE Europec featured
+at 79th EAGE conference and exhibition (OnePetro, 2017).
+106R. Liu, D. Du, W. Pu, Q. Peng, Z. Tao, and Y. Pang, “Viscoelastic dis-
+placement and anomalously enhanced oil recovery of a novel star-like
+amphiphilic polyacrylamide,” Chemical Engineering Research and Design
+142, 369–385 (2019).
+107S. De, P. Krishnan, J. Van Der Schaaf, J. Kuipers, E. Peters,
+and
+J. Padding, “Viscoelastic effects on residual oil distribution in flows
+through pillared microchannels,” Journal of Colloid and Interface Science
+510, 262–271 (2018).
+108J. C. Fan, F. C. Wang, J. Chen, Y. B. Zhu, D. T. Lu, H. Liu, and H. A. Wu,
+“Molecular mechanism of viscoelastic polymer enhanced oil recovery in
+nanopores,” Royal Society Open Science 5, 180076 (2018).
+109D. Wang, H. Xia, Z. Liu,
+and Q. Yang, “Study of the mechanism of
+polymer solution with visco-elastic behavior increasing microscopic oil
+displacement efficiency and the forming of steady oil thread flow chan-
+nels,” in SPE Asia Pacific oil and gas conference and exhibition (OnePetro,
+2001).
+110H. Zhong, Y. Li, W. Zhang, H. Yin, J. Lu,
+and D. Guo, “Microflow
+mechanism of oil displacement by viscoelastic hydrophobically associat-
+ing water-soluble polymers in enhanced oil recovery,” Polymers 10, 628
+(2018).
+111B. Vik, A. Kedir, V. Kippe, K. Sandengen, T. Skauge, J. Solbakken, and
+D. Zhu, “Viscous oil recovery by polymer injection; impact of in-situ poly-
+mer rheology on water front stabilization,” in SPE Europec featured at
+80th EAGE Conference and Exhibition (OnePetro, 2018).
+112I. C. Salmo, N. Zamani, T. Skauge, K. Sorbie, and A. Skauge, “Use of
+dynamic pore network modeling to improve our understanding of exper-
+imental observations in viscous oil displacement by polymers,” in SPE
+Improved Oil Recovery Conference (OnePetro, 2020).
+113C. Xie, K. Xu, K. Mohanty, M. Wang, and M. T. Balhoff, “Nonwetting
+droplet oscillation and displacement by viscoelastic fluids,” Physical Re-
+view Fluids 5, 063301 (2020).
+114A. I. Mohamed, M. Khishvand, and M. Piri, “The role of injection fluid
+elasticity in microscopic displacement efficiency of residual non-wetting
+phase: An in-situ experimental investigation,” Fuel 333, 126180 (2023).
+115C. Xie, P. Qi, K. Xu, J. Xu, and M. T. Balhoff, “Oscillative trapping of
+a droplet in a converging channel induced by elastic instability,” Physical
+Review Letters 128, 054502 (2022).
+116P. Qi, D. H. Ehrenfried, H. Koh, and M. T. Balhoff, “Reduction of residual
+oil saturation in sandstone cores by use of viscoelastic polymers,” SPE
+Journal 22, 447–458 (2017).
+117M. Irfan, K. D. Stephen, and C. P. Lenn, “An experimental study to inves-
+tigate novel physical mechanisms that enhance viscoelastic polymer flood-
+ing and further increase desaturation of residual oil saturation,” Upstream
+Oil and Gas Technology 6, 100026 (2021).
+118S. Parsa, E. Santanach-Carreras, L. Xiao, and D. A. Weitz, “Origin of
+anomalous polymer-induced fluid displacement in porous media,” Physical
+Review Fluids 5, 022001 (2020).
+119P. Druetta and F. Picchioni, “Influence of physical and rheological prop-
+erties of sweeping fluids on the residual oil saturation at the micro-and
+macroscale,” Journal of Non-Newtonian Fluid Mechanics 286, 104444
+(2020).
+120P. K. Kundu, I. M. Cohen, and D. R. Dowling, Fluid Mechanics (Aca-
+demic press, 2015).
+121D. F. James, “Boger fluids,” Annual Review of Fluid Mechanics 41, 129–
+142 (2009).
+122K. Jackson, K. Walters, and R. Williams, “A rheometrical study of boger
+fluids,” Journal of Non-Newtonian Fluid Mechanics 14, 173–188 (1984).
+123M. Verhoef, B. Van den Brule, and M. Hulsen, “On the modelling of a
+pib/pb boger fluid in extensional flow,” Journal of Non-Newtonian Fluid
+Mechanics 80, 155–182 (1999).
+124Y. L. Joo and E. S. Shaqfeh, “Observations of purely elastic instabilities
+in the taylor–dean flow of a boger fluid,” Journal of Fluid Mechanics 262,
+27–73 (1994).
+125M. Alves, F. Pinho,
+and P. J. Oliveira, “Visualizations of boger fluid
+flows in a 4: 1 square–square contraction,” AIChE journal 51, 2908–2922
+(2005).
+126E. Bot, M. Hulsen, and B. Van den Brule, “The motion of two spheres
+falling along their line of centres in a boger fluid,” Journal of Non-
+Newtonian Fluid Mechanics 79, 191–212 (1998).
+127D. F. James and C. A. Roos, “Pressure drop of a boger fluid in a converging
+channel,” Journal of Non-Newtonian Fluid Mechanics 293, 104557 (2021).
+128P. Sousa, P. Coelho, M. Oliveira, and M. Alves, “Three-dimensional flow
+of newtonian and boger fluids in square–square contractions,” Journal of
+Non-Newtonian Fluid Mechanics 160, 122–139 (2009).
+129Y. Wei, G. Seevaratnam, S. Garoff, E. Ramé, and L. Walker, “Dynamic
+wetting of boger fluids,” Journal of Colloid and Interface science 313, 274–
+280 (2007).
+
+26
+130E. Mitsoulis, “Extrudate swell of boger fluids,” Journal of Non-Newtonian
+Fluid Mechanics 165, 812–824 (2010).
+131S. L. Anna, “Non-newtonian fluids in microfluidics,” in Encyclopedia of
+Microfluidics and Nanofluidics, edited by D. Li (Springer US, Boston, MA,
+2008) pp. 1480–1488.
+132L. Mei and S. Qian, “Editorial for the special issue on micromachines for
+non-newtonian microfluidics,” (2022).
+133P. Nghe, E. Terriac, M. Schneider, Z. Li, M. Cloitre, B. Abecassis, and
+P. Tabeling, “Microfluidics and complex fluids,” Lab on a Chip 11, 788–
+794 (2011).
+134S. J. Haward, J. A. Odell, M. Berry, and T. Hall, “Extensional rheology of
+human saliva,” Rheologica Acta 50, 869–879 (2011).
+135G. Juarez and P. E. Arratia, “Extensional rheology of dna suspensions in
+microfluidic devices,” Soft Matter 7, 9444–9452 (2011).
+136I. Bloomfield, I. Johnston,
+and L. Bilston, “Effects of proteins, blood
+cells and glucose on the viscosity of cerebrospinal fluid,” Pediatric Neuro-
+surgery 28, 246–251 (1998).
+137R. P. Chhabra and J. F. Richardson, Non-Newtonian Flow and Applied
+Rheology: Engineering Applications (Butterworth-Heinemann, 2011).
+138M. Everts and J. P. Meyer, “Heat transfer of developing and fully devel-
+oped flow in smooth horizontal tubes in the transitional flow regime,” In-
+ternational Journal of Heat and Mass Transfer 117, 1331–1351 (2018).
+139A. Varshney and V. Steinberg, “Elastic alfven waves in elastic turbulence,”
+Nature Communications 10, 1–7 (2019).
+140A. Varshney and V. Steinberg, “Mixing layer instability and vorticity am-
+plification in a creeping viscoelastic flow,” Physical Review Fluids 3,
+103303 (2018).
+141M. B. Khan and C. Sasmal, “Electro-elastic instability in electroosmotic
+flows of viscoelastic fluids through a model porous system,” European
+Journal of Mechanics-B/Fluids (2022).
+142S. Kandlikar, S. Garimella, D. Li, S. Colin, and M. R. King, Heat transfer
+and fluid flow in minichannels and microchannels (elsevier, 2005).
+143S. M. Ghiaasiaan, Two-phase flow, boiling, and condensation: in conven-
+tional and miniature systems (Cambridge University Press, 2007).
+144S.-M. Kim and I. Mudawar, “Review of databases and predictive meth-
+ods for heat transfer in condensing and boiling mini/micro-channel flows,”
+International Journal of Heat and Mass Transfer 77, 627–652 (2014).
+145L. Cheng, D. Mewes, and A. Luke, “Boiling phenomena with surfactants
+and polymeric additives: a state-of-the-art review,” International Journal
+of Heat and Mass Transfer 50, 2744–2771 (2007).
+146J. Wang, J. Wang, L. Feng, and T. Lin, “Fluid mixing in droplet-based
+microfluidics with a serpentine microchannel,” RSC Advances 5, 104138–
+104144 (2015).
+147H. Song, D. L. Chen, and R. F. Ismagilov, “Reactions in droplets in mi-
+crofluidic channels,” Angewandte Chemie International Edition 45, 7336–
+7356 (2006).
+148R. Chen, Z. Sun, and D. Chen, “Droplet-based microfluidics for cell en-
+capsulation and delivery,” in Microfluidics for Pharmaceutical Applica-
+tions (Elsevier, 2019) pp. 307–335.
+149J. Clausell-Tormos, D. Lieber, J.-C. Baret, A. El-Harrak, O. J. Miller,
+L. Frenz, J. Blouwolff, K. J. Humphry, S. Köster, H. Duan, et al., “Droplet-
+based microfluidic platforms for the encapsulation and screening of mam-
+malian cells and multicellular organisms,” Chemistry & Biology 15, 427–
+437 (2008).
+150L. Mazutis, J. Gilbert, W. L. Ung, D. A. Weitz, A. D. Griffiths, and J. A.
+Heyman, “Single-cell analysis and sorting using droplet-based microflu-
+idics,” Nature Protocols 8, 870–891 (2013).
+151N. Shembekar, C. Chaipan, R. Utharala, and C. A. Merten, “Droplet-based
+microfluidics in drug discovery, transcriptomics and high-throughput
+molecular genetics,” Lab on a Chip 16, 1314–1331 (2016).
+152W.-w. Liu and Y. Zhu, ““development and application of analytical de-
+tection techniques for droplet-based microfluidics”-a review,” Analytica
+Chimica Acta 1113, 66–84 (2020).
+153R. Seemann, M. Brinkmann, T. Pfohl,
+and S. Herminghaus, “Droplet
+based microfluidics,” Reports on Progress in Physics 75, 016601 (2011).
+154Y. Xuan and Q. Li, “Heat transfer enhancement of nanofluids,” Interna-
+tional Journal of heat and fluid flow 21, 58–64 (2000).
+155S. K. Das, S. U. Choi, W. Yu, and T. Pradeep, Nanofluids: science and
+technology (John Wiley & Sons, 2007).
+156A. J. Chamkha, M. Molana, A. Rahnama,
+and F. Ghadami, “On the
+nanofluids applications in microchannels: a comprehensive review,” Pow-
+der Technology 332, 287–322 (2018).
+157H. Mohammed, G. Bhaskaran, N. Shuaib, and R. Saidur, “Heat trans-
+fer and fluid flow characteristics in microchannels heat exchanger using
+nanofluids: a review,” Renewable and Sustainable Energy Reviews 15,
+1502–1512 (2011).
+158K. N. Ramesh, T. K. Sharma, and G. Rao, “Latest advancements in heat
+transfer enhancement in the micro-channel heat sinks: a review,” Archives
+of Computational Methods in Engineering 28, 3135–3165 (2021).
+159J.-C. Yang, F.-C. Li, W.-W. Zhou, Y.-R. He,
+and B.-C. Jiang, “Ex-
+perimental investigation on the thermal conductivity and shear viscosity
+of viscoelastic-fluid-based nanofluids,” International Journal of Heat and
+Mass Transfer 55, 3160–3166 (2012).
+160J.-C. Yang, F.-C. Li, Y.-R. He, Y.-M. Huang, and B.-C. Jiang, “Experi-
+mental study on the characteristics of heat transfer and flow resistance in
+turbulent pipe flows of viscoelastic-fluid-based cu nanofluid,” International
+Journal of Heat and Mass Transfer 62, 303–313 (2013).
+161T. Hayat, F. Haider, T. Muhammad,
+and A. Alsaedi, “On darcy-
+forchheimer flow of viscoelastic nanofluids: A comparative study,” Journal
+of Molecular Liquids 233, 278–287 (2017).
+162J. Yang, F. Li, H. Xu, Y. He, Y. Huang, and B. Jiang, “Heat transfer perfor-
+mance of viscoelastic-fluid-based nanofluid pipe flow at entrance region,”
+Experimental Heat Transfer 28, 125–138 (2015).
+163P. M. Adler and H. Brenner, “Multiphase flow in porous media,” Annual
+Review of Fluid Mechanics 20, 35–59 (1988).
+164A. Gerami, R. T. Armstrong, B. Johnston, M. E. Warkiani, N. Mosavat,
+and P. Mostaghimi, “Coal-on-a-chip: visualizing flow in coal fractures,”
+Energy & Fuels 31, 10393–10403 (2017).
+165K. He, L. Xu, Y. Gao, X. Yin, and K. B. Neeves, “Evaluation of surfactant
+performance in fracturing fluids for enhanced well productivity in uncon-
+ventional reservoirs using rock-on-a-chip approach,” Journal of Petroleum
+Science and Engineering 135, 531–541 (2015).
+166S. Torabzadey, “The effect of temperature and interfacial tension on wa-
+ter/oil relative permeabilities of consolidated sands,” in SPE Enhanced Oil
+Recovery Symposium (OnePetro, 1984).
+167Y. Qin, Y. Wu, P. Liu, F. Zhao, and Z. Yuan, “Experimental studies on
+effects of temperature on oil and water relative permeability in heavy-oil
+reservoirs,” Scientific Reports 8, 1–9 (2018).
+168F. Afolabi, S. M. Mahmood, N. Yekeen, S. Akbari, and H. Sharifigal-
+iuk, “Polymeric surfactants for enhanced oil recovery: A review of re-
+cent progress,” Journal of Petroleum Science and Engineering 208, 109358
+(2022).
+169P. Raffa, A. A. Broekhuis, and F. Picchioni, “Polymeric surfactants for
+enhanced oil recovery: A review,” Journal of Petroleum Science and Engi-
+neering 145, 723–733 (2016).
+170F. Souas, A. Safri, and A. Benmounah, “On the rheological behavior of
+light crude oil: a review,” Petroleum Science and Technology 38, 849–857
+(2020).
+171T. S. T. Ariffin, E. Yahya, and H. Husin, “The rheology of light crude oil
+and water-in-oil-emulsion,” Procedia Engineering 148, 1149–1155 (2016).
+172S. W. Hasan, M. T. Ghannam, and N. Esmail, “Heavy crude oil viscosity
+reduction and rheology for pipeline transportation,” Fuel 89, 1095–1100
+(2010).
+173S. Livescu, “Mathematical modeling of thixotropic drilling mud and crude
+oil flow in wells and pipelines—a review,” Journal of Petroleum Science
+and Engineering 98, 174–184 (2012).
+174H. Liu, Y. Lu, and J. Zhang, “A comprehensive investigation of the vis-
+coelasticity and time-dependent yielding transition of waxy crude oils,”
+Journal of Rheology 62, 527–541 (2018).
+175L. Guo, X. Xu, Y. Lei, L. Wang, P. Yu,
+and Q. Xu, “Study on the
+viscoelastic-thixotropic characteristics of waxy crude oil based on stress
+loading,” Journal of Petroleum Science and Engineering 208, 109159
+(2022).
+176F. Pimenta and M. Alves, “Electro-elastic instabilities in cross-shaped
+microchannels,” Journal of Non-Newtonian Fluid Mechanics 259, 61–77
+(2018).
+177S. H. Sadek, F. T. Pinho, and M. A. Alves, “Electro-elastic flow insta-
+bilities of viscoelastic fluids in contraction/expansion micro-geometries,”
+Journal of Non-Newtonian Fluid Mechanics 283, 104293 (2020).
+178J. H. Masliyah and S. Bhattacharjee, Electrokinetic and Colloid Transport
+
+27
+Phenomena (John Wiley & Sons, 2006).
+179G. Wang, D. Niu, F. Xie, Y. Wang, X. Zhao, and G. Ding, “Experimen-
+tal and numerical investigation of a microchannel heat sink (mchs) with
+micro-scale ribs and grooves for chip cooling,” Applied Thermal Engi-
+neering 85, 61–70 (2015).
+180K. Vafai and L. Zhu, “Analysis of two-layered micro-channel heat sink
+concept in electronic cooling,” International Journal of Heat and Mass
+Transfer 42, 2287–2297 (1999).
+181A. J. Demello, “Control and detection of chemical reactions in microfluidic
+systems,” Nature 442, 394–402 (2006).
+182Y. Alihosseini, M. Z. Targhi, M. M. Heyhat, and N. Ghorbani, “Effect of
+a micro heat sink geometric design on thermo-hydraulic performance: A
+review,” Applied Thermal Engineering 170, 114974 (2020).
+183M. Doi, S. F. Edwards, and S. F. Edwards, The theory of polymer dynam-
+ics, Vol. 73 (oxford university press, 1988).
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf,len=1917
+page_content='Applications of elastic instability and elastic turbulence: Review,  limitations, and future directions  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sasmal*1  Soft Matter Engineering and Microfluidics Lab, Department of Chemical Engineering, Indian Institute of Technology Ropar,  Punjab, India-140001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  (*Electronic mail: csasmal@iitrpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='in)  (Dated: 9 January 2023)  Viscoelastic fluids are a subclass of complex fluids used in widespread applications ranging from biological to large-  scale industrial settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' These fluids are often associated with various complex flow phenomena due to the presence  of non-linear elastic stresses, originating due to the stretching and relaxation phenomena of microstructure (such as  polymer molecules in the case of a viscoelastic polymer solution) in a deformed flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' One such phenomenon is  elastic instability (EI) which emerges due to the interaction between elastic stresses and streamline curvature present  in a flow system at small values of the Renolds number (ratio of the inertial to that of the viscous forces) when the  Weissenberg number (ratio of the microstructure relaxation time to that of the rate of flow deformation) exceeds a  critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' On further increasing this Weissenberg number to higher values, the unstable flow field caused due  to this elastic instability transits to a more chaotic and turbulent-like flow structure called elastic turbulence (ET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The  fluctuating hydrodynamics arising due to this ET flow exhibit many statistical resembles seen for the regular Newtonian  turbulence occurring at large values of the Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Over the last two decades or so, an extensive investigation  has been performed on this particular topic in the complex fluids research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Some excellent articles recently  present this ET phenomenon’s development, understanding, and progress in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This article focuses on the  application perspectives of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In particular, this article aims to provide a comprehensive review of the  investigations conducted so far in the literature to demonstrate the potential of these EI and ET phenomena in three main  application areas, namely, microfluidic mixing, microscale heat transfer, and chemically enhanced oil recovery (EOR)  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Additionally, a detailed discussion of the limitations and future directions of these EI and ET phenomena from  an application point of view is also presented in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  INTRODUCTION  Adding a minute amount of solid polymers or surfactants,  even in parts per million (ppm) amount, into a simple fluid  like water dramatically changes the flow behaviour of the re-  sulting solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This is due to the induction of fluid elasticity  within the solution, resulting from the molecular conforma-  tion changes of the polymer or surfactant molecules due to  their stretching and relaxation phenomena in a deformed flow  field1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It could also be induced naturally in a solution due  to the presence of microstructure that stretches and relaxes  in a deformed flow field, for instance, blood3,4, emulsions5,6,  foams7,8, particle suspensions9,10, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The extent of this fluid  elasticity in a solution is generally expressed in terms of a non-  dimensional time, named Weissenberg number (Wi), defined  as the product of the microstructure relaxation time (λ) and  the flow deformation rate ( ˙γ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=', Wi = λ ˙γ11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The mecha-  nisms through which this fluid elasticity could influence the  coherent flow structures (and the associated transport phe-  nomena such as the transfers of momentum, heat, and mass) of  a flow field also depend on the Reynolds number (Re), defined  as the ratio of the inertial (∼ O(ρU2  ch)) to that of the viscous  forces (∼ O(η Uch  Lch )), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=', Re = LchUchρ  η  where Lch is the char-  acteristic length scale, Uch is the characteristic velocity scale,  ρ and η are the fluid density and viscosity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The  effect of fluid elasticity on the flow dynamics at sufficiently  high values of the Reynolds number (where the flow field is  in the turbulent regime) has been studied extensively over the  past several decades, which is popularly known as the elasto-  inertial turbulent (EIT) flow12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This particular EIT flow has  gained extensive attention from researchers due to its associ-  ation with the turbulent drag reduction (DR) phenomenon13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  It was first reported by Tom in 1948 when he found a signif-  icant reduction in the friction drag in a high Reynolds num-  ber turbulent pipe flow of monochlorobenzene solvent due to  the addition of a minute amount of poly(methyl methacrylate)  (PMMA) solid polymers into it14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Subsequently, many other  experimental studies were conducted with different combina-  tions of polymers and solvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Also, they observed the same  Tom phenomenon (named after Tom) with the reduction in the  drag to various extents15–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' With the advancement of com-  putational techniques and hardware, later on, extensive direct  numerical simulations (DNS) were also performed to investi-  gate the underlying physics behind this phenomenon in more  detail18–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Many sophisticated experimental techniques, such  as Schlieren photographs or particle image velocimetry (PIV),  were also employed to investigate this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' All these  experimental and numerical studies provided detailed infor-  mation on the flow fields and polymer conformations at dif-  ferent lengths and time scales, which facilitated a deeper and  better understanding of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Some excellent re-  view articles are present on the development and progress of  the investigations and understandings of this DR phenomenon  in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, even after almost 75 years of its  findings, the DR phenomenon due to polymer additives ( and  so the EIT flow ) still attracts a lot of attention in the complex  fluids research community12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  On the other hand, when the Reynolds number becomes  vanishingly small, the effect of the fluid elasticity on the flow  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='02395v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='flu-dyn]  6 Jan 2023  2  dynamics gives rise to a new flow regime called elastic turbu-  lence (ET)21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A recent numerical study has shown that there  is a continuous pathway (a single linearly unstable modal  branch) present that connects these ET and EIT flow states22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Before the transition of this ET regime, elastic instability (EI)  first originates within the system, resulting from the interac-  tion between the streamline curvature and the normal elas-  tic stresses present within the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, elastic in-  stability is the precursor of this elastic turbulent flow state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  McKinley and co-workers23,24 proposed criteria for curvilin-  ear geometries for the onset of this purely elastic instability,  as written below  M =  �  τ11  η ˙γ  λU  R ≥ Mcrit  (1)  where τ11 is the normal elastic stress in the flow direction  along a curved streamline, ˙γ is the characteristic value of the  local flow deformation rate, R is the characteristic radius of  the streamline curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' When this M parameter exceeds a  critical value, Mcrit, a purely elastic instability is originated in  a flow system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' McKinley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='24 presented a detailed investi-  gation wherein they calculated the values of this Mcrit param-  eter for various geometries such as Taylor-Couette, lid-driven  cavity, flow past a confined cylinder, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, they  also showed how this critical M parameter should be modified  to account for the solvent viscosity ratio (defined as the ratio  of the solvent to that of the zero-shear viscosity of the poly-  mer solution) and the shear-thinning behaviour of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A  very good agreement was found between this scaling predic-  tion and the corresponding experimental observations for the  onset of elastic instability, for instance, in a lid-driven cav-  ity25,26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  The EI phenomenon in a flow system creates an unstable  flow field, which transits to a further chaotic and turbulent-like  flow state or the ET state as the Weissenberg number further  increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The term "elastic turbulence" was first coined in  1965 by Vinogradov and Manin27 when they studied the flow  of polymer melts through a micro-scale contraction/expansion  geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, Groisman and Steinberg28 were the ones  who investigated this ET flow thoroughly for a system com-  prising a cylindrical cell with a rotating circular plate filled  with polyacrylamide polymer solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' For the first time,  they observed that elastic turbulence could generate a fluc-  tuating flow field with a broad range of spatial and temporal  scales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' likewise, it is seen in regular hydrodynamic Newto-  nian turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In particular, they showed that the turbulence  generated due to these polymer additives at a very low value  of the Reynolds number would be comparable to the corre-  sponding Newtonian turbulence generated in a pipe flow at  a Reynolds number as high as 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Subsequently, Steinberg  and co-workers carried out further experimental studies on  this ET flow in other different geometries such as curvilin-  ear channel29,30, rotating disks31, flow past an obstacle32,33,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Further extensive studies on the statistical analysis of the  velocity and pressure fluctuations and the establishment of the  scaling relations of the exponents of the power-law decay of  elastic energy and pressure fluctuations in the ET regime were  also conducted by the same research group34,35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Many ex-  perimental studies on the ET flow in a microchannel having  single or multiple pore constrictions have also been presented  in the literature36,37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This particular geometry is considered  one of the simplified model porous systems for understanding  the micro-scale flow dynamics in an actual porous media38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Studies on a more complicated model porous system, such as  a microchannel having many cylindrical pillars present in it,  have also been recently carried out in the elastic turbulent flow  regime39–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A comprehensive review on the flows of com-  plex viscoelastic fluids in a porous media has recently pre-  sented by Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The corresponding numerical evi-  dence on the existence of elastic turbulence is also presented  in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, this number is relatively less in  comparison to that of experimental ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This is mainly be-  cause of the existence of the well-known "High Wessenberg  Number Problem (HWNP)" that occurs for viscoelastic fluid  simulations, particularly for simulations of constant shear vis-  cosity viscoelastic fluids46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This problem leads to the loss  of numerical stability at sufficiently high Weissenberg num-  bers where the ET flow is expected to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Despite that,  some studies have been carried out by employing various nu-  merical stabilization techniques such as the log-conformation  tensor approach47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Among a few studies, for instance, Berti  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='48 performed an investigation in a two-dimensional pe-  riodic Kolmogorov flow utilizing the Oldroyd-B viscoelastic  constitutive model and found a disordered and turbulent-like  flow state with increased drag and Lyapunov exponent once  the Weissenberg number exceeded a critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Their en-  ergy spectrum of the velocity fluctuations showed a power-  law decay with an exponent value close to that obtained both  in the experiments and theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ardekani and  co-workers performed full-scale two-dimensional numerical  simulations for various geometries such as microchannel with  pore constrictions49 and flow past obstacles50 to delineate the  mechanisms for originating the elastic instability and turbu-  lence in such micro-geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sasmal and co-workers also  conducted extensive numerical simulations to investigate the  EI and ET phenomena in the flows of viscoelastic micellar so-  lutions51–53 as well as polymer solutions in complex porous  media54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A comprehensive review of the investigations and  understandings of the elastic instability and elastic turbulence  phenomena has been presented recently in some excellent re-  view articles21,55,56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, the readers are requested to  go through them to gain further insights and to be aware of  the latest development and new directions in the EI and ET  phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The main aim of this article is to present the ap-  plication perspectives of these two phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Therefore, it is readily evident that the EI and ET phenom-  ena originate in a chaotic and turbulent-like flow state, like-  wise the regular Newtonian turbulence, even at negligible val-  ues of the Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This low Reynolds number cri-  terion of these two phenomena naturally leads to their appli-  cations in microfluidic geometries where a laminar flow con-  dition persists due to small-scale dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' As a result, the  rate of any transport process, such as heat transfer or mixing in  these micro-scale geometries, is dominated mainly by molec-  ular diffusion, which in turn necessitates a much longer time  3  to carry out these process57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Over the years, a voluminous  amount of studies, in terms of theory, simulations and exper-  iments, have been carried out in the development of various  methods to increase the rate of transport process in these mi-  crofluidic geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' These methods are broadly classified  into two categories, namely, active and passive methods58–60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  In active methods, the micro-scale geometries are fabricated  in such a way that a secondary flow pattern could be generated  inside the flow system (for instance, the Dean flow61), thereby  facilitating a greater rate of either the heat transfer62,63 or the  mixing process64,65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' On the other hand, in passive methods,  the rate of transport processes is increased with the help of ex-  ternal driving forces such as mechanical rotation, electric and  magnetic fields, or an acoustic field66–68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' All these externally  applied fields would help generate chaotic convection inside  a microfluidic geometry, thereby increasing the rate of trans-  port processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In this perspective, the EI and ET phenomena  have a considerable potential in microfluidic applications for  enhancing the rate of various transport processes, such as heat  transfer or mixing, as they can inherently generate chaotic and  turbulent-like flow structures inside a microfluidic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  This has already been demonstrated by a large number of ex-  perimental as well as numerical studies carried out for various  micro-scale geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Apart from heat transfer and mixing  applications in micro-scale geometries, the EI and ET phe-  nomena can also significantly influence the polymer flooding  used in the enhanced oil recovery (EOR) process69,70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In this  process, a polymer solution is injected into an oil reservoir to  displace the oil present in the reservoir’s porous rock struc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This rock is made of billions of interconnected tortu-  ous micropores through which polymer solution and oil flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Therefore, there is a high possibility of inducing these EI  and ET phenomena inside a porous matrix during the poly-  mer flooding, which may significantly influence the oil dis-  placement efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It requires a well understanding of these  two phenomena during the flows of viscoelastic fluids through  a micro-scale porous geometry to increase the effectiveness  of the polymer flooding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  In particular, this article  presents a comprehensive review of these potential applica-  tions utilizing the EI and ET phenomena demonstrated so far  in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In particular, we focus on three application  domains where these two phenomena could contribute signif-  icantly: microfluidic mixing, micro-scale heat transfer, and  chemically enhanced oil recovery (EOR) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In addition,  a detailed discussion of the limitations and future directions  is also presented, which would help in the development and  progress of these two phenomena from an application point of  view in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  APPLICATIONS IN MICROFLUIDIC MIXING  Mixing of fluids in microscale geometries was the first  potential application of the EI and ET phenomena that was  demonstrated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In doing so, Groisman and  Steinberg29 conducted an experimental study in a curvilinear  microchannel wherein they showed the mixing efficiency of  two fluids entering the microchannel through two inlets, as  schematically shown in sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In their experiments,  they used a pure solvent consisting of 65% saccharose and  1% NaCl in water as simple Newtonian fluid and a viscoelas-  tic polymer solution by adding 80 ppm polyacrylamide in the  solvent to show the difference in the mixing behaviour arising  due to the elastic turbulence phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A fluorescent dye  was mixed into the fluid entering through one of the inlets, and  the image was taken using a charge-coupled device (CCD) in  the presence of fluorescent light at a position downstream of  the inlet, as marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The surface plots of the dye  concentration in pure solvent and polymer solutions are de-  picted in sub-Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 1(b) and (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It can be seen  that the two fluids (with finite and zero fluorescent dye con-  centrations) were flowing side by side, and they did not mix  with each other in the case of the pure solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' On the other  hand, efficient mixing of the two fluids occurred in the case of  the polymer solution under the same operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It  was possible due to the presence of the EI and ET phenomena  in the latter solution, which increased the chaotic convection  inside the microchannel and hence the mixing efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It  was confirmed by Groisman and Steinberg29 through the sta-  tistical analysis of the temporal fluctuations of the velocity and  dye concentration at a point inside the microchannel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They  observed several statistical characteristics which ensured the  existence of elastic turbulence in the flow system, such as a  power-law decay with an exponent value of around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='3 in the  power spectrum of the velocity fluctuations and exponential  tails of the probability distributions of the dye concentration  profile fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In another subsequent study, Groisman  and Steinberg71 demonstrated the mixing capability of the ET  phenomenon in a Couette-Taylor (CT) geometry consisting of  a cylindrical cell with a flat bottom and a concentric rotat-  ing upper plate inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Figure 2 depicts the mixing patterns  of an ink drop placed at the center of the CT cell filled with  either viscoelastic polymer solutions (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 2(a)) or with  a Newtonian solvent (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It can be seen that the  ink started to spread over the surface after a certain time by  the toroidal vortex resulting from the ET phenomenon when  it was placed in a viscoelastic polymer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' As time pro-  gressed further, those toroidal structures were split into more  fine structures either due to the excitation of the fluid mo-  tion at small spatial scales or due to a significant stretching of  the fluid elements along their Lagrangian trajectories by ran-  domly fluctuating large-scale eddies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, the con-  trast in the fine structures gradually decreased, suggesting the  progress of mixing with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Finally, at time t = 8 min, the  dye concentration became completely homogeneous in the so-  lution, and a complete mixing was achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This time was  almost four orders of magnitude smaller than the time re-  quired for the mixing due to molecular diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hence, it  was concluded that this mixing was definitely achieved due  to the chaotic and random flow structures resulting from the  elastic turbulence phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' On the other hand, in Newto-  nian solvent (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 2), the ink concentration was inhomo-  geneous even after 9 hr of the experiments due to the absence  of inertial effects and chaotic convection under the same op-  erating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='73 carried out an experimental  study in a similar kind of geometry and showed that the emul-  4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Experimental study of the mixing performance in a curvilinear microchannel utilizing the EI and ET phenomena29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' (a) The experi-  mental setup consisted of a curvilinear microchannel with two inlets and smoothly connected sixty half-rings (numbered 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='., 30) with outer  and inner radii of R1 = 6mm and R2 = 3mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The depth of the channel was 3mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fluids with a fixed concentration of fluorescent  dye (lower inlet) and zero concentration (upper inlet) entered the microchannel through the two inlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Dye concentration profiles in (b) pure  solvent and (c) viscoelastic polymer solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Here the bright white color corresponds to a finite dye concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  sification rate of two immiscible liquids was substantially in-  creased due to a greater mixing caused by the ET phenomenon  when the dispersed medium was viscoelastic Boger fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In  contrast, they remained separated when the dispersed medium  was Newtonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  A study in the same curvilinear microchannel geometry as  used by Groismann and Steinberg29 (however, it was 30 times  smaller than used by Groismann and Steinberg29) was also  carried out by Burghelea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This study also found a  chaotic mixing of two fluids when a minute amount of poly-  mers were added to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They also reported that the mix-  ing time in the chaotic elastic turbulence regime was three  to four orders of magnitude shorter than due to the molecu-  lar diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, the mixing time was found to be  almost independent of the diffusion coefficient of the macro-  molecules that were present in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, they  suggested that this ET phenomenon could efficiently mix flu-  ids with additives of low diffusivities, such as large DNA  molecules, viruses, particles, living cells, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='72 also  performed an experimental study to demonstrate the poten-  tial of elastic turbulence phenomenon in enhancing the mix-  ing phenomenon in a curvilinear microchannel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They used  glycerol water and CTAC/NaSal (cetyltrimethyl ammonium  chloride/sodium salicylate) surfactant solutions in their exper-  iments as Newtonian and viscoelastic fluids, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fig-  ure 3 shows the details of the geometry used in their study  and the corresponding surface plot of the dye distribution pro-  files both in glycerol water and surfactant solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The ge-  ometry had two inlets through which fluids entered into the  microchannel, and the fluid entering through the upper in-  let was colored with a fluorescent dye, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  An almost uniform distribution of dye, particularly down-  stream of the microchannel, was seen at any cross-sectional  area of the microchannel in the case of flows of surfactant  solutions (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 3(c)), resulting from the elastic instability-  induced chaotic mixing of two fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In contrast, a highly  non-uniform distribution of dye concentration profile was ob-  served due to the absence of span-wise flow fluctuations when  glycerol water was used as the working fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Tatsumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='75  did an investigation in a serpentine microchannel with New-  tonian sucrose and viscoelastic polyacrylamide (dissolved in  sucrose) polymer solutions and also found an enhancement in  the mixing phenomenon in the latter solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  The corresponding three-dimensional numerical simula-  Field of view  Lasersheet5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mixing patterns of an ink drop placed at the center of a Couette-Taylor (CT) cell utilizing the ET phenomenon at different times71 in  viscoelastic polymer solutions (a) and a Newtonian solvent (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  tions for showing the mixing performance of both Newtonian  and viscoelastic fluids in a curvilinear microchannel were per-  formed by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They used the Giesekus model to realize  the fluid viscoelasticity and covered a wide range of the Weis-  senberg number at a fixed value of the polymer viscosity ratio  (ratio of the solvent viscosity to that of the zero shear-rate  viscosity of the polymer solution) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Likewise the ex-  periments, they also found a significant enhancement in the  mixing performance in the case of viscoelastic fluids com-  pared to the case of Newtonian fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Apart from the mixing  phenomenon, they also discussed the flow dynamics results  in detail in terms of evaluating the microstructure extension,  secondary flow structures, root mean square velocity fluctu-  ations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This enhancement in the mixing phenomenon of  viscoelastic fluids was even observed in the flow through a  straight microchannel, which was evident in another subse-  quent numerical study from the same research group76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fig-  ure 4 displays the dye concentration profile and the mixing in-  dex (defined in the caption of the figure) inside the microchan-  nel both for Newtonian and viscoelastic fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In the case of  Newtonian fluids (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 4(a)), the dyed and undyed fluids  moved side by side without mixing in the span-wise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  In contrast, a chaotic mixing of the two fluids was evident in  the whole domain in the case of viscoelastic fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It was  demonstrated more quantitatively in sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 4(c), wherein  the temporal variation of the mixing index was presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' For  Newtonian fluids, the value of the mixing index remained con-  stant at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5, and there was almost no mixing in these  simple fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, it gradually decreased with time for  viscoelastic fluids, thereby suggesting the presence of chaotic  mixing in these fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Grilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='78 presented a numerical  study on the mixing performance of viscoelastic fluids in a  microchannel with a periodic array of cylindrical obstacles  present in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They simulated a range of the Weissenberg num-  ber between 0 (Newtonian) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='6 using the Oldroyd-B vis-  coelastic fluid model at a constant fixed value of the polymer  viscosity ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, a stability analysis based  on the dynamic mode decomposition (DMD) technique was  utilized in their study to identify the most energetic mode re-  sponsible for the unsteady chaotic flow behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They also  observed a power-law scaling behaviour both in the flow and  dye concentration fluctuations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' likewise, it was seen in the ex-  periments in the elastic turbulence regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' As proof of the  presence of ET flow and to show its potential in microfluidic  applications such as in the mixing process, a numerical ex-  periment was conducted wherein they observed the mixing of  t=0  30sec  =0  4min  90sec  15min  2 min  4min  8min  2hr  (a)  (b)6  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Experimental study of mixing phenomenon of two viscoelastic fluids in a curvilinear microchannel72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The details of the geometry used  in the study are shown in (a), whereas (b) and (c) depict the dye concentration profile inside the microchannel when the working fluids were  glycerol water and viscoelastic CTAC/NaSal surfactant solutions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In (a), p1, p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='. are the probe lines where the dye concentration  was measured to obtain the mixing efficiency inside the microchannel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The flow direction is shown by the green arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Numerical study of mixing phenomenon in a straight three-dimensional microchannel76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A dye was mixed with the fluid placed  at the lower half of the channel, and its profile inside the microchannel at t∗ = 160 (where t∗ is the non-dimensional time ) is shown for  (a) Newtonian and (b) viscoelastic fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Here crms is the mixing index defined as the root mean square of the dye concentration c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=',  crms =  �  ∑N  i=1(ci −cm)2/N, where ci, cm, and N are the dye concentration at a grid point, mean dye concentration, and the total number  of grid points in the whole computational domain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A zero value of crms corresponds to perfect mixing, whereas a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  corresponds to no mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' (c) Variation of crms with the non-dimensional time both for viscoelastic and Newtonian fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The values of the  Reynolds and Weissenberg numbers were fixed at 1 and 30, respectively, in these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  a dye placed on the upper wall of the microchannel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' As the  Weissenberg number gradually increased in their simulations,  the spreading of the dye also progressively increased inside  the microchannel due to the increase in the chaotic convection  resulting from the elastic turbulence phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='79 performed an experimental study to inves-  tigate the mixing phenomenon of two fluids in a conver-  gent/divergent microfluidic geometry, as schematically shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='79(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The geometry had three inlets, namely, two side  inlets (orange arrow) and one main inlet (green arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The  Total flowrate:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='6uL/min(Re=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='l)  Width:100μum,depth:50μm  (a)  Coloredwithdye  Coloredwithdye  Iniet  Uncolored  Inlet  Uncolored  555  Outlet  Outlet  (c)  (b)0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='50  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='45  viscoelasticfluid  Newtonian fluid  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='40  0  2  4  6  8  10  Crms  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='35  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='30  (c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='25  0  2  4  6  8  10  0  30  60  90  120  150  180  t*7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Experimental study of the mixing performance in a convergent/divergent microfluidic geometry79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' (a) The geometry consisted of two  side inlets (orange arrow) and one main inlet (green arrow), wherein the flow was happening from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The fluid entering through  the main inlet was mixed with a fluorescent dye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The dye concentration profile at the upstream and downstream sections of the geometry  in (b) viscous Newtonian (glycerol water) and (c) viscoelastic fluids (polyethylene oxide (PEO) dissolved in glycerol water) under the same  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  mainstream fluid (1 wt% polyethylene oxide (PEO) dissolved  in 55 wt% glycerol water) entering the geometry had higher  viscoelastic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It was mixed with a finite concentra-  tion of fluorescent dye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In contrast, sidestream fluids having  less viscoelastic properties (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='1 wt% PEO dissolved in wa-  ter) or only viscous properties (glycerol water) entered into  the geometry without any dye in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The dye concentration  profiles inside the geometry both for viscous Newtonian and  viscoelastic polymer solutions are depicted in sub-Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 5(b)  and (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It can be seen that the dye spread very  little when the fluids were viscous Newtonian both upstream  and downstream sections of the geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' however, it became  prominent in the case of viscoelastic polymer solutions, par-  ticularly at the downstream section of the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' There-  fore, it suggested a greater mixing of the fluids in the latter  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In particular, they observed a mixing efficiency (whose  values of 0 and 100 correspond to no mixing and perfect mix-  ing conditions) of as high as 68% downstream of the geom-  etry due to the emergence of viscoelastic instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In an-  other study80, they demonstrated the use of this viscoelastic  instability for efficient mixing of fluids in an abruptly con-  tracted microchip (8:1 contraction ratio) made out of poly-  methyl methacrylate (PMMA) polymer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' According to them,  the device was cheap, disposable, and straightforward to fab-  ricate yet effective for mixing over a short length with a rela-  tively high flow rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='81 recently carried out an experimental inves-  tigation to demonstrate the use of EI and ET phenomena in  the enhancement of fluids mixing in a novel gear-shaped mi-  crochannel setup, as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 6(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  In particular, they made this setup by combining two geome-  tries (which were already proven effective in inducing elastic  instability and turbulence in a system), namely, a serpentine  microchannel and a microchannel with step expansion and  contraction or with a side-well structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They found that the  gear-shaped microchannel provides a more effective mixing  of the two fluids than the serpentine microchannel under iden-  tical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 6(c)-(d), wherein both  the surface plot of dye concentration (left) and its fluorescent  intensity along a plane (right) are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' From both plots,  it can be seen that the dye was more uniformly distributed  in the case of a gear-shaped microchannel than in a serpen-  tine microchannel, whereas the pressure drop was almost the  same in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It led to an increased mixing efficiency  of the two fluids in the former microchannel with a smaller  mixing length than needed for the latter one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Their study fur-  ther demonstrated that this enhancement in the mixing of vis-  coelastic fluids ultimately facilitated the production of silica  nanoparticles with more uniform size and shape distributions  than that obtained with Newtonian fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='82 per-  formed an experimental study on a serpentine microchannel  and showed that the orthogonal injection of the fluid into the  primary flow in the microchannel enhanced the mixing effi-  ciency in a short mixing length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  All the above-mentioned studies have used pressure-driven  flows for originating the elastic instability and, subsequently,  the elastic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, very few corresponding stud-  ies are present for the electrokinetic-driven flows to induce  these EI and ET phenomena inside a micro-scale system and  their influence on the mixing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Among a few studies,  Bryce and Freeman83 performed an experimental study us-  ing a long microchannel with many constrictions (2:1 ratio)  present in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They showed that the mixing efficiency was de-  creased in viscoelastic fluids than in Newtonian fluids under  the same operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This starkly contrasts with that  observed in pressure-driven flows, where two to three orders  of magnitude enhancement was observed despite the develop-  ment of large-scale elastic instabilities inside the system in the  EK-driven flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' According to them, it was attributed to the  absence of shear-driven or any other deformations in this par-  ticular geometry wherein mostly extensional-dominated elas-  tic instabilities were present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, a very recent numeri-  cal study by Khan and Sasmal84 found an increase in the mix-  ing efficiency in the same kind of geometry as that used by  Downstream  Upstream  (a)  0001  1000  1000  (b)  25  C  All dimensions in  Jim  3000  Channel depth 1508  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' (a) Schematic of the gear-shaped microchannel used in the study of Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 81 for efficient mixing of fluids, which is a combination  of serpentine and expansion and contraction (or side-well structure) microchannels (b) Schematic of the flow arrangement wherein dyed and  dye-free solutions were simultaneously injected through two inlets with the same flow rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The mixing zone consisted of 22 connected half  rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The mixing pattern inside the system at two flow rates for only serpentine (c and d) and gear-shaped (e and f) microchannels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Here the  right-hand sub-figures represent the fluorescent intensity profiles (at a plane marked by the green arrow in (c)) normalized by the maximum  intensity after background subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Bryce and Freeman83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They used the Oldroyd-B viscoelas-  tic constitutive equation to mimic the rheological behaviour  of a Boger fluid and the Poisson–Boltzmann (PB) equation to  calculate the ion distribution in the system for a wide range  of electric field strength and viscosity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Once the Weis-  senberg number exceeded a critical value, an electro-elastic  instability was developed, resulting in a chaotic and fluctu-  ating convective flow field inside the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This eventually  led to the mixing of two fluids present in two halves of the sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 7(i)-(ii) wherein the fluid present in the  upper half of the channel was mixed with a dye, and the lower  half was dye-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' When the fluids were in Newtonian nature  (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 7i(a)), they moved side by side without any mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  However, as fluid viscoelasticity was introduced into the flu-  ids, the mixing of the two fluids started (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 7i(b)), which  was further incremented as the Weissenberg number further  increased (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 7i(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It is further evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 7(ii),  where the variation of the mixing efficiency parameter η (val-  ues of 0 and 100 corresponded to no-mixing and perfectly  mixing conditions) with the Weissenberg number was pre-  sented, and a drastic increase in its value was observed once  the Weissenberg number exceeded a critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This dif-  ference in the observations between the studies of Bryce and  Freeman83 and Khan and Sasmal84 may be attributed to the  difference in the rheological behaviour of the working fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Khan and Sasmal84 used a perfectly Boger fluid in their sim-  ulations, whereas Bryce and Freeman83 did not provide the  rheological details of their working fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In another study by  Sasmal85, it has been shown that the mixing of two viscoelas-  tic fluids could even be achieved in a straight microchannel  utilizing the electro-elastic instability and turbulence phenom-  ena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' These were locally generated inside the system by plac-  ing patches of constant wall zeta potential on both the top  and bottom walls of the microchannel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Figure 7(iii) shows  the distribution of dye concentration inside the microchannel  (the upper half was filled with a dyed fluid, whereas the lower  half was filled with the same fluid with no dye) at different  values of the Weissenberg number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The fluids traveled side by  side without cross-convective mixing in the cases of Newto-  nian (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 7iii(a)) and viscoelastic fluids with low values  of the Weissenberg number (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 7iii(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' As the Weis-  senberg number increased to higher values, mixing between  the two fluids was observed, which was seen both in the dye  distribution presented in sub-Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 7iii(c) and (d) at two differ-  ent values of the Weissenberg number and in the variation of  the mixing efficiency with the Weissenberg number depicted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 7(iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, this study proposed a very simple  and effective approach for mixing two viscoelastic fluids in a  straight microchannel under the influence of an electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  However, it should be experimentally verified so that the po-  tential of this proposed approach and the phenomena of EI and  ET can be further established for mixing fluids in micro-scale  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  APPLICATIONS IN MICROSCALE HEAT TRANSFER  The corresponding potential in applying EI and ET phe-  nomena for the microscale heat transfer process was investi-  gated much later than the mixing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Whalley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='86 was  probably the first who showed this potential in 2015, almost  24 years later than the experiment carried out for the mixing  process by Groisman and Steinberg29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They used a square ser-  pentine microchannel and Boger polymer solutions comprised  Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5ml/h  Q=12ml/h  (a)  Serpentine Microchannel  (Re=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='7, Wi=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='8)  00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='51 (e)  (Re=41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='7,Wi=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='2)  "Gear-Shaped"Microchannel  (c)  Serpentine  200μm/250μm  Serpentine  +SideWellStructure  H50μm  (b)  MixingZone  100 μm  Dye-Free  (d)  f  Solution  Outlet  Gear  Gear  Junction  Dyed  Solution  Stable  Unstable9  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Electrokinetically driven flows in a step expansion and contraction microchannel84 wherein the fluid present in the upper half was  mixed with a dye, and that present in the lower half was dye free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' (i) Dye distribution profile for Newtonian (a) and viscoelastic fluids at  two values of the Weissenberg number, namely, 6 (b) and 15 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' (ii) Variation of the mixing efficiency parameter (η) with the Weissenberg  number in the same geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mixing two viscoelastic fluids in a straight microchannel with patches of constant wall zeta potential on both  top and bottom walls of the microchannel85 in electrokinetic-driven flows utilizing the electro-elastic instability and turbulence phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  (iii) Dye distribution profile for Newtonian (a) and viscoelastic fluids with values of the Weissenberg number, namely, 1 (b), 2(c), and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' (iv)  The corresponding variation of the mixing efficiency parameter with the Weissenberg number in this geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  of high molecular-weight polyacrylamide polymers dissolved  in a Newtonian solvent consisting of 65% sucrose, 1% NaCl,  and 34% water in their investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Based on the varia-  tions of the non-dimensional heat transfer rate and/or Nus-  selt number and the non-dimensional pressure drop and/or  friction factor with the Weissenberg number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' they identified  three regimes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' regime I (Wi < 5) wherein the heat  transfer rate and pressure drop in viscoelastic polymer solu-  tions were almost the same as that obtained with a Newto-  nian solvent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' regime II (5 < Wi < 25) wherein the pressure  drop showed a significant enhancement in polymer solutions  compared to that in a Newtonian solvent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' however, the Nus-  selt number was hardly influenced, and regime III (Wi > 25)  wherein the pressure drop reached a plateau value, whereas  (a)  (b)  60  40 Steady  (c)  20  Unsteadyregime  (Electro-elastic instability)  0  5  10  15  Wi  (i)  m(i)  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='00  (a)  Nomixing  Mixing  40  (b)  20  Steady and  stableregime  [(c)  Unstableand  chaoticregime  (d)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  3  Wi  (ili)  (iv)10  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Variation of (a) the normalized friction factor ( f Re) and (b) the surface averaged Nusselt number (Nuavg) with the Weissenberg  number in a square serpentine microchannel86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Here ’Vis’ and ’New’ stand for the results of viscoelastic polymer solutions and Newtonian  solvent, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' f and Re are the friction factor and Reynolds number, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  the Nusselt number showed a dramatic augmentation in its  value, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' At the highest Weissenberg number consid-  ered in their study, an enhancement of up to 300% in the  heat transfer rate was achieved in viscoelastic polymer solu-  tions compared to that in a Newtonian solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' For the same  square serpentine microchannel geometry, Abed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='87 also  performed a further detailed study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In this study, in addition  to constant viscosity Boger viscoelastic polymer solutions (as  used by Whalley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='86 in their study), they also consid-  ered shear-thinning viscoelastic polymer solutions comprised  of the same polyacrylamide polymers but dissolved in a dif-  ferent solvent, namely, a mixture of water and glycerine to  investigate the effect of the polymer solution type on the heat  transfer rate and pressure drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They also proposed the def-  inition of a modified Weissenberg number (Wi∗) to collapse  the Nusselt number data obtained at various polymer concen-  trations onto a master plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It basically incorporates the ef-  fects of geometric dimensions, Prandtl number (Pr) and Weis-  senberg number (Wi) defined as Wi∗ = W  L PrWi, where W and  L are the depth and length of the square serpentine channel,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The physical significance of this modified num-  ber is that it is the ratio of elastic stress to thermal diffusion  stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Figure 9 depicts the variation of the normalized sur-  face averaged Nusselt number with the modified Weissenberg  number both for Boger and shear-thinning polymer solutions  along with the Newtonian limit under otherwise identical con-  ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It can be seen from this figure that one needs to span  more range of the modified Weissenberg number for shear-  thinning polymer solutions than that for Boger polymer so-  lutions for the same relative increase in the normalized sur-  face averaged Nusselt number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, it suggests that the  surface-averaged Nusselt number is a strong function of not  only the modified Weissenberg number but also the degree of  shear-thinning behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, they also observed a  substantial increase in the heat transfer rate due to the pres-  ence of EI and ET phenomena in viscoelastic polymer so-  lutions, for instance, approximately 200% and 380% at low  and high polymer concentrations, respectively, than that ob-  tained in Newtonian solvents alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='92 also conducted  an experimental study for this square serpentine microchan-  nel with viscoelastic polyacrylamide solutions (at two differ-  ent concentrations, namely, 100 and 200 ppm) and Newtonian  sucrose solutions (50 wt%) flowing into it at different Weis-  senberg and Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Once again, the heat transfer  rate was greater in viscoelastic solutions than in Newtonian  fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, it was increased with the polymer con-  centrations at any Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, detailed anal-  ysis and explanation of the results were absent in this study as  the main motivation was to show the potential of a Titanium-  Platinum (Ti-Pt) film that they developed to measure the tem-  perature for investigating heat transfer in microfluidic applica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Later, they performed another detailed investigation us-  ing this Ti-Pt film for the same square serpentine microchan-  nel geometry93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Copeland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='88 conducted an experimental investigation  on how the elastic turbulence phenomenon could influence the  convective heat transfer phenomena inside a miniature viscous  disk pump (VDP)94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It is easy to fabricate, and it has sim-  ple maintenance and good flow control capability compared  to other mechanical pumps available for transporting fluids in  various microfluidic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This pump consists of a ro-  tating disk, C shaped microchannel, and two ports, one for the  fluid inlet and the other for its outlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They performed both  the heat transfer and mixing experiments on this microdevice  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  80ppm PA in sucrosesolution  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0  120 ppmAA in sucrose solution  00  f Re (Vis) / fRe (New)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  Regime III  C  Regime II  Regime II  8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0  RegimeI  Regimei  Regime II  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  00  00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0  lewt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='limit  Newt limit  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  [a)  OS0 ppmPAA insuxrose solution  (b)  O120 ppmPAA in sucrose solution  0  0  20  40  60  80  100  0  20  40  60  80  100  Wi  Wi11  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Variation of the surface averaged normalized Nusselt num-  ber with the modified Weissenberg number in a square serpentine  microchannel87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Here the results presented as blue symbols are for  constant viscosity Boger fluids comprised of polyacrylamide poly-  mers dissolved in a mixture of water (W) and sucrose (SUC) sol-  vents at different concentrations (in ppm), whereas red symbols are  for shear-thinning polymer solutions obtained by dissolving the same  polymers in a mixture of water and glycerine (GLY) solvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Variation of the normalized average Nusselt number with  the modified Reynolds number (ReETC)88 in a microfluidic viscous  disk pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The definition of ReETC is provided in the texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Here  Nu and Nu0 are the average Nusselt numbers obtained in viscoelastic  polymer solutions and Newtonian solvent, respectively, and ˙γ is the  shear rate originating due to the rotation of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  over a wide range of shear rates originating due to the rotation  of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Figure 10 shows the variation of the normalized  Nusselt number with the modified Reynolds number defined  as ReETC = ˙γ ρ  µ L2  c  �  ρc  ρco  �m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This definition of the Reynolds  number included the effects of the local shear rate ( ˙γ), stream-  wise development length scale (Lc), flow static density (ρco),  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Variation of the normalized average Nusselt number  (Nu/Nus) with the Weissenberg number (Wi)89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Here ’HPAM’  stands for hydrolyzed polyacrylamide polymer solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Variation of the average Nusselt number with the Weis-  senberg number in viscoelastic polymer solutions comprised of hy-  drolyzed polyacrylamide polymers (200 ppm) dissolved in 65% su-  crose and 1% NaCl solutions90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  absolute viscosity (µ), and polymer concentration (ρc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The  values of m and ρco were used as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='38 and 336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='1 ppm in this  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' All of these parameters could greatly influence the  transition and development of the ET phenomenon, and hence  this definition of the Reynolds number could explain the re-  sults better, as suggested by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The figure clearly shows  that the normalized Nusselt number increases with the mod-  ified Reynolds number due to the presence of the elastic tur-  bulence phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In particular, they observed an aug-  mentation of around 240% in viscoelastic polymer solutions  ReETC  3000  146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0  △  292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='1  x  2000  △438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='1  △  口  X 584.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='2  8  1000  Nu  0  Nuo  0  1  2  3300ppmHPAM65%sucrose  200ppmHPAM_65%sucrose  6  100ppmHPAM_65%sucrose  LinearFitofConcatenatedData  5  4  n  Nu oc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='2 Wi  2  Newt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' liquid  0  0  1  2  3  4  5  6  7  8  WiPower-law dependence:  NucWio  名  10  M  Exponential dependence:  Nu oc ewi  Elasticturbulence  regime  2  Elastic instability  regime  Wi  105  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  4  口口  口  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  3  Shear-thinning solutions  O50-W/GLY  0  100-W/GLY  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  200-W/GLY  2  口  O80-W/SUC  口  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  120-W/SUC  Newt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='limit  500-W/SUC  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  0  0  500  1000  1500  2000  2500  3000  3500  Wi*12  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' (a) Variations of the average Nusselt number and (b) pressure drop gradient with the Weissenberg number for geometries with  different dimensions91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  (polyacrylamide + sucrose + NaCl) than in Newtonian sucrose  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Traore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='95 conducted an experimental study for a von-  Karman swirling flow geometry consisting of a cylindrical cup  (of radius 40 mm) with two disks (of radius 39 mm) placed  at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The top disk was allowed to rotate at a higher  temperature, whereas the bottom was kept fixed and main-  tained at a lower temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The distance between them  was 60 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They used polyacrylamide polymers dissolved in  an aqueous solution of sucrose as the working fluids for their  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Their analysis of the temperature fluctuations  revealed characteristics similar to that observed for a passive  scalar in the case of the mixing process in many earlier exper-  iments carried out in the elastic turbulent regime29,71,96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' For  instance, the probability distribution functions of the tempera-  ture fluctuations showed the presence of exponential tails and  exponential decay of the second-order moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore,  the power spectrum of the temperature fluctuations obeyed a  power-law decay with an exponent value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They ob-  served an enhancement in the heat transfer rate of up to four  times in polymer solutions as compared to that obtained in  solvent alone (without polymers) under the same conditions  due to the presence of elastic turbulence in the former flu-  ids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, they calculated that a comparable increase  in the heat transfer rate could be obtained by inertial turbu-  lence at a Reynolds number of 1600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, they noticed  that the relative increase in the efficiency of the heat trans-  fer rate was significantly lower than that obtained in the mix-  ing process utilizing this ET phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='89 pre-  sented a further detailed investigation for the same geometry  and polymer solutions by varying the polymer concentration,  sucrose proportion in the solvent, and degree of salinity in the  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They found the occurrence of elastic instability at  earlier values of the swirling velocity and Weissenberg num-  ber as the polymer concentration increases and the salinity de-  creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The heat transfer rate was higher in viscoelastic poly-  mer solutions than in Newtonian sucrose solutions, and it in-  creased with the Weissenberg number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They found a linear re-  lationship between the normalized Nusselt number and Weis-  senberg number as Nu/Nus ∝ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='2Wi in the elastic turbulence  regime, where Nu and Nus are the average Nusselt numbers  obtained in viscoelastic polymer solutions and Newtonian sol-  vents, respectively, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The normalized average Nusselt  number was seen to be almost independent of the polymer  concentration in the ET regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, at low rotation  speeds, the extent of heat transfer enhancement increased with  the reduction in the salinity of the polymer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Once  the rotation speed exceeded a critical value, it became inde-  pendent of the salinity of the polymer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In another  study by the same authors90 for the same geometry and poly-  mer solutions, once again, they found an enhancement in the  heat transfer rate in viscoelastic polymer solutions compared  to Newtonian solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This study also found the critical value  of the Weissenberg number (∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='14) for the onset of the elas-  tic instability and a power-law decay in the injected power  spectrum with an exponent of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Moreover, the variation  of the Nusselt number with the Weissenberg number showed  an exponential dependence in the elastic instability regime,  whereas a power-law dependence in the fully-developed elas-  tic turbulent regime, as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  The influence of the geometry dimension on the elastic tur-  bulence and subsequent heat transfer phenomena was recently  studied by Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They used three different geometries,  namely, straight microchannel, serpentine microchannel, and  helically coiled microchannel, to realize one (1D), two (2D),  and three-dimensional (3D) effects, respectively, of the flow  field on these phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The variations of the pressure drop  gradient and average Nusselt number with the Weissenberg  number in geometries with different dimensions are depicted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Both the average Nusselt number and pressure drop  (a) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0  1D,200ppm  (b)  400  2D,200ppm  1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='200ppm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  3D,200 ppm  2D,200ppm  nN  dropgradient  Wi2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='2  300  3D,200ppm  (Pa/mm)  W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  Nusselt r  200  Wi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='9  AP/AL  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0  Wil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  100  Wi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='s  Wil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0  0  0  5  10  15  20  25  30  0  5  10  15  20  Weissenbergnumber,Wi  Weissenbergnumber,Wi13  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Surface plots of the non-dimensional temperature distri-  bution along with the velocity vector plots at three different cross-  sectional areas (C1, C2, and C3) of the microchannel97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Here the  first row represents the results for a Newtonian fluid, whereas the  second and third rows depict the results for viscoelastic fluids with  Wi = 5 and 20, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  gradient increased with the Weissenberg number irrespective  of the geometry dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' At a fixed Weissenberg number,  the average Nusselt number increased as the geometry dimen-  sion increased from 1D to 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In contrast, the pressure drop  gradient first increased as the geometry dimension increased  from 1D to 2D and then decreased upon further increasing to  3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, they suggested that 3D geometry is best suit-  able for increasing the heat transfer performance by utilizing  the elastic turbulence phenomenon as it showed reduced pres-  sure drop gradient and higher heat transfer performance than  2D geometry under identical flow conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore,  a non-linear dependence of both the average Nusselt number  and pressure drop gradient with the Weissenberg number was  observed, as can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Although a consid-  erable number of experimental studies have been carried out  on how the EI and ET phenomena influence the heat transfer  aspects in various geometries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' however, the number of cor-  responding numerical studies is very limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This is mainly  due to the ’High Weissenberg Number Problem (HWNP)’ en-  countered in viscoelastic fluid simulations, as mentioned ear-  lier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Among very few studies, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='97 was probably the first  who numerically simulated the heat transfer performance in  a three-dimensional square serpentine microchannel (whose  wall is maintained at a higher temperature than the fluid inlet  temperature) in the elastic turbulence regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They used the  Oldroyd-B fluid model to realize the fluid viscoelasticity and  the log-conformation approach to stabilize the numerical sim-  ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Figure 14 shows the non-dimensional temperature  distribution along with the velocity vectors at three different  cross-sectional areas of the microchannel both for Newtonian  (first row) and viscoelastic fluids with Wi = 5 (second row)  and 20 (third row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It can be evident that for Newtonian flu-  ids, the temperature distribution showed a perfect symmetry  around the center of the channel, and the isotherms adopted a  circular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' All these suggest that heat transfer in New-  tonian fluids primarily occurred by the conduction mode in  this microchannel geometry due to the absence of fluid ad-  vection at these low Reynolds number flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, as  viscoelasticity was gradually introduced into the Newtonian  fluid, a dramatic change happened both in the temperature dis-  tribution and velocity vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' First of all, the symmetry that  was seen for Newtonian fluids was completely lost, and the  temperature distribution became more uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' These ten-  dencies became more prominent as the fluid viscoelasticity  further increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This happened due to the increased chaotic  convection inside the microchannel resulting from the elas-  tic turbulence phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' As expected, the heat transfer  rate also increased inside the microchannel, as evident from  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 15(b), wherein the variation of the surface-averaged Nus-  selt number with the Weissenberg number is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Fig-  ure 15(a) shows the temporal variation of the non-dimensional  temperature at various values of the Weissenberg number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' For  Newtonian fluids, the temperature did not show any fluctua-  tion and remained at a steady value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In contrast, it became in-  creasingly fluctuating as the fluid viscoelasticity gradually in-  creased due to the increased intensity of the ET phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  A similar observation was also found in their other study99,  which mainly focused on developing the numerical algorithm  for simulating high Weissenberg number problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Recently, Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='98 performed a numerical study on the  mixed convective heat transfer phenomena inside a lid-driven  cavity filled with viscoelastic fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This geometry is con-  sidered to be one of the widely studied benchmark problems  in the domain of flow and heat transfer phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They  used a large range of pertinent non-dimensional numbers like  the Weissenberg and Reynolds numbers and presented exten-  sive results and discussion on both the flow dynamics and  heat transfer phenomena inside the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In their simula-  tions, two viscoelastic fluid models, namely, Oldroyd-B and  FENE-P were used to show the competitive effect of the fluid  elasticity and shear-thinning behaviours on the generation of  the elastic turbulence phenomenon and subsequent influence  on the heat transfer enhancement inside the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Figure 16  presents the variation of the time and surface-averaged Nus-  selt number with the Weissenberg number for both fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It  can be observed that the flow inside the cavity transited from a  steady to unsteady chaotic regime after a critical value of the  Weissenberg number due to the establishment of the EI and  ET phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The corresponding heat transfer rate was also  drastically increased for Oldroyd-B viscoelastic fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They  suggested that the heat transfer rate inside the cavity could  be increased by more than 100% using the ET phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  However, it was not observed to that extent for FENE-P vis-  coelastic fluids, which show shear-thinning behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It was  because the shear-thinning behaviours tend to suppress the  elastic instability100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A similar kind of observation was also  seen in the experiments of Abed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='87 for a square serpen-  tine microchannel, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Temperature  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='2  02  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='05  C1  C2  C314  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' (a) Temporal variation of the non-dimensional temperature at a probe location inside the microchannel97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Note that here Wi = 0  stands for Newtonian fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' (b) Variation of the surface-averaged Nusselt number with the Weissenberg number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The inset figure shows the  variation of the RMS value of non-dimensional temperature, and ’NF’ represents Newtonian fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Variation of the time and surface-averaged Nusselt number  with the Weissenberg number in a lid-driven cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The results  are presented for two viscoelastic fluid models, namely, Oldroyd-B  and FENE-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  APPLICATIONS IN ENHANCED OIL RECOVERY  (EOR) PROCESS  In the chemical EOR process, particularly in the polymer  flooding EOR process, the flow dynamics occurring within  the micron-sized rock pores could significantly influence the  macroscopic performance of this process due to the induc-  tion of elastic instability and elastic turbulence phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Therefore, the study of the flow dynamics of single-phase vis-  coelastic fluids in a microfluidic porous geometry has recently  received immense attention38,42,44,54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, in an actual  EOR process, multi-phases are always present, for instance,  oil and a polymer solution in the case of polymer flood-  ing EOR process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, several studies were also con-  ducted with multi-phases to investigate the oil displacement  efficiency utilizing the EI and ET phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' For instance,  Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='101 performed a detailed experimental study on  the capability of a viscoelastic polymer solution (partially hy-  drolyzed polyacrylamide (HPAM)) to displace a synthetic oil  in a model fabricated porous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Some other solutions,  such as glycerol and xanthan, were also used to compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  They provided both microscopic flow details (utilizing streak  photography and particle image velocimetry techniques) and  macroscopic flow behaviours such as pressure drop and appar-  ent viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Figure 17(a1) shows the temporal variation of  the average velocity at a sampled region inside the porous ma-  trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It can be seen that viscoelastic HPAM solution showed  much larger flow fluctuations than glycerol under the same  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Also, the velocity speed in the sampled region in-  creased gradually for the HPAM solution after a critical value  of the flow rate (filled squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In contrast, it remained al-  most at the same value for the glycerol solution, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 17(a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  The apparent viscosity (which is proportional to the pressure  drop) also increased abruptly in HPAM solution after a criti-  cal value of the flow rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Both these signatures suggest the  presence of elastic turbulence in the HPAM solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fig-  ure 17(a3) depicts the distribution of the oil (red colour re-  gion) and displacing fluid phases for both water-wet and oil-  wet conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In both cases, it has been observed that the  oil droplet had a moving meniscus (the bright halos) when the  HPAM solution was used as the displacing fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore,  the oil droplets were in a moving condition due to the pres-  ence of higher velocity fluctuations resulting from the elastic  turbulence phenomenon in HPAM solutions, resulting in the  higher oil displacement using these viscoelastic polymer so-  lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Similar results and observations were also presented  in another study by the same group102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The direct evidence of  the oil droplet fluctuations and the movement of its meniscus  caused due to this elastic turbulence in HPAM polymer solu-  tions was presented in their another subsequent study103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It  was obtained with the help of the nuclear magnetic resonance  (NMR) pulsed field gradient (PFG) diffusion measurements  (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='15  (b)  9  Wi=0  Wi=5  Wi=10  Wi=20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='10  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='00  nN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='1  10  Wi  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='05  7  Viscoelasticfluid  NF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='Nu=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='00  0  100200300400  50060070080090010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='1  1  10  t  IM80  Oldroyd-B  FENE-P  60  Unstable region  (Elastic instabilities and  Steady region  elasticturbulence)  40  O  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='.0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='.0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='.0  20  8  0  10-1  100  101  102  Wi15  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' (a1) Temporal variation of the averaged velocity measured within a sampled region of 100 µm square at the center of a pore inside  the porous geometry (a2) Variation of the apparent viscosity (left side) and fractional velocity spread (right side) with the flow rate at the same  sampled region (a3) Multiphase flows through the model fabricated porous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Here the red regions represent the oil phase and the presence  of bright halos on each oil droplet indicates the moving menisci101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Two-phase flow of synthetic oil and dispensing (a) PEO  and (b) HPAM polymer solutions under the same flow conditions at  a particular probe area inside a porous matrix104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  conducted in a three-dimensional (3D) opaque porous struc-  ture (sandstone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, this in-situ experimental study,  for the first time, established the presence of elastic turbulence  in flows of viscoelastic polymer solutions once the flow rate  exceeds a critical value and its subsequent influence on the  enhancement in the breakup and mobilization of trapped oil  droplets inside the porous matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This ultimately leads to a  higher displacement efficiency of oil in viscoelastic polymer  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Hincapie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='104 presented an experimental study on de-  tailed pore-scale flow visualization of both single and two-  phase flow dynamics inside a micromodel (composed of three  layers wherein the middle layer was made of silicon and had  the porous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It was then sandwiched with top and bot-  tom layers made of glass for easy visualization purpose) that  mimics a real porous matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Their flow visualization exper-  iments revealed different micro-scale phenomena originating  due to the viscoelastic instability during the flooding of vis-  coelastic HPAM polymer solutions into this porous matrix,  namely, streamline crossing, changing flow direction, flow  penetration into small corners, and formation of local vor-  tices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' All these micro-scale phenomena collectively resulted  in a larger displacement of synthetic oil saturated initially in  the porous matrix when an HPAM polymer solution was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  This is also evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 18 wherein the distribution of oil  and dispensing phases is depicted under the same flow con-  ditions at a particular probe area inside the porous matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It  can be easily seen that the HPAM polymer solution displaced  more oils from the porous matrix due to the establishment of  elastic turbulence inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They also observed other probe  areas and found the same trend104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, in their ex-  periments, an enhancement in the apparent viscosity was also  seen after a critical value of the flow rate likewise Clarke et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='101,102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In another study from the same research group, they  provided more analysis on how polymer concentration, salin-  ity, pre-shearing of polymer solutions, molecular weight, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=',  would tend to influence the shear-thickening and elastic tur-  bulence phenomena inside the porous matrix105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='106  2000  Waterwet (hydrophilic)  Oil wet (hydrophobic)  1800  1600  1400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='24wt%Xanthan  1200  1000  800  600  400  Glycerol 84%  (al)  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='12% HPAM (3630S)  b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='00  Time,(s)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='24wt%HPAM3630S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='00  (a2)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='50  spread  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='50  (a3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='00  1  Flow rate, q (ul s-1)  10  100(a)  (b)  Oil  Oil  Oil16  developed a novel polymer, named star-like amphiphilic poly-  acrylamide (SHPAM), consisting of nano-SiO2 as the core  and a layer of amphiphilic chains as the shell using a facile  free radical polymerization method, to demonstrate its higher  displacement efficiency than HPAM polymer solutions from a  geological rock core (sandstone) initially saturated with crude  oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Their nuclear magnetic resonance (NMR) spectroscopy  results revealed that SHPAM polymers have higher displace-  ment efficiency than HPAM polymers under the same oper-  ating conditions (and even at a lower concentration) due to  the higher shear-thickening and viscoelastic properties in the  former polymers owing to the presence of cross-linked mi-  crostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  De et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='107 presented a detailed and systematic experi-  mental study on the mechanism of residue oil displacement  in a model porous media consisting of a microchannel hav-  ing several cylindrical micropillars placed in it using dis-  placing fluids of different rheological characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Fig-  ure 19 represents the steady-state snapshots of the distribu-  tion of oil and different displacing fluid phases (namely, wa-  ter, xanthan, HPAM, and viscoelastic surfactant (VES) so-  lution comprised of cationic surfactant cetyltrimethylammo-  nium bromide (CTAB), sodium salicylate (NaSal) and sodium  chloride (NaCl) dissolved in de-mineralized water) almost at  the same values of the capillary number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' When the capillary  number was small, large oil blobs were seen to be present ir-  respective of the displacing fluid type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, as this num-  ber was gradually incremented, oil ganglia of large sizes were  still present when the displacing fluids were water and xan-  than, sub-Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 19(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' On the other hand, in the case  of viscoelastic HPAM and VES solutions (sub-Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 19(c) and  (d)), those became considerably small in size compared to that  seen in water and xanthan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This was due to the presence of the  viscoelastic instability effect in these two displacing fluids,  which disrupted large oil blobs into small ones and facilitated  a larger displacement of oils from the porous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This can  be seen in sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 19(e) wherein the remaining oil saturation  was presented against the value of the capillary number for  different displacing fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It decreased as the capillary num-  ber increased for all displacing fluids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' however, the extent of  this decrease was more for HPAM and VES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Due to the ab-  sence of elastic instability in water and xanthan (inelastic and  weakly elastic shear-thinning fluids, respectively), the remain-  ing oil saturation was comparatively high in these two fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='110 performed both experiments and numeri-  cal simulations (based on the volume of fluid (VOF) method)  using a quartz sand epoxy resin as the model porous me-  dia and hydrophobically associating water-soluble polymers  (HAWP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Their simulation results revealed that the fluid vis-  coelasticity in the case of polymer flooding leads to a larger  sweep area and stable front than water flooding, resulting in  a decrease of the residual oil saturation for polymer flooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  However, an additional pressure drop was also observed in the  case of polymer flooding in their simulations, as was seen in  the corresponding experiments101,102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' These observations of  a larger sweep area and stable front in the case of viscoelas-  tic polymer flooding were also seen in the experiments per-  formed by Vik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='111 and the dynamic pore network mod-  eling by Salmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Molecular-scale simulations were  also performed to understand the mechanism of the trapped  oil displacement from a dead micro-pore zone in porous me-  dia by Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  They conducted molecular dynamics  (MD) simulations with various values of the polymer chain  length (N) and injected pore volume (PV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Figure 20(a-b)  shows the snapshots of the distribution of oil (black-coloured  molecules) and displacing fluid molecules (red and green-  coloured molecules represent water and polymers, respec-  tively) inside the nanopore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It can be seen that in the case  of water flooding (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 20(a)), the oil molecules remained  at the dead end of the nanopore even at higher values of PV,  whereas they came out from the dead zone in the case of poly-  mer flooding (sub-Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 20(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The latter tendency further  incremented as the polymer chain length increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They pro-  posed a mechanism for this enhanced displacement efficiency  of the oil during the polymer flooding based on the pulling ef-  fect of elastic polymer molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Similar findings were also  seen in the experiments of Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='109 in a microscopic  pore of a porous media, sub-Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 20(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore,  they observed the formation of "oil thread" during the poly-  mer flooding (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 20(f)), resulting in the origin of a new  mechanism for the high displacement efficiency of this flood-  ing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A detailed theoretical analysis was presented to  explain this phenomenon, and the presence of elastic stresses  in polymer solutions was found to be responsible for the for-  mation and stabilization of this oil thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  To understand the transport mechanism of oil blobs in an  actual porous media during the polymer flooding process, fur-  ther studies were conducted with a simple model porous sys-  tem consisting of an expansion/contraction microchannel and  placing one oil droplet inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' For instance, Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='113  conducted an extensive two-dimensional Lattice-Boltzmann  method (LBM) based numerical investigation using the sim-  ple Maxwell model to account for the fluid viscoelasticity of  the displacing fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In the case of simple Newtonian dis-  placing fluid, the droplet was seen to pass through the con-  stricted (or the pore-throat) region easily as time progressed,  sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 21(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, in the case of viscoelastic displac-  ing fluid, the droplet started to oscillate in front of the en-  trance of the constricted region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It also did not pass through  this region, sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 21(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It was due to the presence of elas-  tic instability in the polymer flooding case, which generated  large and fluctuating vortices around the entrance of the con-  stricted region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This, in turn, blocked the movement of the dis-  persed droplet into this constricted region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' No such vortices  were formed for Newtonian fluid flooding, and as a result, the  droplet passed smoothly through the constricted region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In a  subsequent experimental study, Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='115 also observed this  oscillating trap of the droplet likewise seen in their numerical  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Once again, the droplet passed easily through the  constricted region when the displacing fluids were Newtonian  (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 21(c)) and inelastic shear-thinning (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 21(d));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  however, it was inhibited when the displacing fluids were vis-  coelastic (sub-Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 21(e) and (f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They also proposed scal-  ing relationships for the amplitude of droplet oscillation and  droplet length and found a good agreement with the corre-  sponding experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Both these were found to in-  17  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Snapshots of the distribution of oil and different displacing fluids, namely, (a) water, (b) xanthan, (c) HPAM, (d) VES, inside the  porous matrix at different capillary numbers (Ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The latter was defined as the ratio of the viscous to that of the surface tension forces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=',  Ca = µu  σ where µ, u and σ are the displacing fluid viscosity, Darcy’s velocity, and the interfacial tension between the displacing and displaced  fluids, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' (e) Variation of the remaining percentage oil saturation with the capillary number for various displacing fluids107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  crease with fluid viscoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='113 also performed simulations for a system com-  prising a straight microchannel with a side dead zone where  the droplet was present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  The droplet was displaced from  the dead zone and merged with the main flow during the  viscoelastic fluid flooding due to elastic instability-induced  chaotic convection, whereas it remained trapped inside the  dead end during the Newtonian fluid flooding, as was also  seen in earlier experiments109 and molecular-scale simula-  tions108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This further establishes the role of elastic instability  and elastic turbulence phenomena in displacing the trapped  oil ganglia in a porous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A very recent study by Mo-  hamed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='114 also proved it by examining the morphologies  of oil globules in a three-dimensional porous structure with  the help of an in-situ high-resolution microcomputed tomog-  raphy (µ-CT) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The snapshots of oil globules inside  the three-dimensional micro-pore structure are presented in  Figure 22 both for water and polymer solution flooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It  can be observed that a big oil blob that was present in the cir-  cled area of the pore structure during the water flooding (sub-  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 22) was not present during the polymer flooding (sub-  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 22(b)) under the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They proposed that the  elastic turbulence phenomenon in the latter case fragmented  and mobilized the oil globule, and hence a higher displace-  ment of oil was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This was also reflected in their cal-  culation of the residual oil saturation, which decreased with  the increased fluid viscoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A correlation between the  residual oil saturation and the Weissenberg number was also  Ca:210-5  Ca:4-10  110-3  Ca:8-104  Water  Hpam  Oil phase  Oil phase  (al)  (a3)  (e1)  (a)  (c)  Xanthari10  Ca:1-10  Ca:3-105  Ca:1-10  Ca:1103  VES  Oil phase  Oil phase  (d2)  (b)  (d)  50  HPAM  45  VES  Water  40  Xanthan  saturation  35  30-  25  20-  15-  10-  (e)  5  10-5  104  10-3  Ca18  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Molecular dynamics (MD) simulations of (a) water and (b) polymer flooding through a nanopore with a dead zone where an oil  droplet (black-coloured molecules) is trapped108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Here N and PV are the polymer chain length and injected pore volume, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The  corresponding experimental results of (c) water and (d) polymer flooding in a microscopic pore of a porous media by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The  distribution of (e) oil and water and (f) oil and polymer solutions inside the porous media109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Here the red-dyed regions represent the oil phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  proposed in their study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A similar observation was also seen  in earlier experiments of Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='116, and they also proposed a  correlation between the residual oil saturation and the Debo-  rah number, as provided by Mohamed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Irfan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='117  also conducted a recent experimental investigation on a three-  dimensional porous structure made of Berea sandstone and  found an enhancement in the residual oil displacement from it  by the use of HPAM polymer solution as the displacing fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  They also concluded that elastic turbulence was responsible  for induced pressure and velocity fluctuations at small pores  of the porous media, increasing the oil displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Druetta and Picchioni119 performed a numerical study us-  ing the upper convected Maxwell (UCM) and Oldroyd-B  viscoelastic fluid models on a two-dimensional porous rock  structure at relatively low values of the Weissenberg num-  ber where the elastic turbulence phenomenon was not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  However, still, they observed an increase in the oil displace-  ment efficiency of around 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='4% during the viscoelastic fluid  flooding compared to the traditional water flooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This was  attributed to a larger sweep area (without local channels for  flow) in viscoelastic fluid flooding than in water flooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  A larger sweep area was caused due to more penetration of  polymer solutions into small pores of the porous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  This was evident both in their numerical solutions and experi-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, they also suggested that apart from the fluid  viscoelasticity, the interfacial tension (IFT) between the dis-  placing and displaced fluids also plays an essential role in the  sweeping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Parsa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='118 conducted an experimental  study to investigate the pore-scale interaction between an oil  blob and displacing fluid in a three-dimensional micromodel  porous media consisting of a square quartz capillary filled  with randomly and loosely packed monodisperse borosilicate  glass beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They used confocal microscopy to configure the  displacement of oil within the porous media and also to ob-  tain a detailed velocity field within the displacing fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fig-  ure 23 represents the snapshot of an oil ganglion trapped (pur-  (a)  (b)  waterflooding(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='25PV)  N=250(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='85PV)  (d)  (e)  (f)  Oilthread19  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Numerical simulations of different states of a non-wetting oil droplet during its transport through an expansion/contraction mi-  crochannel when the displacing fluids were (a) Newtonian and (b) viscoelastic polymer solutions113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The corresponding experimental results  when the displacing fluids were (c) Newtonian, (d) inelastic shear-thinning, (e) viscoelastic with lower relaxation time, and (f) viscoelastic  with higher relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Here Ca and De are the capillary and Deborah numbers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The arrows show the direction of droplet  motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Visualization of trapped oil globules (red) in a three-  dimensional porous structure during (a) water flooding and (b) vis-  coelastic polymer solution flooding114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  ple) inside the micromodel porous media along with the veloc-  ity vector fields (blue arrows) in the displacing fluid at three  different cases, namely, after initial water flooding, polymer  solution flooding, and the chase water flooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  They ob-  served that the oil globule was present inside the porous media  even after polymer solution flooding (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 23(b)), which  was only completely removed after flooding with the chase  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Snapshot of an oil ganglion (purple) trapped in three-  dimensional micromodel porous media along with the velocity vec-  tor field (blue arrows) immediately after the (a) initial water flooding,  (b) polymer flooding, and (c) chase water flooding118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Here the size  of the arrows dictates the velocity field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  water (sub-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 23(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, they showed that polymer  solution flooding will not always facilitate oil displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  However, it should be mentioned here that there was no elas-  tic turbulence present in the system, as was confirmed by their  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, they noticed significant local changes in the  velocity field due to polymer solution flooding, leading to the  origin of sufficiently large viscous forces at the interface of the  immiscible fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' They proposed that these large and hetero-  geneous local changes in the flow field resulted in increased  (a)  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='08s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='259s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='133s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='309s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='357s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='315s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='387s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='320s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='392s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='328s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='400s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='3325  (a)  (b)  Flow direction of displacing fluids  L=0  t=0  =23  t=0  t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='6s  t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5s  =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='53  (=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='95  =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='5s  (c)  (d)  (e)  (f)(a)  (b)(a)  (b)  (c)  500um20  oil displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  LIMITATIONS  As mentioned earlier and discussed, the elastic instability  and turbulence phenomena are generated in a system where  the effect of inertial forces is negligible compared to that of  viscous and elastic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In other words, these unstable and  chaotic flow regimes are generated in a system when the elas-  ticity number (El), defined as the ratio of the Weissenberg  (Wi) to that of the Reynolds number (Re), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=', El = Wi  Re, be-  comes much larger than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, it is only possible  to generate in a micro-scale system whose dimensions are of  micron or millimeter sizes if we take the realistic values of  the physical properties of a fluid, such as density, viscosity,  relaxation time of polymer molecules, or any other micro-  scopic structure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This is the reason why all the studies  regarding the potential applications of these two phenomena  were so far carried out for microfluidic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' How-  ever, this requirement of small-scale dimensions for generat-  ing these two phenomena may limit their use in many prac-  tical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' To understand this, we can take the ex-  ample of flow through a straight pipe for which the pressure  drop (∆p) varies inversely with the fourth power of the ra-  dius of the pipe (R), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=', ∆p ∼  1  R4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' We know this from the  well-known Hagen-Poiseuille equation120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It suggests that as  the radius of the pipe gradually decreases, the pressure drop  increases non-linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hence, the mechanical power needed  to pump the fluid inside the system also increases abruptly if  all other parameters in the Hagen-Poiseuille equation remain  fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' On top of that, an additional abrupt pressure drop is  created in a system once the elastic instability and turbulence  phenomena set in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In fact, this behaviour is considered one  of the characteristic features of these phenomena, which has  been found in many earlier experimental as well as numeri-  cal studies74,78,86,87,92,95,96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The requirement of this extra sub-  stantial mechanical power for pumping the viscoelastic fluids  in the elastic turbulence regime may preclude its application  (either in the enhancement of the rate of heat transfer or mix-  ing process) from the viewpoint of the operational limit of  an instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, one may need to specially de-  sign the whole system to sustain such a huge pressure drop in  such small-scale microsystems, which in turn, may increase  the operational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' One needs to be more cautious when ap-  plying these EI and ET phenomena to an application system  consisting of sophisticated and flexible microfluidic compo-  nents, which may be damaged due to the presence of this high-  pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In the case of electrokinetic-driven flows,  one needs to apply a high-voltage difference across a system  to pump the fluid and generate the electro-elastic turbulence,  which may also become problematic in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  The next limitation in applying the EI and ET phenomena  to any practical application is the rheological property of the  fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Most of the studies performed so far on the potential of  the application of these phenomena used the Boger fluid121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  It shows a constant shear viscosity but exhibits high exten-  sional viscosity122,123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This is a special type of fluid that is  made manually in the lab to investigate the explicit effect of  fluid elasticity on various flow phenomena124–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' One has  to be careful in choosing the polymers and its concentration  as well as the solvent to make such type of fluid121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' There-  fore, the working fluids in any application should be of Boger  fluid type so that an unstable and chaotic flow field could be  created inside the system to enhance the rate of any transport  phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Although many fluids that are routinely encoun-  tered in many practical microfluidic applications such as poly-  mer solutions, emulsions, suspensions, many biofluids includ-  ing blood, saliva, DNA and protein suspensions, cerebrospinal  fluid, suspensions of cells and bioparticles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=',4,131–136 ex-  hibit non-Newtonian behaviours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' however, they hardly show  the Boger fluid type behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' All these fluids exhibit elas-  tic behaviours along with other non-linear behaviours such as  shear-thinning, shear-thickening, viscoplasticity, thixotropic,  etc137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' These other rheological behaviours significantly influ-  ence the EI and ET phenomena in a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' For instance,  the shear-thinning behaviour of a viscoelastic fluid has been  shown to suppress the onset of the elastic instability in a sys-  tem100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This, in turn, may inhibit the generation (or the in-  tensity) of elastic turbulence in a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It has been, in fact,  observed both in experimental and numerical studies wherein  a reduction in the rate of the heat transfer process occurred due  to the suppression of the elastic turbulence phenomenon ow-  ing to the shear-thinning properties of a viscoelastic fluid87,98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Therefore, one has to opt for a Boger fluid to utilize the full  potential of elastic turbulence in any practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  This may be easy for heat transfer applications for which a  coolant could be made in such a way that it should exhibit the  same rheological behaviours as that of a Boger fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' How-  ever, problems may arise in the case of mixing applications  wherein the making and rheological behaviours of working  fluids are not on our hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Of course, one can add a minute  amount of solid polymers or surfactants into the working fluid  to make it viscoelastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, it does not guarantee that  the resulting solution would behave like a constant viscosity  Boger fluid, as the making of such fluid depends on many fac-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Moreover, a particular application may not allow such  addition of polymers or surfactants into the main working flu-  ids (although they are present in parts per million (ppm) quan-  tities) as it may create problems for their further downstream  applications or processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Therefore, one has to perform  a rigorous investigation before applying the EI and ET phe-  nomena to any particular application, particularly related to  enhancing the mixing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  The requirement of geometrical configuration may also  sometimes limit the application of these two phenomena to  any practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' As already mentioned earlier and  also seen in most of the studies, the onset of elastic instabil-  ity happens due to the interaction between the normal elas-  tic stresses and the streamline curvature present in a sys-  tem23,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The latter requires a curved geometry, which some-  times may become difficult to fabricate for microfluidic ap-  plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The type and number of curved surfaces and their  arrangement can significantly influence the onset and gener-  ation of elastic turbulence phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hence, one has to  perform a thorough optimization study to select a particular  21  geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' All these may become expensive from a practical  perspective as compared to other options that are available to  perform the same duty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  CONCLUSIONS AND FUTURE DIRECTIONS  The elastic instability and elastic turbulence phenomena,  indeed, have the potential to increase the rate of transport pro-  cesses such as heat transfer or mixing processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However,  the application of these phenomena is limited to only micro-  scale systems wherein the effect of inertial forces is negligi-  ble compared to viscous and elastic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, the  working fluid has to be non-Newtonian viscoelastic in nature,  which in turn, precludes the applicability of these phenom-  ena for an application wherein a simple Newtonian fluid is  handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Among various non-linear rheological characteris-  tics that a fluid can show, elasticity should be the dominant  one, which promotes the generation of these phenomena in  a microfluidic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The constant shear viscosity and high  extensional viscosity Boger fluid121 should be the ideal choice  for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Based on the discussion presented in the  preceding section, it can be readily acknowledged that these  phenomena have a higher potential in microfluidic heat trans-  fer applications than micro mixing applications due to lesser  restrictions in the former applications for applying these phe-  nomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Although a considerable number of studies have al-  ready been conducted to show the potential of these phenom-  ena in increasing the rate of either the heat transfer or the mix-  ing process or in enhancing the oil displacement efficiency in  the EOR process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' however, still a large scope is present as  far as the application point of view is concerned of these two  phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Most studies on the applications of the EI and ET phe-  nomena were carried out for curvilinear or serpentine  microchannels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The presence of a highly curved sur-  face in this geometry produces high streamline curva-  ture in the flow field, which in turn, facilitates the gen-  eration of elastic turbulence in this geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  How-  ever, a detailed study on how the number and angle of  curving could influence these phenomena and, subse-  quently, the rate of transport processes is still missing  in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A direct relationship between the pres-  sure drop and the rate of transport processes should be  established;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' likewise, it was done for the regular hy-  drodynamic turbulence138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' More investigations should  be carried out for other micro-scale geometries, for in-  stance, a microchannel with in-built obstacles present  in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Although this geometry has already been used  to induce elastic instability in a number of experimen-  tal and numerical studies32,50,53,78,139,140;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' however, the  corresponding study on the potential in enhancing the  rate of transport processes in this geometry is not in-  vestigated yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Several factors, such as the shape of the  obstacle, the number of obstacles, and the gap between  two consecutive obstacles, could influence the rate of  transport processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hence, a detailed investigation is  needed on the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A microchannel with either step  expansion and contraction or micro constrictions could  also be investigated to see its potential in enhancing the  rate of transport processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This particular geometry  has also shown the potential to create elastic instability  and turbulence36,49,53,141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Moreover, the fabrication of  this latter geometry would be relatively easier than that  of the curvilinear or serpentine microchannel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Research efforts should be spent on establishing the cor-  relations for the average Nusselt number (in the case of  heat transfer applications) as a function of the relevant  dimensionless numbers such as Weissenberg, Reynolds,  Prandtl, and Richardson numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Although some stud-  ies have already attempted to establish such correla-  tions89,90;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' however, physical insights and/or scaling ar-  guments behind the selection of a particular form of the  correlation and the power-law exponents of different  dimensionless numbers (particularly, the Weissenberg  number) was somehow missing in those studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' More  detailed and rigorous investigations are needed to estab-  lish such correlations, including the effect of geomet-  ric parameters and polymer concentration along with  other thermo-physical properties (such as specific heat,  thermal conductivity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=') and flow conditions prop-  erly and systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Furthermore, the studies on  heat transfer applications of the EI and ET phenom-  ena are mostly limited to forced convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' In con-  trast, almost no study (either experimental or numeri-  cal) is present (except one recent numerical study by  Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='98) on how these phenomena could influ-  ence the other two modes of heat transfer, namely, nat-  ural or free convection and mixed convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' These  two modes of heat transfer are also used in many mi-  crofluidic applications142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, the phenomena  of boiling and condensation happening in micro-scale  geometries are also widely used in many microfluidic  applications143,144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Many studies have been conducted  on how adding polymer or surfactant molecules into a  solvent like water could influence these phenomena145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  However, those studies did not look into the problem  from a perspective of elastic instability and turbulence  phenomena, which could be generated inside a drop  and could influence these processes significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' All  these previous studies investigated how adding either  polymer or surfactant molecules influenced the surface  tension and dynamic viscosity and, subsequently, the  boiling and condensation heat transfer phenomena in a  micro-scale geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, a large scope for fu-  ture investigations is present in this particular area of  heat transfer phenomena utilizing the EI and ET phe-  nomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Droplet-based microfluidics has become a promising  technique in past decades in many cutting-edge tech-  nological applications, such as fluid mixing146,147, cell  encapsulation and delivery148,149, cell sorting150, drug  discovery and genetic applications151, sensing152, and  many others153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A great potential is present in this par-  ticular area of microfluidics wherein the introduction  22  of elastic instability and turbulence could dramatically  influence the transport phenomena inside a drop, such  as mixing, which in turn, could significantly influence  further downstream processes such as chemical reac-  tions147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  In recent days, lots of investigations in terms of both ex-  periments and simulations have been conducted on the  heat transfer enhancement capability of nanofluids154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  These fluids are formed by adding a small amount of  nanoparticles (made of metals, oxides, carbides, carbon  nanotubes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=') into a base fluid like water, glycerol,  or oil155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Using such nanofluids, many experimental,  as well as numerical studies, have found a significant  enhancement in the heat transfer rate as compared to  that achieved in base fluids only in a microfluidic sys-  tem such as microchannel or heat sink156–158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This en-  hancement in the heat transfer rate in nanofluids is ba-  sically due to an increase in the effective thermal con-  ductivity of these fluids owing to the higher values of  the thermal conductivity of the nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A gen-  eration of elastic turbulence in these nanofluids flowing  in a microfluidic system could increase the heat transfer  rate by many-fold than that achieved either only using  nanofluids or the elastic turbulence phenomenon alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Although some studies are present in the literature on  viscoelastic nanofluids159–162;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' however, no study has  so far attempted to investigate the elastic turbulence  phenomenon and subsequently the heat transfer rate in  these fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Although a substantial number of studies have been  conducted to show the potential of the ET phenomenon  in increasing the oil displacement efficiency during the  polymer flooding process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' however, still, a complete un-  derstanding of the mechanism behind this enhancement  is missing in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  All those earlier studies  have used a microporous model structure to carry out  the investigation, which may not mimic the actual situ-  ation under an oil reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' For instance, the materials  used for making these micromodels are often silicon,  glass, polymers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=', which may not exhibit the same  surface characteristics as the minerals that are present  in the rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' It can significantly influence the contact  angle and hence the corresponding multiphase flow dy-  namics inside a porous media163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, the micro-  models prepared for this kind of multiphase flow dy-  namics study (in particular, the influence of the ET phe-  nomenon) should mimic the surface wettability, miner-  alogy, and roughness parameters of natural rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The  Rock-on-a-Chip (ROC) approach164,165 should be con-  sidered in future studies, which will facilitate a better  understanding of the influence of the ET phenomenon  on the oil displacement mechanism in a porous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  A three-dimensional ROC instead of a two-dimensional  one should be employed in understanding the trans-  port mechanism, along with sophisticated experimental  techniques (such as confocal microscopy) to visualize  and analyze the flow fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Furthermore, almost all  previous studies were carried out at room temperature  and pressure, whereas the oil reservoirs are primarily  present at elevated temperatures and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' These  two parameters could significantly impact the perme-  ability and the interfacial tension between two immis-  cible fluids and, subsequently, the multiphase flow dy-  namics inside the porous media166,167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Although sev-  eral studies have emphasized that the ET phenomenon  is responsible for higher oil displacement efficiency  during polymer flooding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' however, no detailed statis-  tical analysis on the temporal and spatial fluctuations of  either the velocity or the pressure (at different probe lo-  cations) was presented, which could firmly establish the  claim further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Polymeric surfactants are believed to be promising in  the chemically enhanced oil recovery process168,169 due  to their ability to reduce the interfacial tension and in-  crease the solution viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Both these tend to facili-  tate more oil displacement in a porous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However,  most studies on these polymeric surfactants related to  the EOR process focused on their synthesis and char-  acterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' There is almost no study present in the  literature which focuses on the oil displacement mech-  anism (as well as the ET phenomenon) at the pore level  in these displacing fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' This is particularly impor-  tant to investigate as this is still a debatable subject  whether these polymeric surfactants should be used in  the EOR process due to their high synthesis and han-  dling costs168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Therefore, the study of the ET phe-  nomenon in the presence of these polymeric surfactants  and other phases, such as oil, deserves significant atten-  tion in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  All the previous studies on the ET phenomenon during  the EOR process considered the rheological properties  of the displacing fluid but not the displaced fluid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=',  the crude oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However, it can be readily acknowledged  that crude oil exhibits various non-Newtonian charac-  teristics, such as shear-thinning, yield stress, thixotropy,  or even viscoelastic170–175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' The rheological properties  of the displaced fluid could also significantly regulate  the ET phenomenon and the subsequent oil displace-  ment efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Therefore, careful pore-scale inves-  tigations (comprising both numerical simulations and  experiments) should be conducted in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fur-  thermore, thorough and systematic studies of the effect  of polymer type, polymer concentration, and molecular  weight on the ET phenomenon should also be carried  out so that it can be appropriately utilized during the  EOR process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Most studies related to either the generation of EI  and ET phenomena or to demonstrate their potential  in heat transfer rate or mixing enhancement applica-  tions have been conducted in pressure-driven flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  In comparison, very few studies were conducted for  electrokinetically-driven generation and applications of  the EI and ET phenomena83–85,176,177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  However, it  23  can be readily acknowledged that electrokinetic-driven  flows are often used to transport fluids in micro-scale  geometries for the following reasons i) the EK-based  microdevices do not have any moving mechanical parts  as they rely on the application of an electric field, and  hence, they are easy to handle ii) the electrokinetic  flows offer less resistance to the flow than pressure-  driven flows due to almost plug-like velocity profile in  the former flows178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, a huge scope is present  for further future studies in these areas of electro-elastic  instability and electro-elastic turbulence both from the  application and fundamental understanding point of  view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  From the discussions presented herein, it is clear that a great  potential for the applications of elastic instability and elas-  tic turbulence phenomena is present in micro-scale systems  to increase the rate of various transport processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' However,  in applying so, we should also keep in mind the limitations  that are discussed in the preceding section of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' So  far, the studies carried out to show the application potential of  these two phenomena were limited to lab-scale experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Furthermore, the problem setup used either in experiments  or simulations was not related to any direct application;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' in-  stead, it was a prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, in the future, experiments  should be conducted for a direct application to show the real  potential of these two phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' For example, we could uti-  lize these phenomena in the micro heat sink applications for  cooling electronic chips179,180 or enhancing the reaction rate  (and ultimately increasing the percentage of desired products  in a chemical reaction) in a microsystem by enhancing the cor-  responding fluid mixing181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Therefore, such practical studies  are definitely needed in the future to establish the potential of  the EI and ET phenomena for real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  1R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bird, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Curtiss, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Armstrong, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hassager, Dynamics of  Polymeric Liquids, volume 2: Kinetic Theory (Wiley, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Schroeder, “Single polymer dynamics for molecular rheology,” Jour-  nal of Rheology 62, 371–403 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  3G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Thurston, “Viscoelasticity of human blood,” Biophysical Journal 12,  1205–1217 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  4A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Beris, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Horner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Jariwala, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Armstrong, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wagner,  “Recent advances in blood rheology: A review,” Soft Matter (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  5H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhang, “Advances in the rheology  of emulsion explosive,” Journal of Molecular Liquids 336, 116854 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  6P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fuhrmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Breunig, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sala, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sagis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Stieger, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Scholten,  “Rheological behaviour of attractive emulsions differing in droplet-droplet  interaction strength,” Journal of Colloid and Interface Science 607, 389–  400 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  7Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Briceño-Ahumada, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mikhailovskaya, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Staton, “The role of  continuous phase rheology on the stabilization of edible foams: A review,”  Physics of Fluids 34, 031302 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  8D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Seol,  and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kim, “Numerical study on rheology of two-  dimensional dry foam,” Physics of Fluids 33, 052111 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  9A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Jain and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Shaqfeh, “Transient and steady shear rheology of  particle-laden viscoelastic suspensions,” Journal of Rheology 65, 1269–  1295 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  10H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Shewan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yakubov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bonilla, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Stokes, “Vis-  coelasticity of non-colloidal hydrogel particle suspensions at the liquid–  solid transition,” Soft Matter 17, 5073–5083 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  11R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bird, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Armstrong, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hassager, “Dynamics of polymeric  liquids, volume 1: Fluid mechanics,” (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  12Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Dubief, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Terrapon, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hof, “Elasto-inertial turbulence,” An-  nual Review of Fluid Mechanics 55, 2023 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  13C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' White and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mungal, “Mechanics and prediction of turbulent  drag reduction with polymer additives,” Annual Review of Fluid Mechan-  ics 40, 235–256 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  14B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Toms, “Some observations on the flow of linear polymer solutions  through straight tubes at large reynolds numbers,” in Proceedings of the 1st  International Congress on Rheology, North-Holland, Amsterdam 2, 135–  141 (1948).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  15C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wells Jr and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Spangler, “Injection of a drag-reducing fluid into  turbulent pipe flow of a newtonian fluid,” Physics of Fluids 10, 1890–1894  (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  16P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Virk, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Merrill, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mickley, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Smith, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mollo-Christensen,  “The Toms phenomenon: turbulent pipe flow of dilute polymer solutions,”  Journal of Fluid Mechanics 30, 305–328 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  17H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hershey and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zakin, “Existence of two types of drag reduction in  pipe flow of dilute polymer solutions,” Industrial & Engineering Chemistry  Fundamentals 6, 381–387 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  18T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Min, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yoo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Choi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Joseph, “Drag reduction by polymer  additives in a turbulent channel flow,” Journal of Fluid Mechanics 486,  213–238 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  19B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Owolabi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Dennis, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Poole, “Turbulent drag reduction  by polymer additives in parallel-shear flows,” Journal of Fluid Mechanics  827 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  20C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sureshkumar,  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Khomami, “Influence of rheological  parameters on polymer induced turbulent drag reduction,” Journal of Non-  Newtonian Fluid Mechanics 140, 23–40 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  21S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Datta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ardekani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Arratia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Beris, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bischofberger,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' McKinley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Eggers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' López-Aguilar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fielding, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fr-  ishman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=', “Perspectives on viscoelastic flow instabilities and elastic  turbulence,” Physical Review Fluids 7, 080701 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  22M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Khalid, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Shankar, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Subramanian, “Continuous pathway be-  tween the elasto-inertial and elastic turbulent states in viscoelastic channel  flow,” Physical Review Letters 127, 134502 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  23P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pakdel and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' McKinley, “Elastic instability and curved streamlines,”  Physical Review Letters 77, 2459 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  24G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' McKinley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pakdel, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Öztekin, “Rheological and geomet-  ric scaling of purely elastic flow instabilities,” Journal of Non-Newtonian  Fluid Mechanics 67, 19–47 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  25P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pakdel and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' McKinley, “Cavity flows of elastic liquids: purely  elastic instabilities,” Physics of Fluids 10, 1058–1070 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  26P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pakdel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Spiegelberg, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' McKinley, “Cavity flows of elastic  liquids: two-dimensional flows,” Physics of Fluids 9, 3123–3140 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  27G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Vinogradov and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Manin, “An experimental study of elastic tur-  bulence,” Kolloid-Zeitschrift und Zeitschrift für Polymere 201, 93–98  (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  28A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Groisman and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “Elastic turbulence in a polymer solution  flow,” Nature 405, 53–55 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  29A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Groisman and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “Efficient mixing at low reynolds numbers  using polymer additives,” Nature 410, 905–908 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  30Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Jun and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “Elastic turbulence in a curvilinear channel flow,”  Physical Review E 84, 056325 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  31T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Burghelea, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Segre,  and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “Elastic turbulence in von-  Karman swirling flow between two disks,” Physics of fluids 19, 053104  (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  32A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Varshney and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “Drag enhancement and drag reduction in  viscoelastic flow,” Physical Review Fluids 3, 103302 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  33A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Varshney and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “Elastic wake instabilities in a creeping flow  between two obstacles,” Physical Review Fluids 2, 051301 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  34Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Jun and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “Power and pressure fluctuations in elastic turbu-  lence over a wide range of polymer concentrations,” Physical review letters  102, 124503 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  35S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yamani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Keshavarz, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Raj, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zaki, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' McKinley,  and  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bischofberger, “Spectral universality of elastoinertial turbulence,” Phys-  ical Review Letters 127, 074501 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  36C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Browne, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Shih, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Datta, “Bistability in the unstable flow  of polymer solutions through pore constriction arrays,” Journal of Fluid  Mechanics 890 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  37E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ekanem, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Berg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' De, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fadili, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bultreys, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Rücker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' South-  wick, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Crawshaw, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Luckham, “Signature of elastic turbulence  of viscoelastic fluid flow in a single pore throat,” Physical Review E 101,  042605 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  24  38C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Browne, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Shih, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Datta, “Pore-scale flow characterization  of polymer solutions in microfluidic porous media,” Small 16, 1903944  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  39C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Browne and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Datta, “Elastic turbulence generates anomalous  flow resistance in porous media,” Science Advances 7, eabj2619 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  40D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Carlson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Toda-Peters, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Shen, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Haward, “Volu-  metric evolution of elastic turbulence in porous media,” Journal of Fluid  Mechanics 950, A36 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  41D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kawale, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Marques, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zitha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kreutzer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Rossen, and  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Boukany, “Elastic instabilities during the flow of hydrolyzed poly-  acrylamide solution in porous media: effect of pore-shape and salt,” Soft  Matter 13, 765–775 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  42S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Haward, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hopkins, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Shen, “Stagnation points control  chaotic fluctuations in viscoelastic porous media flow,” Proceedings of the  National Academy of Sciences 118, e2111651118 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  43S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Datta, “Patches of patches: Spatial fluctuations of elastic turbulence  in porous media,” Science Talks 3, 100044 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  44D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Walkama, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Waisbord, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Guasto, “Disorder suppresses  chaos in viscoelastic flows,” Physical Review Letters 124, 164501 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  45M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kumar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Guasto, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ardekani, “Transport of complex and  active fluids in porous media,” Journal of Rheology 66, 375–397 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  46R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Keunings, “On the high weissenberg number problem,” Journal of Non-  Newtonian Fluid Mechanics 20, 209–226 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  47A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Afonso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Oliveira, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pinho, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Alves, “The log-conformation  tensor approach in the finite-volume method framework,” Journal of Non-  Newtonian Fluid Mechanics 157, 55–65 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  48S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Berti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bistagnino, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Boffetta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Celani, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Musacchio, “Two-  dimensional elastic turbulence,” Physical Review E 77, 055306 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  49M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Aramideh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Browne, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Datta, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ardekani,  “Numerical investigation of multistability in the unstable flow of a polymer  solution through porous media,” Physical Review Fluids 6, 033304 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  50M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kumar and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ardekani, “Elastic instabilities between two cylin-  ders confined in a channel,” Physics of Fluids 33, 074107 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  51M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Khan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sasmal, “Effect of micelle breakage rate on flows  of wormlike micellar solutions through pore throats,” Journal of Non-  Newtonian Fluid Mechanics 307, 104853 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  52M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Khan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sasmal, “Elastic instabilities and bifurcations in flows  of wormlike micellar solutions past single and two vertically aligned mi-  crocylinders: Effect of blockage and gap ratios,” Physics of Fluids 33,  033109 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  53M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Khan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sasmal, “Effect of chain scission on flow characteristics  of wormlike micellar solutions past a confined microfluidic cylinder: A  numerical analysis,” Soft Matter 16, 5261–5272 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  54A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Chauan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Gupta, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sasmal, “Effect of geometric disorder on  chaotic viscoelastic porous media flows,” Physics of Fluids 34, 093105  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  55V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “Elastic turbulence: an experimental view on inertialess ran-  dom flow,” Annual Review of Fluid Mechanics 53, 27–58 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  56V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “New direction and perspectives in elastic instability and tur-  bulence in various viscoelastic flow geometries without inertia,” Low Tem-  perature Physics 48, 492–507 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  57H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bruus, Theoretical Microfluidics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 18 (Oxford University Press,  2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  58N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kockmann, Transport phenomena in micro process engineering  (Springer Science & Business Media, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  59P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Panigrahi, Transport Phenomena in Microfluidic Systems (John Wiley  & Sons, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  60H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Meijer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Singh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Den Toonder, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Anderson, “Passive and active mixing in microfluidic devices,” in Macro-  molecular Symposia, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 279 (Wiley Online Library, 2009) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 201–209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  61D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Di Carlo, “Inertial microfluidics,” Lab on a Chip 9, 3038–3046 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  62M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinke and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kandlikar, “Single-phase heat transfer enhance-  ment techniques in microchannel and minichannel flows,” in Interna-  tional Conference on Nanochannels, Microchannels, and Minichannels,  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 41642 (2004) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 141–148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  63L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Léal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Miscevic, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lavieille, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Amokrane, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pigache, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Topin,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Nogarède, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Tadrist, “An overview of heat transfer enhancement  methods and new perspectives: Focus on active methods using electroac-  tive materials,” International Journal of heat and mass transfer 61, 505–524  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  64K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ward and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fan, “Mixing in microfluidic devices and enhancement  methods,” Journal of Micromechanics and Microengineering 25, 094001  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  65L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Capretto, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Cheng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hill, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhang, “Micromixing within mi-  crofluidic devices,” Microfluidics , 27–68 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  66V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hessel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Löwe, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Schönfeld, “Micromixers—a review on passive  and active mixing principles,” Chemical Engineering Science 60, 2479–  2501 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  67M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sheikholeslami, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Gorji-Bandpy, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ganji, “Review of heat  transfer enhancement methods: Focus on passive methods using swirl flow  devices,” Renewable and Sustainable Energy Reviews 49, 444–469 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  68S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Cheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Cai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, “A state-of-  the-art overview on the developing trend of heat transfer enhancement by  single-phase flow at micro scale,” International Journal of Heat and Mass  Transfer 143, 118476 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  69D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Shah, Improved oil recovery by surfactant and polymer flooding (El-  sevier, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  70A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Firozjaii and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Saghafi, “Review on chemical enhanced oil re-  covery using polymer flooding: Fundamentals, experimental and numeri-  cal simulation,” Petroleum 6, 115–122 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  71A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Groisman and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “Elastic turbulence in curvilinear flows of  polymer solutions,” New Journal of Physics 6, 29 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  72F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kinoshita, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Oishi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fujii, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Oshima, “Cre-  ation of very-low-reynolds-number chaotic fluid motions in microchannels  using viscoelastic surfactant solution,” Experimental Thermal and Fluid  Science 34, 20–27 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  73R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Poole, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Budhiraja, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Cain, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Scott, “Emulsification using elas-  tic turbulence,” Journal of Non-Newtonian Fluid Mechanics 177, 15–18  (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  74T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Burghelea, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Segre, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bar-Joseph, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Groisman, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg,  “Chaotic flow and efficient mixing in a microchannel with a polymer so-  lution,” Physical Review E 69, 066305 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  75K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Tatsumi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Takeda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Suga, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Nakabe, “Turbulence character-  istics and mixing performances of viscoelastic fluid flow in a serpentine  microchannel,” in Journal of Physics: Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 318 (IOP  Publishing, 2011) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 092020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  76H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Cao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Tomoaki, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yu, “Direct numerical  simulation of elastic turbulence and its mixing-enhancement effect in a  straight channel flow,” Chinese Physics B 22, 024703 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  77F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Cao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Tomoaki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Haruyuki, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Marie,  “A purely elastic instability and mixing enhancement in a 3d curvilinear  channel flow,” Chinese Physics Letters 29, 094704 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  78M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Grilli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Vázquez-Quesada, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ellero, “Transition to turbulence  and mixing in a viscoelastic fluid flowing inside a channel with a periodic  array of cylindrical obstacles,” Physical review letters 110, 174501 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  79H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Gan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lam, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Nguyen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Tam, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yang, “Efficient  mixing of viscoelastic fluids in a microchannel at low reynolds number,”  Microfluidics and Nanofluidics 3, 101–108 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  80H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Gan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lam,  and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Nguyen, “Polymer-based device for  efficient mixing of viscoelastic fluids,” Applied Physics Letters 88, 224103  (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  81S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Park, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lee, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kim,  “Gear-shaped micromixer for synthesis of silica particles utilizing inertio-  elastic flow instability,” Lab on a Chip 21, 513–520 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  82H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yao, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wen, “Efficient mixing enhancement by orthog-  onal injection of shear-thinning fluids in a micro serpentine channel at low  reynolds numbers,” Chemical Engineering Science 235, 116368 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  83R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bryce and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Freeman, “Abatement of mixing in shear-free elonga-  tionally unstable viscoelastic microflows,” Lab on a Chip 10, 1436–1441  (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  84M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Khan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sasmal, “Electro-elastic instability in electroosmotic  flows of viscoelastic fluids through a model porous system,” European  Journal of Mechanics-B/Fluids 97, 173–186 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  85C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sasmal, “A simple yet efficient approach for electrokinetic mixing of  viscoelastic fluids in a straight microchannel,” Scientific Reports 12, 1–13  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  86R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Whalley, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Abed, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Dennis, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Poole, “Enhancing heat trans-  fer at the micro-scale using elastic turbulence,” Theoretical and Applied  Mechanics Letters 5, 103–106 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  87W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Abed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Whalley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Dennis, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Poole, “Experimental  25  investigation of the impact of elastic turbulence on heat transfer in a ser-  pentine channel,” Journal of Non-Newtonian Fluid Mechanics 231, 68–78  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  88D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Copeland, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ren, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Su, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ligrani, “Elastic turbulence influ-  ences and convective heat transfer within a miniature viscous disk pump,”  International Journal of Heat and Mass Transfer 108, 1764–1774 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  89G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Shen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wen, “Effects of rheologi-  cal properties on heat transfer enhancements by elastic instability in von-  karman swirling flow,” International Journal of Heat and Mass Transfer  152, 119535 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  90G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhao, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wen, “Experimental study on flow and  heat transfer enhancement by elastic instability in swirling flow,” Interna-  tional Journal of Thermal Sciences 157, 106504 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  91H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yao, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wen, “Flow resistance and convective heat trans-  fer by elastic turbulence in 1d/2d/3d geometries,” International Journal of  Thermal Sciences 176, 107512 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  92D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Qian,  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Joo,  “Measuring heat transfer performance of viscoelastic fluid flow in curved  microchannel using Ti–Pt film temperature sensor,” Experimental Thermal  and Fluid Science 77, 226–233 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  93D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Qian, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Joo, “Efficient heat trans-  fer enhancement by elastic turbulence with polymer solution in a curved  microchannel,” Microfluidics and Nanofluidics 21, 10 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  94D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Blanchard, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ligrani,  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Gale, “Miniature single-disk viscous  pump (single-dvp), performance characterization,” (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  95B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Traore, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Castelain, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Burghelea, “Efficient heat transfer in a  regime of elastic turbulence,” Journal of Non-Newtonian Fluid Mechanics  223, 62–76 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  96T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Burghelea, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Segre, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “Mixing by polymers: Experi-  mental test of decay regime of mixing,” Physical review letters 92, 164501  (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  97D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Cheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Qian, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Joo,  “Numerical simulation of heat transfer enhancement by elastic turbulence  in a curvy channel,” Microfluidics and Nanofluidics 21, 1–16 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  98S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Chauhan,  and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sasmal, “Influence of elastic instabil-  ity and elastic turbulence on mixed convection of viscoelastic fluids in  a lid-driven cavity,” International Journal of Heat and Mass Transfer 186,  122469 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  99H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Cai, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, “Numerical  simulation of heat transfer process of viscoelastic fluid flow at high weis-  senberg number by log-conformation reformulation,” Journal of Fluids En-  gineering 139 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  100L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Casanellas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Alves, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Poole, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lerouge, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lindner, “The  stabilizing effect of shear thinning on the onset of purely elastic instabili-  ties in serpentine microflows,” Soft Matter 12, 6167–6175 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  101A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Clarke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Howe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mitchell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Staniland,  and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hawkes,  “How viscoelastic-polymer flooding enhances displacement efficiency,”  SPE Journal 21, 0675–0687 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  102A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Clarke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Howe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mitchell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Staniland, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hawkes,  and  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Leeper, “Mechanism of anomalously increased oil displacement with  aqueous viscoelastic polymer solutions,” Soft Matter 11, 3536–3541  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  103J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mitchell, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lyons, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Howe, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Clarke, “Viscoelastic polymer  flows and elastic turbulence in three-dimensional porous structures,” Soft  Matter 12, 460–468 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  104R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hincapie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Rock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wegner, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ganzer, “Oil mobilization by  viscoelastic flow instabilities effects during polymer eor: A pore-scale vi-  sualization approach,” in SPE Latin America and Caribbean Petroleum  Engineering Conference (OnePetro, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  105A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Rock, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hincapie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wegner, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ganzer, “Advanced flow behav-  ior characterization of enhanced oil recovery polymers using glass-silicon-  glass micromodels that resemble porous media,” in SPE Europec featured  at 79th EAGE conference and exhibition (OnePetro, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  106R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Du, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Peng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Tao, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pang, “Viscoelastic dis-  placement and anomalously enhanced oil recovery of a novel star-like  amphiphilic polyacrylamide,” Chemical Engineering Research and Design  142, 369–385 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  107S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' De, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Krishnan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Van Der Schaaf, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kuipers, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Peters,  and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Padding, “Viscoelastic effects on residual oil distribution in flows  through pillared microchannels,” Journal of Colloid and Interface Science  510, 262–271 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  108J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Fan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Liu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wu,  “Molecular mechanism of viscoelastic polymer enhanced oil recovery in  nanopores,” Royal Society Open Science 5, 180076 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  109D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xia, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Liu,  and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yang, “Study of the mechanism of  polymer solution with visco-elastic behavior increasing microscopic oil  displacement efficiency and the forming of steady oil thread flow chan-  nels,” in SPE Asia Pacific oil and gas conference and exhibition (OnePetro,  2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  110H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lu,  and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Guo, “Microflow  mechanism of oil displacement by viscoelastic hydrophobically associat-  ing water-soluble polymers in enhanced oil recovery,” Polymers 10, 628  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  111B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Vik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kedir, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kippe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sandengen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Skauge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Solbakken, and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhu, “Viscous oil recovery by polymer injection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' impact of in-situ poly-  mer rheology on water front stabilization,” in SPE Europec featured at  80th EAGE Conference and Exhibition (OnePetro, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  112I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Salmo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zamani, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Skauge, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sorbie, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Skauge, “Use of  dynamic pore network modeling to improve our understanding of exper-  imental observations in viscous oil displacement by polymers,” in SPE  Improved Oil Recovery Conference (OnePetro, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  113C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xie, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mohanty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wang, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Balhoff, “Nonwetting  droplet oscillation and displacement by viscoelastic fluids,” Physical Re-  view Fluids 5, 063301 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  114A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mohamed, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Khishvand, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Piri, “The role of injection fluid  elasticity in microscopic displacement efficiency of residual non-wetting  phase: An in-situ experimental investigation,” Fuel 333, 126180 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  115C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xie, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Qi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Balhoff, “Oscillative trapping of  a droplet in a converging channel induced by elastic instability,” Physical  Review Letters 128, 054502 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  116P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Qi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ehrenfried, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Koh, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Balhoff, “Reduction of residual  oil saturation in sandstone cores by use of viscoelastic polymers,” SPE  Journal 22, 447–458 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  117M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Irfan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Stephen, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lenn, “An experimental study to inves-  tigate novel physical mechanisms that enhance viscoelastic polymer flood-  ing and further increase desaturation of residual oil saturation,” Upstream  Oil and Gas Technology 6, 100026 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  118S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Parsa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Santanach-Carreras, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xiao, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Weitz, “Origin of  anomalous polymer-induced fluid displacement in porous media,” Physical  Review Fluids 5, 022001 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  119P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Druetta and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Picchioni, “Influence of physical and rheological prop-  erties of sweeping fluids on the residual oil saturation at the micro-and  macroscale,” Journal of Non-Newtonian Fluid Mechanics 286, 104444  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  120P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kundu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Cohen, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Dowling, Fluid Mechanics (Aca-  demic press, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  121D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' James, “Boger fluids,” Annual Review of Fluid Mechanics 41, 129–  142 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  122K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Jackson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Walters, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Williams, “A rheometrical study of boger  fluids,” Journal of Non-Newtonian Fluid Mechanics 14, 173–188 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  123M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Verhoef, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Van den Brule, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hulsen, “On the modelling of a  pib/pb boger fluid in extensional flow,” Journal of Non-Newtonian Fluid  Mechanics 80, 155–182 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  124Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Joo and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Shaqfeh, “Observations of purely elastic instabilities  in the taylor–dean flow of a boger fluid,” Journal of Fluid Mechanics 262,  27–73 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  125M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Alves, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pinho,  and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Oliveira, “Visualizations of boger fluid  flows in a 4: 1 square–square contraction,” AIChE journal 51, 2908–2922  (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  126E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bot, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hulsen, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Van den Brule, “The motion of two spheres  falling along their line of centres in a boger fluid,” Journal of Non-  Newtonian Fluid Mechanics 79, 191–212 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  127D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' James and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Roos, “Pressure drop of a boger fluid in a converging  channel,” Journal of Non-Newtonian Fluid Mechanics 293, 104557 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  128P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sousa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Coelho, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Oliveira, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Alves, “Three-dimensional flow  of newtonian and boger fluids in square–square contractions,” Journal of  Non-Newtonian Fluid Mechanics 160, 122–139 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  129Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wei, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Seevaratnam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Garoff, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ramé, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Walker, “Dynamic  wetting of boger fluids,” Journal of Colloid and Interface science 313, 274–  280 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  26  130E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mitsoulis, “Extrudate swell of boger fluids,” Journal of Non-Newtonian  Fluid Mechanics 165, 812–824 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  131S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Anna, “Non-newtonian fluids in microfluidics,” in Encyclopedia of  Microfluidics and Nanofluidics, edited by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li (Springer US, Boston, MA,  2008) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 1480–1488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  132L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mei and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Qian, “Editorial for the special issue on micromachines for  non-newtonian microfluidics,” (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  133P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Nghe, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Terriac, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Schneider, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Cloitre, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Abecassis, and  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Tabeling, “Microfluidics and complex fluids,” Lab on a Chip 11, 788–  794 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  134S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Haward, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Odell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Berry, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hall, “Extensional rheology of  human saliva,” Rheologica Acta 50, 869–879 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  135G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Juarez and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Arratia, “Extensional rheology of dna suspensions in  microfluidic devices,” Soft Matter 7, 9444–9452 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  136I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bloomfield, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Johnston,  and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bilston, “Effects of proteins, blood  cells and glucose on the viscosity of cerebrospinal fluid,” Pediatric Neuro-  surgery 28, 246–251 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  137R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Chhabra and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Richardson, Non-Newtonian Flow and Applied  Rheology: Engineering Applications (Butterworth-Heinemann, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  138M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Everts and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Meyer, “Heat transfer of developing and fully devel-  oped flow in smooth horizontal tubes in the transitional flow regime,” In-  ternational Journal of Heat and Mass Transfer 117, 1331–1351 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  139A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Varshney and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “Elastic alfven waves in elastic turbulence,”  Nature Communications 10, 1–7 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  140A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Varshney and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Steinberg, “Mixing layer instability and vorticity am-  plification in a creeping viscoelastic flow,” Physical Review Fluids 3,  103303 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  141M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Khan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sasmal, “Electro-elastic instability in electroosmotic  flows of viscoelastic fluids through a model porous system,” European  Journal of Mechanics-B/Fluids (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  142S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kandlikar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Garimella, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Colin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' King, Heat transfer  and fluid flow in minichannels and microchannels (elsevier, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  143S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ghiaasiaan, Two-phase flow, boiling, and condensation: in conven-  tional and miniature systems (Cambridge University Press, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  144S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Kim and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mudawar, “Review of databases and predictive meth-  ods for heat transfer in condensing and boiling mini/micro-channel flows,”  International Journal of Heat and Mass Transfer 77, 627–652 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  145L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Cheng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mewes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Luke, “Boiling phenomena with surfactants  and polymeric additives: a state-of-the-art review,” International Journal  of Heat and Mass Transfer 50, 2744–2771 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  146J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Feng, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lin, “Fluid mixing in droplet-based  microfluidics with a serpentine microchannel,” RSC Advances 5, 104138–  104144 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  147H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Song, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Chen, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ismagilov, “Reactions in droplets in mi-  crofluidic channels,” Angewandte Chemie International Edition 45, 7336–  7356 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  148R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sun, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Chen, “Droplet-based microfluidics for cell en-  capsulation and delivery,” in Microfluidics for Pharmaceutical Applica-  tions (Elsevier, 2019) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 307–335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  149J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Clausell-Tormos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lieber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Baret, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' El-Harrak, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Miller,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Frenz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Blouwolff, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Humphry, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Köster, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Duan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=', “Droplet-  based microfluidic platforms for the encapsulation and screening of mam-  malian cells and multicellular organisms,” Chemistry & Biology 15, 427–  437 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  150L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mazutis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Gilbert, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ung, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Weitz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Griffiths, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  Heyman, “Single-cell analysis and sorting using droplet-based microflu-  idics,” Nature Protocols 8, 870–891 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  151N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Shembekar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Chaipan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Utharala, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Merten, “Droplet-based  microfluidics in drug discovery, transcriptomics and high-throughput  molecular genetics,” Lab on a Chip 16, 1314–1331 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  152W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Liu and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhu, ““development and application of analytical de-  tection techniques for droplet-based microfluidics”-a review,” Analytica  Chimica Acta 1113, 66–84 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  153R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Seemann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Brinkmann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pfohl,  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Herminghaus, “Droplet  based microfluidics,” Reports on Progress in Physics 75, 016601 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  154Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xuan and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, “Heat transfer enhancement of nanofluids,” Interna-  tional Journal of heat and fluid flow 21, 58–64 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  155S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Das, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Choi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yu, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pradeep, Nanofluids: science and  technology (John Wiley & Sons, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  156A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Chamkha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Molana, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Rahnama,  and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ghadami, “On the  nanofluids applications in microchannels: a comprehensive review,” Pow-  der Technology 332, 287–322 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  157H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mohammed, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bhaskaran, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Shuaib, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Saidur, “Heat trans-  fer and fluid flow characteristics in microchannels heat exchanger using  nanofluids: a review,” Renewable and Sustainable Energy Reviews 15,  1502–1512 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  158K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ramesh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sharma, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Rao, “Latest advancements in heat  transfer enhancement in the micro-channel heat sinks: a review,” Archives  of Computational Methods in Engineering 28, 3135–3165 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  159J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' He,  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Jiang, “Ex-  perimental investigation on the thermal conductivity and shear viscosity  of viscoelastic-fluid-based nanofluids,” International Journal of Heat and  Mass Transfer 55, 3160–3166 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  160J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' He, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Huang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Jiang, “Experi-  mental study on the characteristics of heat transfer and flow resistance in  turbulent pipe flows of viscoelastic-fluid-based cu nanofluid,” International  Journal of Heat and Mass Transfer 62, 303–313 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  161T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hayat, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Haider, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Muhammad,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Alsaedi, “On darcy-  forchheimer flow of viscoelastic nanofluids: A comparative study,” Journal  of Molecular Liquids 233, 278–287 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  162J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' He, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Huang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Jiang, “Heat transfer perfor-  mance of viscoelastic-fluid-based nanofluid pipe flow at entrance region,”  Experimental Heat Transfer 28, 125–138 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  163P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Adler and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Brenner, “Multiphase flow in porous media,” Annual  Review of Fluid Mechanics 20, 35–59 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  164A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Gerami, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Armstrong, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Johnston, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Warkiani, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mosavat,  and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mostaghimi, “Coal-on-a-chip: visualizing flow in coal fractures,”  Energy & Fuels 31, 10393–10403 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  165K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' He, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Gao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yin, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Neeves, “Evaluation of surfactant  performance in fracturing fluids for enhanced well productivity in uncon-  ventional reservoirs using rock-on-a-chip approach,” Journal of Petroleum  Science and Engineering 135, 531–541 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  166S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Torabzadey, “The effect of temperature and interfacial tension on wa-  ter/oil relative permeabilities of consolidated sands,” in SPE Enhanced Oil  Recovery Symposium (OnePetro, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  167Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Qin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Liu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhao, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yuan, “Experimental studies on  effects of temperature on oil and water relative permeability in heavy-oil  reservoirs,” Scientific Reports 8, 1–9 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  168F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Afolabi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Mahmood, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yekeen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Akbari, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sharifigal-  iuk, “Polymeric surfactants for enhanced oil recovery: A review of re-  cent progress,” Journal of Petroleum Science and Engineering 208, 109358  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  169P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Raffa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Broekhuis, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Picchioni, “Polymeric surfactants for  enhanced oil recovery: A review,” Journal of Petroleum Science and Engi-  neering 145, 723–733 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  170F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Souas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Safri, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Benmounah, “On the rheological behavior of  light crude oil: a review,” Petroleum Science and Technology 38, 849–857  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  171T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ariffin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yahya, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Husin, “The rheology of light crude oil  and water-in-oil-emulsion,” Procedia Engineering 148, 1149–1155 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  172S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Hasan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ghannam, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Esmail, “Heavy crude oil viscosity  reduction and rheology for pipeline transportation,” Fuel 89, 1095–1100  (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  173S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Livescu, “Mathematical modeling of thixotropic drilling mud and crude  oil flow in wells and pipelines—a review,” Journal of Petroleum Science  and Engineering 98, 174–184 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  174H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhang, “A comprehensive investigation of the vis-  coelasticity and time-dependent yielding transition of waxy crude oils,”  Journal of Rheology 62, 527–541 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  175L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Guo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Lei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Yu,  and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xu, “Study on the  viscoelastic-thixotropic characteristics of waxy crude oil based on stress  loading,” Journal of Petroleum Science and Engineering 208, 109159  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  176F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pimenta and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Alves, “Electro-elastic instabilities in cross-shaped  microchannels,” Journal of Non-Newtonian Fluid Mechanics 259, 61–77  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  177S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Sadek, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Pinho, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Alves, “Electro-elastic flow insta-  bilities of viscoelastic fluids in contraction/expansion micro-geometries,”  Journal of Non-Newtonian Fluid Mechanics 283, 104293 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  178J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Masliyah and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Bhattacharjee, Electrokinetic and Colloid Transport  27  Phenomena (John Wiley & Sons, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  179G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Niu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhao, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ding, “Experimen-  tal and numerical investigation of a microchannel heat sink (mchs) with  micro-scale ribs and grooves for chip cooling,” Applied Thermal Engi-  neering 85, 61–70 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  180K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Vafai and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Zhu, “Analysis of two-layered micro-channel heat sink  concept in electronic cooling,” International Journal of Heat and Mass  Transfer 42, 2287–2297 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  181A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Demello, “Control and detection of chemical reactions in microfluidic  systems,” Nature 442, 394–402 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  182Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Alihosseini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Targhi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Heyhat, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Ghorbani, “Effect of  a micro heat sink geometric design on thermo-hydraulic performance: A  review,” Applied Thermal Engineering 170, 114974 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content='  183M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Doi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Edwards, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' Edwards, The theory of polymer dynam-  ics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
+page_content=' 73 (oxford university press, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E0T4oBgHgl3EQfewCK/content/2301.02395v1.pdf'}
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+MORPHOLOGY-BASED NON-RIGID REGISTRATION OF
+CORONARY COMPUTED TOMOGRAPHY AND INTRAVASCULAR
+IMAGES THROUGH VIRTUAL CATHETER PATH OPTIMIZATION
+Karim Kadry∗
+Institute of Medical Engineering and Science
+Massachusetts Institute of Technology
+Cambridge, MA 02139
+kkadry@mit.edu
+Abhishek Karmakar
+Meinig School of Biomedical Engineering
+Cornell University
+Ithaca, NY 14850
+ak944@cornell.edu
+Andreas Schuh
+Biomedical Image Analysis Group
+Imperial College London
+HeartFlow, Inc., USA
+London, UK
+aschuh@heartflow.com
+Kersten Peterson
+HeartFlow, Inc., USA
+Redwood City, CA, 94063, USA
+kpetersen@heartflow.com
+Michiel Schaap
+HeartFlow, Inc., USA
+Redwood City, CA, 94063, USA
+mschaap@heartflow.com
+David Marlevi
+Department of Molecular Medicine and Surgery
+Karolinska Institute
+Stockholm, Sweden
+david.marlevi@ki.se
+Charles Taylor
+Department of Electrical Engineering
+HeartFlow, Inc., USA
+Redwood City, CA, 94063, USA
+ctaylor@heartflow.com
+Elazer Edelman
+Institute of Medical Engineering and Science
+Massachusetts Institute of Technology
+Cambridge, MA 02139
+ere@mit.edu
+Farhad Nezami
+Department of Surgery
+Brigham and Women’s Hospital Harvard Medical School
+Boston, MA 02115
+frikhtegarnezami@bwh.harvard.edu
+ABSTRACT
+Coronary Computed Tomography Angiography (CCTA) provides information on the presence, extent,
+and severity of obstructive coronary artery disease. Large-scale clinical studies analyzing CCTA-
+derived metrics typically require ground-truth validation in the form of high-fidelity 3D intravascular
+imaging. However, manual rigid alignment of intravascular images to corresponding CCTA images is
+both time consuming and user-dependent. Moreover, intravascular modalities suffer from several
+non-rigid motion-induced distortions arising from distortions in the imaging catheter path. To address
+these issues, we here present a semi-automatic segmentation-based framework for both rigid and
+non-rigid matching of intravascular images to CCTA images. We formulate the problem in terms
+of finding the optimal virtual catheter path that samples the CCTA data to recapitulate the coronary
+artery morphology found in the intravascular image. We validate our co-registration framework on a
+cohort of n = 40 patients using bifurcation landmarks as ground truth for longitudinal and rotational
+registration. Our results indicate that our non-rigid registration significantly outperforms other co-
+registration approaches for luminal bifurcation alignment in both longitudinal (mean mismatch: 3.3
+frames) and rotational directions (mean mismatch: 28.6 degrees). By providing a differentiable
+framework for automatic multi-modal intravascular data fusion, our developed co-registration modules
+∗Corresponding Author
+arXiv:2301.00060v1  [cs.CV]  30 Dec 2022
+
+arXiv Template
+A PREPRINT
+significantly reduces the manual effort required to conduct large-scale multi-modal clinical studies
+while also providing a solid foundation for the development of machine learning-based co-registration
+approaches.
+1
+Introduction
+Coronary computed tomography angiography (CCTA) is a three dimensional image modality that provides information
+on the presence, extent and severity of obstructive coronary artery disease (CAD) (Tzimas et al. [2022]). As such, CCTA
+allows for the detection of stenotic atherosclerotic sections and assists clinicians in diagnosing CAD and planning
+treatment. CCTA Images can also be used to create computational models of coronary blood flow, allowing for
+the non-invasive estimation of fractional flow reserve (FFR-CT); a key diagnostic parameter in assessing functional
+impairment (Uzu et al. [2019]).
+Albeit widespread in use, CCTA provides primary information on luminal anatomy, with limited capacity in assessing
+soft-tissue intraplaque tissue components. CCTA also suffers from blooming artifacts in the presence of highly
+attenuating calcium deposits (Kim et al. [2015], Budoff et al. [2008]), which, combined with comparably low image
+resolution, creates difficulties in resolving highly calcified arteries. Multiple studies have also been conducted to
+quantify the degree to which CCTA can accurately assess CAD-related diagnostic metrics such as luminal area (Uzu
+et al. [2019]), calcium morphology (Takahashi et al. [2021]), and plaque burden (Fischer et al. [2013], De Graaf et al.
+[2013], Brodoefel et al. [2009]). The majority of such studies (Takahashi et al. [2021], Fischer et al. [2013], Uzu et al.
+[2019], Brodoefel et al. [2009]) validate the performance of CCTA by manually co-registering image slices taken along
+the CCTA artery to intravascular imaging modalities such as intravascular ultrasound (IVUS) and optical coherence
+tomography (OCT); both providing higher-fidelity visualization of the lumen and surrounding tissue. There is also an
+increasing interest in validating CCTA-derived segmentation algorithms against co-registered intravascular imaging
+frames, again necessitating such multimodal image assessment (Lin et al. [2021], van Assen et al. [2019]).
+Manual co-registeration of CCTA and intravascular images is, however, a challenging and time consuming task.
+Typically, cross-sectional frames of the artery of interest are extracted from the CCTA images which then have to be
+matched with corresponding frames from an intravascular acquisition through an imaging catheter pullback procedure.
+Rigid registration in the longitudinal and rotational directions is usually achieved by matching single landmarks in both
+modalities, such as a large bifurcation (Takahashi et al. [2021]). However, the beating of the heart, the irregular motion
+of the imaging catheter, and the rotation of the catheter about its own axis create non-rigid distortions that accumulate
+along the length of the pullback (Tsiknakis et al. [2021]). Manually correcting for such artifacts is prohibitively
+time-consuming, requiring a cardiologist to manually mark fiduciary points in both images and shift images such that
+the annotated points sufficiently align (Carlier et al. [2014], Tu et al. [2011], Hebsgaard et al. [2015]). Although such
+techniques are accurate up to rigid translation, they require time investment from a trained expert to find matching
+features in both modalities, creating a need for computational algorithms that non-rigidly register CCTA images to
+corresponding intravascular data in an automatic fashion.
+Automatic co-registration techniques typically consist of discretely optimizing a constructed cost function over a set of
+longitudinal or rotational image shifts, where the cost function varies depending on the modalities being registered.
+Some proposed cost functions include metrics such as lumen diameters (Qin et al. [2021]), lumen contours (Molony
+et al. [2016], Karmakar et al. [2020]), calcium thickness (Gharaibeh et al. [2020], Molony et al. [2016]), and image
+pixel intensities (Tsiknakis et al. [2021]). Similarly, rigid rotational registration for intravascular pullbacks has also
+been based on extracted features such as luminal contours (Karmakar et al. [2020]), and calcium angle (Molony et al.
+[2016]). However, the registration accuracy of all rigid registration methods is compromised by inconsistent motor
+pullback speeds and rotational drift, which introduce non-rigid longitudinal and rotational distortions that misalign
+image features such as diseased plaque and bifurcations.
+To compensate for the longitudinal, rotational, and transverse motion of the catheter, several non-rigid registration
+approaches have been proposed, typically to be employed after initial rigid alignment. Currently, non-rigid registration
+of multiple intravascular imaging datasets has been predominantly performed through Dynamic Time Warping (DTW)
+and Dynamic Programming (DP) (Tsiknakis et al. [2021], Molony et al. [2016]). However, DTW introduces non-
+physiological assumptions into the registration process by discretely skipping or repeating intravascular frames, assumed
+to be evenly spaced along the longitudinal direction. As a result, DTW is not well suited for use for intravascular images,
+with pullback acquisitions sometimes rendering up to 10 repeated intravascular imaging frames at a time (Molony et al.
+[2016]). On the contrary, continuous non-rigid registration methods have been developed to model the longitudinal
+stretch and rotational drift between intravascular imaging frames using affine transforms and spline interpolation (Zhang
+et al. [2014], Uzu et al. [2019]). While such continuous non-rigid methods are more realistic, they extensively rely on
+manual annotations of all bifurcation zones for image registration, severely limiting their scalability. As such, there is
+2
+
+arXiv Template
+A PREPRINT
+no continuous non-rigid registration method as of yet that does not explicitly require fiduciary landmarks for rotational
+and longitudinal alignment. Further, there has been an increasing interest in machine learning approaches to image
+co-registration in which a neural network is trained to predict a spatial transform that maps a moving image onto a
+static target image (Balakrishnan et al. [2019], Fu et al. [2020]). Such approaches critically rely on a differentiable and
+continuous spatial transform allowing for back-propagation of gradients to adjust the neural network weights (Jaderberg
+et al. [2015]). While such continuous and differential spatial transforms are available for co-registration of 3D and 2D
+medical images, a similar framework that accounts for the unique variation in intravascular catheter motion has not
+been developed.
+Given the previous limitations noted in prior co-registration algorithms, we here propose a novel semi-automatic
+framework that takes as input an intravascular imaging pullback and a CCTA 3D image and aligns each intravascular
+image frame along the artery to the equivalent frame in the CCTA image. The proposed continuous registration
+methodology does not require manual matching of landmarks, with the only manual effort being the selection of viable
+intravascular imaging frames and the provision of a rough centerline within the CT image. Specifically, we explore the
+problem of reconstructing the path of a virtual catheter moving through and sampling from a 3D CCTA image such that
+the set of frames produced by the motion of the catheter optimally reflect the equivalent target intravascular pullback.
+Key contributions of this framework include:
+• We present the first continuous co-registration framework for rigid and non-rigid matching of CCTA images
+and intravascular images up to pixelwise alignment, with segmentations of the lumen and vessel wall as sole
+input.
+• We introduce a rigid registration approach that consists of our published longitudinal rigid registration
+algorithm, which uses lumen area in a multi-step decision process, and a rotational registration step that
+leverages the segmentation of the vessel wall to produce an initial rotational configuration for subsequent
+registration.
+• We introduce a novel non-rigid registration step, based only on the lumen segmentation, which is robust to
+physiological catheter motions. The registration is formulated in terms of finding the path of a virtual catheter,
+which translates the CCTA image into an intravascular-like image by sampling the segmentation along the
+virtual catheter path. The virtual catheter path is reconstructed by spatially deforming the CCTA centerline by
+B-spline deformations formulated in the longitudinal, rotational, and transverse directions, ensuring a smooth
+and physiological reconstruction of catheter motion.
+• Our non-rigid registration module being both continuous and differentiable, allows for easy integration into
+future machine-learning-based approaches for intravascular image registration.
+• We validate in a direct clinical setting, evaluating performance across a multimodal cohort of cardiac
+patients(n = 40) and benchmarking performance against previously developed state-of-the-art approaches.
+2
+Methodology
+An overview of the co-registration pipeline is detailed in Figure 1. In brief, bi-modality images are processed to produce
+binary segmentations of the lumen and vessel wall (section 2.1.1), which are first used in a rigid registration step,
+involving both longitudinal and rotational alignment (section 2.1.2). The rigid registration is then used as an initial
+estimate of a virtual catheter path forming the basis for a non-rigid registration (section 2.1.3). The virtual catheter
+path initially samples the geometry of the CT lumen to produce a virtual imaging pullback that is then compared to a
+Signed Distance Field (SDF) derived from the intravascular equivalent. A non-rigid transformation for the longitudinal,
+rotational, and transverse motion distortions is applied on the virtual catheter path and optimized to align the SDF’s in
+both modalities. The performance of our proposed co-registration algorithm is then validated on a clinical cohort of
+relevant cardiac patients (section 2.2.2)
+2.1
+Co-registration framework
+2.1.1
+Preprocessing
+As the basis for our co-registration pipeline, luminal segmentations from the two different image modalities are provided.
+Starting with the intravascular image set, luminal frame-by-frame segmentations are used to produce an SDF using
+a fast marching method (Treister and Haber [2016]), clamped to only have negative values (indicating that a pixel is
+inside the lumen). Further, the SDF is smoothed in the axial direction with a Gaussian convolutional kernel of size 3
+and standard deviation 0.1 in order to regularize the optimization process.
+3
+
+arXiv Template
+A PREPRINT
+OCT image
+CT image
+Rigid registration
+Non-rigid registration
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+OCT
+Figure 1: Overview of the proposed registration pipeline. The imaging modalities are rigidly co-registered in the
+longitudinal and rotational directions, serving as the basis for the initialization of the virtual pullback trajectory. The
+virtual pullback trajectory is then used to sample a CT lumen signed distance field (SDF), used in direct comparison to
+the equivalent OCT SDF.
+Arc angle (degrees)
+Rigid longitudinal registration
+OCT image
+Lumen
+CT image
+Rigid rotational registration
+Vessel thickness (pixels)
+0
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+OCT
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+OCT
+Figure 2: Overview of the proposed rigid registration pipeline. The lumen segmentation area vectors from both
+modalities are used to rigidly register the modalities in the longitudinal direction using a sliding window approach. The
+longitudinal registration is then used to match each equivalent frame for the rotational registration. The vessel wall
+segmentations are then converted to vessel thickness-arc angle plots and are used to determine an optimal rigid rotation.
+Coupled to the intravascular image set, a corresponding 3D SDF from the CCTA images is generated. Although several
+methods could potentially be applied for such, a convenient approach is to derive the SDF from a computational mesh
+of the coronary tree. Herein, to create an SDF a narrow band is defined within the object mesh boundary, subsequently
+used to compute exact Euclidean distances from each voxel center to the boundary. Outside the object boundary,
+the distance field values are then set to zero. Corresponding binary segmentations can then be produced by simple
+thresholding operations. Using these computational meshes, vessel centerlines are obtained using VMTK (Antiga et al.
+[2008]), generating an array ¯r representing n spatial positions with an axial spacing of 0.2mm. A spatial derivative
+is then applied to the centerline points ¯r, defining a tangent vector T for each point. The two vectors U and V that
+are orthogonal to the tangent vector can then be obtained through the parallel transport method (Guo et al. [2013]),
+ensuring that the vectors V and U remain stable between frames placed along the axial direction. The centerline points
+and the orthogonal vectors hence define a set of frames(¯r,T,U,and V) in 3D space that are used to sample the CCTA
+SDF along an equivalent virtual catheter pullback, with dimensions equalling the intravascular dataset (in our case:
+96x96xNframes with an in-plane resolution of 80 micrometers), all using a curved-planar reformation procedure
+(Kanitsar et al. [2002]). The resulting SDF is then smoothed in the axial direction with a Gaussian convolutional
+kernel of size 3 and standard deviation of 0.1. Through this method, virtual pullbacks of both the lumen and vessel
+segmentations were produced.
+4
+
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+80arXiv Template
+A PREPRINT
+2.1.2
+Rigid registration
+An overview of the rigid registration step can be seen in Figure 2. For the rigid longitudinal registration, the processed
+lumen segmentations are used to create an area vector of equal lengths, sampling the CT virtual pullback to correspond
+to the acquired intravascualr set. Here, We leverage our previous work to rigidly align the pullbacks using a multi-step
+sliding window method, minimizing the difference in area vectors (for details see (Karmakar et al. [2020])). Before
+registration, continuous segments of the OCT pullback with poor lumen segmentations due to residual blood or catheter
+housing were manually excluded.
+For rigid rotational registration, the luminal profiles were deemed unreliable for producing good alignment. Therefore,
+the vessel border segmentations were instead used for rotationally aligning the pullbacks. For each CT and intravascular
+frame, respectively, a thickness-arc angle vector is extracted by tracing a set of radial rays from the centroid of the
+vessel segmentation in increments of 12 degrees. The thickness vectors are then matched according to the result of the
+longitudinal registration, with non-overlapping frames subsequently cropped. The optimal rigid rotation angle is then
+obtained by sliding the set of CT thickness vectors over each equivalent intravascular image vector and minimizing the
+mean squared error across all frames.
+2.1.3
+Non-rigid registration
+The non-rigid registration process (Figure 3) consists of optimizing a set of frame variables (¯r, T, U, and V) representing
+a virtual catheter path moving through the CCTA image. The loss function to be optimized is defined as the mean squared
+error between the 3D SDF generated from the two image sets, with the CCTA-SDF sampled along the aforementioned
+virtual catheter path. The virtual pullback is initialized as the centerline that was calculated from the CCTA 3D model
+and longitudinally cropped and rotated according to the output of the rigid registration. After rigid registration a spline
+is defined based on the centerline points ¯r where the centerline points are fully described by their arclength values ¯s
+along the spline. Accordingly, every i-th frame can be manipulated by 4 variables, representing the arclength along
+the virtual catheter path si, the rotation angle of the frame θi about the catheter path T, and the in-plane transverse
+displacements du and dv along the frame vectors U and V respectively (see Figure 3).
+To regularize the motion of the virtual catheter to be smooth and physiological, the 4 frame manipulation variable
+sets are parametrized by a sparse set of control points controlling a B-spline deformation (Rueckert et al. [1999])
+independently acting on 4 nx1 vectors representing the frame manipulation variables ¯s, ¯θ, ¯du, and ¯dv. Thus, for a 1D
+control point grid of size N, the relation between a frame manipulation variable v and the control points p can be
+described by:
+v(s) =
+N
+�
+i=0
+Bi(s)pi,
+(1)
+where Bi(s) is a polynomial basis function of order 2. In matrix form, the same can be represented by:
+V = BP,
+(2)
+in which V ∈ Rn×1, B ∈ Rn×N, P ∈ RN×1 where n is the number of frames and N is the number of control points. B
+is the univariate B-spline tensor and can be pre-computed from the initial frame manipulation variable vectors, while P
+is the deformed control point grid vector that is optimized during co-registration.
+Instead of directly optimizing for the set of Ns = 30 control points P s controlling the arclength variables s for each
+frame, the control point deformations ∆P s
+i can be parametrized by a deformation vector Xs of size Ns − 1. dictating
+the relative displacement of each control point from its proximal neighbor, with the most proximal control point being
+fixed. This is done to account for the cumulative effect of catheter motor speed variations on the rest of the pullback.
+Therefore, the deformation of each control point can be defined as the cumulative sum of the relative deformations
+along the proximal control points. Moreover, to regularize the catheter motion and prevent backwards movement, the
+relative deformation of each control point is limited to a fraction (0.35) of the distance between control points.
+∆P s
+i = Xs
+i +
+i−1
+�
+j=0
+Xs
+j
+(3)
+Once the control points are deformed into a new configuration, the new arclength values for each frame ¯s is calculated
+through equation 2 and the frame vectors (T,U, and V) are then recalculated.
+¯s = BsP s
+(4)
+5
+
+arXiv Template
+A PREPRINT
+Rigid initialization
+Non-rigid transform
+Rotational transform
+Longitudinal transform
+Transverse transform
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+V-displacement
+OCT: Target
+Loss
+CT: Moving
+Figure 3: Overview of the spatial deformation acting on the virtual catheter path. The longitudinal transform stretches
+and compresses the space between adjacent frames, at which point the frame vectors (T,U, and V) are recalculated. The
+rotational transform is then applied to the frame vectors orthogonal to the tangent (U and V) about T, and the transverse
+transform is then applied to shift the centerline points in the direction of the new frame vectors (U and V).
+2.1.4
+Non-rigid rotational registration
+Similar to the longitudinal registration, the set of Nθ = 20 control points P θ controlling the rotation of each frame
+about the catheter axis can be parameterized by a relative rotation vector Xθ of size Nθ. The rotation value for each
+control points is defined by:
+∆P θ
+i = Xθ
+i +
+i−1
+�
+j=0
+Xθ
+j
+(5)
+The rotation correction for each frame is applied after the non-rigid longitudinal transformation but before the non-rigid
+transverse transformation. Once the control points are deformed into a new configuration, the new rotation values for
+each frame ¯θ can be calculated through equation 2 and used to rotate frame vectors U and V about the tangent vectors T.
+¯θ = BθP θ
+(6)
+2.1.5
+Non-rigid transverse registration
+The virtual catheter was biased to stay close to the centerline by optimizing the Nd = 60 control points determining the
+in-plane transverse displacements du and dv directly. Thus the 2 orthogonal transverse displacements for each frame
+was calculated from the matrix relation:
+¯d = BdP d
+(7)
+Where for each frame the displacements along the vectors U and V were applied as a final step after the non-rigid
+longitudinal and rotational transforms.
+2.2
+Performance evaluation
+2.2.1
+Image data
+To evalute our proposed co-registration framework, a dataset consisting of n = 40 matched OCT and CT image pairs
+from 5 different clinical centers were selected, all originating from the Precise Percutaneous Coronary Intervention Plan
+(P3) study (Nagumo et al. [2021]). As each OCT pullback image consisted of 375 frames, the intravascular imaging
+dataset comprised of approximately 15,000 image frames. The OCT lumen in every frame was manually annotated by
+6
+
+arXiv Template
+A PREPRINT
+trained cardiologists. Further, the vessel wall borders in every OCT frame were segmented using a convolutional neural
+network, using the previously published U-net as base architecture (Ronneberger et al. [2015]). Details of the network,
+training, and validation can be found in Supplementary Material A. The lumen and vessel wall segmentations were then
+re-sampled to represent a 3D image of dimensions 96x96xNframes with an in-frame resolution of 80 micrometers
+and an out-of-frame resolution of 0.4 mm. All utilized intravascular pullbacks were manually deemed as of sufficient
+image quality, with appropriate quality lumen segmentations. For the CCTA data, a 3D model of the coronary tree for
+each patient was produced by HeartFlow using the CCTA image (Sonck et al. [2022]). The 3D model was then used to
+produce a 3D SDF with a resolution of 0.25mm along each axis with an image dimension of 768x768x482.
+2.2.2
+Co-registration accuracy
+In order to evaluate the performance of the non-rigid registration, 114 bifurcations were manually marked in the OCT
+pullback as well as in the rigid and non-rigid virtual pullback segmentations generated from the CCTA data. Bifurcations
+were defined as the last image frame before a visual coronary artery split into two branches could be seen. Bifurcations
+that were common to both modalities had their frame numbers recorded for validation of the non-rigid registration
+algorithm. Longitudinal validation was conducted by comparing the frame number of a bifurcation in the OCT data
+with the equivalent bifurcation frame number in the virtual pullback before and after non-rigid registration. In order to
+validate the non-rigid rotational registration, the bifurcation angle difference between OCT pullback and the virtual
+pullback was compared before and after rotational registration. As the bifurcation angle between bifurcation sections
+that were not longitudinally matched is expected to be uncorrelated, a separate analysis was conducted to characterize
+how angular mismatch varies when the bifurcations are longitudinally matched. Furthermore, only bifurcations that had
+a frame mismatch below 6 frames were considered for extensive analysis of rotational accuracy.
+2.2.3
+Comparison to alternative approaches
+The most common co-registration methodology employed for coronary artery registration has been discrete optimization
+approaches such as DTW and Dynamic Programming. Therefore, in order to evaluate the performance of our
+longitudinal and rotational co-registration framework against state-of-the-art discrete approaches, we applied the
+methodology described in Karmakar et. al (Karmakar et al. [2022]) on the same dataset used in this study. The approach
+utilizes DTW to longitudinally align two coronary imaging modalities and Dynamic Programming to rotationally align
+each frame. We utilized a window length of 4 (0.8mm) as implemented in the previous study and recorded identical
+alignment metrics for 114 matched bifurcations in the dataset. The non-rigid registration algorithm was applied after
+the rigid longitudinal registration step described in section 2.1.2. A substudy was also conducted in which the angular
+alignment of all bifurcations was compared to the angular alignment of longitudinally matched bifurcations.
+2.2.4
+Optimization details
+The gradient descent-based optimization procedure was implemented in PyTorch with the Adam optimizer (Kingma
+and Ba [2014]). A learning rate of 0.001 was used for the non-rigid longitudinal parameters and a rate of 0.01 was
+used for both the rotational and transverse parameters. Each co-registration procedure was run for a minimum of 200
+iterations to ensure convergence.
+3
+Results
+3.1
+Longitudinal Registration
+Longitudinal registration plots in Figures 4 and 7A1-2 show that using rigid registration alone (Figure 7A1), few
+bifurcations were longitudinally aligned within 6 (dotted line), 4 (dashed line), or 2 (solid line) frame distances.
+However, after non-rigid alignment (Figure 7A2), distinct improvement can be observed with a majority of bifurcations
+are aligned within 6 frames. These results are visualized by the longitudinal mismatch plot (Figure 5A), revealing
+that after rigid alignment, the percentage of bifurcations matched within 2, 4, and 6 frames are 26.3, 42.1, and 57.9%,
+respectively, while after non-rigid alignment, these values increase to 60.5, 78.9, and 86.8%. Moreover, the scatterplot
+for non-rigid registration (A2) demonstrates that the majority of bifurcations (86% shown in green) were enhanced
+in terms of frame alignment, while a negligible number of bifurcations had slightly (11.4 % shown in orange) or
+significantly (2.6 % shown in red) worse alignment after non-rigid registration. Table 2 further demonstrates the effect
+of non-rigid registration, in which the mean frame difference after rigid registration was 7.9 frames and subsequently
+decreased to 3.3 frames after non-rigid registration.
+7
+
+arXiv Template
+A PREPRINT
+A1
+B1
+C1
+D1
+E1
+F1
+G1
+A2
+B2
+C2
+D2
+E2
+F2
+G2
+A3
+B3
+C3
+D3
+E3
+F3
+G3
+Bifurcation Frames
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Frame number
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+Area (mm^2)
+CT
+OCT
+A
+B
+C D
+E F
+G
+Figure 4: Qualitative results for a single co registered case. Top row shows area plot along the artery for the non-rigidly
+registered CT (green) and the OCT images. The bifurcation zones (Sections A-G) are marked and labeled for further
+analysis. Bifurcation frames from the CT, OCT, and overlapped segmentation maps are presented in the bottom row for
+qualitative analysis of the rotational and transverse co-registration.
+A
+B
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Maximum frame mismatch
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Matched bifucations (%)
+Rigid
+Rigid+Non-rigid
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Maximum angular mismatch (degrees)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Matched bifucations (%)
+Rigid
+Rigid+Non-rigid
+Figure 5: Quantitative results comparing the quality of
+rigid and non-rigid co registration in longitudinal and ro-
+tational directions with varying degrees of misalignment.
+The mismatch plots exhibit the % of matched bifurca-
+tions with increasing longitudinal (A) and rotational (B)
+alignment mismatch criteria (x-axis).
+~10 degrees
+~20 degrees
+~30 degrees
+Angular Mismatch
+Figure 6: Grid plot showing multiple aligned bifurcation
+segmentations using an SDF-based loss. Angular mis-
+matches up to 10, 20, and 30 degrees are shown in the
+first, second, and third columns respectively.
+3.2
+Rotational Registration
+Examination of the individual bifurcating frames in figure 4 for the CT (row 1) and OCT (row 2) frames indicates
+excellent rotational and transverse alignment between both imaging modalities as evident from the raw images and the
+overlapped segmentations (row 3). Rotational registration plots in figure 7B1-2 demonstrate that few bifurcations are
+rotationally aligned within 30 (dotted line), 20 (dashed line), or 10 (solid line) degrees after rigid alignment (B1). After
+non-rigid alignment (Figure 7B2), a majority of bifurcations were aligned within 30 degrees, with a significant amount
+aligned within 20 and 10 degrees. Examination of the rotational mismatch plot (Figure 5B) quantitatively demonstrates
+an increase in the percentage of bifurcations aligned up to an angular mismatch of 10, 20, and 30 degrees from %
+values of 25.3, 40.4, and 52.3 to 51.5, 69.7, and 79.8% respectively. Similarly, the non-rigid registration scatterplot
+8
+
+arXiv Template
+A PREPRINT
+A1
+A2
+B1
+B2
+Bifurcation number
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Longitudinal misalignment
+Bifurcation number
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Longitudinal misalignment
+Non-rigid<0
+Non-rigid<2
+Non-rigid>=2
+Bifurcation number
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Angular misalignment
+Non-rigid<0
+Non-rigid<20
+Non-rigid>=20
+Bifurcation number
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Angular misalignment
+Figure 7: Quantitative results comparing the quality of rigid and non-rigid co-registration in longitudinal and rotational
+directions. The first row compares bifurcation frame mismatch before (A1) and after (A2) non-rigid registration in
+the form of scatterplots. The second row compares bifurcation angular mismatch before (B1) and after (B2) non-rigid
+registration in the form of scatterplots. The scatterplot for the longitudinal and rotational non-rigid registration (A2 and
+B2) are color-coded to exhibit the change in alignment metric after non-rigid registration, where green represents an
+increase in alignment, orange represents a mild decrease in alignment, and red represents a strong decrease in alignment.
+Only bifurcations that were longitudinally matched within 6 OCT frames were analyzed for rotational alignment.
+demonstrates that the majority of bifurcations had their angular mismatch decreased after non-rigid alignment (66%
+shown in green) and only a minority had their angular mismatch values slightly (22% shown in orange) or significantly
+(12% shown in red) increased. The mean value of the angular mismatch before and after non-rigid alignment is reported
+in Table 2, in which the mean angular mismatch decreases from 36.0 to 28.6 degrees.
+3.3
+Comparison with previous approaches
+A direct comparison of the virtual catheter method with state-of-the-art discrete optimization approaches can be seen in
+Tables 2 and 3. Comparing the virtual catheter method to a discrete optimization approach for longitudinal registration,
+it was shown that DTW produces significantly poorer results in longitudinal registration, with the longitudinal mismatch
+of 11.7 frames being higher than rigid longitudinal registration average of 7.9 frames. Comparing the virtual catheter
+method to using Dynamic Programming for rotational registration, it was shown that such discrete optimization
+algorithms exhibit poor performance for CT-OCT rotational registration (angular mismatch of 77.9 degrees) which
+is higher than the angular mismatch after rigid rotational registration alone. Table 3 quantifies the angular mismatch
+in the case where non-rigid longitudinal registration is successful. For bifurcations with a maximum frame mismatch
+of 6 after non-rigid registration, the angular mismatch decreases from 77.9 to 65.2 for the Dynamic Programming
+approach, while for the virtual catheter method, the angular mismatch decreases from 28.6 to 24.8. In contrast, the
+angular mismatch for the rigid registration is unchanged after excluding non-matching bifurcations.
+9
+
+arXiv Template
+A PREPRINT
+Figure 8: Qualitative results comparing the alignment of calcium annotations between OCT (first row) and CT (third
+row) for selected frames with good luminal alignment. The middle row shows the calcium annotations for OCT (red)
+and CT (green) superimposed on each other.
+Table 1: Accuracy of alternative co-registration approaches, proposed for intravascular-intravascular image registration.
+Data is presented from left to right including evaluated co-registered modalities, dataset size, and overall methodological
+approach. Further, average errors are presented in both longitudinal (frames) and rotational (degree) directions.
+Ref.
+Modalities
+Dataset Size
+Methodology
+Longitudinal mismatch
+Angular mismatch
+Karmakar et al. [2022]
+OCT-OCT
+9 patients
+DTW + Dynamic Programming
+0.9 ± 0.8
+7.7 ± 6.7
+Tsiknakis et al. [2023]
+OCT-OCT
+21 patients
+DTW + Harmony Search
+5.6 ± 6.7
+1.2 ± 0.81
+Karmakar et al. [2022]
+OCT-IVUS
+7 patients
+DTW + Dynamic Programming
+1.45 ± 0.7
+29.1 ± 23.2
+Molony et al. [2016]
+OCT-IVUS
+12 patients
+DTW + Dynamic Programming
+5.0 ± 6.2
+17.8 ± 21.9
+Table 2: Accuracy of co-registration approaches applied to CT-OCT image registration. Data is presented from left to
+right including evaluated co-registered modalities, dataset size, and overall methodological approach. Further, average
+errors are presented in both longitudinal (frames) and rotational (degree) directions. All approaches in this table have
+been evaluated on the same dataset
+Ref.
+Modalities
+Dataset Size
+Methodology
+Longitudinal mismatch
+Angular mismatch
+Karmakar et al. [2022]
+CT-OCT
+40 patients
+DTW + Dynamic Programming
+11.7 ± 12.1
+77.9 ± 61.0
+Ours (Rigid)
+CT-OCT
+40 patients
+Virtual Catheter Method
+7.9 ± 7.1
+36.0 ± 31.9
+Ours (Rigid+Non-rigid)
+CT-OCT
+40 patients
+Virtual Catheter Method
+3.3 ± 3.9
+28.6 ± 40.9
+Table 3: Accuracy of co-registration approaches applied to CT-OCT image registration for bifurcations that are
+longitudinally matched. Data is presented from left to right including methodological approach and number of
+longitudinally matched bifurcations. Bifurcations are considered longitudinally matched when they have a maximum
+frame difference of 6 after non-rigid longitudinal registration. Further, average errors are presented for rotational
+direction in degrees.
+Ref.
+Methodology
+Matched Bifurcations
+Angular Mismatch
+Karmakar et al. [2022]
+DTW + Dynamic Programming
+52/114
+65.2 ± 72.9
+Ours (Rigid)
+Virtual Catheter Method
+99/114
+36.0 ± 33.0
+Ours (Rigid+Non-rigid)
+Virtual Catheter Method
+99/114
+24.8 ± 39.0
+10
+
+OUarXiv Template
+A PREPRINT
+4
+Discussion
+The aim of the current study was to develop a fully automatic registration algorithm to align CCTA and intravascular
+images. Specifically, we propose a novel registration process finding the optimal rigid and non-rigid spatial transforms
+using a virtual catheter path in the CCTA data, aligning the non-invasive modality to its invasive counterpart. Our results
+indicate excellent co-registration accuracy, with excellent agreement with reference manual landmark annotations
+(Figure 4). Further, our results underline the critical importance of a non-rigid registration step, with significant
+enhancement in both longitudinal and rotational alignments observed when comparing rigid vs. non-rigid alignments in
+Table 2. We demonstrate that for the vast majority of bifurcations, our framework is able to improve the longitudinal
+and rotational alignment of common bifurcations within the CT and OCT images. Lastly, we demonstrate the added
+value of our approach as compared to state-of-the-art alternatives, with a head-to-head comparison to previously
+developed discrete optimization alignment algorithms (Table 1). A head-to-head comparison demonstrates that discrete
+optimization approaches for longitudinal and rotational alignment suffer a significant drop in alignment quality when
+applied for the task of CT-OCT co-registration. Meanwhile, our approach maintains performance accuracy in line with
+simpler tasks such as intravascular-intravascular image registration.
+4.1
+Related work
+Currently, a majority of CCTA studies that validate their CT findings with intravascular images have used rigid manual
+registration based on fiduciary landmarks such as bifurcations or large calcifications (Carlier et al. [2014], Tu et al.
+[2011], Hebsgaard et al. [2015]). One of the few studies that attempted to align CT and intravascular data up to a non-
+rigid level is by Uzu et al. [2019] in which a B-spline deformation model was used to optimize the alignment of manually
+annotated bifurcation landmarks. Though powerful, such an approach is time-consuming due to the significant amount
+of manual processing required to process the OCT images, rigidly align the geometric models, and mark bifurcations
+within every artery. In comparison, our approach implicitly matches nearby bifurcations using longitudinally smoothed
+SDFs representing the CT and OCT lumens, respectively. Other approaches that register intravascular-to-intravascular
+modalities have in the past relied on DTW (Molony et al. [2016], Karmakar et al. [2022]), discretely optimizing the
+frame-wise progression of one intravascular pullback to maximize longitudinal and rotational alignment with another.
+Such methods, nevertheless, attempt to recapitulate the continuous motion distortions introduced by the catheter path
+with discrete non-physiological frames or repeats (Molony et al. [2016]). However, skipping or repeating several frames
+that the catheter motion is not smooth or continuous, which is an unrealistic assumption about the catheter path.
+Direct numerical comparison of reported co-registration accuracy across published approaches is inherently difficult
+as co-registration accuracy is highly dependent on the specific dataset explored as well as which modalities are being
+coregistered. For example, the simplest co-registration task would be represented by the alignment of same-modality
+images such as OCT-OCT image pairs. For such tasks, our previously published DTW and dynamic programming
+approach (Karmakar et al. [2022]) exhibits similar perfomance compared to other state-of-the-art algorithms (Tsiknakis
+et al. [2023]) (See Table 1). When applied to multimodality datasets, such as IVUS-OCT image pairs, our previously
+developed approach (Karmakar et al. [2022]) suffers distinctive drops in both longitudinal and rotational accuracy (see
+Table 1), however, still maintains comparative performance to similar Dynamic Programming approaches (Molony et al.
+[2016]). Thus, in order to facilitate a head-to-head comparison on the more challenging task of registering CT-OCT
+image pairs, we applied our previously developed discrete optimization algorithm (Karmakar et al. [2022]) on our
+multi-modal dataset of 40 patients. Doing so, we found that our previous approach produced significantly worse
+longitudinal and rotational alignment compared to the virtual catheter method, with both longitudinal and angular
+alignment being worse than simple rigid registration (Table 2). In contrast, our developed methodology achieves
+competitive results with even intravascular-intravascular registration studies (Table 1 and Table 2).
+4.2
+Methodological Adaptations
+From the above it can be seen that the task of co-registering CT and OCT images presents several unique difficulties for
+discrete registration algorithms. Our framework has several features that were designed to mitigate such challenges.
+First, the low resolution of CT images induces a circular bias in the lumen segmentations (see Figure 4), as well as a
+tendency to miss small bifurcations. Such circularly symmetric regions hence create zones of longitudinal and rotational
+ambiguity along the pullback. Our approach tries to minimize the dependency of such by formulating the longitudinal
+and rotational transforms acting on the virtual path in terms of a regularized and smooth B spline deformation. As
+such, the optimization procedure is mainly dominated by the alignment of prominent non-symmetric features such
+as bifurcations, rather than the circularly symmetric lumen segments. This ensures that the rotational alignment of
+all non-bifurcating lumen frames that are in proximity to matched bifurcations are properly matched due to B spline
+interpolation (Figure 4). Another significant issue faced in previous rotational co-registration algorithms (Karmakar
+11
+
+arXiv Template
+A PREPRINT
+et al. [2022], Molony et al. [2016]) is that lumen bifurcations are only able to contribute to rotational alignment if they
+exist within the same frame. As such, poor longitudinal alignment of bifurcations was a significant contributing factor
+to the poor performance of our previously developed dynamic programming algorithm for rotational co-registration
+(Table 2). Our framework, in contrast, minimizes this dependency through the use of a Gaussian smoothing kernel
+applied longitudinally over the SDF. Longitudinal smoothing allows single-frame bifurcations to appear in adjacent
+frames and smooths the loss surface such that bifurcations in the different modalities can be better aligned (Figures 4
+and 7). Another design choice that was found to increase training stability and co-registration quality was the use of
+SDF’s to determine alignment, as opposed to using a segmentation loss such as cross-entropy or Dice. This was due to
+the fact that when the lumen segmentations were fully overlapping, multiple rotational and transverse configurations
+contribute equally to a segmentation loss function, preventing the algorithm from making fine adjustments in the spatial
+transform. Figure 6 further demonstrates how the quality of rotational registration varies with angular mismatch, where
+the angular mismatch tends to occur due to the limitations of local optimization of pixel alignment. At less than 10
+degrees mismatch, the difference in alignment is minimal, while under 30 degrees, the difference in alignment can
+be attributed to differing lumen bifurcation shapes in the CT and OCT data. Lastly, many co-registration methods
+normalize the position of the lumen by the artery centroid (Uzu et al. [2019], Karmakar et al. [2020, 2022], Molony
+et al. [2016]). While such an approach manages to align CT and OCT frames with circularly symmetric lumen, it fails
+to effectively align equivalent frames with bifurcations as the segmentation can have different maximum diameters
+between the modalities and thus different centroids. Moreover, centering the image around the lumen centroids can
+cause the algorithm to align bifurcations 180-degrees from the correct orientation. It was empirically found that this
+phenomenon was found to be a significant contributing factor to the degradation of the co-registration performance of
+our previously developed discrete co-registration algorithm. In this framework, we instead choose to jointly optimize
+for the transverse displacements of the virtual path in addition to the longitudinal and rotational displacements, which
+allows for the bifurcations in both modalities to be anchored around the OCT catheter location and enables near
+pixelwise alignment of the lumen (Figures 4,6) and plaque constituents such as calcium (Figure 8).
+4.3
+Limitations
+Though very promising for clinical applications, our developed approach has a number of limitations. First, the
+non-rigid spatial transform acting on the virtual catheter path is found through gradient-based optimization, requiring
+that landmarks lie sufficiently close such that proper matching is ensured. For example, common bifurcations that
+have a frame mismatch of more than 6 frames (corresponding to the longitudinal smoothing kernel) are expected to
+be uncorrelated in terms of orientation. This issue can be mitigated by integrating deep learning networks which can
+accurately predict the spatial transform needed to align the two modalities. Another limitation is the dependence of
+non-rigid registration on the lumen. The lumen estimation is expected to be accurate for both modalities and as such,
+ensures good registration accuracy for regions that include many bifurcations. However, due to the poor resolution of
+CCTA images, the lumen estimation tends to be highly circular. Accordingly, it is expected that rotational co-registration
+certainty increases with bifurcation proximity but decreases in stenotic regions that contain highly circular CT luminal
+profiles. In the future, co-registration accuracy can likely be improved by including contextual information relating
+to the vessel wall such as lesion content and morphology as a supervisory signal in the loss function. Third, the use
+of a pixel-wise loss as a surrogate for luminal alignment may not necessarily result in optimal alignment of lumen
+bifurcations. As seen in Figure 6, a pixel-wise loss function can occasionally bias the spatial transform to align the
+central lumen body over aligning the bifurcation in scenarios where the bifurcation shapes are not perfectly matching.
+In the future, this issue can be mitigated by introducing an orientation loss to bias the spatial transform to rotationally
+align bifurcations. Lastly, regularizing the spatial transform and smoothing the SDF’s can create difficulties in localizing
+landmarks up to frame-wise precision. This can be seen in the area curve in Figure 4 section B with the slightly
+mismatched bifurcation and in Figure 7 A2 and B2 with a minor amount of bifurcations with increased frame and
+angular mismatch values. The localization capabilities of the algorithm can be improved by introducing multi-scale
+deformation steps where finer control point grids can be recursively used as the basis for the spatial transform.
+4.4
+Translational Benefits
+The development of automatic frame-wise matching algorithms for CT-OCT data fusion would enable the development
+of several research-based applications. First, intravascular imaging data can act as ground truth to validate the reliability
+of CT in delineating several morphological metrics of atherosclerosis, such as luminal area, lipid content, and calcium
+volume. Understanding when CT-derived morphological metrics are reliable is critical for both therapy planning and
+deciding when to rely on intravascular imaging. For example, studying the interaction between calcium blooming and
+measured lumen size in CT images necessitates that ground truth lumen measurements be available, which can only be
+provided by frame-wise co-registration algorithms. Figure 8 demonstrates that such a frame-by-frame comparison can
+be done provided that longitudinal and rotational co-registration is of sufficient quality. Second, multi-modal data fusion
+12
+
+arXiv Template
+A PREPRINT
+would allow for the enhanced generation of patient-specific digital twins from coronary images. There has been an
+increasing interest in the use of intravascular-images to create computational digital twins for the prediction of coronary
+pathophysiology and clinical decision-making (Kadry et al. [2021]). However, intravascular images, while providing
+excellent resolution within the imaging frame, do not provide sufficient information to create a fully physiological
+artery model. Intravascular images typically suffer from intra-frame motion drift artifacts in the longitudinal and
+rotational directions and cannot capture information on the three-dimensional centerline of the artery. On the other
+hand, CT suffers from poor resolution but is able to capture the three-dimensional nature of the artery with high
+accuracy. Combining both modalities would allow researchers to investigate the importance of longitudinal and
+rotational distortions, as well as modeling arterial tortuosity.
+References
+Georgios Tzimas, Gaurav S Gulsin, Hidenobu Takagi, Niya Mileva, Jeroen Sonck, Olivier Muller, Jonathon A Leipsic,
+and Carlos Collet. Coronary ct angiography to guide percutaneous coronary intervention. Radiology: Cardiothoracic
+Imaging, 4(1):e210171, 2022.
+Kenzo Uzu, Hiromasa Otake, Gilwoo Choi, Takayoshi Toba, Hyun Jin Kim, Arjun Roy, Michiel Schaap, Leo Grady,
+Masahito Kawata, Toshiro Shinke, et al. Lumen boundaries extracted from coronary computed tomography angiogra-
+phy on computed fractional flow reserve (ffrct): validation with optical coherence tomography. Eurointervention:
+Journal of Europcr in Collaboration with the Working Group on Interventional Cardiology of the European Society
+of Cardiology, 14(15):e1609–e1618, 2019.
+Choongki Kim, Sung-Jin Hong, Dong-Ho Shin, Jung-Sun Kim, Byeong-Keuk Kim, Young-Guk Ko, Donghoon Choi,
+Yangsoo Jang, and Myeong-Ki Hong. Limitations of coronary computed tomographic angiography for delineating the
+lumen and vessel contours of coronary arteries in patients with stable angina. European Heart Journal-Cardiovascular
+Imaging, 16(12):1358–1365, 2015.
+Matthew J Budoff, David Dowe, James G Jollis, Michael Gitter, John Sutherland, Edward Halamert, Markus Scherer,
+Raye Bellinger, Arthur Martin, Robert Benton, et al. Diagnostic performance of 64-multidetector row coronary
+computed tomographic angiography for evaluation of coronary artery stenosis in individuals without known coronary
+artery disease: results from the prospective multicenter accuracy (assessment by coronary computed tomographic
+angiography of individuals undergoing invasive coronary angiography) trial. Journal of the American College of
+Cardiology, 52(21):1724–1732, 2008.
+Yu Takahashi, Takayoshi Toba, Hiromasa Otake, Yusuke Fukuyama, Shinsuke Nakano, Yoichiro Matsuoka, Kosuke
+Tanimura, Yu Izawa, Hiroyuki Kawamori, Atsushi K Kono, et al. Feasibility of morphological assessment of coronary
+artery calcification with electrocardiography-gated non-contrast computed tomography: a comparative study with
+optical coherence tomography. The International Journal of Cardiovascular Imaging, 37(4):1445–1453, 2021.
+Collin Fischer, Edward Hulten, Pallavi Belur, Ryan Smith, Szilard Voros, and Todd C Villines. Coronary ct angiography
+versus intravascular ultrasound for estimation of coronary stenosis and atherosclerotic plaque burden: a meta-analysis.
+Journal of cardiovascular computed tomography, 7(4):256–266, 2013.
+Michiel A De Graaf, Alexander Broersen, Pieter H Kitslaar, Cornelis J Roos, Jouke Dijkstra, Boudewijn PF Lelieveldt,
+J Wouter Jukema, Martin J Schalij, Victoria Delgado, Jeroen J Bax, et al. Automatic quantification and characterization
+of coronary atherosclerosis with computed tomography coronary angiography: cross-correlation with intravascular
+ultrasound virtual histology. The international journal of cardiovascular imaging, 29(5):1177–1190, 2013.
+H Brodoefel, C Burgstahler, M Heuschmid, A Reimann, F Khosa, A Kopp, S Schroeder, CD Claussen, and ME Clouse.
+Accuracy of dual-source ct in the characterisation of non-calcified plaque: use of a colour-coded analysis compared
+with virtual histology intravascular ultrasound. The British journal of radiology, 82(982):805–812, 2009.
+A Lin, N Manral, P McElhinney, A Killekar, H Matsumoto, S Cadet, S Achenbach, SJ Nicholls, DT Wong, D Berman,
+et al. Deep learning-based plaque quantification from coronary computed tomography angiography: external
+validation and comparison with intravascular ultrasound. European Heart Journal, 42(Supplement_1):ehab724–0161,
+2021.
+Marly van Assen, Akos Varga-Szemes, U Joseph Schoepf, Taylor M Duguay, H Todd Hudson, Svetlana Egorova,
+Kjell Johnson, Samantha St Pierre, Beatrice Zaki, Matthijs Oudkerk, et al. Automated plaque analysis for the
+prognostication of major adverse cardiac events. European Journal of Radiology, 116:76–83, 2019.
+Nikos Tsiknakis, Constantinos Spanakis, Panagiota Tsompou, Georgia Karanasiou, Gianna Karanasiou, Antonis
+Sakellarios, George Rigas, Savvas Kyriakidis, Michael Papafaklis, Sotirios Nikopoulos, et al. Ivus longitudinal and
+axial registration for atherosclerosis progression evaluation. Diagnostics, 11(8):1513, 2021.
+13
+
+arXiv Template
+A PREPRINT
+Stéphane Carlier, Rich Didday, Tristan Slots, Peter Kayaert, Jeroen Sonck, Mike El-Mourad, Nicolas Preumont,
+Dany Schoors, and Guy Van Camp. A new method for real-time co-registration of 3d coronary angiography and
+intravascular ultrasound or optical coherence tomography. Cardiovascular Revascularization Medicine, 15(4):
+226–232, 2014.
+Shengxian Tu, Niels R Holm, Gerhard Koning, Zheng Huang, and Johan HC Reiber. Fusion of 3d qca and ivus/oct.
+The international journal of cardiovascular imaging, 27(2):197–207, 2011.
+Lasse Hebsgaard, Troels Munck Nielsen, Shengxian Tu, Lars Romer Krusell, Michael Maeng, Karsten Tange Veien,
+Bent Raungaard, Christian Juhl Terkelsen, Anne Kaltoft, Johan HC Reiber, et al. Co-registration of optical coherence
+tomography and x-ray angiography in percutaneous coronary intervention. the does optical coherence tomography
+optimize revascularization (doctor) fusion study. International journal of cardiology, 182:272–278, 2015.
+Hui Qin, Chunming Li, Yingguang Li, Jiayue Huang, Fan Yang, Takashi Kubo, Takashi Akasaka, Changyan Xiao,
+Juan Luis Gutiérrez-Chico, and Shengxian Tu.
+Automatic coregistration between coronary angiography and
+intravascular optical coherence tomography: Feasibility and accuracy. JACC: Asia, 1(2):274–278, 2021.
+David S Molony, Lucas H Timmins, Emad Rasoul-Arzrumly, Habib Samady, and Don P Giddens. Evaluation of a
+framework for the co-registration of intravascular ultrasound and optical coherence tomography coronary artery
+pullbacks. Journal of biomechanics, 49(16):4048–4056, 2016.
+Abhishek Karmakar, Max L Olender, Farhad Rikhtegar Nezami, David Marlevi, Evan Shlofmitz, Richard A Shlofmitz,
+and Elazer R Edelman. Detailed investigation of lumen-based tomographic co-registration. In 2020 IEEE International
+Conference on Bioinformatics and Biomedicine (BIBM), pages 1038–1042. IEEE, 2020.
+Yazan Gharaibeh, Juhwan Lee, David Prabhu, Pengfei Dong, Vladislav N Zimin, Luis A Dallan, Hiram Bezerra, Linxia
+Gu, and David Wilson. Co-registration of pre-and post-stent intravascular oct images for validation of finite element
+model simulation of stent expansion. In Medical Imaging 2020: Biomedical Applications in Molecular, Structural,
+and Functional Imaging, volume 11317, pages 306–316. SPIE, 2020.
+Ling Zhang, Richard Downe, Zhi Chen, Shanhui Sun, T Masiarov, Tomas Kovarnik, John Lopez, Milan Sonka, and
+Andreas Wahle. Side-branch guided registration of intravascular ultrasound pullbacks in coronary arteries. In
+MICCAI Workshop in Computing and Visualization for IntraVascular Imaging and Computer Assisted Stenting
+(CVII-STENT), pages 44–51, 2014.
+Guha Balakrishnan, Amy Zhao, Mert R Sabuncu, John Guttag, and Adrian V Dalca. Voxelmorph: a learning framework
+for deformable medical image registration. IEEE transactions on medical imaging, 38(8):1788–1800, 2019.
+Yabo Fu, Yang Lei, Tonghe Wang, Walter J Curran, Tian Liu, and Xiaofeng Yang. Deep learning in medical image
+registration: a review. Physics in Medicine & Biology, 65(20):20TR01, 2020.
+Max Jaderberg, Karen Simonyan, Andrew Zisserman, et al. Spatial transformer networks. Advances in neural
+information processing systems, 28, 2015.
+Eran Treister and Eldad Haber. A fast marching algorithm for the factored eikonal equation. Journal of Computational
+physics, 324:210–225, 2016.
+Luca Antiga, Marina Piccinelli, Lorenzo Botti, Bogdan Ene-Iordache, Andrea Remuzzi, and David A Steinman.
+An image-based modeling framework for patient-specific computational hemodynamics. Medical & biological
+engineering & computing, 46(11):1097–1112, 2008.
+Jixiang Guo, Shun Li, Yim Pan Chui, Jing Qin, and Pheng Ann Heng. Mesh quality oriented 3d geometric vascular
+modeling based on parallel transport frame. Computers in Biology and Medicine, 43(7):879–888, 2013.
+Armin Kanitsar, Dominik Fleischmann, Rainer Wegenkittl, Petr Felkel, and Eduard Groller. CPR-curved planar
+reformation. IEEE, 2002.
+Daniel Rueckert, Luke I Sonoda, Carmel Hayes, Derek LG Hill, Martin O Leach, and David J Hawkes. Nonrigid
+registration using free-form deformations: application to breast mr images. IEEE transactions on medical imaging,
+18(8):712–721, 1999.
+Sakura Nagumo, Carlos Collet, Bjarne L Norgaard, Hiromasa Otake, Brian Ko, Bon-kwon Koo, Jonathon Leipsic,
+Daniele Andreini, Ward Heggermont, Jesper M Jensen, et al. Rationale and design of the precise percutaneous
+coronary intervention plan (p3) study: Prospective evaluation of a virtual computed tomography-based percutaneous
+intervention planner. Clinical cardiology, 44(4):446–454, 2021.
+Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmenta-
+tion. In International Conference on Medical image computing and computer-assisted intervention, pages 234–241.
+Springer, 2015.
+14
+
+arXiv Template
+A PREPRINT
+Jeroen Sonck, Sakura Nagumo, Bjarne L Norgaard, Hiromasa Otake, Brian Ko, Jinlong Zhang, Takuya Mizukami,
+Michael Maeng, Daniele Andreini, Yu Takahashi, et al. Clinical validation of a virtual planner for coronary
+interventions based on coronary ct angiography. Cardiovascular Imaging, 15(7):1242–1255, 2022.
+Abhishek Karmakar, Max L Olender, David Marlevi, Evan Shlofmitz, Richard A Shlofmitz, Elazer R Edelman, and
+Farhad R Nezami. Framework for lumen-based nonrigid tomographic coregistration of intravascular images. Journal
+of Medical Imaging, 9(4):044006, 2022.
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980,
+2014.
+Nikos Tsiknakis, Constantinos Spanakis, Panagiota Tsoumpou, Georgia Karanasiou, Gianna Karanasiou, Antonis
+Sakellarios, George Rigas, Savvas Kyriakidis, Michail I Papafaklis, Sotirios Nikopoulos, et al. Oct sequence
+registration before and after percutaneous coronary intervention (stent implantation). Biomedical Signal Processing
+and Control, 79:104251, 2023.
+Karim Kadry, Max L Olender, David Marlevi, Elazer R Edelman, and Farhad R Nezami. A platform for high-fidelity
+patient-specific structural modelling of atherosclerotic arteries: from intravascular imaging to three-dimensional
+stress distributions. Journal of the Royal Society Interface, 18(182):20210436, 2021.
+15
+
diff --git a/99AyT4oBgHgl3EQfRPYW/content/tmp_files/load_file.txt b/99AyT4oBgHgl3EQfRPYW/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d3c3661f183d2399e60edd076c2400f1a4ab778d
--- /dev/null
+++ b/99AyT4oBgHgl3EQfRPYW/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf,len=627
+page_content='MORPHOLOGY-BASED NON-RIGID REGISTRATION OF  CORONARY COMPUTED TOMOGRAPHY AND INTRAVASCULAR  IMAGES THROUGH VIRTUAL CATHETER PATH OPTIMIZATION  Karim Kadry∗  Institute of Medical Engineering and Science  Massachusetts Institute of Technology  Cambridge, MA 02139  kkadry@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='edu  Abhishek Karmakar  Meinig School of Biomedical Engineering  Cornell University  Ithaca, NY 14850  ak944@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='edu  Andreas Schuh  Biomedical Image Analysis Group  Imperial College London  HeartFlow, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=', USA  London, UK  aschuh@heartflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='com  Kersten Peterson  HeartFlow, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=', USA  Redwood City, CA, 94063, USA  kpetersen@heartflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='com  Michiel Schaap  HeartFlow, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=', USA  Redwood City, CA, 94063, USA  mschaap@heartflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='com  David Marlevi  Department of Molecular Medicine and Surgery  Karolinska Institute  Stockholm, Sweden  david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='marlevi@ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='se  Charles Taylor  Department of Electrical Engineering  HeartFlow, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=', USA  Redwood City, CA, 94063, USA  ctaylor@heartflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='com  Elazer Edelman  Institute of Medical Engineering and Science  Massachusetts Institute of Technology  Cambridge, MA 02139  ere@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='edu  Farhad Nezami  Department of Surgery  Brigham and Women’s Hospital Harvard Medical School  Boston, MA 02115  frikhtegarnezami@bwh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='edu  ABSTRACT  Coronary Computed Tomography Angiography (CCTA) provides information on the presence, extent,  and severity of obstructive coronary artery disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Large-scale clinical studies analyzing CCTA-  derived metrics typically require ground-truth validation in the form of high-fidelity 3D intravascular  imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' However, manual rigid alignment of intravascular images to corresponding CCTA images is  both time consuming and user-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Moreover, intravascular modalities suffer from several  non-rigid motion-induced distortions arising from distortions in the imaging catheter path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' To address  these issues, we here present a semi-automatic segmentation-based framework for both rigid and  non-rigid matching of intravascular images to CCTA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' We formulate the problem in terms  of finding the optimal virtual catheter path that samples the CCTA data to recapitulate the coronary  artery morphology found in the intravascular image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' We validate our co-registration framework on a  cohort of n = 40 patients using bifurcation landmarks as ground truth for longitudinal and rotational  registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Our results indicate that our non-rigid registration significantly outperforms other co-  registration approaches for luminal bifurcation alignment in both longitudinal (mean mismatch: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='3  frames) and rotational directions (mean mismatch: 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='6 degrees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' By providing a differentiable  framework for automatic multi-modal intravascular data fusion, our developed co-registration modules  ∗Corresponding Author  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='00060v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='CV]  30 Dec 2022  arXiv Template  A PREPRINT  significantly reduces the manual effort required to conduct large-scale multi-modal clinical studies  while also providing a solid foundation for the development of machine learning-based co-registration  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  1  Introduction  Coronary computed tomography angiography (CCTA) is a three dimensional image modality that provides information  on the presence, extent and severity of obstructive coronary artery disease (CAD) (Tzimas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2022]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' As such, CCTA  allows for the detection of stenotic atherosclerotic sections and assists clinicians in diagnosing CAD and planning  treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' CCTA Images can also be used to create computational models of coronary blood flow, allowing for  the non-invasive estimation of fractional flow reserve (FFR-CT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' a key diagnostic parameter in assessing functional  impairment (Uzu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2019]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Albeit widespread in use, CCTA provides primary information on luminal anatomy, with limited capacity in assessing  soft-tissue intraplaque tissue components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' CCTA also suffers from blooming artifacts in the presence of highly  attenuating calcium deposits (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2015], Budoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2008]), which, combined with comparably low image  resolution, creates difficulties in resolving highly calcified arteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Multiple studies have also been conducted to  quantify the degree to which CCTA can accurately assess CAD-related diagnostic metrics such as luminal area (Uzu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2019]), calcium morphology (Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2021]), and plaque burden (Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2013], De Graaf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  [2013], Brodoefel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2009]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The majority of such studies (Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2021], Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2013], Uzu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  [2019], Brodoefel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2009]) validate the performance of CCTA by manually co-registering image slices taken along  the CCTA artery to intravascular imaging modalities such as intravascular ultrasound (IVUS) and optical coherence  tomography (OCT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' both providing higher-fidelity visualization of the lumen and surrounding tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' There is also an  increasing interest in validating CCTA-derived segmentation algorithms against co-registered intravascular imaging  frames, again necessitating such multimodal image assessment (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2021], van Assen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2019]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Manual co-registeration of CCTA and intravascular images is, however, a challenging and time consuming task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Typically, cross-sectional frames of the artery of interest are extracted from the CCTA images which then have to be  matched with corresponding frames from an intravascular acquisition through an imaging catheter pullback procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Rigid registration in the longitudinal and rotational directions is usually achieved by matching single landmarks in both  modalities, such as a large bifurcation (Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' However, the beating of the heart, the irregular motion  of the imaging catheter, and the rotation of the catheter about its own axis create non-rigid distortions that accumulate  along the length of the pullback (Tsiknakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Manually correcting for such artifacts is prohibitively  time-consuming, requiring a cardiologist to manually mark fiduciary points in both images and shift images such that  the annotated points sufficiently align (Carlier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2014], Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2011], Hebsgaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2015]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Although such  techniques are accurate up to rigid translation, they require time investment from a trained expert to find matching  features in both modalities, creating a need for computational algorithms that non-rigidly register CCTA images to  corresponding intravascular data in an automatic fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Automatic co-registration techniques typically consist of discretely optimizing a constructed cost function over a set of  longitudinal or rotational image shifts, where the cost function varies depending on the modalities being registered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Some proposed cost functions include metrics such as lumen diameters (Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2021]), lumen contours (Molony  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2016], Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2020]), calcium thickness (Gharaibeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2020], Molony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2016]), and image  pixel intensities (Tsiknakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Similarly, rigid rotational registration for intravascular pullbacks has also  been based on extracted features such as luminal contours (Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2020]), and calcium angle (Molony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  [2016]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' However, the registration accuracy of all rigid registration methods is compromised by inconsistent motor  pullback speeds and rotational drift, which introduce non-rigid longitudinal and rotational distortions that misalign  image features such as diseased plaque and bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  To compensate for the longitudinal, rotational, and transverse motion of the catheter, several non-rigid registration  approaches have been proposed, typically to be employed after initial rigid alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Currently, non-rigid registration  of multiple intravascular imaging datasets has been predominantly performed through Dynamic Time Warping (DTW)  and Dynamic Programming (DP) (Tsiknakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2021], Molony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2016]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' However, DTW introduces non-  physiological assumptions into the registration process by discretely skipping or repeating intravascular frames, assumed  to be evenly spaced along the longitudinal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' As a result, DTW is not well suited for use for intravascular images,  with pullback acquisitions sometimes rendering up to 10 repeated intravascular imaging frames at a time (Molony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  [2016]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' On the contrary, continuous non-rigid registration methods have been developed to model the longitudinal  stretch and rotational drift between intravascular imaging frames using affine transforms and spline interpolation (Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2014], Uzu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2019]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' While such continuous non-rigid methods are more realistic, they extensively rely on  manual annotations of all bifurcation zones for image registration, severely limiting their scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' As such, there is  2  arXiv Template  A PREPRINT  no continuous non-rigid registration method as of yet that does not explicitly require fiduciary landmarks for rotational  and longitudinal alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Further, there has been an increasing interest in machine learning approaches to image  co-registration in which a neural network is trained to predict a spatial transform that maps a moving image onto a  static target image (Balakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2019], Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2020]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Such approaches critically rely on a differentiable and  continuous spatial transform allowing for back-propagation of gradients to adjust the neural network weights (Jaderberg  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2015]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' While such continuous and differential spatial transforms are available for co-registration of 3D and 2D  medical images, a similar framework that accounts for the unique variation in intravascular catheter motion has not  been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Given the previous limitations noted in prior co-registration algorithms, we here propose a novel semi-automatic  framework that takes as input an intravascular imaging pullback and a CCTA 3D image and aligns each intravascular  image frame along the artery to the equivalent frame in the CCTA image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The proposed continuous registration  methodology does not require manual matching of landmarks, with the only manual effort being the selection of viable  intravascular imaging frames and the provision of a rough centerline within the CT image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Specifically, we explore the  problem of reconstructing the path of a virtual catheter moving through and sampling from a 3D CCTA image such that  the set of frames produced by the motion of the catheter optimally reflect the equivalent target intravascular pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Key contributions of this framework include:  We present the first continuous co-registration framework for rigid and non-rigid matching of CCTA images  and intravascular images up to pixelwise alignment, with segmentations of the lumen and vessel wall as sole  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  We introduce a rigid registration approach that consists of our published longitudinal rigid registration  algorithm, which uses lumen area in a multi-step decision process, and a rotational registration step that  leverages the segmentation of the vessel wall to produce an initial rotational configuration for subsequent  registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  We introduce a novel non-rigid registration step, based only on the lumen segmentation, which is robust to  physiological catheter motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The registration is formulated in terms of finding the path of a virtual catheter,  which translates the CCTA image into an intravascular-like image by sampling the segmentation along the  virtual catheter path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The virtual catheter path is reconstructed by spatially deforming the CCTA centerline by  B-spline deformations formulated in the longitudinal, rotational, and transverse directions, ensuring a smooth  and physiological reconstruction of catheter motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Our non-rigid registration module being both continuous and differentiable, allows for easy integration into  future machine-learning-based approaches for intravascular image registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  We validate in a direct clinical setting, evaluating performance across a multimodal cohort of cardiac  patients(n = 40) and benchmarking performance against previously developed state-of-the-art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  2  Methodology  An overview of the co-registration pipeline is detailed in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' In brief, bi-modality images are processed to produce  binary segmentations of the lumen and vessel wall (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1), which are first used in a rigid registration step,  involving both longitudinal and rotational alignment (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The rigid registration is then used as an initial  estimate of a virtual catheter path forming the basis for a non-rigid registration (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The virtual catheter  path initially samples the geometry of the CT lumen to produce a virtual imaging pullback that is then compared to a  Signed Distance Field (SDF) derived from the intravascular equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' A non-rigid transformation for the longitudinal,  rotational, and transverse motion distortions is applied on the virtual catheter path and optimized to align the SDF’s in  both modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The performance of our proposed co-registration algorithm is then validated on a clinical cohort of  relevant cardiac patients (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1  Co-registration framework  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1  Preprocessing  As the basis for our co-registration pipeline, luminal segmentations from the two different image modalities are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Starting with the intravascular image set, luminal frame-by-frame segmentations are used to produce an SDF using  a fast marching method (Treister and Haber [2016]), clamped to only have negative values (indicating that a pixel is  inside the lumen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Further, the SDF is smoothed in the axial direction with a Gaussian convolutional kernel of size 3  and standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1 in order to regularize the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  3  arXiv Template  A PREPRINT  OCT image  CT image  Rigid registration  Non-rigid registration  0  20  40  60  80  100  120  140  160  Frame number  2  4  6  8  10  12  14  16  Area (mm^2)  CT rigid  OCT  OCT  CT  Aligned frames  0  20  40  60  80  100  120  140  160  Frame number  2  4  6  8  10  12  14  16  Area (mm^2)  CT non-rigid  OCT  Figure 1: Overview of the proposed registration pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The imaging modalities are rigidly co-registered in the  longitudinal and rotational directions, serving as the basis for the initialization of the virtual pullback trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The  virtual pullback trajectory is then used to sample a CT lumen signed distance field (SDF), used in direct comparison to  the equivalent OCT SDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Arc angle (degrees)  Rigid longitudinal registration  OCT image  Lumen  CT image  Rigid rotational registration  Vessel thickness (pixels)  0  50  100  150  200  250  300  350  1  2  3  4  5  6  7  Lumen  Vessel  Vessel  0  20  40  60  80  100  120  140  160  Frame number  2  4  6  8  10  12  14  16  Area (mm^2)  CT rigid  OCT  CT rigid  OCT  Figure 2: Overview of the proposed rigid registration pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The lumen segmentation area vectors from both  modalities are used to rigidly register the modalities in the longitudinal direction using a sliding window approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The  longitudinal registration is then used to match each equivalent frame for the rotational registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The vessel wall  segmentations are then converted to vessel thickness-arc angle plots and are used to determine an optimal rigid rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Coupled to the intravascular image set, a corresponding 3D SDF from the CCTA images is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Although several  methods could potentially be applied for such, a convenient approach is to derive the SDF from a computational mesh  of the coronary tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Herein, to create an SDF a narrow band is defined within the object mesh boundary, subsequently  used to compute exact Euclidean distances from each voxel center to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Outside the object boundary,  the distance field values are then set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Corresponding binary segmentations can then be produced by simple  thresholding operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Using these computational meshes, vessel centerlines are obtained using VMTK (Antiga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  [2008]), generating an array ¯r representing n spatial positions with an axial spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' A spatial derivative  is then applied to the centerline points ¯r, defining a tangent vector T for each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The two vectors U and V that  are orthogonal to the tangent vector can then be obtained through the parallel transport method (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2013]),  ensuring that the vectors V and U remain stable between frames placed along the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The centerline points  and the orthogonal vectors hence define a set of frames(¯r,T,U,and V) in 3D space that are used to sample the CCTA  SDF along an equivalent virtual catheter pullback, with dimensions equalling the intravascular dataset (in our case:  96x96xNframes with an in-plane resolution of 80 micrometers), all using a curved-planar reformation procedure  (Kanitsar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2002]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The resulting SDF is then smoothed in the axial direction with a Gaussian convolutional  kernel of size 3 and standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Through this method, virtual pullbacks of both the lumen and vessel  segmentations were produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  4  0  20  40  60  80  0  20  40  60  800  20  40  60  80  0  20  40  60  800  20  40  60  80  0  20  40  60  800  20  40  60  80  0  20  40  60  80arXiv Template  A PREPRINT  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2  Rigid registration  An overview of the rigid registration step can be seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' For the rigid longitudinal registration, the processed  lumen segmentations are used to create an area vector of equal lengths, sampling the CT virtual pullback to correspond  to the acquired intravascualr set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Here, We leverage our previous work to rigidly align the pullbacks using a multi-step  sliding window method, minimizing the difference in area vectors (for details see (Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2020])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Before  registration, continuous segments of the OCT pullback with poor lumen segmentations due to residual blood or catheter  housing were manually excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  For rigid rotational registration, the luminal profiles were deemed unreliable for producing good alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Therefore,  the vessel border segmentations were instead used for rotationally aligning the pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' For each CT and intravascular  frame, respectively, a thickness-arc angle vector is extracted by tracing a set of radial rays from the centroid of the  vessel segmentation in increments of 12 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The thickness vectors are then matched according to the result of the  longitudinal registration, with non-overlapping frames subsequently cropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The optimal rigid rotation angle is then  obtained by sliding the set of CT thickness vectors over each equivalent intravascular image vector and minimizing the  mean squared error across all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='3  Non-rigid registration  The non-rigid registration process (Figure 3) consists of optimizing a set of frame variables (¯r, T, U, and V) representing  a virtual catheter path moving through the CCTA image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The loss function to be optimized is defined as the mean squared  error between the 3D SDF generated from the two image sets, with the CCTA-SDF sampled along the aforementioned  virtual catheter path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The virtual pullback is initialized as the centerline that was calculated from the CCTA 3D model  and longitudinally cropped and rotated according to the output of the rigid registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' After rigid registration a spline  is defined based on the centerline points ¯r where the centerline points are fully described by their arclength values ¯s  along the spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Accordingly, every i-th frame can be manipulated by 4 variables, representing the arclength along  the virtual catheter path si, the rotation angle of the frame θi about the catheter path T, and the in-plane transverse  displacements du and dv along the frame vectors U and V respectively (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  To regularize the motion of the virtual catheter to be smooth and physiological, the 4 frame manipulation variable  sets are parametrized by a sparse set of control points controlling a B-spline deformation (Rueckert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [1999])  independently acting on 4 nx1 vectors representing the frame manipulation variables ¯s, ¯θ, ¯du, and ¯dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Thus, for a 1D  control point grid of size N, the relation between a frame manipulation variable v and the control points p can be  described by:  v(s) =  N  �  i=0  Bi(s)pi,  (1)  where Bi(s) is a polynomial basis function of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' In matrix form, the same can be represented by:  V = BP,  (2)  in which V ∈ Rn×1, B ∈ Rn×N, P ∈ RN×1 where n is the number of frames and N is the number of control points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' B  is the univariate B-spline tensor and can be pre-computed from the initial frame manipulation variable vectors, while P  is the deformed control point grid vector that is optimized during co-registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Instead of directly optimizing for the set of Ns = 30 control points P s controlling the arclength variables s for each  frame, the control point deformations ∆P s  i can be parametrized by a deformation vector Xs of size Ns − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' dictating  the relative displacement of each control point from its proximal neighbor, with the most proximal control point being  fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' This is done to account for the cumulative effect of catheter motor speed variations on the rest of the pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Therefore, the deformation of each control point can be defined as the cumulative sum of the relative deformations  along the proximal control points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Moreover, to regularize the catheter motion and prevent backwards movement, the  relative deformation of each control point is limited to a fraction (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='35) of the distance between control points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  ∆P s  i = Xs  i +  i−1  �  j=0  Xs  j  (3)  Once the control points are deformed into a new configuration, the new arclength values for each frame ¯s is calculated  through equation 2 and the frame vectors (T,U, and V) are then recalculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  ¯s = BsP s  (4)  5  arXiv Template  A PREPRINT  Rigid initialization  Non-rigid transform  Rotational transform  Longitudinal transform  Transverse transform  0  20  40  60  80  100  Frame number  0  5  10  15  20  25  30  35  40  Arclength (mm)  Rigid  Non Rigid  0  20  40  60  80  100  Frame number  60  50  40  30  20  10  Theta (degrees)  Rigid  Non Rigid  0  20  40  60  80  100  Frame number  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='5  Displacement (mm)  U-displacement  V-displacement  OCT: Target  Loss  CT: Moving  Figure 3: Overview of the spatial deformation acting on the virtual catheter path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The longitudinal transform stretches  and compresses the space between adjacent frames, at which point the frame vectors (T,U, and V) are recalculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The  rotational transform is then applied to the frame vectors orthogonal to the tangent (U and V) about T, and the transverse  transform is then applied to shift the centerline points in the direction of the new frame vectors (U and V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='4  Non-rigid rotational registration  Similar to the longitudinal registration, the set of Nθ = 20 control points P θ controlling the rotation of each frame  about the catheter axis can be parameterized by a relative rotation vector Xθ of size Nθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The rotation value for each  control points is defined by:  ∆P θ  i = Xθ  i +  i−1  �  j=0  Xθ  j  (5)  The rotation correction for each frame is applied after the non-rigid longitudinal transformation but before the non-rigid  transverse transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Once the control points are deformed into a new configuration, the new rotation values for  each frame ¯θ can be calculated through equation 2 and used to rotate frame vectors U and V about the tangent vectors T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  ¯θ = BθP θ  (6)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='5  Non-rigid transverse registration  The virtual catheter was biased to stay close to the centerline by optimizing the Nd = 60 control points determining the  in-plane transverse displacements du and dv directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Thus the 2 orthogonal transverse displacements for each frame  was calculated from the matrix relation:  ¯d = BdP d  (7)  Where for each frame the displacements along the vectors U and V were applied as a final step after the non-rigid  longitudinal and rotational transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2  Performance evaluation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1  Image data  To evalute our proposed co-registration framework, a dataset consisting of n = 40 matched OCT and CT image pairs  from 5 different clinical centers were selected, all originating from the Precise Percutaneous Coronary Intervention Plan  (P3) study (Nagumo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' As each OCT pullback image consisted of 375 frames, the intravascular imaging  dataset comprised of approximately 15,000 image frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The OCT lumen in every frame was manually annotated by  6  arXiv Template  A PREPRINT  trained cardiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Further, the vessel wall borders in every OCT frame were segmented using a convolutional neural  network, using the previously published U-net as base architecture (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2015]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Details of the network,  training, and validation can be found in Supplementary Material A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The lumen and vessel wall segmentations were then  re-sampled to represent a 3D image of dimensions 96x96xNframes with an in-frame resolution of 80 micrometers  and an out-of-frame resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='4 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' All utilized intravascular pullbacks were manually deemed as of sufficient  image quality, with appropriate quality lumen segmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' For the CCTA data, a 3D model of the coronary tree for  each patient was produced by HeartFlow using the CCTA image (Sonck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2022]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The 3D model was then used to  produce a 3D SDF with a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='25mm along each axis with an image dimension of 768x768x482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2  Co-registration accuracy  In order to evaluate the performance of the non-rigid registration, 114 bifurcations were manually marked in the OCT  pullback as well as in the rigid and non-rigid virtual pullback segmentations generated from the CCTA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Bifurcations  were defined as the last image frame before a visual coronary artery split into two branches could be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Bifurcations  that were common to both modalities had their frame numbers recorded for validation of the non-rigid registration  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Longitudinal validation was conducted by comparing the frame number of a bifurcation in the OCT data  with the equivalent bifurcation frame number in the virtual pullback before and after non-rigid registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' In order to  validate the non-rigid rotational registration, the bifurcation angle difference between OCT pullback and the virtual  pullback was compared before and after rotational registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' As the bifurcation angle between bifurcation sections  that were not longitudinally matched is expected to be uncorrelated, a separate analysis was conducted to characterize  how angular mismatch varies when the bifurcations are longitudinally matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Furthermore, only bifurcations that had  a frame mismatch below 6 frames were considered for extensive analysis of rotational accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='3  Comparison to alternative approaches  The most common co-registration methodology employed for coronary artery registration has been discrete optimization  approaches such as DTW and Dynamic Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Therefore, in order to evaluate the performance of our  longitudinal and rotational co-registration framework against state-of-the-art discrete approaches, we applied the  methodology described in Karmakar et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' al (Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2022]) on the same dataset used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The approach  utilizes DTW to longitudinally align two coronary imaging modalities and Dynamic Programming to rotationally align  each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' We utilized a window length of 4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='8mm) as implemented in the previous study and recorded identical  alignment metrics for 114 matched bifurcations in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The non-rigid registration algorithm was applied after  the rigid longitudinal registration step described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' A substudy was also conducted in which the angular  alignment of all bifurcations was compared to the angular alignment of longitudinally matched bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='4  Optimization details  The gradient descent-based optimization procedure was implemented in PyTorch with the Adam optimizer (Kingma  and Ba [2014]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' A learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='001 was used for the non-rigid longitudinal parameters and a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='01 was  used for both the rotational and transverse parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Each co-registration procedure was run for a minimum of 200  iterations to ensure convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  3  Results  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1  Longitudinal Registration  Longitudinal registration plots in Figures 4 and 7A1-2 show that using rigid registration alone (Figure 7A1), few  bifurcations were longitudinally aligned within 6 (dotted line), 4 (dashed line), or 2 (solid line) frame distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  However, after non-rigid alignment (Figure 7A2), distinct improvement can be observed with a majority of bifurcations  are aligned within 6 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' These results are visualized by the longitudinal mismatch plot (Figure 5A), revealing  that after rigid alignment, the percentage of bifurcations matched within 2, 4, and 6 frames are 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='3, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1, and 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9%,  respectively, while after non-rigid alignment, these values increase to 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='5, 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9, and 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Moreover, the scatterplot  for non-rigid registration (A2) demonstrates that the majority of bifurcations (86% shown in green) were enhanced  in terms of frame alignment, while a negligible number of bifurcations had slightly (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='4 % shown in orange) or  significantly (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='6 % shown in red) worse alignment after non-rigid registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Table 2 further demonstrates the effect  of non-rigid registration, in which the mean frame difference after rigid registration was 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9 frames and subsequently  decreased to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='3 frames after non-rigid registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  7  arXiv Template  A PREPRINT  A1  B1  C1  D1  E1  F1  G1  A2  B2  C2  D2  E2  F2  G2  A3  B3  C3  D3  E3  F3  G3  Bifurcation Frames  0  20  40  60  80  100  120  140  160  Frame number  0  2  4  6  8  10  12  14  16  18  Area (mm^2)  CT  OCT  A  B  C D  E F  G  Figure 4: Qualitative results for a single co registered case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Top row shows area plot along the artery for the non-rigidly  registered CT (green) and the OCT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The bifurcation zones (Sections A-G) are marked and labeled for further  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Bifurcation frames from the CT, OCT, and overlapped segmentation maps are presented in the bottom row for  qualitative analysis of the rotational and transverse co-registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  A  B  0  5  10  15  20  25  30  35  40  Maximum frame mismatch  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  Matched bifucations (%)  Rigid  Rigid+Non-rigid  0  20  40  60  80  100  120  140  160  180  Maximum angular mismatch (degrees)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  Matched bifucations (%)  Rigid  Rigid+Non-rigid  Figure 5: Quantitative results comparing the quality of  rigid and non-rigid co registration in longitudinal and ro-  tational directions with varying degrees of misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  The mismatch plots exhibit the % of matched bifurca-  tions with increasing longitudinal (A) and rotational (B)  alignment mismatch criteria (x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  ~10 degrees  ~20 degrees  ~30 degrees  Angular Mismatch  Figure 6: Grid plot showing multiple aligned bifurcation  segmentations using an SDF-based loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Angular mis-  matches up to 10, 20, and 30 degrees are shown in the  first, second, and third columns respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2  Rotational Registration  Examination of the individual bifurcating frames in figure 4 for the CT (row 1) and OCT (row 2) frames indicates  excellent rotational and transverse alignment between both imaging modalities as evident from the raw images and the  overlapped segmentations (row 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Rotational registration plots in figure 7B1-2 demonstrate that few bifurcations are  rotationally aligned within 30 (dotted line), 20 (dashed line), or 10 (solid line) degrees after rigid alignment (B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' After  non-rigid alignment (Figure 7B2), a majority of bifurcations were aligned within 30 degrees, with a significant amount  aligned within 20 and 10 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Examination of the rotational mismatch plot (Figure 5B) quantitatively demonstrates  an increase in the percentage of bifurcations aligned up to an angular mismatch of 10, 20, and 30 degrees from %  values of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='3, 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='4, and 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='3 to 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='5, 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='7, and 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='8% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' the non-rigid registration scatterplot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='arXiv Template  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='A PREPRINT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='A1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='A2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='B1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='B2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Bifurcation number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='35  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Longitudinal misalignment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Bifurcation number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='35  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Longitudinal misalignment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Non-rigid<0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Non-rigid<2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Non-rigid>=2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Bifurcation number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='125  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='175  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Angular misalignment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Non-rigid<0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Non-rigid<20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Non-rigid>=20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Bifurcation number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='125  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='175  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Angular misalignment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='Figure 7: Quantitative results comparing the quality of rigid and non-rigid co-registration in longitudinal and rotational  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The first row compares bifurcation frame mismatch before (A1) and after (A2) non-rigid registration in  the form of scatterplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The second row compares bifurcation angular mismatch before (B1) and after (B2) non-rigid  registration in the form of scatterplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The scatterplot for the longitudinal and rotational non-rigid registration (A2 and  B2) are color-coded to exhibit the change in alignment metric after non-rigid registration, where green represents an  increase in alignment, orange represents a mild decrease in alignment, and red represents a strong decrease in alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Only bifurcations that were longitudinally matched within 6 OCT frames were analyzed for rotational alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  demonstrates that the majority of bifurcations had their angular mismatch decreased after non-rigid alignment (66%  shown in green) and only a minority had their angular mismatch values slightly (22% shown in orange) or significantly  (12% shown in red) increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The mean value of the angular mismatch before and after non-rigid alignment is reported  in Table 2, in which the mean angular mismatch decreases from 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0 to 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='6 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='3  Comparison with previous approaches  A direct comparison of the virtual catheter method with state-of-the-art discrete optimization approaches can be seen in  Tables 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Comparing the virtual catheter method to a discrete optimization approach for longitudinal registration,  it was shown that DTW produces significantly poorer results in longitudinal registration, with the longitudinal mismatch  of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='7 frames being higher than rigid longitudinal registration average of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Comparing the virtual catheter  method to using Dynamic Programming for rotational registration, it was shown that such discrete optimization  algorithms exhibit poor performance for CT-OCT rotational registration (angular mismatch of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9 degrees) which  is higher than the angular mismatch after rigid rotational registration alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Table 3 quantifies the angular mismatch  in the case where non-rigid longitudinal registration is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' For bifurcations with a maximum frame mismatch  of 6 after non-rigid registration, the angular mismatch decreases from 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9 to 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2 for the Dynamic Programming  approach, while for the virtual catheter method, the angular mismatch decreases from 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='6 to 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' In contrast, the  angular mismatch for the rigid registration is unchanged after excluding non-matching bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  9  arXiv Template  A PREPRINT  Figure 8: Qualitative results comparing the alignment of calcium annotations between OCT (first row) and CT (third  row) for selected frames with good luminal alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The middle row shows the calcium annotations for OCT (red)  and CT (green) superimposed on each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Table 1: Accuracy of alternative co-registration approaches, proposed for intravascular-intravascular image registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Data is presented from left to right including evaluated co-registered modalities, dataset size, and overall methodological  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Further, average errors are presented in both longitudinal (frames) and rotational (degree) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Modalities  Dataset Size  Methodology  Longitudinal mismatch  Angular mismatch  Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2022]  OCT-OCT  9 patients  DTW + Dynamic Programming  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='7 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='7  Tsiknakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2023]  OCT-OCT  21 patients  DTW + Harmony Search  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='6 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='81  Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2022]  OCT-IVUS  7 patients  DTW + Dynamic Programming  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='7  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1 ± 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2  Molony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2016]  OCT-IVUS  12 patients  DTW + Dynamic Programming  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='8 ± 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9  Table 2: Accuracy of co-registration approaches applied to CT-OCT image registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Data is presented from left to  right including evaluated co-registered modalities, dataset size, and overall methodological approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Further, average  errors are presented in both longitudinal (frames) and rotational (degree) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' All approaches in this table have  been evaluated on the same dataset  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Modalities  Dataset Size  Methodology  Longitudinal mismatch  Angular mismatch  Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2022]  CT-OCT  40 patients  DTW + Dynamic Programming  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='7 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9 ± 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  Ours (Rigid)  CT-OCT  40 patients  Virtual Catheter Method  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0 ± 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9  Ours (Rigid+Non-rigid)  CT-OCT  40 patients  Virtual Catheter Method  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='6 ± 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9  Table 3: Accuracy of co-registration approaches applied to CT-OCT image registration for bifurcations that are  longitudinally matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Data is presented from left to right including methodological approach and number of  longitudinally matched bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Bifurcations are considered longitudinally matched when they have a maximum  frame difference of 6 after non-rigid longitudinal registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Further, average errors are presented for rotational  direction in degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Methodology  Matched Bifurcations  Angular Mismatch  Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2022]  DTW + Dynamic Programming  52/114  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2 ± 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='9  Ours (Rigid)  Virtual Catheter Method  99/114  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0 ± 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  Ours (Rigid+Non-rigid)  Virtual Catheter Method  99/114  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='8 ± 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='0  10  OUarXiv Template  A PREPRINT  4  Discussion  The aim of the current study was to develop a fully automatic registration algorithm to align CCTA and intravascular  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Specifically, we propose a novel registration process finding the optimal rigid and non-rigid spatial transforms  using a virtual catheter path in the CCTA data, aligning the non-invasive modality to its invasive counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Our results  indicate excellent co-registration accuracy, with excellent agreement with reference manual landmark annotations  (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Further, our results underline the critical importance of a non-rigid registration step, with significant  enhancement in both longitudinal and rotational alignments observed when comparing rigid vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' non-rigid alignments in  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' We demonstrate that for the vast majority of bifurcations, our framework is able to improve the longitudinal  and rotational alignment of common bifurcations within the CT and OCT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Lastly, we demonstrate the added  value of our approach as compared to state-of-the-art alternatives, with a head-to-head comparison to previously  developed discrete optimization alignment algorithms (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' A head-to-head comparison demonstrates that discrete  optimization approaches for longitudinal and rotational alignment suffer a significant drop in alignment quality when  applied for the task of CT-OCT co-registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Meanwhile, our approach maintains performance accuracy in line with  simpler tasks such as intravascular-intravascular image registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='1  Related work  Currently, a majority of CCTA studies that validate their CT findings with intravascular images have used rigid manual  registration based on fiduciary landmarks such as bifurcations or large calcifications (Carlier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2014], Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  [2011], Hebsgaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2015]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' One of the few studies that attempted to align CT and intravascular data up to a non-  rigid level is by Uzu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2019] in which a B-spline deformation model was used to optimize the alignment of manually  annotated bifurcation landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Though powerful, such an approach is time-consuming due to the significant amount  of manual processing required to process the OCT images, rigidly align the geometric models, and mark bifurcations  within every artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' In comparison, our approach implicitly matches nearby bifurcations using longitudinally smoothed  SDFs representing the CT and OCT lumens, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Other approaches that register intravascular-to-intravascular  modalities have in the past relied on DTW (Molony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2016], Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2022]), discretely optimizing the  frame-wise progression of one intravascular pullback to maximize longitudinal and rotational alignment with another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Such methods, nevertheless, attempt to recapitulate the continuous motion distortions introduced by the catheter path  with discrete non-physiological frames or repeats (Molony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2016]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' However, skipping or repeating several frames  that the catheter motion is not smooth or continuous, which is an unrealistic assumption about the catheter path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Direct numerical comparison of reported co-registration accuracy across published approaches is inherently difficult  as co-registration accuracy is highly dependent on the specific dataset explored as well as which modalities are being  coregistered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' For example, the simplest co-registration task would be represented by the alignment of same-modality  images such as OCT-OCT image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' For such tasks, our previously published DTW and dynamic programming  approach (Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2022]) exhibits similar perfomance compared to other state-of-the-art algorithms (Tsiknakis  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2023]) (See Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' When applied to multimodality datasets, such as IVUS-OCT image pairs, our previously  developed approach (Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2022]) suffers distinctive drops in both longitudinal and rotational accuracy (see  Table 1), however, still maintains comparative performance to similar Dynamic Programming approaches (Molony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  [2016]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Thus, in order to facilitate a head-to-head comparison on the more challenging task of registering CT-OCT  image pairs, we applied our previously developed discrete optimization algorithm (Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2022]) on our  multi-modal dataset of 40 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Doing so, we found that our previous approach produced significantly worse  longitudinal and rotational alignment compared to the virtual catheter method, with both longitudinal and angular  alignment being worse than simple rigid registration (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' In contrast, our developed methodology achieves  competitive results with even intravascular-intravascular registration studies (Table 1 and Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='2  Methodological Adaptations  From the above it can be seen that the task of co-registering CT and OCT images presents several unique difficulties for  discrete registration algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Our framework has several features that were designed to mitigate such challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  First, the low resolution of CT images induces a circular bias in the lumen segmentations (see Figure 4), as well as a  tendency to miss small bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Such circularly symmetric regions hence create zones of longitudinal and rotational  ambiguity along the pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Our approach tries to minimize the dependency of such by formulating the longitudinal  and rotational transforms acting on the virtual path in terms of a regularized and smooth B spline deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' As  such, the optimization procedure is mainly dominated by the alignment of prominent non-symmetric features such  as bifurcations, rather than the circularly symmetric lumen segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' This ensures that the rotational alignment of  all non-bifurcating lumen frames that are in proximity to matched bifurcations are properly matched due to B spline  interpolation (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Another significant issue faced in previous rotational co-registration algorithms (Karmakar  11  arXiv Template  A PREPRINT  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2022], Molony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2016]) is that lumen bifurcations are only able to contribute to rotational alignment if they  exist within the same frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' As such, poor longitudinal alignment of bifurcations was a significant contributing factor  to the poor performance of our previously developed dynamic programming algorithm for rotational co-registration  (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Our framework, in contrast, minimizes this dependency through the use of a Gaussian smoothing kernel  applied longitudinally over the SDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Longitudinal smoothing allows single-frame bifurcations to appear in adjacent  frames and smooths the loss surface such that bifurcations in the different modalities can be better aligned (Figures 4  and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Another design choice that was found to increase training stability and co-registration quality was the use of  SDF’s to determine alignment, as opposed to using a segmentation loss such as cross-entropy or Dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' This was due to  the fact that when the lumen segmentations were fully overlapping, multiple rotational and transverse configurations  contribute equally to a segmentation loss function, preventing the algorithm from making fine adjustments in the spatial  transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Figure 6 further demonstrates how the quality of rotational registration varies with angular mismatch, where  the angular mismatch tends to occur due to the limitations of local optimization of pixel alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' At less than 10  degrees mismatch, the difference in alignment is minimal, while under 30 degrees, the difference in alignment can  be attributed to differing lumen bifurcation shapes in the CT and OCT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Lastly, many co-registration methods  normalize the position of the lumen by the artery centroid (Uzu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2019], Karmakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2020, 2022], Molony  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2016]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' While such an approach manages to align CT and OCT frames with circularly symmetric lumen, it fails  to effectively align equivalent frames with bifurcations as the segmentation can have different maximum diameters  between the modalities and thus different centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Moreover, centering the image around the lumen centroids can  cause the algorithm to align bifurcations 180-degrees from the correct orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' It was empirically found that this  phenomenon was found to be a significant contributing factor to the degradation of the co-registration performance of  our previously developed discrete co-registration algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' In this framework, we instead choose to jointly optimize  for the transverse displacements of the virtual path in addition to the longitudinal and rotational displacements, which  allows for the bifurcations in both modalities to be anchored around the OCT catheter location and enables near  pixelwise alignment of the lumen (Figures 4,6) and plaque constituents such as calcium (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='3  Limitations  Though very promising for clinical applications, our developed approach has a number of limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' First, the  non-rigid spatial transform acting on the virtual catheter path is found through gradient-based optimization, requiring  that landmarks lie sufficiently close such that proper matching is ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' For example, common bifurcations that  have a frame mismatch of more than 6 frames (corresponding to the longitudinal smoothing kernel) are expected to  be uncorrelated in terms of orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' This issue can be mitigated by integrating deep learning networks which can  accurately predict the spatial transform needed to align the two modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Another limitation is the dependence of  non-rigid registration on the lumen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The lumen estimation is expected to be accurate for both modalities and as such,  ensures good registration accuracy for regions that include many bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' However, due to the poor resolution of  CCTA images, the lumen estimation tends to be highly circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Accordingly, it is expected that rotational co-registration  certainty increases with bifurcation proximity but decreases in stenotic regions that contain highly circular CT luminal  profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' In the future, co-registration accuracy can likely be improved by including contextual information relating  to the vessel wall such as lesion content and morphology as a supervisory signal in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Third, the use  of a pixel-wise loss as a surrogate for luminal alignment may not necessarily result in optimal alignment of lumen  bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' As seen in Figure 6, a pixel-wise loss function can occasionally bias the spatial transform to align the  central lumen body over aligning the bifurcation in scenarios where the bifurcation shapes are not perfectly matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  In the future, this issue can be mitigated by introducing an orientation loss to bias the spatial transform to rotationally  align bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Lastly, regularizing the spatial transform and smoothing the SDF’s can create difficulties in localizing  landmarks up to frame-wise precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' This can be seen in the area curve in Figure 4 section B with the slightly  mismatched bifurcation and in Figure 7 A2 and B2 with a minor amount of bifurcations with increased frame and  angular mismatch values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The localization capabilities of the algorithm can be improved by introducing multi-scale  deformation steps where finer control point grids can be recursively used as the basis for the spatial transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='4  Translational Benefits  The development of automatic frame-wise matching algorithms for CT-OCT data fusion would enable the development  of several research-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' First, intravascular imaging data can act as ground truth to validate the reliability  of CT in delineating several morphological metrics of atherosclerosis, such as luminal area, lipid content, and calcium  volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Understanding when CT-derived morphological metrics are reliable is critical for both therapy planning and  deciding when to rely on intravascular imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' For example, studying the interaction between calcium blooming and  measured lumen size in CT images necessitates that ground truth lumen measurements be available, which can only be  provided by frame-wise co-registration algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Figure 8 demonstrates that such a frame-by-frame comparison can  be done provided that longitudinal and rotational co-registration is of sufficient quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Second, multi-modal data fusion  12  arXiv Template  A PREPRINT  would allow for the enhanced generation of patient-specific digital twins from coronary images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' There has been an  increasing interest in the use of intravascular-images to create computational digital twins for the prediction of coronary  pathophysiology and clinical decision-making (Kadry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' [2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' However, intravascular images, while providing  excellent resolution within the imaging frame, do not provide sufficient information to create a fully physiological  artery model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Intravascular images typically suffer from intra-frame motion drift artifacts in the longitudinal and  rotational directions and cannot capture information on the three-dimensional centerline of the artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' On the other  hand, CT suffers from poor resolution but is able to capture the three-dimensional nature of the artery with high  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Combining both modalities would allow researchers to investigate the importance of longitudinal and  rotational distortions, as well as modeling arterial tortuosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  References  Georgios Tzimas, Gaurav S Gulsin, Hidenobu Takagi, Niya Mileva, Jeroen Sonck, Olivier Muller, Jonathon A Leipsic,  and Carlos Collet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Coronary ct angiography to guide percutaneous coronary intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Radiology: Cardiothoracic  Imaging, 4(1):e210171, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Kenzo Uzu, Hiromasa Otake, Gilwoo Choi, Takayoshi Toba, Hyun Jin Kim, Arjun Roy, Michiel Schaap, Leo Grady,  Masahito Kawata, Toshiro Shinke, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Lumen boundaries extracted from coronary computed tomography angiogra-  phy on computed fractional flow reserve (ffrct): validation with optical coherence tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Eurointervention:  Journal of Europcr in Collaboration with the Working Group on Interventional Cardiology of the European Society  of Cardiology, 14(15):e1609–e1618, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Choongki Kim, Sung-Jin Hong, Dong-Ho Shin, Jung-Sun Kim, Byeong-Keuk Kim, Young-Guk Ko, Donghoon Choi,  Yangsoo Jang, and Myeong-Ki Hong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Limitations of coronary computed tomographic angiography for delineating the  lumen and vessel contours of coronary arteries in patients with stable angina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' European Heart Journal-Cardiovascular  Imaging, 16(12):1358–1365, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Matthew J Budoff, David Dowe, James G Jollis, Michael Gitter, John Sutherland, Edward Halamert, Markus Scherer,  Raye Bellinger, Arthur Martin, Robert Benton, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Diagnostic performance of 64-multidetector row coronary  computed tomographic angiography for evaluation of coronary artery stenosis in individuals without known coronary  artery disease: results from the prospective multicenter accuracy (assessment by coronary computed tomographic  angiography of individuals undergoing invasive coronary angiography) trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Journal of the American College of  Cardiology, 52(21):1724–1732, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Yu Takahashi, Takayoshi Toba, Hiromasa Otake, Yusuke Fukuyama, Shinsuke Nakano, Yoichiro Matsuoka, Kosuke  Tanimura, Yu Izawa, Hiroyuki Kawamori, Atsushi K Kono, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Feasibility of morphological assessment of coronary  artery calcification with electrocardiography-gated non-contrast computed tomography: a comparative study with  optical coherence tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The International Journal of Cardiovascular Imaging, 37(4):1445–1453, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Collin Fischer, Edward Hulten, Pallavi Belur, Ryan Smith, Szilard Voros, and Todd C Villines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Coronary ct angiography  versus intravascular ultrasound for estimation of coronary stenosis and atherosclerotic plaque burden: a meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Journal of cardiovascular computed tomography, 7(4):256–266, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Michiel A De Graaf, Alexander Broersen, Pieter H Kitslaar, Cornelis J Roos, Jouke Dijkstra, Boudewijn PF Lelieveldt,  J Wouter Jukema, Martin J Schalij, Victoria Delgado, Jeroen J Bax, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Automatic quantification and characterization  of coronary atherosclerosis with computed tomography coronary angiography: cross-correlation with intravascular  ultrasound virtual histology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The international journal of cardiovascular imaging, 29(5):1177–1190, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  H Brodoefel, C Burgstahler, M Heuschmid, A Reimann, F Khosa, A Kopp, S Schroeder, CD Claussen, and ME Clouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Accuracy of dual-source ct in the characterisation of non-calcified plaque: use of a colour-coded analysis compared  with virtual histology intravascular ultrasound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' The British journal of radiology, 82(982):805–812, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  A Lin, N Manral, P McElhinney, A Killekar, H Matsumoto, S Cadet, S Achenbach, SJ Nicholls, DT Wong, D Berman,  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Deep learning-based plaque quantification from coronary computed tomography angiography: external  validation and comparison with intravascular ultrasound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' European Heart Journal, 42(Supplement_1):ehab724–0161,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Marly van Assen, Akos Varga-Szemes, U Joseph Schoepf, Taylor M Duguay, H Todd Hudson, Svetlana Egorova,  Kjell Johnson, Samantha St Pierre, Beatrice Zaki, Matthijs Oudkerk, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Automated plaque analysis for the  prognostication of major adverse cardiac events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' European Journal of Radiology, 116:76–83, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Nikos Tsiknakis, Constantinos Spanakis, Panagiota Tsompou, Georgia Karanasiou, Gianna Karanasiou, Antonis  Sakellarios, George Rigas, Savvas Kyriakidis, Michael Papafaklis, Sotirios Nikopoulos, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Ivus longitudinal and  axial registration for atherosclerosis progression evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Diagnostics, 11(8):1513, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  13  arXiv Template  A PREPRINT  Stéphane Carlier, Rich Didday, Tristan Slots, Peter Kayaert, Jeroen Sonck, Mike El-Mourad, Nicolas Preumont,  Dany Schoors, and Guy Van Camp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' A new method for real-time co-registration of 3d coronary angiography and  intravascular ultrasound or optical coherence tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Cardiovascular Revascularization Medicine, 15(4):  226–232, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Shengxian Tu, Niels R Holm, Gerhard Koning, Zheng Huang, and Johan HC Reiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Fusion of 3d qca and ivus/oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  The international journal of cardiovascular imaging, 27(2):197–207, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Lasse Hebsgaard, Troels Munck Nielsen, Shengxian Tu, Lars Romer Krusell, Michael Maeng, Karsten Tange Veien,  Bent Raungaard, Christian Juhl Terkelsen, Anne Kaltoft, Johan HC Reiber, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Co-registration of optical coherence  tomography and x-ray angiography in percutaneous coronary intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' the does optical coherence tomography  optimize revascularization (doctor) fusion study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' International journal of cardiology, 182:272–278, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Hui Qin, Chunming Li, Yingguang Li, Jiayue Huang, Fan Yang, Takashi Kubo, Takashi Akasaka, Changyan Xiao,  Juan Luis Gutiérrez-Chico, and Shengxian Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Automatic coregistration between coronary angiography and  intravascular optical coherence tomography: Feasibility and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' JACC: Asia, 1(2):274–278, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  David S Molony, Lucas H Timmins, Emad Rasoul-Arzrumly, Habib Samady, and Don P Giddens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Evaluation of a  framework for the co-registration of intravascular ultrasound and optical coherence tomography coronary artery  pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Journal of biomechanics, 49(16):4048–4056, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Abhishek Karmakar, Max L Olender, Farhad Rikhtegar Nezami, David Marlevi, Evan Shlofmitz, Richard A Shlofmitz,  and Elazer R Edelman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Detailed investigation of lumen-based tomographic co-registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' In 2020 IEEE International  Conference on Bioinformatics and Biomedicine (BIBM), pages 1038–1042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Yazan Gharaibeh, Juhwan Lee, David Prabhu, Pengfei Dong, Vladislav N Zimin, Luis A Dallan, Hiram Bezerra, Linxia  Gu, and David Wilson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Co-registration of pre-and post-stent intravascular oct images for validation of finite element  model simulation of stent expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' In Medical Imaging 2020: Biomedical Applications in Molecular, Structural,  and Functional Imaging, volume 11317, pages 306–316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' SPIE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Ling Zhang, Richard Downe, Zhi Chen, Shanhui Sun, T Masiarov, Tomas Kovarnik, John Lopez, Milan Sonka, and  Andreas Wahle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Side-branch guided registration of intravascular ultrasound pullbacks in coronary arteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' In  MICCAI Workshop in Computing and Visualization for IntraVascular Imaging and Computer Assisted Stenting  (CVII-STENT), pages 44–51, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Guha Balakrishnan, Amy Zhao, Mert R Sabuncu, John Guttag, and Adrian V Dalca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Voxelmorph: a learning framework  for deformable medical image registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' IEEE transactions on medical imaging, 38(8):1788–1800, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Yabo Fu, Yang Lei, Tonghe Wang, Walter J Curran, Tian Liu, and Xiaofeng Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Deep learning in medical image  registration: a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Physics in Medicine & Biology, 65(20):20TR01, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Max Jaderberg, Karen Simonyan, Andrew Zisserman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Spatial transformer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Advances in neural  information processing systems, 28, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Eran Treister and Eldad Haber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' A fast marching algorithm for the factored eikonal equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Journal of Computational  physics, 324:210–225, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Luca Antiga, Marina Piccinelli, Lorenzo Botti, Bogdan Ene-Iordache, Andrea Remuzzi, and David A Steinman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  An image-based modeling framework for patient-specific computational hemodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Medical & biological  engineering & computing, 46(11):1097–1112, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Jixiang Guo, Shun Li, Yim Pan Chui, Jing Qin, and Pheng Ann Heng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Mesh quality oriented 3d geometric vascular  modeling based on parallel transport frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Computers in Biology and Medicine, 43(7):879–888, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Armin Kanitsar, Dominik Fleischmann, Rainer Wegenkittl, Petr Felkel, and Eduard Groller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' CPR-curved planar  reformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' IEEE, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Daniel Rueckert, Luke I Sonoda, Carmel Hayes, Derek LG Hill, Martin O Leach, and David J Hawkes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Nonrigid  registration using free-form deformations: application to breast mr images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' IEEE transactions on medical imaging,  18(8):712–721, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Sakura Nagumo, Carlos Collet, Bjarne L Norgaard, Hiromasa Otake, Brian Ko, Bon-kwon Koo, Jonathon Leipsic,  Daniele Andreini, Ward Heggermont, Jesper M Jensen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Rationale and design of the precise percutaneous  coronary intervention plan (p3) study: Prospective evaluation of a virtual computed tomography-based percutaneous  intervention planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Clinical cardiology, 44(4):446–454, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Olaf Ronneberger, Philipp Fischer, and Thomas Brox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' U-net: Convolutional networks for biomedical image segmenta-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' In International Conference on Medical image computing and computer-assisted intervention, pages 234–241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  14  arXiv Template  A PREPRINT  Jeroen Sonck, Sakura Nagumo, Bjarne L Norgaard, Hiromasa Otake, Brian Ko, Jinlong Zhang, Takuya Mizukami,  Michael Maeng, Daniele Andreini, Yu Takahashi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Clinical validation of a virtual planner for coronary  interventions based on coronary ct angiography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Cardiovascular Imaging, 15(7):1242–1255, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Abhishek Karmakar, Max L Olender, David Marlevi, Evan Shlofmitz, Richard A Shlofmitz, Elazer R Edelman, and  Farhad R Nezami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Framework for lumen-based nonrigid tomographic coregistration of intravascular images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Journal  of Medical Imaging, 9(4):044006, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Diederik P Kingma and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Adam: A method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='6980,  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Nikos Tsiknakis, Constantinos Spanakis, Panagiota Tsoumpou, Georgia Karanasiou, Gianna Karanasiou, Antonis  Sakellarios, George Rigas, Savvas Kyriakidis, Michail I Papafaklis, Sotirios Nikopoulos, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Oct sequence  registration before and after percutaneous coronary intervention (stent implantation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Biomedical Signal Processing  and Control, 79:104251, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  Karim Kadry, Max L Olender, David Marlevi, Elazer R Edelman, and Farhad R Nezami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' A platform for high-fidelity  patient-specific structural modelling of atherosclerotic arteries: from intravascular imaging to three-dimensional  stress distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content=' Journal of the Royal Society Interface, 18(182):20210436, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
+page_content='  15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf'}
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+1
+Transferable Energy Storage Bidder
+Yousuf Baker, Ningkun Zheng, Student Member, IEEE, Bolun Xu, Member, IEEE
+Abstract—Energy storage resources must consider both price
+uncertainties and their physical operating characteristics when
+participating in wholesale electricity markets. This is a challeng-
+ing problem as electricity prices are highly volatile, and energy
+storage has efficiency losses, power, and energy constraints. This
+paper presents a novel, versatile, and transferable approach
+combining model-based optimization with a convolutional long
+short-term memory network for energy storage to respond to
+or bid into wholesale electricity markets. We apply transfer
+learning to the ConvLSTM network to quickly adapt the trained
+bidding model to new market environments. We test our proposed
+approach using historical prices from New York State, showing
+it achieves state-of-the-art results, achieving between 70% to
+near 90% profit ratio compared to perfect foresight cases,
+in both price response and wholesale market bidding setting
+with various energy storage durations. We also test a transfer
+learning approach by pre-training the bidding model using
+New York data and applying it to arbitrage in Queensland,
+Australia. The result shows transfer learning achieves exceptional
+arbitrage profitability with as little as three days of local training
+data, demonstrating its significant advantage over training from
+scratch in scenarios with very limited data availability.
+Index Terms—Energy storage; Deep learning; Transfer learn-
+ing; Power system economics.
+I. INTRODUCTION
+Successful participation of energy storage resources in com-
+petitive electricity markets benefits storage investors and social
+welfare. Ancillary services such as frequency regulation have
+been the primary sources of profit for energy storage owners,
+but these markets have quickly saturated due to surging storage
+deployments and small market size [1]. In the meantime, the
+share of storage arbitraging in wholesale markets has tripled
+from a little less than 20% in 2016 to almost 60% in 2021 [1].
+Thus price arbitrage in wholesale markets will be the main
+focus for future grid-scale energy storage projects.
+Energy storage arbitrages price differences and earns rev-
+enues in wholesale energy markets, i.e., charging during low-
+price periods and discharging during high-price periods. At
+the same time, arbitrage from energy storage helps to reduce
+renewable curtailments, meet peak demands, mitigate extreme
+events, and reduce the cost of electricity [2], [3]. As countries
+and regions ramp up decarbonization efforts, energy storage
+resources are taking on an increasingly important role in future
+electricity markets and are becoming a cornerstone for cost-
+effective decarbonization [4], [5]. Thus, both energy storage
+owners and market organizers have significant economic and
+welfare drivers to evolve models and algorithms for energy
+storage arbitraging robustly and profitably.
+However, energy storage arbitrage is non-trivial due to
+highly volatile electricity prices and limited storage capacity.
+Y. Baker, N. Zheng, and B. Xu are with Columbia University, NY, USA
+(e-mail: {ykb2105, nz2343, bx2177}@columbia.edu).
+Various methods have been proposed in the literature to ad-
+dress energy storage participation in wholesale markets based
+on different theories, they require dedicated location-specific
+tuning and excessive computing power to achieve competitive
+arbitrage performance [6]. This paper proposes a novel end-to-
+end system for opportunity value calculation, prediction, and
+control, combining model-based dynamic programming with
+neural networks. Our approach innovates and provides several
+advantages as follows:
+• Our approach has reliable performance as it uses model-
+based dynamic programming to address physical con-
+straints in both training and control stages;
+• Our approach is extremely computation efficient as it uses
+dynamic programming to pre-process the training data,
+reducing the complexity of the learning module;
+• Our approach is transferable to different market en-
+vironments while maintaining competitive performance
+because of the integration of transfer learning;
+• Our approach is founded on dynamic programming value
+functions and adapts to different storage market designs
+and participation scenarios, including price response and
+market economic bidding;
+• Our approach achieves state-of-the-art arbitrage perfor-
+mance, achieving 70% to near 90% profit ratio compared
+to perfect foresight with various storage durations when
+tested using price data from New York, US, and Queens-
+land, Australia.
+The rest of the paper is organized as follows: Section II
+summarizes energy storage market participation and previous
+work using the learning method, Section III and IV elaborates
+on the arbitrage formulation and solution method, Section V
+presents the case study for price response and economic bid
+market rules in New York and the application of transfer
+learning for Queensland, and Section VII concludes the paper.
+II. LITERATURE REVIEW
+A. Energy Storage Price Response and Self-Schedule
+Energy storage price response assumes the storage partici-
+pant can observe the real-time price realization first and then
+decide on the operation privately without informing the system
+operator. The price response participation option primarily
+applies to small-scale behind-the-meter (BTM) storage re-
+sources (< 1 MW) [7]. Plenty of prior works have investigated
+energy storage price response using a variety of methods,
+including model-predictive control (MPC) [8], stochastic pro-
+gramming [9], approximate dynamic programming [10], and
+reinforcement learning [11]. Price response is comparably an
+easier problem than economic bids as the storage operator is
+not limited to market clearing models and can act after observ-
+ing new price signals. However, since price response mostly
+arXiv:2301.01233v1  [cs.LG]  2 Jan 2023
+
+2
+applies to small BTM storage projects, the revenue generated
+from arbitrage will unlikely justify any specialized computing
+hardware investments. Hence the arbitrage algorithm must be
+slim and efficient to minimize the computation cost.
+Alternatively, some markets allow energy storage operators
+to self-schedule and submit the operational schedule to the
+market operator. Still, this option is less frequently used in
+practice compared to participating by economic bids [12].
+Self-scheduled storage cannot update the operation based on
+the system clearing price, a key difference compared to price-
+response or economic bids, which often causes the storage to
+miss price spike opportunities and deliver fewer market profits.
+B. Energy Storage Economic Bids
+FERC Order 841, issued in 2018, ordered all system oper-
+ators in the US must allow storage to submit bids and cleared
+in spot markets [13]. In this case, the storage participant must
+submit charge and discharge bids to the system operator at a
+specific time period ahead of the market clearing, usually one
+hour (also called hour-ahead bidding). The storage participant
+must follow market clearing results to charge or discharge,
+unlike in the price response case in which the storage can
+privately decide the control decision after observing the price.
+The bid design adds another layer of complexity in arbitrag-
+ing, as optimal bid design requires mathematical tools due to
+storage SoC constraints. Wang et al. [14] formulate the energy
+storage look-ahead profit maximization problem as a bi-level
+optimization problem. A second approach for energy storage
+arbitrage control is backward dynamic programming [15],
+and then the evolution is approximate-dynamic programming.
+Jiang and Powell outline a general approximate-dynamic pro-
+gramming framework for policy generation for energy storage
+operating with a stochastic generation source in response to
+stochastic demand [16], and further introduce a “distribution-
+free” variant of the previous algorithm that does not make any
+assumption on the price process[10]. However, all of these
+methods are held back by large computational costs that make
+them hard to implement in real-world applications of arbitrage.
+There are other algorithms for energy storage real-time
+arbitrage control: Wang and Zhang [17] solve the arbitrage
+problem using reinforcement learning to come to an optimal
+arbitrage policy, and Zheng et al. [18] outline a computa-
+tionally efficient analytical stochastic dynamic programming
+algorithm (SDP) for the problem of real-time price arbitrage
+of energy storage. Krishnamurthy et al. [19] also propose an
+SDP algorithm for arbitrage under day-ahead and real-time
+price uncertainties. However, none of the methods outlined
+above demonstrate or address transferability between different
+ISO zones and geographic locations, or the hour-ahead bid
+submission requirements in most real-time markets.
+C. Machine Learning for Storage Arbitrage
+Recent efforts to apply machine learning for storage ar-
+bitrage can be grouped into two thrusts: the first is to use
+machine learning to generate price predictions and then in-
+tegrate them with MPC. In this case, the learning module
+is independent of the storage model. Sarafraz et al. [20]
+and Nwulu and Fahrioglu [21] outline two machine learning
+approaches for predicting locational marginal price (LMP)
+prediction using neuro-fuzzy logic and soft computing re-
+spectively, and Chaweewat and Singh [22] propose a residual
+neural network approach to price interval prediction. The
+main difficulty in combining price prediction with storage
+optimization is storage arbitrage requires a look-ahead of at
+least 24 hours to capture the daily price cycles [8], while most
+real-time prediction methods may only accurately generate a
+few steps ahead of time. To this end, existing MPC approaches
+rely on pre-scheduling storage using day-ahead prices but have
+to neglect the real-time price variability, which is significantly
+higher than in day-ahead prices [9].
+The second approach is to directly use machine learning,
+mainly reinforcement learning (RL), to learn the optimal
+control policy for storage arbitrage directly. Wang et al. [11]
+developed the first RL approach to arbitrage storage in real-
+time markets. Cao et al. [23] propose a deep reinforcement
+learning approach to learn an optimal control policy for energy
+storage arbitrage with consideration of battery degradation.
+Kwon et al. [24] demonstrated RL could optimize more
+sophisticated storage models in arbitrage by integrating battery
+degradation into the model. Yet, a common disadvantage of
+RL-based approaches is transferability, as the model must
+undergo time-consuming training to be adapted to a new price
+zone or market environment. Transferability is a crucial aspect
+of storage arbitrage due to spatial and temporal variations: a
+typical system consists of hundreds of price nodes, and system
+price behaviors evolve with changes in system resource mix
+and ambient climate conditions. While previous efforts have
+looked into combining transfer learning with RL [25] and its
+application in selected energy-related issues, including demand
+response prediction [26], event identification [27], and battery
+health forecast [28]. Yet, the transferability of the storage
+arbitrage model has not been previously studied.
+III. PROBLEM STATEMENT AND SYSTEM OUTLINE
+Our algorithm aims to predict the opportunity value at the
+current state of charge (SoC) of energy storage to maximize
+the price arbitrage profit. Our system is composed of three
+components: valuation, forecasting, and arbitrage. We will
+first present our methods for valuation and arbitrage and then
+combine them with our forecasting model to form our bidding
+algorithm. We define Qt(e) as the opportunity value function
+representing the monetary value of the SoC e at time step
+t. The problem formulation is adapted from
+[29], [30], in
+which the solution is formulated using dynamic programming
+as follows:
+max
+bt,pt,et
+∈E(et−1)
+λt(pt − bt) − cpt + ˆQ
+�
+et|θ, X)
+(1a)
+where the first term is arbitrage revenue which is the product of
+the real-time market price λt and the energy storage dispatch
+decision (pt − bt), where pt is the discharge power and bt
+is the charge power. The second term is the discharge cost,
+where c is the marginal discharge cost. The third term ˆQ is
+the predicted storage opportunity value function with respect
+
+3
+Fig. 1. The proposed structure of training opportunity value function prediction model.
+to SoC et. The dynamic programming approach evaluates the
+energy storage by back-propagation, which is not viable in the
+real-time market where we do not have price realization ahead
+of time. Thus, we need to directly predict the value function
+ˆQ using historical (and current) price data. ˆQ is dependent
+on the prediction model parameters θ and the prediction input
+features X over a look-back period.
+We denote that the storage charge and discharge power and
+the final storage SoC belong to a feasibility set E(et−1) which
+is dependent on the storage starting SoC et−1 at the start of
+time period t (same as by the end of time period t−1). E(et−1)
+is described with the following constraints:
+0 ≤ bt ≤ P, 0 ≤ pt ≤ P
+(1b)
+pt = 0 if λt < 0
+(1c)
+et − et−1 = −pt/η + btη
+(1d)
+0 ≤ et ≤ E
+(1e)
+where (1b) models the upper bound, P, and lower bound, 0,
+constraints on the storage charge and discharge power. (1c) is a
+relaxed form of the constraint that enforces the energy storage
+not charging and discharging simultaneously. Negative price
+is the necessary condition for storage to charge and discharge
+simultaneously in price arbitrage, hence by enforcing the stor-
+age to not discharge when the price is negative we eliminate
+simultaneous charging and discharging [29]. (1d) models the
+energy storage SoC evolution constraint with efficiency η and
+(1e) models the upper bound E and lower bound (we assume
+as 0) of the storage SoC level.
+Creating our proposed system amounts to solving the
+problem of optimizing the prediction model parameters θ
+to maximize storage arbitrage profit over a set of training
+price data and physical storage parameters. Intuitively, this
+problem can be formulated as a bi-level problem in which
+the upper level maximizes the total profit over the entire
+training time horizon. At the same time, the lower-level
+enforces a non-anticipatory decision-making process in which
+the storage dispatch decision only depends on the current
+price and the predicted value function as in (1). However, this
+problem quickly becomes computationally intractable since
+the prediction model is embedded in the lower-level problem,
+formulated as a constrained optimization problem. Therefore,
+strong duality is required to convert the bi-level problem into a
+single-level equivalent problem or to derive partial derivatives
+and calculate the back-propagation gradients for gradient-
+based approaches. However, gradient-based approaches are
+complicated by the inclusion of SoC constraints [31]. In
+either case, the computational complexity quickly becomes
+overwhelming as the lower-level can include thousands of
+problems representing the arbitrage over a particular price data
+point.
+Problem Statement. We consider an alternative two-stage
+training approach in which we first generate the optimal
+opportunity value function and then train the learning model
+to predict the generated value function. This is formulated as
+min
+θ
+�
+e∈S
+���
+���ˆqt
+�
+e|θ, X) − qt(e)
+���
+���
+2
+2
+(2a)
+subject to
+qt(e) = ∂
+∂eQt(e)
+(2b)
+Qt−1(et−1) =
+max
+bt,pt,et
+∈E(et−1)
+λt(pt − bt) − cpt + Qt(et)
+(2c)
+Note that (2c) is also subject to the storage operation con-
+straint set E(et−1) as described in (1b)–(1e). (2c) is a dynamic
+programming energy storage price arbitrage formulation in
+which the storage opportunity value is defined recursively
+as the maximized storage arbitrage profit including the profit
+from the current time step and the future opportunity values.
+This formulation fits a piece-wise linear approximation of
+the value function qt(e) based on the first order derivative
+of the optimal value function Qt, and e is from the set of
+SoC segments S. Note that in this formulation the prediction
+model parameters θ are not involved in (2c), hence this is a
+two-stage model in which we solve (2c) first and obtain all
+optimal value function results from Qt, and more specifically,
+their derivatives qt. We are then able to use (2a) to solve for
+the optimal value function at each time step, which we use to
+train the prediction model.
+IV. SOLUTION AND SYSTEM SETUP
+Our approach includes three steps: first, we use the deter-
+ministic price arbitrage dynamic programming approach from
+the previous section to generate the optimal storage opportu-
+nity value function segments using historical price data. We
+then train a learning model to predict the optimal storage
+opportunity value segments from past price data. Finally, we
+
+qe(e)
+Analytical Dynamic
+Programming
+Algorithm (model
+based)
+[X, Y] = [ADAP/AkP] Q] 
+Model-Based
+Arbitrage
+CNN-LSTM
+Price Pre-
+Processing
+Opportu
+qt (el0, X) - qt(e4
+test the learned model over unseen (future) price datasets.
+The system structure is shown in Fig. 1, which includes
+the dynamic programming solution and training method, with
+specifics on the data engineering in Section IV-A.
+A. Feature and Label Formatting
+In general, the spot price for energy exhibits long-term
+and short-term cycles according to cycling demand: the daily
+cycling between peak and non-peak hours and the long-term
+seasonal cycles; though events and the stochastic nature of
+price create differences in between. Thus we chose to use
+a convolutional long short-term memory (ConvLSTM) neural
+net, which can learn patterns in time series data. For learning
+timestep t, our network input/target pair could be [λt, qt] (or
+qt+hr, where +hr represents an hour time shift for the HA
+case). However, to better capture daily cycling, we elaborate
+our single-step input-output pair by constructing the following
+input-output matrices:
+{X, Y} = {[ΛDAP|ΛRTP], Q}
+ΛDAP =
+�
+����
+λDAP,t−m
+λDAP,t−m+1
+. . .
+λDAP,t
+λDAP,t−m−1
+λDAP,t−m
+. . .
+λDAP,t−1
+...
+...
+...
+λDAP,t−m−5hr
+λDAP,t−m−5hr+1
+. . .
+λDAP,t−5hr
+�
+����
+ΛRTP =
+�
+����
+λRTP,t−n
+λRTP,t−n+1
+. . .
+λRTP,t
+λRTP,t−n−1
+λRTP,t−n
+. . .
+λRTP,t−1
+...
+...
+...
+λRTP,t−n−5hr
+λRTP,t−n−5hr+1
+. . .
+λRTP,t−5hr
+�
+����
+Q =
+�
+����
+qt
+qt−1
+...
+qt−5hr
+�
+����
+where ΛDAP and ΛRTP are matrices made up of our day ahead
+and real-time price data, and m, n are a lookback window for
+the day ahead and real-time prices respectively, and 5hr is the
+number of timesteps that make up five hours in a given market
+resolution (60 in a 5 min resolution market). This allows the
+network to capture not only the information on past prices for
+the current value function but also the relationship between
+past value functions in a 5-hour lookback. We chose five hours
+here as it is long enough to capture cycles within a single day
+(e.g. peak vs non-peak demand and the transition between
+them). The inclusion of the day ahead price here serves as
+a more stable price reference for the corresponding hour’s
+spot price. Also of note is the cyclic symmetry of the price
+matrices along the diagonal, which allows the network to learn
+better the equivariant properties of the dataset [32]. Finally, the
+choice of a ConvLSTM, as opposed to a traditional LSTM, is
+to allow the network to capture the ”vertical” temporal relation
+between the five hours of data in each data block.
+Note that for DAP, the shift applied to t across rows
+corresponds to a step shift in the resolution of the real-time
+market (RTM). Meaning that if it is a 5-minute resolution
+RTM, the first 12 rows of ΛDAP will be the same since the
+day-ahead market (DAM) is hourly resolution.
+B. Model Selection and Transfer Learning
+The focus of this paper is to demonstrate the robustness
+of the approach across different market conditions and bat-
+tery durations and to show its transferability between zones.
+Thus we chose one general network architecture for testing.
+However, initial experimentation showed minimal gain-loss
+in network performance on minor parameter changes across
+cases. Further, to guarantee that the training converges to a
+well-performing set of weights, multiple networks were trained
+for each case. The weights achieving the most consistent and
+low validation error were saved for evaluation. Of those, the
+best model was chosen by the highest arbitrage profit. The
+network is trained over 100 epochs in the case where it is
+trained from scratch, and 25 epochs for the transfer learning
+training, with a learning rate of 10−3. Further, we use a
+callback function that saves the model weights only when the
+validation error improves, ensuring that the weights loaded for
+training are not overfitted. This callback also allows us to set
+our epochs with significant overhead to ensure convergence
+without over-fitting in all cases.
+Furthermore, we apply transfer learning to quickly adapt a
+trained model from one price zone to another. Our transfer
+learning approach freezes all model layers except the output
+layer and retraining on the dataset of the task to be transferred
+to [33]. The underlying assumption is that the output layer is
+more sensitive to data variability while the rest of the network
+captures persistent patterns in the data.
+C. Full Algorithm
+We lay out our workflow, which is a sequence of three
+algorithms. As a prerequisite for model training, we generate
+all value functions Q according to the dynamic programming
+solution in VII-A. After this, we construct our data set and
+train our LSTM prediction model according to Algorithm 1,
+which produces our trained model weights θ.
+Algorithm 1 Value Function Prediction Model Training
+1: Dataset Preparation: Pre-Process data according to IV-A
+2: Initialization: Initialize model parameters θ using random
+seed.
+3: i ← 0
+4: while stop criteria not true do
+5:
+for t ∈ [1, t] do
+6:
+x ← [ΛDAP|ΛRTP]
+7:
+y ← Q
+8:
+Calculate Loss Components by Eq. (2a)
+9:
+Update θ by backpropagation
+10:
+end for
+11:
+i ← i + 1
+12: end while
+13: return θ
+▷ Parameters of the prediction model
+After this, if the trained LSTM model produced by Algo-
+rithm 1 is to then be used by another zone, it can be retrained
+using the transfer learning approach outlined in Algorithm
+2. This is largely the same as the workflow of algorithm
+1, save that the training dataset is of the new zone and the
+
+5
+newly trained model weights are denoted θ∗. We differentiate
+between the two sets of model weights since we compare
+the two approaches of transferring (transfer learning, applying
+the model on new zones without retraining) later in the
+paper. Finally, algorithm 3 outlines the process of simulating
+Algorithm 2 Transfer Learning
+1: Initialization: Initialize model parameters θ∗ using ran-
+dom seed
+2: θ∗ ← θ (trained model parameters)
+3: Freeze all parameters except output layer parameters
+4: Repeat
+training
+loop
+using
+new
+region’s
+data
+set
+{[Λ∗
+DAP|Λ∗
+RTP], Q∗}
+5: return θ∗
+▷ Parameters of the prediction model
+arbitrage using our prediction model. The arbitrage simulation
+is as follows: use the prediction model trained in algorithm
+1 and/or algorithm 2 to predict value functions using the
+current real-time price and a look-back window (including the
+day ahead look-back) and then generate the bids according
+to VII-B. Once the bids are generated, use them to simulate
+arbitrage and market clearing as outlined in VII-C.
+Algorithm 3 Arbitrage with Value Function Prediction
+1: Initialization:
+2: Set energy storage parameters c, P, ηp, ηb, E.
+3: Initialize et−1 ← e0.
+4: for t ∈ [1, T] do
+5:
+Predict ˆv
+�
+et|θ, x
+�
+6:
+Solve single-period optimization (1)
+7:
+Return et, pt, dt
+8: end for
+V. CASE STUDY SET-UPS
+A. Market Participation Setting and Storage Parameters
+We consider the following four market designs and par-
+ticipation settings to demonstrate that our proposed approach
+fits a wide range of storage participation options and market
+designs:
+• HA-1 Energy storage owner submits single-segment bids
+one hour-ahead to real-time markets. This represents the
+current storage bidding model in most wholesale real-
+time markets in the US [10], [34] where energy storage
+submits one charge bid and one discharge bid one hour
+ahead of the market clearing. The storage can update its
+bid for each hour, but the bids must stay the same within
+each hour for multiple market clearings (for example,
+real-time markets clear every five minutes in NYISO, so
+one hour includes 12 real-time clearings).
+• HA-10 Same to HA-1 except the storage submits10-
+segment SoC-dependent charge and discharge bids. This
+is a new market design proposed by CAISO to econom-
+ically manage storage SoC in real-time [35], [36].
+• PR-10 The storage conducts price response in real-time,
+deciding the storage control after observing the published
+real-time price, instead of submitting bids [37]. The price
+response option is limited to behind-the-meter storage
+in which the associated demand is cleared in real-time
+market prices. In this case, the storage is not limited to
+any bidding models and can use any decision-making
+models. Yet, we assume the storage uses a 10-segment
+approximation of its opportunity value as it provides a
+good enough approximation to the actual value function.
+This also enables us to benchmark HA-10 and PR-10
+cases to demonstrate the economic cost of the hour-ahead
+bidding requirement.
+• PR-1 Same as PR-10 except the storage uses the average
+opportunity value (i.e., one segment approximation) for
+arbitrage control. This is not a realistic case as there is no
+motivation for the storage operator to limit itself to using
+a single-segment, less accurate approximation of its value
+function to conduct arbitrage. However, we include this
+case with the sole purpose to benchmark against the HA-
+1 case and PR-10 case.
+In all case studies, we consider storage with a 90% one-
+way efficiency and a 10$/MWh cost of discharge (excluding
+the opportunity cost), unless otherwise specified. We consider
+three storage durations including 2-hour, 4-hour, and 12-hour.
+Further, we adapt our base prediction model to predict the hour
+ahead case by adding an hour time shift to our ground truth
+training target value function, which corresponds to 12-time
+steps in 5-minute price resolution.
+We conduct the majority of our case studies over price
+data from New York ISO (NYISO) [38] for four price zones:
+NYC (Zone J), LONGIL (Zone K), NORTH (Zone D), and
+WEST (Zone A). We also use data from the Australian Energy
+Market Operator (AEMO) for Queensland to demonstrate the
+transferability of our approach using transfer learning [39].
+B. Market and Price Data
+We observe differences in price statistics and generation
+mix across zones from the same ISO, and in between zones
+from ISO’s in other states and even countries, summarized in
+Table I. In New York zones, these differences can be attributed
+to significant transmission congestion when comparing the
+two main zone groups in NY [40]. QUEENSLAND has the
+highest price volatility, which can potentially be attributed to
+the absence of a day-ahead market. Further, we see a clear
+tie between penetration rates of renewables into the zones and
+the price volatility [41]. We also see the highest occurrence
+of negative prices in NORTH (NY), which is due to the
+significantly higher penetration of wind when compared to
+NYC and LONGIL.
+TABLE I
+PRICE DATA STATISTICS
+,
+Zone
+Negative Price #
+STD
+Renewable %
+NYC (NY)
+208
+28.82
+0.93
+LONGIL (NY)
+190
+50.17
+0.93
+NORTH (NY)
+6334
+40.25
+13.06
+WEST (NY)
+633
+37.55
+13.06
+QNSLND (AUS)
+522
+243.00
+13.19
+
+6
+Fig. 2. Accumulated Profit over 2019 test set for NYISO Zones
+All code for valuation, network training, and arbitrage are
+written in python with Jupyter notebook and is available on
+GitHub1. All trials are run on a desktop computer with AMD
+Ryzen 9 processor and Nvidia GPU on Tensorflow 2.9.1 and
+with cuDNN and CUDA versions 8.1 and 11.2, respectively.
+All case studies using price data from NYISO were trained
+using data from 2017 to 2018, and tested over 2019 data.
+Each year of price data for each price zone has 8760 day-
+ahead price data points (hourly resolution) and 105,120 real-
+time price points (5-minute resolution). The look-back price
+window includes the last 36 real-time prices (3 hours) and
+24-day-ahead prices (one day). The maximum training time
+over two years of training price data, including the generation
+of historical optimal value functions and training of the neural
+network, is 390 seconds, a bit more than five minutes. The net-
+work consists of a Convolutional Block with three sequential
+time-distributed Convolutions+MaxPool layers, then an LSTM
+block with two sets of bi-directional LSTM+drop out layers,
+and then finally a Dense layer at the output end. The specific
+model hyperparameters and details can be found on GitHub.
+VI. RESULTS
+A. Benchmark with Competing Methods
+We first benchmark our proposed approach with other
+competing energy storage price arbitrage methods in a price
+response setting, in which storage can observe price first
+and act accordingly, without having to bid ahead into mar-
+kets. We benchmark the proposed method (DP-ConvLSTM)
+with a reinforcement learning method (RL) [17], a modified
+stochastic dynamic programming with day-ahead price updates
+(SDP) [37], the proposed method but implemented with a
+multilayer perceptron (DP-MLP) network [42], and perfect
+price predictions which provide the highest profit possible.
+In RL, we have 11 actions, 103 price states, and 121 SoC
+states, which takes more than 1 hour to train for 5-min
+resolution arbitrage. The RL approach uses a Markov decision
+process (MDP) model by discretizing the storage SoC, and it
+only works with perfect efficiency (100%). To provide a fair
+comparison, all methods in this case consider storage with
+perfect efficiency.
+1https://github.com/ybaker661/LSTM-Value-Prediction
+Fig. 2 shows the comparison result when trained using price
+data from 2017-2018 and tested in 2019 at price zones in NY-
+ISO. The result shows DP-ConvLSTM has a clear advantage
+over other methods in terms of profitability. Notably, the DP-
+ConvLSTM approach performs exceptionally well in capturing
+low-frequency extreme events, such as the surge in profits
+around June in LONGIL and WEST, where the ConvLSTM
+captures profit spikes that the RL benchmark misses, and the
+difference between the ConvLSTM profit value and the perfect
+prediction comes from the difference in arbitrage decision as
+a result of numerical saturation. In this context, numerical
+saturation means that the network learns to predict numerical
+values in the range of data it most frequently sees (value
+functions of stable prices), and so when it predicts on anomaly
+data (price spike value functions) that are numerically much
+larger, the network prediction saturates at the largest common
+numerical value it sees.
+B. Price Response
+In this subsection, we compare the price response arbitrage
+performance (PR-1 and PR-10) with different storage dura-
+tions. Table III shows the arbitrage profit ratio results.
+Overall, the result shows stable performance over the four
+price zones and three storage durations. In comparison, our
+previous work using SDP [37] and DP-MLP [42] have worse
+performance in LONGIL (more frequent price spikes) and
+NORTH (more frequent negative prices). The comparison
+between PR-1 and PR-10 shows that increasing the value
+function approximation from one to ten segments increased
+the profit ratio by around 3%. Considering different storage
+durations, the profit ratio is lower for long-duration energy
+storage (12hr), as the longer storage duration leads to a
+longer temporal correlation into the future, leading to higher
+prediction difficulties. Still, our method achieved around 75%
+profit ratio (HA-10) in the worst-case scenario in the NORTH
+zone.
+C. Hour-ahead Bidding
+We now investigate hour-ahead bidding which is the most
+common market design for energy storage owners operating in
+the real-time market, where the storage submits bids an hour
+
+NYC
+LONGIL
+NORTH
+WEST
+16
+... Perfect Prediction
+.... Perfect Prediction
+..... Perfect Prediction
+.... Perfect Prediction
+DP+LSTM
+DP+LSTM
+DP+LSTM
+DP+LSTM
+14
+14
+DP+MLP
+DP+MLP
+DP+MLP
+DP+MLP
+-- SDP
+25
+-- SDP
+--- SDP
+---- SDP
+- RL
+RL
+RL
+2
+ 20
+ 20
+Profit
+Cumulative F
+6
+8
+10 
+12
+0
+2
+6
+10
+12
+2
+6
+8
+10
+12
+0
+2
+6
+8
+10
+12
+months
+months
+months
+months7
+TABLE II
+PROFIT RATIO FOR HA PREDICTION QUEENSLAND AUS WITH DIFFERENT AMOUNTS OF TRAINING DATA
+HA-1
+HA-10
+Duration
+Training
+No Data
+3 Days
+1 Week
+1 Month
+1 Year
+No Data
+3 Days
+1 Week
+1 Month
+1 Year
+2hr
+T.L.
+77.24
+82.76
+82.85
+81.30
+85.35
+78.90
+84.00
+81.54
+79.93
+83.42
+No T.L.
+X
+48.22
+51.36
+78.59
+83.88
+X
+44.59
+44.88
+78.10
+86.95
+4hr
+T.L.
+81.21
+81.29
+81.31
+78.11
+79.12
+84.06
+87.44
+84.34
+77.79
+82.65
+No T.L.
+X
+62.40
+65.45
+74.92
+80.42
+X
+55.99
+55.99
+74.36
+83.11
+12hr
+T.L.
+92.43
+83.96
+81.32
+78.74
+82.73
+90.69
+80.78
+79.11
+78.84
+81.35
+No T.L
+X
+74.79
+75.53
+74.39
+75.43
+X
+73.65
+73.59
+74.56
+82.36
+TABLE III
+CAPTURED PROFIT RATIOS: PRICE RESPONSE
+Zone
+PR-1
+PR-10
+2hr
+4hr
+12hr
+2hr
+4hr
+12hr
+NYC
+80.83
+79.96
+73.54
+83.69
+83.63
+75.67
+LONGIL
+82.33
+80.61
+79.10
+82.98
+83.94
+82.38
+NORTH
+78.24
+75.87
+71.02
+79.52
+79.96
+74.29
+WEST
+84.43
+80.37
+83.65
+87.97
+87.44
+84.43
+TABLE IV
+CAPTURED PROFIT RATIOS: HOUR AHEAD
+Zone
+HA-1
+HA-10
+2hr
+4hr
+12hr
+2hr
+4hr
+12hr
+NYC
+73.99
+74.82
+77.00
+78.79
+80.61
+74.47
+LONGIL
+74.26
+76.56
+82.01
+75.30
+79.63
+81.89
+NORTH
+73.22
+71.71
+70.17
+75.83
+77.21
+73.16
+WEST
+78.79
+80.13
+84.17
+83.12
+83.94
+84.60
+ahead of time. Table IV shows the hour-ahead bidding profit
+ratio in the NYC case study. The profit ratio is lower than the
+price response as the storage owner must decide on the bids
+one hour before the actual time of arbitrage. The short-duration
+storage (2hr) cases have higher profit ratio reductions (up to
+7%) as the value function is more sensitive to recent market
+prices due to it’s shorter duration. On the other hand, the long-
+duration storage (12hr) is more resilient and the hour-ahead
+bidding has little impact on the profit ratio.
+Hour-ahead bidding results also restate our observation from
+the price response case, that multi-segment SoC bids are
+more beneficial for short-duration storage to better manage
+their SoC, but the improvement is not obvious for long-
+duration storage. Overall, our approach achieved a higher
+than 70% profit ratio in all hour-ahead cases, showing ro-
+bust performance under different market designs and storage
+technologies.
+D. Transfer Learning in AEMO
+We now demonstrate the effectiveness of applying transfer
+learning to quickly adapt a pre-trained value function predic-
+tion model from one market to a new market. In this case
+study, we pre-train the prediction model using NYC price
+data from 2017-2018 and conduct arbitrage in Queensland,
+Australia. In Queensland, we use selected data from 2019
+for training and the first 6 months of 2021 for evaluation.
+We skipped the year 2020 because of COVID-19’s impact.
+To present the sensitivity of transfer learning over a limited
+amount of data, we consider various durations of training
+datasets ranging from 3 days to one year. We present a
+sensitivity analysis comparing the performance of transfer
+learning versus training a model from scratch for the situations
+where we have access to training data for only 3 days, 1 week,
+1 month, and 1 year of data for the target zone. Thus this case
+study has the following steps:
+1) Use a pre-trained network (transfer learning) or a ran-
+domly initialized network (training from scratch).
+2) Use a limited duration of Queensland price data from
+2019, ranging from 3 days to 1 year, to train the model
+using transfer learning as outlined in algorithm 2, or
+normal training outlined in algorithm 1.
+3) Test the arbitrage performance to arbitrage, as outlined
+in algorithm 3, using the first six-month of data in
+Queensland, 2021.
+Table II shows the arbitrage profit ratio results for Queens-
+land. The main finding is that in the data scare scenarios,
+the transfer learning approach vastly outperforms training a
+model from scratch. We also see that adding more data to
+the transfer learning case does not necessarily increase perfor-
+mance, whereas training the model from scratch only becomes
+a viable option once a certain amount of data is available.
+For the 2-hour storage, training from scratch becomes viable
+around the point where you have 1 month of data available for
+the target zone. For the 4-hour storage, the model still needs
+about 1 month of data to reasonably perform when trained
+from scratch, though the model is able to capture higher profit
+ratios for 3 and 1 week of data than the 2-hour case. Compared
+to both of these, the 12-hour storage seems to be the easiest for
+the model to learn, only needing 3 days of data when training
+from scratch to achieve reasonable performance; however, the
+12-hour storage shows that the transfer learning approach
+outperforms training from scratch for all data scenarios for 1
+and 10 segments. However, since the 12-hr storage has lower
+opportunity cost and less significant change in the opportunity
+cost between sequential time steps, predicting the opportunity
+value might not be an effective method.
+The takeaway is that transfer learning beats out training the
+model from scratch when data scarcity is an issue. However,
+when the dataset size increases to a general size of 1 month,
+training from scratch becomes a viable option. Additionally,
+adding extra data, past 3 days or 1 week for transfer learning
+and past 6 months for training from scratch, does not necessar-
+ily yield better performance. As such, it is more useful to focus
+on stabilizing ConvLSTM’s volatile and initialization-sensitive
+training as well as other changes to the training process. We
+
+8
+see that in almost all cases, using the model trained on NY
+data without any retraining performs comparably or even better
+than transfer learning and even training a model from scratch.
+This indicates statistical robustness and generality in the NY
+zone data, and it also points to a unified generating distribution
+behind the price data/opportunity value of zones. However, we
+cannot conclude this for certain without further analysis of
+other permutations of transfer learning, and on testing different
+data duration permutations (including different data scarcity
+scenarios in the training of the base model along with in
+transfer learning).
+VII. CONCLUSION
+In this paper, we propose a computation-efficient, versatile,
+and transferable energy storage arbitrage model that fits both
+price response and market bidding. Our proposed approach
+achieves state-of-the-art profits compared to other methods and
+is both computation and data-efficient. We also demonstrate
+that by incorporating transfer learning, we can quickly adapt
+our bidding model to a new location with very limited training
+data. Our model suits a variety of arbitrage settings, including
+behind-the-meter price response and economic bids for utility-
+scale storage, and can be implemented using non-proprietary
+software and regular computing hardware. Our work would fa-
+cilitate storage participation in electricity markets and promote
+economic decarbonization of the electric power system.
+REFERENCES
+[1] G. McGrath and O. Comstock, “Battery systems on the u.s. power
+grid are increasingly used to respond to price,” Jul 2022. [Online].
+Available: https://www.eia.gov/todayinenergy/detail.php?id=53199
+[2] N. Srianandarajah, S. J. Wilson, and A. C. Chapman, “From green to
+amber: is australia’s national electricity market signalling a financial
+warning for wind and solar
+power?” Energy Policy, vol. 167,
+p. 113052, 2022. [Online]. Available: https://www.sciencedirect.com/
+science/article/pii/S0301421522002774
+[3] X. Yan, C. Gu, F. Li, and Z. Wang, “Lmp-based pricing for energy
+storage in local market to facilitate pv penetration,” IEEE Transactions
+on Power Systems, vol. PP, pp. 1–1, 12 2017.
+[4] W. H. B. Room, “Fact sheet: President biden sets 2030 greenhouse gas
+pollution reduction target aimed at creating good-paying union jobs and
+securing us leadership on clean energy technologies,” The White House,
+2021.
+[5] “2030
+climate
+&
+energy
+framework.”
+[Online].
+Avail-
+able:
+https://climate.ec.europa.eu/eu-action/climate-strategies-targets/
+2030-climate-energy-framework en
+[6] R. Sioshansi, P. Denholm, J. Arteaga, S. Awara, S. Bhattacharjee, A. Bot-
+terud, W. Cole, A. Cort´es, A. De Queiroz, J. DeCarolis et al., “Energy-
+storage modeling: State-of-the-art and future research directions,” IEEE
+Transactions on Power Systems, vol. 37, no. 2, pp. 860–875, 2021.
+[7] M. Roozbehani, M. A. Dahleh, and S. K. Mitter, “Volatility of power
+grids under real-time pricing,” IEEE Transactions on Power Systems,
+vol. 27, no. 4, pp. 1926–1940, 2012.
+[8] K. Abdulla, J. De Hoog, V. Muenzel, F. Suits, K. Steer, A. Wirth,
+and S. Halgamuge, “Optimal operation of energy storage systems
+considering forecasts and battery degradation,” IEEE Transactions on
+Smart Grid, vol. 9, no. 3, pp. 2086–2096, 2016.
+[9] D. Krishnamurthy, C. Uckun, Z. Zhou, P. R. Thimmapuram, and
+A. Botterud, “Energy storage arbitrage under day-ahead and real-time
+price uncertainty,” IEEE Transactions on Power Systems, vol. 33, no. 1,
+pp. 84–93, 2017.
+[10] D. R. Jiang and W. B. Powell, “Optimal hour-ahead bidding in the real-
+time electricity market with battery storage using approximate dynamic
+programming,” INFORMS Journal on Computing, vol. 27, no. 3, pp.
+525–543, 2015.
+[11] H. Wang and B. Zhang, “Energy storage arbitrage in real-time markets
+via reinforcement learning,” in 2018 IEEE Power & Energy Society
+General Meeting (PESGM).
+IEEE, 2018, pp. 1–5.
+[12] H. Mohsenian-Rad, “Optimal bidding, scheduling, and deployment of
+battery systems in california day-ahead energy market,” IEEE Transac-
+tions on Power Systems, vol. 31, no. 1, pp. 442–453, 2016.
+[13] S. Bhattacharjee, R. Sioshansi, and H. Zareipour, “Energy storage
+participation in wholesale markets: The impact of state-of-energy man-
+agement,” IEEE Open Access Journal of Power and Energy, vol. 9, pp.
+173–182, 2022.
+[14] Y. Wang, Y. Dvorkin, R. Fernandez-Blanco, B. Xu, T. Qiu, and D. S.
+Kirschen, “Look-ahead bidding strategy for energy storage,” IEEE
+Transactions on Sustainable Energy, vol. 8, no. 3, pp. 1106–1117, 2017.
+[15] M. L. Puterman, Markov decision processes: discrete stochastic dynamic
+programming.
+John Wiley & Sons, 2014.
+[16] D. R. Jiang and W. B. Powell, “An approximate dynamic programming
+algorithm for monotone value functions,” 2014. [Online]. Available:
+https://arxiv.org/abs/1401.1590
+[17] H. Wang and B. Zhang, “Energy storage arbitrage in real-time
+markets via reinforcement learning,” CoRR, vol. abs/1711.03127, 2017.
+[Online]. Available: http://arxiv.org/abs/1711.03127
+[18] N. Zheng, J. Jaworski, and B. Xu, “Arbitraging variable efficiency
+energy storage using analytical stochastic dynamic programming,” IEEE
+Transactions on Power Systems, vol. 37, no. 6, pp. 4785–4795, 2022.
+[19] D. Krishnamurthy, C. Uckun, Z. Zhou, P. R. Thimmapuram, and
+A. Botterud, “Energy storage arbitrage under day-ahead and real-time
+price uncertainty,” IEEE Transactions on Power Systems, vol. 33, no. 1,
+pp. 84–93, 2018.
+[20] F. Sarafraz, H. Ghasemi, and H. Monsef, “Locational marginal price
+forecasting by locally linear neuro-fuzzy model,” in 2011 10th Inter-
+national Conference on Environment and Electrical Engineering, 2011,
+pp. 1–4.
+[21] N. I. Nwulu and M. Fahrioglu, “A soft computing approach to
+projecting locational marginal price,” NEURAL COMPUTING &
+APPLICATIONS, p. 1115–1124, 2013. [Online]. Available: https:
+//hdl.handle.net/11511/64998
+[22] P. Chaweewat and J. G. Singh, “An electricity price interval forecasting
+by using residual neural network,” International Transactions on Electri-
+cal Energy Systems, vol. 30, no. 9, p. e12506, 2020. [Online]. Available:
+https://onlinelibrary.wiley.com/doi/abs/10.1002/2050-7038.12506
+[23] J. Cao, D. Harrold, Z. Fan, T. Morstyn, D. Healey, and K. Li, “Deep
+reinforcement learning-based energy storage arbitrage with accurate
+lithium-ion battery degradation model,” IEEE Transactions on Smart
+Grid, vol. 11, no. 5, pp. 4513–4521, 2020.
+[24] K.-b. Kwon and H. Zhu, “Reinforcement learning based optimal battery
+control under cycle-based degradation cost,” IEEE Transactions on
+Smart Grid, 2022.
+[25] M. E. Taylor and P. Stone, “Transfer learning for reinforcement learning
+domains: A survey.” Journal of Machine Learning Research, vol. 10,
+no. 7, 2009.
+[26] T. Peirelinck, H. Kazmi, B. V. Mbuwir, C. Hermans, F. Spiessens,
+J. Suykens, and G. Deconinck, “Transfer learning in demand response:
+A review of algorithms for data-efficient modelling and control,”
+Energy
+and
+AI,
+vol.
+7,
+p.
+100126,
+2022.
+[Online].
+Available:
+https://www.sciencedirect.com/science/article/pii/S2666546821000732
+[27] H. Li, Z. Ma, and Y. Weng, “A transfer learning framework for power
+system event identification,” IEEE Transactions on Power Systems,
+vol. 37, no. 6, pp. 4424–4435, 2022.
+[28] S. Kim, Y. Y. Choi, K. J. Kim, and J.-I. Choi, “Forecasting
+state-of-health of lithium-ion batteries using variational long short-term
+memory with transfer learning,” Journal of Energy Storage, vol. 41,
+p. 102893, 2021. [Online]. Available: https://www.sciencedirect.com/
+science/article/pii/S2352152X21006101
+[29] B. Xu, M. Korp˚as, and A. Botterud, “Operational valuation of energy
+storage under multi-stage price uncertainties,” in 2020 59th IEEE
+Conference on Decision and Control (CDC).
+IEEE, 2020, pp. 55–60.
+[30] N. Zheng and B. Xu, “Impact of bidding and dispatch models over
+energy storage utilization in bulk power systems,” IREP Symposium on
+Bulk Power System Dynamics and Control 2022, 2022.
+[31] H. Cui, F. Li, X. Fang, H. Chen, and H. Wang, “Bilevel arbitrage
+potential evaluation for grid-scale energy storage considering wind
+power and lmp smoothing effect,” IEEE Transactions on Sustainable
+Energy, vol. 9, no. 2, pp. 707–718, 2017.
+[32] S. Dieleman, J. De Fauw, and K. Kavukcuoglu, “Exploiting cyclic
+symmetry in convolutional neural networks,” 2016. [Online]. Available:
+https://arxiv.org/abs/1602.02660
+
+9
+[33] F. Chollet, “Transfer learning & fine-tuning,” https://keras.io/guides/
+transfer learning/, 2020.
+[34] A. Sakti, A. Botterud, and F. O’Sullivan, “Review of wholesale markets
+and regulations for advanced energy storage services in the united states:
+Current status and path forward,” Energy policy, vol. 120, pp. 569–579,
+2018.
+[35] N. Zheng, X. Qin, D. Wu, G. Murtaugh, and B. Xu, “Energy storage
+state-of-charge market model,” arXiv preprint arXiv:2207.07221, 2022.
+[36] C.
+Chen
+and
+L.
+Tong,
+“Convexifying
+market
+clearing
+of
+soc-
+dependent bids from merchant storage participants,” arXiv preprint
+arXiv:2209.02107, 2022.
+[37] N. Zheng, J. J. Jaworski, and B. Xu, “Arbitraging variable efficiency
+energy storage using analytical stochastic dynamic programming,” IEEE
+Transactions on Power Systems, 2022.
+[38] “Energy
+market
+&
+operational
+data.”
+[Online].
+Available:
+https:
+//www.nyiso.com/energy-market-operational-data
+[39] “Engineering precinct battery.” [Online]. Available: http://dashboards.
+sustainability.uq.edu.au/engineering-precinct-battery/interactive/#/
+[40] D. B. Patton, P. LeeVanSchaick, J. Chen, and M. M. Unit, “2014 state
+of the market report for the new york iso markets,” Potomac Economics,
+2016.
+[41] M. Waite and V. Modi, “Electricity load implications of space heating
+decarbonization pathways,” Joule, vol. 4, no. 2, pp. 376–394, 2020.
+[42] N. Zheng, X. Liu, B. Xu, and Y. Shi, “Energy storage price ar-
+bitrage via opportunity value function prediction,” arXiv preprint
+arXiv:2211.07797, 2022.
+APPENDIX
+A. Dynamic Programming Solution Algorithm
+We first solve the dynamic programming problem as listed
+in (2c) subject to constraints (1b)–(1e). We use results from our
+prior work [29] to solve the dynamic programming problem
+(2c) and obtain the full piece-wise linear approximation of the
+opportunity value function Qt for all time periods (i.e., one
+value function for each time step for an entire year, 105120
+for 5 min price resolution 35040 for 20 min price resolution).
+We start by defining qt as the derivative of storage opportunity
+value function Qt, which represents the marginal opportunity
+value of energy stored in the storage. Then we can use an
+analytical formulation to calculate the opportunity value qt(e)
+at any given energy storage SoC level.
+Our prior work proved qt−1 can be recursively calculated
+with next period value function qt, power rating P, and effi-
+ciency η. The value function calculated using the deterministic
+formulation is thus
+qt−1(e) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+qt(e + Pη)
+if λt ≤ qt(e + Pη)η
+λt/η
+if qt(e + Pη)η < λt ≤ qt(e)η
+qt(e)
+if qt(e)η < λt ≤ [qt(e)/η + c]+
+(λt − c)η
+if [qt(e)/η + c]+ < λt
+≤ [qt(e − P/η)/η + c]+
+qt(e − P/η)
+if λt > [qt(e − P/η)/η + c]+
+(3)
+and calculates the opportunity value function assuming the
+price follows a recursive computation framework. This deter-
+ministic formulation is what we will use in our investigation,
+and from this we are able to calculate opportunity value
+function qt(e) at any time step using backwards recursion by
+defining an end period value function qT . We then discretize qt
+by splitting the energy storage SoC level e into small equally
+spaced segments, which must be far smaller than power rating
+P. For any SoC level et, we can find the nearest segment and
+return the corresponding value.
+B. Bid generation
+We now design discharge and charge bids using the oppor-
+tunity valuation results based on our prior work [30], [35]. We
+consider generating time-varying SoC-dependent bids with a
+total number of J segments for charge bids Bt,j and discharge
+bids Ct,j. Note that these bids represent the combination of
+the discharge cost and the change in the opportunity value. We
+assume each bid segment j is associated with an SoC range
+Ej−1 to Ej. The discharge bids are thus calculated based on
+the average value function between the internal Ej−1 and Ej
+Ct,j = 1
+E
+� Ej
+Ej−1
+∂
+∂pt
+(cpt − Qt(et−1 − pt/η + btη))det−1
+= c + 1
+E
+� Ej
+Ej−1
+qt(et−1 − pt/η + btη)det−1/η
+≈ c + 1
+ηE
+� Ej
+Ej−1
+qt(e)de
+Similarly for charge bids
+Bt,j = 1
+E
+� Ej
+Ej−1
+∂
+∂bt
+(cpt − Qt(et−1 − pt/η + btη))det−1
+= 1
+E
+� Ej
+Ej−1
+qt(et−1 − pt/η + btη)det−1η
+≈ η
+E
+� Ej
+Ej−1
+qt(e)de
+In the special case of one segment, i.e., bids are not
+dependent on SoC (the current energy storage bidding model
+in most wholesale markets), Ej−1 is zero or the lowest allowed
+SoC and Ej is the highest allowed SoC value or the energy
+capacity. In this case the bids are simply based on the average
+marginal opportunity value ¯qt
+¯qt = 1
+E
+� E
+0
+qt(e)de
+(4)
+and the discharge bid is c + ¯qt/η, and the charge bid is ¯qtη.
+C. Real-time market clearing and arbitrage simulation
+We consider the following simplified real-time market clear-
+ing model with a generalized multi-segment energy storage
+bids
+min
+pt,j,s,dt,j,s
+Jt(gt) +
+�
+s
+�
+j
+(Ct,j,sdt,j,s − Bt,j,sbt,j,s) (5a)
+s.t.
+et,j,s − et−1,j,s = bt,j,sη − pt,j,s/η
+(5b)
+0 ≤ et,j,s ≤ Ej,s − Ej−1,s
+(5c)
+gt +
+�
+s
+�
+j
+pt,j,s = Dt +
+�
+s
+�
+j
+bt,j,s : λt
+(5d)
+where (5a) is the objective function minimizing total bidding
+costs. Note that we use aggregated generator supply curve
+Jt(gt) and total generation gt instead of modeling the bids
+from each individual generator for simplicity to focus on
+energy storage. The second term of the objective is the
+discharge bids and charge bids for each energy storage s and
+
+10
+each SoC segment j. (5b) models the SoC evolution under
+single-trip efficiency η for each SoC segment. (5c) models
+the upper and lower energy limit for each SoC segment, note
+that the minimum energy is always zero while the maximum
+energy for each segment is the difference between the upper
+and lower SoC range Ej,s − Ej−1,s. Finally, (5d) is the
+power balance constraint enforcing the sum of generation and
+storage charge/discharge equals to the total demand Dt over
+time period t, the associated dual variable is thus the market
+clearing price λt.
+Now in price-taker analysis, we use historical price data to
+simulate how the energy storage would have been cleared in
+the market. In this case, we perform a Lagrangian relaxation
+of (5d) and move it to the objective. This decomposes the
+optimization into independent sub-problems for each energy
+storage, and for each storage, the price-taker market clearing
+problem is equivalent to the following price arbitrage problem
+max
+pt,j,dt,j λt
+�
+j
+(dt,j − bt,j) −
+�
+j
+(Ct,jdt,j − Bt,jbt,j)
+(6)
+subject to the same storage unit constraints (5b) and (5c).
+Note that for this problem we omit the storage unit index
+s as the problem formulation is the same for each storage.
+Hence, price-taker market clearing simulation is equivalent to
+arbitrage using the same bidding cost model. While we did
+not consider the network model in this formulation, the price-
+taker market clearing model is the same should we use nodal
+prices.
+Note that the formulation in (6) applies to both price-taker
+market bidding (HA-1 and HA-10) and price response (PR-1
+and PR-10). The difference is that in HA cases, storage has
+to decide the bids (Ct,j and Bt,j) one hour before the market
+clearing period t, while in PR cases storage updates bids at
+the same time when observing the price.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf,len=843
+page_content='1  Transferable Energy Storage Bidder  Yousuf Baker, Ningkun Zheng, Student Member, IEEE, Bolun Xu, Member, IEEE  Abstract—Energy storage resources must consider both price  uncertainties and their physical operating characteristics when  participating in wholesale electricity markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' This is a challeng-  ing problem as electricity prices are highly volatile, and energy  storage has efficiency losses, power, and energy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' This  paper presents a novel, versatile, and transferable approach  combining model-based optimization with a convolutional long  short-term memory network for energy storage to respond to  or bid into wholesale electricity markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We apply transfer  learning to the ConvLSTM network to quickly adapt the trained  bidding model to new market environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We test our proposed  approach using historical prices from New York State, showing  it achieves state-of-the-art results, achieving between 70% to  near 90% profit ratio compared to perfect foresight cases,  in both price response and wholesale market bidding setting  with various energy storage durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We also test a transfer  learning approach by pre-training the bidding model using  New York data and applying it to arbitrage in Queensland,  Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The result shows transfer learning achieves exceptional  arbitrage profitability with as little as three days of local training  data, demonstrating its significant advantage over training from  scratch in scenarios with very limited data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Index Terms—Energy storage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Deep learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Transfer learn-  ing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Power system economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' INTRODUCTION  Successful participation of energy storage resources in com-  petitive electricity markets benefits storage investors and social  welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Ancillary services such as frequency regulation have  been the primary sources of profit for energy storage owners,  but these markets have quickly saturated due to surging storage  deployments and small market size [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' In the meantime, the  share of storage arbitraging in wholesale markets has tripled  from a little less than 20% in 2016 to almost 60% in 2021 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Thus price arbitrage in wholesale markets will be the main  focus for future grid-scale energy storage projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Energy storage arbitrages price differences and earns rev-  enues in wholesale energy markets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=', charging during low-  price periods and discharging during high-price periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' At  the same time, arbitrage from energy storage helps to reduce  renewable curtailments, meet peak demands, mitigate extreme  events, and reduce the cost of electricity [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' As countries  and regions ramp up decarbonization efforts, energy storage  resources are taking on an increasingly important role in future  electricity markets and are becoming a cornerstone for cost-  effective decarbonization [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Thus, both energy storage  owners and market organizers have significant economic and  welfare drivers to evolve models and algorithms for energy  storage arbitraging robustly and profitably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  However, energy storage arbitrage is non-trivial due to  highly volatile electricity prices and limited storage capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Baker, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Zheng, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Xu are with Columbia University, NY, USA  (e-mail: {ykb2105, nz2343, bx2177}@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Various methods have been proposed in the literature to ad-  dress energy storage participation in wholesale markets based  on different theories, they require dedicated location-specific  tuning and excessive computing power to achieve competitive  arbitrage performance [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' This paper proposes a novel end-to-  end system for opportunity value calculation, prediction, and  control, combining model-based dynamic programming with  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Our approach innovates and provides several  advantages as follows:  Our approach has reliable performance as it uses model-  based dynamic programming to address physical con-  straints in both training and control stages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Our approach is extremely computation efficient as it uses  dynamic programming to pre-process the training data,  reducing the complexity of the learning module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Our approach is transferable to different market en-  vironments while maintaining competitive performance  because of the integration of transfer learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Our approach is founded on dynamic programming value  functions and adapts to different storage market designs  and participation scenarios, including price response and  market economic bidding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Our approach achieves state-of-the-art arbitrage perfor-  mance, achieving 70% to near 90% profit ratio compared  to perfect foresight with various storage durations when  tested using price data from New York, US, and Queens-  land, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  The rest of the paper is organized as follows: Section II  summarizes energy storage market participation and previous  work using the learning method, Section III and IV elaborates  on the arbitrage formulation and solution method, Section V  presents the case study for price response and economic bid  market rules in New York and the application of transfer  learning for Queensland, and Section VII concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' LITERATURE REVIEW  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Energy Storage Price Response and Self-Schedule  Energy storage price response assumes the storage partici-  pant can observe the real-time price realization first and then  decide on the operation privately without informing the system  operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The price response participation option primarily  applies to small-scale behind-the-meter (BTM) storage re-  sources (< 1 MW) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Plenty of prior works have investigated  energy storage price response using a variety of methods,  including model-predictive control (MPC) [8], stochastic pro-  gramming [9], approximate dynamic programming [10], and  reinforcement learning [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Price response is comparably an  easier problem than economic bids as the storage operator is  not limited to market clearing models and can act after observ-  ing new price signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' However, since price response mostly  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='01233v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='LG]  2 Jan 2023  2  applies to small BTM storage projects, the revenue generated  from arbitrage will unlikely justify any specialized computing  hardware investments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Hence the arbitrage algorithm must be  slim and efficient to minimize the computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Alternatively, some markets allow energy storage operators  to self-schedule and submit the operational schedule to the  market operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Still, this option is less frequently used in  practice compared to participating by economic bids [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Self-scheduled storage cannot update the operation based on  the system clearing price, a key difference compared to price-  response or economic bids, which often causes the storage to  miss price spike opportunities and deliver fewer market profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Energy Storage Economic Bids  FERC Order 841, issued in 2018, ordered all system oper-  ators in the US must allow storage to submit bids and cleared  in spot markets [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' In this case, the storage participant must  submit charge and discharge bids to the system operator at a  specific time period ahead of the market clearing, usually one  hour (also called hour-ahead bidding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The storage participant  must follow market clearing results to charge or discharge,  unlike in the price response case in which the storage can  privately decide the control decision after observing the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  The bid design adds another layer of complexity in arbitrag-  ing, as optimal bid design requires mathematical tools due to  storage SoC constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [14] formulate the energy  storage look-ahead profit maximization problem as a bi-level  optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' A second approach for energy storage  arbitrage control is backward dynamic programming [15],  and then the evolution is approximate-dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Jiang and Powell outline a general approximate-dynamic pro-  gramming framework for policy generation for energy storage  operating with a stochastic generation source in response to  stochastic demand [16], and further introduce a “distribution-  free” variant of the previous algorithm that does not make any  assumption on the price process[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' However, all of these  methods are held back by large computational costs that make  them hard to implement in real-world applications of arbitrage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  There are other algorithms for energy storage real-time  arbitrage control: Wang and Zhang [17] solve the arbitrage  problem using reinforcement learning to come to an optimal  arbitrage policy, and Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [18] outline a computa-  tionally efficient analytical stochastic dynamic programming  algorithm (SDP) for the problem of real-time price arbitrage  of energy storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Krishnamurthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [19] also propose an  SDP algorithm for arbitrage under day-ahead and real-time  price uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' However, none of the methods outlined  above demonstrate or address transferability between different  ISO zones and geographic locations, or the hour-ahead bid  submission requirements in most real-time markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Machine Learning for Storage Arbitrage  Recent efforts to apply machine learning for storage ar-  bitrage can be grouped into two thrusts: the first is to use  machine learning to generate price predictions and then in-  tegrate them with MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' In this case, the learning module  is independent of the storage model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Sarafraz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [20]  and Nwulu and Fahrioglu [21] outline two machine learning  approaches for predicting locational marginal price (LMP)  prediction using neuro-fuzzy logic and soft computing re-  spectively, and Chaweewat and Singh [22] propose a residual  neural network approach to price interval prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The  main difficulty in combining price prediction with storage  optimization is storage arbitrage requires a look-ahead of at  least 24 hours to capture the daily price cycles [8], while most  real-time prediction methods may only accurately generate a  few steps ahead of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' To this end, existing MPC approaches  rely on pre-scheduling storage using day-ahead prices but have  to neglect the real-time price variability, which is significantly  higher than in day-ahead prices [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  The second approach is to directly use machine learning,  mainly reinforcement learning (RL), to learn the optimal  control policy for storage arbitrage directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [11]  developed the first RL approach to arbitrage storage in real-  time markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [23] propose a deep reinforcement  learning approach to learn an optimal control policy for energy  storage arbitrage with consideration of battery degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [24] demonstrated RL could optimize more  sophisticated storage models in arbitrage by integrating battery  degradation into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Yet, a common disadvantage of  RL-based approaches is transferability, as the model must  undergo time-consuming training to be adapted to a new price  zone or market environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Transferability is a crucial aspect  of storage arbitrage due to spatial and temporal variations: a  typical system consists of hundreds of price nodes, and system  price behaviors evolve with changes in system resource mix  and ambient climate conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' While previous efforts have  looked into combining transfer learning with RL [25] and its  application in selected energy-related issues, including demand  response prediction [26], event identification [27], and battery  health forecast [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Yet, the transferability of the storage  arbitrage model has not been previously studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' PROBLEM STATEMENT AND SYSTEM OUTLINE  Our algorithm aims to predict the opportunity value at the  current state of charge (SoC) of energy storage to maximize  the price arbitrage profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Our system is composed of three  components: valuation, forecasting, and arbitrage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We will  first present our methods for valuation and arbitrage and then  combine them with our forecasting model to form our bidding  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We define Qt(e) as the opportunity value function  representing the monetary value of the SoC e at time step  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The problem formulation is adapted from  [29], [30], in  which the solution is formulated using dynamic programming  as follows:  max  bt,pt,et  ∈E(et−1)  λt(pt − bt) − cpt + ˆQ  �  et|θ, X)  (1a)  where the first term is arbitrage revenue which is the product of  the real-time market price λt and the energy storage dispatch  decision (pt − bt), where pt is the discharge power and bt  is the charge power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The second term is the discharge cost,  where c is the marginal discharge cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The third term ˆQ is  the predicted storage opportunity value function with respect  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The proposed structure of training opportunity value function prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  to SoC et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The dynamic programming approach evaluates the  energy storage by back-propagation, which is not viable in the  real-time market where we do not have price realization ahead  of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Thus, we need to directly predict the value function  ˆQ using historical (and current) price data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' ˆQ is dependent  on the prediction model parameters θ and the prediction input  features X over a look-back period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  We denote that the storage charge and discharge power and  the final storage SoC belong to a feasibility set E(et−1) which  is dependent on the storage starting SoC et−1 at the start of  time period t (same as by the end of time period t−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' E(et−1)  is described with the following constraints:  0 ≤ bt ≤ P, 0 ≤ pt ≤ P  (1b)  pt = 0 if λt < 0  (1c)  et − et−1 = −pt/η + btη  (1d)  0 ≤ et ≤ E  (1e)  where (1b) models the upper bound, P, and lower bound, 0,  constraints on the storage charge and discharge power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' (1c) is a  relaxed form of the constraint that enforces the energy storage  not charging and discharging simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Negative price  is the necessary condition for storage to charge and discharge  simultaneously in price arbitrage, hence by enforcing the stor-  age to not discharge when the price is negative we eliminate  simultaneous charging and discharging [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' (1d) models the  energy storage SoC evolution constraint with efficiency η and  (1e) models the upper bound E and lower bound (we assume  as 0) of the storage SoC level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Creating our proposed system amounts to solving the  problem of optimizing the prediction model parameters θ  to maximize storage arbitrage profit over a set of training  price data and physical storage parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Intuitively, this  problem can be formulated as a bi-level problem in which  the upper level maximizes the total profit over the entire  training time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' At the same time, the lower-level  enforces a non-anticipatory decision-making process in which  the storage dispatch decision only depends on the current  price and the predicted value function as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' However, this  problem quickly becomes computationally intractable since  the prediction model is embedded in the lower-level problem,  formulated as a constrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Therefore,  strong duality is required to convert the bi-level problem into a  single-level equivalent problem or to derive partial derivatives  and calculate the back-propagation gradients for gradient-  based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' However, gradient-based approaches are  complicated by the inclusion of SoC constraints [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' In  either case, the computational complexity quickly becomes  overwhelming as the lower-level can include thousands of  problems representing the arbitrage over a particular price data  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Problem Statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We consider an alternative two-stage  training approach in which we first generate the optimal  opportunity value function and then train the learning model  to predict the generated value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' This is formulated as  min  θ  �  e∈S  ���  ���ˆqt  �  e|θ, X) − qt(e)  ���  ���  2  2  (2a)  subject to  qt(e) = ∂  ∂eQt(e)  (2b)  Qt−1(et−1) =  max  bt,pt,et  ∈E(et−1)  λt(pt − bt) − cpt + Qt(et)  (2c)  Note that (2c) is also subject to the storage operation con-  straint set E(et−1) as described in (1b)–(1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' (2c) is a dynamic  programming energy storage price arbitrage formulation in  which the storage opportunity value is defined recursively  as the maximized storage arbitrage profit including the profit  from the current time step and the future opportunity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  This formulation fits a piece-wise linear approximation of  the value function qt(e) based on the first order derivative  of the optimal value function Qt, and e is from the set of  SoC segments S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Note that in this formulation the prediction  model parameters θ are not involved in (2c), hence this is a  two-stage model in which we solve (2c) first and obtain all  optimal value function results from Qt, and more specifically,  their derivatives qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We are then able to use (2a) to solve for  the optimal value function at each time step, which we use to  train the prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' SOLUTION AND SYSTEM SETUP  Our approach includes three steps: first, we use the deter-  ministic price arbitrage dynamic programming approach from  the previous section to generate the optimal storage opportu-  nity value function segments using historical price data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We  then train a learning model to predict the optimal storage  opportunity value segments from past price data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Finally, we  qe(e)  Analytical Dynamic  Programming  Algorithm (model  based)  [X, Y] = [ADAP/AkP] Q]  Model-Based  Arbitrage  CNN-LSTM  Price Pre-  Processing  Opportu  qt (el0, X) - qt(e4  test the learned model over unseen (future) price datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  The system structure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 1, which includes  the dynamic programming solution and training method, with  specifics on the data engineering in Section IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Feature and Label Formatting  In general, the spot price for energy exhibits long-term  and short-term cycles according to cycling demand: the daily  cycling between peak and non-peak hours and the long-term  seasonal cycles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' though events and the stochastic nature of  price create differences in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Thus we chose to use  a convolutional long short-term memory (ConvLSTM) neural  net, which can learn patterns in time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' For learning  timestep t, our network input/target pair could be [λt, qt] (or  qt+hr, where +hr represents an hour time shift for the HA  case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' However, to better capture daily cycling, we elaborate  our single-step input-output pair by constructing the following  input-output matrices:  {X, Y} = {[ΛDAP|ΛRTP], Q}  ΛDAP =  �  ����  λDAP,t−m  λDAP,t−m+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  λDAP,t  λDAP,t−m−1  λDAP,t−m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  λDAP,t−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  λDAP,t−m−5hr  λDAP,t−m−5hr+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  λDAP,t−5hr  �  ����  ΛRTP =  �  ����  λRTP,t−n  λRTP,t−n+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  λRTP,t  λRTP,t−n−1  λRTP,t−n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  λRTP,t−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  λRTP,t−n−5hr  λRTP,t−n−5hr+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  λRTP,t−5hr  �  ����  Q =  �  ����  qt  qt−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  qt−5hr  �  ����  where ΛDAP and ΛRTP are matrices made up of our day ahead  and real-time price data, and m, n are a lookback window for  the day ahead and real-time prices respectively, and 5hr is the  number of timesteps that make up five hours in a given market  resolution (60 in a 5 min resolution market).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' This allows the  network to capture not only the information on past prices for  the current value function but also the relationship between  past value functions in a 5-hour lookback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We chose five hours  here as it is long enough to capture cycles within a single day  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' peak vs non-peak demand and the transition between  them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The inclusion of the day ahead price here serves as  a more stable price reference for the corresponding hour’s  spot price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Also of note is the cyclic symmetry of the price  matrices along the diagonal, which allows the network to learn  better the equivariant properties of the dataset [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Finally, the  choice of a ConvLSTM, as opposed to a traditional LSTM, is  to allow the network to capture the ”vertical” temporal relation  between the five hours of data in each data block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Note that for DAP, the shift applied to t across rows  corresponds to a step shift in the resolution of the real-time  market (RTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Meaning that if it is a 5-minute resolution  RTM, the first 12 rows of ΛDAP will be the same since the  day-ahead market (DAM) is hourly resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Model Selection and Transfer Learning  The focus of this paper is to demonstrate the robustness  of the approach across different market conditions and bat-  tery durations and to show its transferability between zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Thus we chose one general network architecture for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  However, initial experimentation showed minimal gain-loss  in network performance on minor parameter changes across  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Further, to guarantee that the training converges to a  well-performing set of weights, multiple networks were trained  for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The weights achieving the most consistent and  low validation error were saved for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Of those, the  best model was chosen by the highest arbitrage profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The  network is trained over 100 epochs in the case where it is  trained from scratch, and 25 epochs for the transfer learning  training, with a learning rate of 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Further, we use a  callback function that saves the model weights only when the  validation error improves, ensuring that the weights loaded for  training are not overfitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' This callback also allows us to set  our epochs with significant overhead to ensure convergence  without over-fitting in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Furthermore, we apply transfer learning to quickly adapt a  trained model from one price zone to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Our transfer  learning approach freezes all model layers except the output  layer and retraining on the dataset of the task to be transferred  to [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The underlying assumption is that the output layer is  more sensitive to data variability while the rest of the network  captures persistent patterns in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Full Algorithm  We lay out our workflow, which is a sequence of three  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' As a prerequisite for model training, we generate  all value functions Q according to the dynamic programming  solution in VII-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' After this, we construct our data set and  train our LSTM prediction model according to Algorithm 1,  which produces our trained model weights θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Algorithm 1 Value Function Prediction Model Training  1: Dataset Preparation: Pre-Process data according to IV-A  2: Initialization: Initialize model parameters θ using random  seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  3: i ← 0  4: while stop criteria not true do  5:  for t ∈ [1, t] do  6:  x ← [ΛDAP|ΛRTP]  7:  y ← Q  8:  Calculate Loss Components by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' (2a)  9:  Update θ by backpropagation  10:  end for  11:  i ← i + 1  12: end while  13: return θ  ▷ Parameters of the prediction model  After this, if the trained LSTM model produced by Algo-  rithm 1 is to then be used by another zone, it can be retrained  using the transfer learning approach outlined in Algorithm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' This is largely the same as the workflow of algorithm  1, save that the training dataset is of the new zone and the  5  newly trained model weights are denoted θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We differentiate  between the two sets of model weights since we compare  the two approaches of transferring (transfer learning, applying  the model on new zones without retraining) later in the  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Finally, algorithm 3 outlines the process of simulating  Algorithm 2 Transfer Learning  1: Initialization: Initialize model parameters θ∗ using ran-  dom seed  2: θ∗ ← θ (trained model parameters)  3: Freeze all parameters except output layer parameters  4: Repeat  training  loop  using  new  region’s  data  set  {[Λ∗  DAP|Λ∗  RTP], Q∗}  5: return θ∗  ▷ Parameters of the prediction model  arbitrage using our prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The arbitrage simulation  is as follows: use the prediction model trained in algorithm  1 and/or algorithm 2 to predict value functions using the  current real-time price and a look-back window (including the  day ahead look-back) and then generate the bids according  to VII-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Once the bids are generated, use them to simulate  arbitrage and market clearing as outlined in VII-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Algorithm 3 Arbitrage with Value Function Prediction  1: Initialization:  2: Set energy storage parameters c, P, ηp, ηb, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  3: Initialize et−1 ← e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  4: for t ∈ [1, T] do  5:  Predict ˆv  �  et|θ, x  �  6:  Solve single-period optimization (1)  7:  Return et, pt, dt  8: end for  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' CASE STUDY SET-UPS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Market Participation Setting and Storage Parameters  We consider the following four market designs and par-  ticipation settings to demonstrate that our proposed approach  fits a wide range of storage participation options and market  designs:  HA-1 Energy storage owner submits single-segment bids  one hour-ahead to real-time markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' This represents the  current storage bidding model in most wholesale real-  time markets in the US [10], [34] where energy storage  submits one charge bid and one discharge bid one hour  ahead of the market clearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The storage can update its  bid for each hour, but the bids must stay the same within  each hour for multiple market clearings (for example,  real-time markets clear every five minutes in NYISO, so  one hour includes 12 real-time clearings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  HA-10 Same to HA-1 except the storage submits10-  segment SoC-dependent charge and discharge bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' This  is a new market design proposed by CAISO to econom-  ically manage storage SoC in real-time [35], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  PR-10 The storage conducts price response in real-time,  deciding the storage control after observing the published  real-time price, instead of submitting bids [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The price  response option is limited to behind-the-meter storage  in which the associated demand is cleared in real-time  market prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' In this case, the storage is not limited to  any bidding models and can use any decision-making  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Yet, we assume the storage uses a 10-segment  approximation of its opportunity value as it provides a  good enough approximation to the actual value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  This also enables us to benchmark HA-10 and PR-10  cases to demonstrate the economic cost of the hour-ahead  bidding requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  PR-1 Same as PR-10 except the storage uses the average  opportunity value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=', one segment approximation) for  arbitrage control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' This is not a realistic case as there is no  motivation for the storage operator to limit itself to using  a single-segment, less accurate approximation of its value  function to conduct arbitrage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' However, we include this  case with the sole purpose to benchmark against the HA-  1 case and PR-10 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  In all case studies, we consider storage with a 90% one-  way efficiency and a 10$/MWh cost of discharge (excluding  the opportunity cost), unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We consider  three storage durations including 2-hour, 4-hour, and 12-hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Further, we adapt our base prediction model to predict the hour  ahead case by adding an hour time shift to our ground truth  training target value function, which corresponds to 12-time  steps in 5-minute price resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  We conduct the majority of our case studies over price  data from New York ISO (NYISO) [38] for four price zones:  NYC (Zone J), LONGIL (Zone K), NORTH (Zone D), and  WEST (Zone A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We also use data from the Australian Energy  Market Operator (AEMO) for Queensland to demonstrate the  transferability of our approach using transfer learning [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Market and Price Data  We observe differences in price statistics and generation  mix across zones from the same ISO, and in between zones  from ISO’s in other states and even countries, summarized in  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' In New York zones, these differences can be attributed  to significant transmission congestion when comparing the  two main zone groups in NY [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' QUEENSLAND has the  highest price volatility, which can potentially be attributed to  the absence of a day-ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Further, we see a clear  tie between penetration rates of renewables into the zones and  the price volatility [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We also see the highest occurrence  of negative prices in NORTH (NY), which is due to the  significantly higher penetration of wind when compared to  NYC and LONGIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  TABLE I  PRICE DATA STATISTICS  ,  Zone  Negative Price #  STD  Renewable %  NYC (NY)  208  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='93  LONGIL (NY)  190  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='93  NORTH (NY)  6334  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='25  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='06  WEST (NY)  633  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='55  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='06  QNSLND (AUS)  522  243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='00  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='19  6  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Accumulated Profit over 2019 test set for NYISO Zones  All code for valuation, network training, and arbitrage are  written in python with Jupyter notebook and is available on  GitHub1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' All trials are run on a desktop computer with AMD  Ryzen 9 processor and Nvidia GPU on Tensorflow 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='1 and  with cuDNN and CUDA versions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='1 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  All case studies using price data from NYISO were trained  using data from 2017 to 2018, and tested over 2019 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Each year of price data for each price zone has 8760 day-  ahead price data points (hourly resolution) and 105,120 real-  time price points (5-minute resolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The look-back price  window includes the last 36 real-time prices (3 hours) and  24-day-ahead prices (one day).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The maximum training time  over two years of training price data, including the generation  of historical optimal value functions and training of the neural  network, is 390 seconds, a bit more than five minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The net-  work consists of a Convolutional Block with three sequential  time-distributed Convolutions+MaxPool layers, then an LSTM  block with two sets of bi-directional LSTM+drop out layers,  and then finally a Dense layer at the output end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The specific  model hyperparameters and details can be found on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Benchmark with Competing Methods  We first benchmark our proposed approach with other  competing energy storage price arbitrage methods in a price  response setting, in which storage can observe price first  and act accordingly, without having to bid ahead into mar-  kets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We benchmark the proposed method (DP-ConvLSTM)  with a reinforcement learning method (RL) [17], a modified  stochastic dynamic programming with day-ahead price updates  (SDP) [37], the proposed method but implemented with a  multilayer perceptron (DP-MLP) network [42], and perfect  price predictions which provide the highest profit possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  In RL, we have 11 actions, 103 price states, and 121 SoC  states, which takes more than 1 hour to train for 5-min  resolution arbitrage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The RL approach uses a Markov decision  process (MDP) model by discretizing the storage SoC, and it  only works with perfect efficiency (100%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' To provide a fair  comparison, all methods in this case consider storage with  perfect efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='com/ybaker661/LSTM-Value-Prediction  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 2 shows the comparison result when trained using price  data from 2017-2018 and tested in 2019 at price zones in NY-  ISO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The result shows DP-ConvLSTM has a clear advantage  over other methods in terms of profitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Notably, the DP-  ConvLSTM approach performs exceptionally well in capturing  low-frequency extreme events, such as the surge in profits  around June in LONGIL and WEST, where the ConvLSTM  captures profit spikes that the RL benchmark misses, and the  difference between the ConvLSTM profit value and the perfect  prediction comes from the difference in arbitrage decision as  a result of numerical saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' In this context, numerical  saturation means that the network learns to predict numerical  values in the range of data it most frequently sees (value  functions of stable prices), and so when it predicts on anomaly  data (price spike value functions) that are numerically much  larger, the network prediction saturates at the largest common  numerical value it sees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Price Response  In this subsection, we compare the price response arbitrage  performance (PR-1 and PR-10) with different storage dura-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Table III shows the arbitrage profit ratio results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Overall, the result shows stable performance over the four  price zones and three storage durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' In comparison, our  previous work using SDP [37] and DP-MLP [42] have worse  performance in LONGIL (more frequent price spikes) and  NORTH (more frequent negative prices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The comparison  between PR-1 and PR-10 shows that increasing the value  function approximation from one to ten segments increased  the profit ratio by around 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Considering different storage  durations, the profit ratio is lower for long-duration energy  storage (12hr), as the longer storage duration leads to a  longer temporal correlation into the future, leading to higher  prediction difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Still, our method achieved around 75%  profit ratio (HA-10) in the worst-case scenario in the NORTH  zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Hour-ahead Bidding  We now investigate hour-ahead bidding which is the most  common market design for energy storage owners operating in  the real-time market, where the storage submits bids an hour  NYC  LONGIL  NORTH  WEST  16  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Perfect Prediction  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='. Perfect Prediction  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Perfect Prediction  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Perfect Prediction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='DP+LSTM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='DP+LSTM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='DP+LSTM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='DP+LSTM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
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+page_content='-- SDP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
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+page_content='RL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='RL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
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+page_content='Cumulative F  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
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+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
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+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
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+page_content='months  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
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+page_content='36  TABLE III  CAPTURED PROFIT RATIOS: PRICE RESPONSE  Zone  PR-1  PR-10  2hr  4hr  12hr  2hr  4hr  12hr  NYC  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='83  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='96  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='54  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='69  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='63  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='67  LONGIL  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='33  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='61  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='10  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='98  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='94  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='38  NORTH  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='24  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='87  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='02  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='52  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='96  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='29  WEST  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='43  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='37  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='65  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='97  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='44  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='43  TABLE IV  CAPTURED PROFIT RATIOS: HOUR AHEAD  Zone  HA-1  HA-10  2hr  4hr  12hr  2hr  4hr  12hr  NYC  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='99  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='82  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='00  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='79  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='61  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='47  LONGIL  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='26  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='56  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='01  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='30  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='63  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='89  NORTH  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='22  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='71  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='17  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='83  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='21  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='16  WEST  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='79  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='13  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='17  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='12  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='94  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='60  ahead of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Table IV shows the hour-ahead bidding profit  ratio in the NYC case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The profit ratio is lower than the  price response as the storage owner must decide on the bids  one hour before the actual time of arbitrage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The short-duration  storage (2hr) cases have higher profit ratio reductions (up to  7%) as the value function is more sensitive to recent market  prices due to it’s shorter duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' On the other hand, the long-  duration storage (12hr) is more resilient and the hour-ahead  bidding has little impact on the profit ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Hour-ahead bidding results also restate our observation from  the price response case, that multi-segment SoC bids are  more beneficial for short-duration storage to better manage  their SoC, but the improvement is not obvious for long-  duration storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Overall, our approach achieved a higher  than 70% profit ratio in all hour-ahead cases, showing ro-  bust performance under different market designs and storage  technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Transfer Learning in AEMO  We now demonstrate the effectiveness of applying transfer  learning to quickly adapt a pre-trained value function predic-  tion model from one market to a new market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' In this case  study, we pre-train the prediction model using NYC price  data from 2017-2018 and conduct arbitrage in Queensland,  Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' In Queensland, we use selected data from 2019  for training and the first 6 months of 2021 for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  We skipped the year 2020 because of COVID-19’s impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  To present the sensitivity of transfer learning over a limited  amount of data, we consider various durations of training  datasets ranging from 3 days to one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We present a  sensitivity analysis comparing the performance of transfer  learning versus training a model from scratch for the situations  where we have access to training data for only 3 days, 1 week,  1 month, and 1 year of data for the target zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Thus this case  study has the following steps:  1) Use a pre-trained network (transfer learning) or a ran-  domly initialized network (training from scratch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  2) Use a limited duration of Queensland price data from  2019, ranging from 3 days to 1 year, to train the model  using transfer learning as outlined in algorithm 2, or  normal training outlined in algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  3) Test the arbitrage performance to arbitrage, as outlined  in algorithm 3, using the first six-month of data in  Queensland, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Table II shows the arbitrage profit ratio results for Queens-  land.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The main finding is that in the data scare scenarios,  the transfer learning approach vastly outperforms training a  model from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We also see that adding more data to  the transfer learning case does not necessarily increase perfor-  mance, whereas training the model from scratch only becomes  a viable option once a certain amount of data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  For the 2-hour storage, training from scratch becomes viable  around the point where you have 1 month of data available for  the target zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' For the 4-hour storage, the model still needs  about 1 month of data to reasonably perform when trained  from scratch, though the model is able to capture higher profit  ratios for 3 and 1 week of data than the 2-hour case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Compared  to both of these, the 12-hour storage seems to be the easiest for  the model to learn, only needing 3 days of data when training  from scratch to achieve reasonable performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' however, the  12-hour storage shows that the transfer learning approach  outperforms training from scratch for all data scenarios for 1  and 10 segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' However, since the 12-hr storage has lower  opportunity cost and less significant change in the opportunity  cost between sequential time steps, predicting the opportunity  value might not be an effective method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  The takeaway is that transfer learning beats out training the  model from scratch when data scarcity is an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' However,  when the dataset size increases to a general size of 1 month,  training from scratch becomes a viable option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Additionally,  adding extra data, past 3 days or 1 week for transfer learning  and past 6 months for training from scratch, does not necessar-  ily yield better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' As such, it is more useful to focus  on stabilizing ConvLSTM’s volatile and initialization-sensitive  training as well as other changes to the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We  8  see that in almost all cases, using the model trained on NY  data without any retraining performs comparably or even better  than transfer learning and even training a model from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  This indicates statistical robustness and generality in the NY  zone data, and it also points to a unified generating distribution  behind the price data/opportunity value of zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' However, we  cannot conclude this for certain without further analysis of  other permutations of transfer learning, and on testing different  data duration permutations (including different data scarcity  scenarios in the training of the base model along with in  transfer learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' CONCLUSION  In this paper, we propose a computation-efficient, versatile,  and transferable energy storage arbitrage model that fits both  price response and market bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Our proposed approach  achieves state-of-the-art profits compared to other methods and  is both computation and data-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We also demonstrate  that by incorporating transfer learning, we can quickly adapt  our bidding model to a new location with very limited training  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Our model suits a variety of arbitrage settings, including  behind-the-meter price response and economic bids for utility-  scale storage, and can be implemented using non-proprietary  software and regular computing hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Our work would fa-  cilitate storage participation in electricity markets and promote  economic decarbonization of the electric power system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  REFERENCES  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' McGrath and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Comstock, “Battery systems on the u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' power  grid are increasingly used to respond to price,” Jul 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='eia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='gov/todayinenergy/detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='id=53199  [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Srianandarajah, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Wilson, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Chapman, “From green to  amber: is australia’s national electricity market signalling a financial  warning for wind and solar  power?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Energy Policy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 167,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 113052, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='com/  science/article/pii/S0301421522002774  [3] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Yan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Gu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Li, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Wang, “Lmp-based pricing for energy  storage in local market to facilitate pv penetration,” IEEE Transactions  on Power Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' PP, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 1–1, 12 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [4] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Room, “Fact sheet: President biden sets 2030 greenhouse gas  pollution reduction target aimed at creating good-paying union jobs and  securing us leadership on clean energy technologies,” The White House,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [5] “2030  climate  &  energy  framework.”  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Avail-  able:  https://climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='eu/eu-action/climate-strategies-targets/  2030-climate-energy-framework en  [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Sioshansi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Denholm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Arteaga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Awara, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Bhattacharjee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Bot-  terud, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Cole, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Cort´es, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' De Queiroz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' DeCarolis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=', “Energy-  storage modeling: State-of-the-art and future research directions,” IEEE  Transactions on Power Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 860–875, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Roozbehani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Dahleh, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Mitter, “Volatility of power  grids under real-time pricing,” IEEE Transactions on Power Systems,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 1926–1940, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [8] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Abdulla, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' De Hoog, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Muenzel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Suits, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Steer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Wirth,  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Halgamuge, “Optimal operation of energy storage systems  considering forecasts and battery degradation,” IEEE Transactions on  Smart Grid, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 2086–2096, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Krishnamurthy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Uckun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Zhou, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Thimmapuram, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Botterud, “Energy storage arbitrage under day-ahead and real-time  price uncertainty,” IEEE Transactions on Power Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 1,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 84–93, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Jiang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Powell, “Optimal hour-ahead bidding in the real-  time electricity market with battery storage using approximate dynamic  programming,” INFORMS Journal on Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  525–543, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Wang and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Zhang, “Energy storage arbitrage in real-time markets  via reinforcement learning,” in 2018 IEEE Power & Energy Society  General Meeting (PESGM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  IEEE, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Mohsenian-Rad, “Optimal bidding, scheduling, and deployment of  battery systems in california day-ahead energy market,” IEEE Transac-  tions on Power Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 442–453, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Bhattacharjee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Sioshansi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Zareipour, “Energy storage  participation in wholesale markets: The impact of state-of-energy man-  agement,” IEEE Open Access Journal of Power and Energy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  173–182, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [14] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Dvorkin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Fernandez-Blanco, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Xu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Qiu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Kirschen, “Look-ahead bidding strategy for energy storage,” IEEE  Transactions on Sustainable Energy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 1106–1117, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Puterman, Markov decision processes: discrete stochastic dynamic  programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  John Wiley & Sons, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Jiang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Powell, “An approximate dynamic programming  algorithm for monotone value functions,” 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Available:  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='org/abs/1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='1590  [17] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Wang and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Zhang, “Energy storage arbitrage in real-time  markets via reinforcement learning,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' abs/1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='03127, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Available: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='org/abs/1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='03127  [18] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Jaworski, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Xu, “Arbitraging variable efficiency  energy storage using analytical stochastic dynamic programming,” IEEE  Transactions on Power Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 4785–4795, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [19] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Krishnamurthy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Uckun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Zhou, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Thimmapuram, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Botterud, “Energy storage arbitrage under day-ahead and real-time  price uncertainty,” IEEE Transactions on Power Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 1,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 84–93, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [20] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Sarafraz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Ghasemi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Monsef, “Locational marginal price  forecasting by locally linear neuro-fuzzy model,” in 2011 10th Inter-  national Conference on Environment and Electrical Engineering, 2011,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [21] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Nwulu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Fahrioglu, “A soft computing approach to  projecting locational marginal price,” NEURAL COMPUTING &  APPLICATIONS, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 1115–1124, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Available: https:  //hdl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='net/11511/64998  [22] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Chaweewat and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Singh, “An electricity price interval forecasting  by using residual neural network,” International Transactions on Electri-  cal Energy Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 30, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' e12506, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Available:  https://onlinelibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='com/doi/abs/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='1002/2050-7038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='12506  [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Cao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Harrold, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Fan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Morstyn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Healey, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Li, “Deep  reinforcement learning-based energy storage arbitrage with accurate  lithium-ion battery degradation model,” IEEE Transactions on Smart  Grid, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 4513–4521, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [24] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Kwon and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Zhu, “Reinforcement learning based optimal battery  control under cycle-based degradation cost,” IEEE Transactions on  Smart Grid, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Taylor and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Stone, “Transfer learning for reinforcement learning  domains: A survey.” Journal of Machine Learning Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 10,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 7, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [26] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Peirelinck, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Kazmi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Mbuwir, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Hermans, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Spiessens,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Suykens, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Deconinck, “Transfer learning in demand response:  A review of algorithms for data-efficient modelling and control,”  Energy  and  AI,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  7,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  100126,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Available:  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='com/science/article/pii/S2666546821000732  [27] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Ma, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Weng, “A transfer learning framework for power  system event identification,” IEEE Transactions on Power Systems,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 4424–4435, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Choi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Kim, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Choi, “Forecasting  state-of-health of lithium-ion batteries using variational long short-term  memory with transfer learning,” Journal of Energy Storage, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 41,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 102893, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='com/  science/article/pii/S2352152X21006101  [29] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Korp˚as, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Botterud, “Operational valuation of energy  storage under multi-stage price uncertainties,” in 2020 59th IEEE  Conference on Decision and Control (CDC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 55–60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [30] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Zheng and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Xu, “Impact of bidding and dispatch models over  energy storage utilization in bulk power systems,” IREP Symposium on  Bulk Power System Dynamics and Control 2022, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [31] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Cui, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Fang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Chen, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Wang, “Bilevel arbitrage  potential evaluation for grid-scale energy storage considering wind  power and lmp smoothing effect,” IEEE Transactions on Sustainable  Energy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 707–718, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Dieleman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' De Fauw, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Kavukcuoglu, “Exploiting cyclic  symmetry in convolutional neural networks,” 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Available:  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='org/abs/1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='02660  9  [33] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Chollet, “Transfer learning & fine-tuning,” https://keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='io/guides/  transfer learning/, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Sakti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Botterud, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' O’Sullivan, “Review of wholesale markets  and regulations for advanced energy storage services in the united states:  Current status and path forward,” Energy policy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 120, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 569–579,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [35] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Zheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Qin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Wu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Murtaugh, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Xu, “Energy storage  state-of-charge market model,” arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='07221, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [36] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Chen  and  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Tong,  “Convexifying  market  clearing  of  soc-  dependent bids from merchant storage participants,” arXiv preprint  arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='02107, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [37] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Jaworski, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Xu, “Arbitraging variable efficiency  energy storage using analytical stochastic dynamic programming,” IEEE  Transactions on Power Systems, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [38] “Energy  market  &  operational  data.”  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Available:  https:  //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='nyiso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='com/energy-market-operational-data  [39] “Engineering precinct battery.” [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Available: http://dashboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='au/engineering-precinct-battery/interactive/#/  [40] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Patton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' LeeVanSchaick, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Chen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Unit, “2014 state  of the market report for the new york iso markets,” Potomac Economics,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [41] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Waite and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Modi, “Electricity load implications of space heating  decarbonization pathways,” Joule, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' 376–394, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  [42] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Zheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Xu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Shi, “Energy storage price ar-  bitrage via opportunity value function prediction,” arXiv preprint  arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='07797, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Dynamic Programming Solution Algorithm  We first solve the dynamic programming problem as listed  in (2c) subject to constraints (1b)–(1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We use results from our  prior work [29] to solve the dynamic programming problem  (2c) and obtain the full piece-wise linear approximation of the  opportunity value function Qt for all time periods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=', one  value function for each time step for an entire year, 105120  for 5 min price resolution 35040 for 20 min price resolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  We start by defining qt as the derivative of storage opportunity  value function Qt, which represents the marginal opportunity  value of energy stored in the storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Then we can use an  analytical formulation to calculate the opportunity value qt(e)  at any given energy storage SoC level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Our prior work proved qt−1 can be recursively calculated  with next period value function qt, power rating P, and effi-  ciency η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The value function calculated using the deterministic  formulation is thus  qt−1(e) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  qt(e + Pη)  if λt ≤ qt(e + Pη)η  λt/η  if qt(e + Pη)η < λt ≤ qt(e)η  qt(e)  if qt(e)η < λt ≤ [qt(e)/η + c]+  (λt − c)η  if [qt(e)/η + c]+ < λt  ≤ [qt(e − P/η)/η + c]+  qt(e − P/η)  if λt > [qt(e − P/η)/η + c]+  (3)  and calculates the opportunity value function assuming the  price follows a recursive computation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' This deter-  ministic formulation is what we will use in our investigation,  and from this we are able to calculate opportunity value  function qt(e) at any time step using backwards recursion by  defining an end period value function qT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We then discretize qt  by splitting the energy storage SoC level e into small equally  spaced segments, which must be far smaller than power rating  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' For any SoC level et, we can find the nearest segment and  return the corresponding value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Bid generation  We now design discharge and charge bids using the oppor-  tunity valuation results based on our prior work [30], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We  consider generating time-varying SoC-dependent bids with a  total number of J segments for charge bids Bt,j and discharge  bids Ct,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Note that these bids represent the combination of  the discharge cost and the change in the opportunity value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' We  assume each bid segment j is associated with an SoC range  Ej−1 to Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The discharge bids are thus calculated based on  the average value function between the internal Ej−1 and Ej  Ct,j = 1  E  � Ej  Ej−1  ∂  ∂pt  (cpt − Qt(et−1 − pt/η + btη))det−1  = c + 1  E  � Ej  Ej−1  qt(et−1 − pt/η + btη)det−1/η  ≈ c + 1  ηE  � Ej  Ej−1  qt(e)de  Similarly for charge bids  Bt,j = 1  E  � Ej  Ej−1  ∂  ∂bt  (cpt − Qt(et−1 − pt/η + btη))det−1  = 1  E  � Ej  Ej−1  qt(et−1 − pt/η + btη)det−1η  ≈ η  E  � Ej  Ej−1  qt(e)de  In the special case of one segment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=', bids are not  dependent on SoC (the current energy storage bidding model  in most wholesale markets), Ej−1 is zero or the lowest allowed  SoC and Ej is the highest allowed SoC value or the energy  capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' In this case the bids are simply based on the average  marginal opportunity value ¯qt  ¯qt = 1  E  � E  0  qt(e)de  (4)  and the discharge bid is c + ¯qt/η, and the charge bid is ¯qtη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Real-time market clearing and arbitrage simulation  We consider the following simplified real-time market clear-  ing model with a generalized multi-segment energy storage  bids  min  pt,j,s,dt,j,s  Jt(gt) +  �  s  �  j  (Ct,j,sdt,j,s − Bt,j,sbt,j,s) (5a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  et,j,s − et−1,j,s = bt,j,sη − pt,j,s/η  (5b)  0 ≤ et,j,s ≤ Ej,s − Ej−1,s  (5c)  gt +  �  s  �  j  pt,j,s = Dt +  �  s  �  j  bt,j,s : λt  (5d)  where (5a) is the objective function minimizing total bidding  costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Note that we use aggregated generator supply curve  Jt(gt) and total generation gt instead of modeling the bids  from each individual generator for simplicity to focus on  energy storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The second term of the objective is the  discharge bids and charge bids for each energy storage s and  10  each SoC segment j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' (5b) models the SoC evolution under  single-trip efficiency η for each SoC segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' (5c) models  the upper and lower energy limit for each SoC segment, note  that the minimum energy is always zero while the maximum  energy for each segment is the difference between the upper  and lower SoC range Ej,s − Ej−1,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' Finally, (5d) is the  power balance constraint enforcing the sum of generation and  storage charge/discharge equals to the total demand Dt over  time period t, the associated dual variable is thus the market  clearing price λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Now in price-taker analysis, we use historical price data to  simulate how the energy storage would have been cleared in  the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' In this case, we perform a Lagrangian relaxation  of (5d) and move it to the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' This decomposes the  optimization into independent sub-problems for each energy  storage, and for each storage, the price-taker market clearing  problem is equivalent to the following price arbitrage problem  max  pt,j,dt,j λt  �  j  (dt,j − bt,j) −  �  j  (Ct,jdt,j − Bt,jbt,j)  (6)  subject to the same storage unit constraints (5b) and (5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Note that for this problem we omit the storage unit index  s as the problem formulation is the same for each storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Hence, price-taker market clearing simulation is equivalent to  arbitrage using the same bidding cost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' While we did  not consider the network model in this formulation, the price-  taker market clearing model is the same should we use nodal  prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content='  Note that the formulation in (6) applies to both price-taker  market bidding (HA-1 and HA-10) and price response (PR-1  and PR-10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
+page_content=' The difference is that in HA cases, storage has  to decide the bids (Ct,j and Bt,j) one hour before the market  clearing period t, while in PR cases storage updates bids at  the same time when observing the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfSft5/content/2301.01233v1.pdf'}
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+Anti-Exploration by Random Network Distillation
+Alexander Nikulin 1 Vladislav Kurenkov 1 Denis Tarasov 1 Sergey Kolesnikov 1
+Abstract
+Despite the success of Random Network Distilla-
+tion (RND) in various domains, it was shown as
+not discriminative enough to be used as an uncer-
+tainty estimator for penalizing out-of-distribution
+actions in offline reinforcement learning. In this
+paper, we revisit these results and show that, with
+a naive choice of conditioning for the RND prior,
+it becomes infeasible for the actor to effectively
+minimize the anti-exploration bonus and discrim-
+inativity is not an issue. We show that this lim-
+itation can be avoided with conditioning based
+on Feature-wise Linear Modulation (FiLM), re-
+sulting in a simple and efficient ensemble-free
+algorithm based on Soft Actor-Critic. We eval-
+uate it on the D4RL benchmark, showing that it
+is capable of achieving performance comparable
+to ensemble-based methods and outperforming
+ensemble-free approaches by a wide margin. 1
+1. Introduction
+In recent years, significant success has been achieved in ap-
+plying Reinforcement Learning (RL) to challenging and
+large-scale tasks such as Atari (Badia et al., 2020), Go
+(Schrittwieser et al., 2020), Dota 2 (Berner et al., 2019),
+and Minecraft (Baker et al., 2022). However, the online na-
+ture of such RL algorithms makes it difficult to apply them
+in the real world, where online collection of large amounts
+of exploratory data may not be feasible for safety or fi-
+nancial reasons. Offline Reinforcement Learning (Levine
+et al., 2020) promises a more controllable and data-driven
+approach, focusing on algorithms that can learn from a fixed,
+pre-recorded dataset without requiring additional environ-
+ment interactions.
+The use of ensembles for uncertainty-based penalization has
+proven to be one of the most effective approaches for offline
+RL. Ensemble-based algorithms, such as SAC-N, EDAC
+(An et al., 2021), and MSG (Ghasemipour et al., 2022)
+1Tinkoff, Moscow, Russia. Correspondence to: Alexander
+Nikulin <a.p.nikulin@tinkoff.ai>.
+1Our implementation is available at https://github.
+com/tinkoff-ai/sac-rnd
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+training steps
+1e6
+0
+20
+40
+60
+80
+100
+average normalized score
+CQL (ensemble-free)
+SAC-N (ensemble-based)
+SAC-RND (Naive)
+SAC-RND (Ours)
+Figure 1. Mean performance of SAC-RND variants on walker and
+hopper medium-* datasets, each averaged over 3 seeds. We plot
+performance for the naive version, which uses concatenation con-
+ditioning, and our final version, which is described in Section 5.
+We also plot the final scores for the ensemble-free CQL (Kumar
+et al., 2020) and the ensemble-based SAC-N (An et al., 2021). It
+can be seen that our version is a significant improvement over the
+naive version, achieving performance comparable to ensembles.
+currently achieve state-of-the-art results on most D4RL (Fu
+et al., 2020) datasets, outperforming ensemble-free methods
+by a wide margin. Unfortunately, in order to achieve the best
+performance, these algorithms may require tens or hundreds
+of ensemble members, leading to significant computational
+and memory overhead, as well as extended training duration
+(Nikulin et al., 2022).
+Recent research (Yang et al., 2022) has successfully reduced
+the ensemble size to tens of Q-networks in the worst-case
+scenarios. However, given the general trend for model scal-
+ing in offline RL (Kumar et al., 2022; Reed et al., 2022; Lee
+et al., 2022), efficiently training even ten Q-networks with
+80 million parameters each is not feasible. Furthermore,
+Ghasemipour et al. (2022) showed that methods for efficient
+ensemble training found in supervised learning literature
+do not deliver performance comparable to naive ensembles
+and can even worsen the results. Thus, further research
+on efficient uncertainty estimation for offline RL is needed,
+with the goal of reducing the size of the ensemble as much
+as possible or even fully removing it.
+In this work, we move away from ensembles and take an
+alternative approach to uncertainty estimation, proposing an
+arXiv:2301.13616v1  [cs.LG]  31 Jan 2023
+
+Anti-Exploration by Random Network Distillation
+efficient offline RL method with ensemble-free uncertainty
+estimation via Random Network Distillation (RND) (Burda
+et al., 2018). RND, a simple and fast ensemble competitor
+for epistemic uncertainty estimation (Ciosek et al., 2019),
+is an attractive choice for offline RL. However, previous
+research (Rezaeifar et al., 2022) found RND to be insuffi-
+ciently discriminative for good results.
+In our preliminary experiment (Section 3), we show that
+RND is discriminative enough to detect OOD actions, which
+contradicts the previous study (Rezaeifar et al., 2022). Nev-
+ertheless, our results show that the naive application of RND
+does indeed not lead to good results (see Figure 1). Building
+upon these findings, we further simplify the problem and
+analyze the reasons for this issue (Section 4). We discover
+that a naive choice of conditioning for the RND prior can
+hinder the minimization of the anti-exploration bonus by
+the actor, and that conditioning based on Feature-wise Lin-
+ear Modulation (FiLM) (Perez et al., 2018) is particularly
+effective in solving this problem.
+Based on our findings, we propose a new ensemble-free of-
+fline RL algorithm called SAC-RND (Section 5). We eval-
+uate our method on the D4RL (Fu et al., 2020) benchmark
+(Section 6), and show that SAC-RND achieves performance
+comparable to ensemble-based methods while outperform-
+ing ensemble-free approaches.
+2. Background
+Offline Reinforcement Learning. Reinforcement learning
+problem can be described as a Markov Decision Process
+(MDP) defined by the {S, A, P, R, γ} tuple with state space
+S ⊂ RN, action space A ⊂ RM, transition dynamics P :
+S × A → S, reward function R : S × A → R, and a
+discount factor γ. The goal of reinforcement learning in
+an infinite horizon setting is to produce a policy π(a|s)
+that maximizes the expected cumulative discounted return
+Eπ[�∞
+t=0 γtr(st, at)].
+In offline reinforcement learning, a policy must be learned
+from a fixed dataset D collected under a different policy or
+mixture of policies, without any environment interaction.
+This setting poses unique fundamental challenges (Levine
+et al., 2020), since the learning policy is unable to explore
+and has to deal with distributional shift and extrapolation
+errors (Fujimoto et al., 2019) for actions not represented in
+the training dataset.
+Offline RL as Anti-Exploration. There are numerous ap-
+proaches for offline RL, a substantial part of which constrain
+the learned policy to stay within the support of the train-
+ing dataset, thus reducing (Kumar et al., 2020) or avoiding
+(Kostrikov et al., 2021) extrapolation errors. For our work,
+it is essential to understand how such a constraint can be
+framed as anti-exploration (Rezaeifar et al., 2022).
+Similarly to online RL, where novelty bonuses are used as
+additive intrinsic rewards for effective exploration, in offline
+RL, novelty bonuses can induce conservatism, reducing the
+reward in unseen state-action pairs. Hence the name anti-
+exploration, since the same approaches from exploration
+can be used, but a bonus is subtracted from the extrinsic
+reward instead of being added to it.
+However, unlike online RL, subtracting a bonus from the
+raw reward would not be as useful, since the novelty bonus
+is, by design, close to zero for in-dataset state-action pairs.
+Therefore, it is more effective to apply it where the overesti-
+mation for OOD actions emerges — the temporal difference
+learning target:
+r + γEa′∼π(·|s′)[Q(s′, a′) − b(s′, a′)]
+(1)
+where the actor is trained to maximize the expected Q-value,
+as is usually done in off-policy actor-critic algorithms (Lil-
+licrap et al., 2015; Haarnoja et al., 2018). It can be shown
+that, theoretically, these approaches are equivalent, but the
+latter is more suited for use in offline RL (Rezaeifar et al.,
+2022).
+An illustrative example of how such framing can be effective
+are ensemble-based approaches such as SAC-N & EDAC
+(An et al., 2021) and MSG (Ghasemipour et al., 2022),
+which currently outperform their ensemble-free counterparts
+by a large margin on most D4RL (Fu et al., 2020) benchmark
+datasets. For the anti-exploration bonus, these methods use
+ensemble disagreement as a proxy for epistemic uncertainty.
+However, a large number of ensemble members is usually
+required for a competitive result.
+Random Network Distillation. Random network distilla-
+tion (RND) was first proposed in online RL (Burda et al.,
+2018) as a simple and effective exploration bonus. To this
+day, RND is still considered a strong baseline for explo-
+ration that can work well even in stochastic environments,
+contrary to some more modern approaches (Jarrett et al.,
+2022).
+RND consists of two neural networks: a fixed and randomly
+initialized prior network ¯f ¯
+ψ, and a predictor network fψ
+which learns to predict the prior outputs on the training data:
+∥fψ(s) − ¯f ¯
+ψ(s)∥2
+2
+(2)
+Both networks map states to embeddings in RK, and the
+gradient through prior is disabled. The interpretation of
+the novelty is straightforward: with the sufficiently diverse
+prior, the predictor must learn to match embeddings on data
+points similar to the training dataset, while failing to predict
+on new examples. A bonus in such a case may simply be a
+prediction error, as in Equation (2).
+
+Anti-Exploration by Random Network Distillation
+In a subsequent work, Ciosek et al. (2019) analyses the
+success of RND in a supervised setting, and shows that
+fitting random priors can be a competitive alternative to
+ensembles for estimating epistemic uncertainty.
+Note that in practice, the choice of predictor and prior having
+the same architecture and the estimation of novelty from
+states only are very common, but arbitrary. Moreover, for
+offline RL, we are interested in estimating the novelty of an
+action conditioned on the state, which is why in our work
+RND depends on both: fψ(s, a).
+Multiplicative Interactions. The most common way to
+fuse two different streams of information is feature con-
+catenation, which is straightforward but can be suboptimal
+(Dumoulin et al., 2018). Jayakumar et al. (2020) shows that
+multiplicative interactions provide a powerful inductive bias
+for fusing or conditioning from multiple streams and are
+superior in practice. We provide a brief review of those used
+in our work (excluding concatenation): gating, bilinear, and
+feature-wise linear modulation (FiLM).
+Gating. Simple conditioning with two linear layers and
+pointwise multiplication of the resulting features (Srivastava
+et al., 2019).
+f(a, s) = tanh(W1a + b1) ⊙ σ(W2s + b2)
+Bilinear. Bilinear layer in its most general form, as pro-
+posed by Jayakumar et al. (2020).
+f(a, s) = sT Wa + sT U + Va + b
+where W is a 3D tensor, U, V are regular matrices and b
+is a vector. However, in our work, we also use the imple-
+mentation as in PyTorch, which does not learn U, V by
+default.
+FiLM. Special case of a bilinear layer with low-rank weight
+matrices (Perez et al., 2018).
+f(h, s) = γ(s) ⊙ h + β(s)
+Usually, FiLM operates on hidden activations h before non-
+linearity between layers. Thus, the main network takes a as
+an input.
+3. Random Network Distillation is
+Discriminative Enough
+To better understand the possible difficulties of applying
+RND to offline RL, we first reproduce the main experiment
+from Rezaeifar et al. (2022), which showed that RND is not
+discriminative enough to be used as a novelty bonus. For
+convenience, we provide the original figure from Rezaeifar
+et al. (2022) in the Appendix A. We also compare RND with
+0.0
+2.5
+5.0
+7.5
+10.0
+standard deviation
+0
+20000
+40000
+Q-Ensemble
+0.0
+2.5
+5.0
+7.5
+10.0
+prediction error
+RND
+dataset
+uniform
+dataset + noise (std .25)
+dataset + noise (std .5)
+Figure 2. Anti-exploration bonus (Rezaeifar et al., 2022) on the
+walker2d-medium dataset for trained SAC-N (An et al., 2021),
+Q-ensemble (N = 25) and RND. Bonus is computed for state-
+action pairs from the original dataset and different perturbations of
+actions: random actions, dataset actions to which Gaussian noise
+is added with different scales. Both RND networks use simple
+state-action concatenation. The result is strikingly different from a
+similar figure in the Rezaeifar et al. (2022) (we provide the original
+figure in the Appendix A for convenience). Contrary to previous
+research, it can be seen that RND is capable of distinguishing ID
+from OOD actions and is comparable to a trained Q-ensemble.
+a trained Q-ensemble (N = 25) from the SAC-N algorithm
+(An et al., 2021). Similarly to Rezaeifar et al. (2022), we
+use simple state-action concatenation. Predictor and prior
+share the identical architecture of 4-layer MLPs.
+The goal of the experiment (see Figure 2) is to visually
+plot the anti-exploration bonus for ID state-action pairs
+and different perturbations of actions to model OOD data:
+random actions sampled from a uniform distribution and
+dataset actions to which Gaussian noise with different scales
+is added.
+To our surprise, the result on Figure 2 is strikingly different
+from previous work. It shows that RND is able to discrim-
+inate between ID and OOD actions with varying degrees
+of distributional shift and is comparable to a trained Q-
+ensemble. In contrast, Rezaeifar et al. (2022) hypothesizes
+that RND can only work well out of the box for discrete ac-
+tion spaces and visual features, and concludes that extending
+it to continuous action spaces is not straightforward.
+After further investigation of the open-sourced codebase2 in
+search of discrepancies with our implementation, we found
+that the only difference is that, contrary to the advice of
+Ciosek et al. (2019), Rezaeifar et al. (2022) sets the predictor
+smaller than prior by two layers during RND pretraining. It
+is important to make the predictor larger or comparable in
+capacity to the prior so that it can minimize the loss to zero
+on the training dataset (Ciosek et al., 2019). However, the
+actual RND hyperparameters used in the final publication
+were not listed, so we cannot draw a definitive conclusion
+about the reason for such different results.
+2https://github.com/shidilrzf/Anti-exploration-RL
+
+Anti-Exploration by Random Network Distillation
+4. Concatenation Prior Hinders Bonus
+Minimization
+A well-behaved anti-exploration bonus for continuous action
+spaces, be it RND or any other, should satisfy at least two
+criteria. First, it should be discriminative enough to detect
+novel actions and downweight their value estimates (see
+Equation (1)). Ideally, the bonus should be close to zero for
+ID data so that we do not bias the Q-function, as this can
+be detrimental to training. Second, it should allow the actor
+to easily minimize the bonus with gradient descent during
+training.
+In Section 3, we showed that RND can detect OOD ac-
+tions. Nevertheless, naive use of RND as an anti-exploration
+bonus on top of the Soft Actor Critic algorithm (Haarnoja
+et al., 2018) still does not provide satisfactory performance
+(see Figure 1) with scores lower than CQL (Kumar et al.,
+2020) and SAC-N (An et al., 2021). This gives us an hint
+that the problem may not be the discriminative power of
+RND, but that the actor cannot effectively minimize the
+anti-exploration bonus during training.
+To test our hypothesis that the actor cannot effectively min-
+imize the anti-exploration bonus, we further simplify the
+problem by removing the critic from the SAC algorithm
+but keeping the entropy bonus (see Algorithm 2 in the Ap-
+pendix). We expect that, in such a setting, the actor will be
+able to successfully minimize the anti-exploration bonus to
+the possible minimum, i.e. comparable to the bonus for the
+ground truth data at the end of the RND pretraining. As a
+consequence, since dataset actions provide the minimum
+bonus by design, we also expect that the distance from the
+agent to dataset actions should be small.
+We set predictor architecture to state-action concatenation.
+Additionally, we explore different conditioning schemes for
+the prior. We use the halfcheetah, walker2d and hopper
+medium datasets, with 3 seeds each. Figure 3 compares
+the anti-exploration bonus for dataset actions during RND
+pretraining (see Figure 3a) and for agent actions during
+training (see Figure 3b).
+As one can see for all prior architectures except one, the
+anti-exploration bonus during actor training is much higher
+than it should be according to the values on the dataset
+actions. These results confirm our hypothesis. Furthermore,
+we can note from Figure 3c that the actor cannot clone the
+behavioral policy, since the distance to the dataset actions
+can even increase during training.
+However, RND with the FiLM prior architecture allows the
+actor to effectively minimize the anti-exploration bonus and
+successfully clone the behavioral policy. This suggests that,
+with the right inductive bias for the prior, we can solve the
+problems of naive RND and possibly achieve better results.
+Table 1. Comparison of different RND predictors. Prior uses FiLM
+conditioning. Predictor uses conditioning in the first layer. All
+scores are averaged over 3 random seeds. Halfcheetah tasks are
+ommited, as we found them non-representative of the final perfor-
+mance on harder tasks.
+Task Name
+Concat
+Gating
+Bilinear
+FiLM
+hopper-medium-v2
+94.8
+39.7
+98.4
+86.3
+hopper-medium-expert-v2
+71.5
+59.3
+110.3
+102.7
+hopper-medium-replay-v2
+100.3
+51.3
+100.8
+100.3
+walker2d-medium-v2
+94.8
+82.3
+92.8
+95.1
+walker2d-medium-expert-v2
+86.1
+84.2
+108.9
+110.0
+walker2d-medium-replay-v2
+90.3
+87.5
+88.3
+75.7
+Average
+89.6
+67.3
+99.9
+95.0
+5. Anti-Exploration by Random Network
+Distillation
+We are now ready to present SAC-RND: a new offline RL
+method for continuous action spaces, based on our findings
+in Section 3 and Section 4. It is simple, ensemble-free and
+achieves state-of-the-art results comparable to ensemble-
+based methods.
+We have chosen the Soft Actor-Critic
+(Haarnoja et al., 2018) algorithm as the backbone of the
+method. In this section, we will explain how the RND is
+trained and how we define the anti-exploration bonus.
+Random Network Distillation. We pretrain RND with
+MSE loss between prior and predictor embeddings, stop-
+ping gradient through prior and freezing both networks after-
+wards during SAC training. We keep both networks similar
+in size to the agent and critic, which are 4 layer MLPs. Con-
+trary to Burda et al. (2018); Ciosek et al. (2019), we do not
+add additional layers to the predictor to prevent undesirable
+results. This is because, when the predictor size is bigger
+than prior on state-based tasks (not image-based as in orig-
+inal work by Burda et al. (2018)), we observe that it can
+sometimes overgeneralize to OOD prior embeddings.
+According to Section 4, for the prior, we use FiLM condi-
+tioning on penultimate layer before nonlinearity. In prin-
+ciple, the predictor can be arbitrary (Ciosek et al., 2019),
+but in practice, its architecture and conditioning type can
+also affect performance. We conduct a preliminary study
+on a small subset of the D4RL Gym tasks to select the best-
+performing conditioning. Based on the results in Table 1,
+we chose a predictor with bilinear conditioning in the first
+layer, as it showed the best performance.
+Anti-Exploration Bonus. We define the anti-exploration
+bonus similarly to RND loss as
+b(s, a) = ∥fψ(s, a) − ¯f ¯
+ψ(s, a)∥2
+2
+(3)
+and additionally divide it by RND loss running standard
+deviation (which is tracked during pretraining phase) to
+increase its scale uniformly among environments. Such
+
+Anti-Exploration by Random Network Distillation
+0
+50000
+100000
+150000
+200000
+250000
+300000
+350000
+training steps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+RND bonus
+Concat
+FiLM
+Bilinear
+Gated
+(a) RND bonus for dataset actions
+0
+20000
+40000
+60000
+80000
+100000
+training steps
+0
+1
+2
+3
+4
+5
+6
+7
+RND bonus
+Concat
+FiLM
+Bilinear
+Gated
+(b) RND bonus for actor actions
+0
+20000
+40000
+60000
+80000
+100000
+training steps
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+mean squared error
+Concat
+FiLM
+Bilinear
+Gating
+(c) Distance to dataset actions
+Figure 3. Effect of different state-action conditioning in the prior of RND on actor training. We use the halfcheetah, walker2d and hopper
+medium datasets, with 3 seeds each. For training procedure, see Algorithm 2 in the Appendix. (a) Anti-exploration bonus for in-dataset
+actions during RND pretraining. We additionally divide the bonus by the RND loss running standard deviation to increase its scale
+(see Section 5) so the anti-exploration bonus increases slightly over time as standard deviation decreases. However, this does not affect
+minimization by the actor and is needed to highlight the differences. (b) Anti-exploration bonus for actor actions during training. Ideally,
+it should converge to values close to the final values in (a). (c) Distance of actor actions to true in-dataset actions during training. Ideally,
+it should decrease, as actions closer to the behavioral policy have the lowest bonus by design.
+scaling simplifies hyperparameter search, shrinking the pos-
+sible range of useful α coefficients that control the level of
+conservatism during training.
+For detailed training procedure and full SAC losses, we
+refer to Algorithm 1 in the Appendix (differences with the
+original SAC algorithm are highlighted in blue).
+6. Experiments
+In this section, we present an empirical evaluation of our
+method using the D4RL benchmark on the Gym domain
+(Section 6.1) and the more challenging AntMaze domain
+(Section 6.2). Next, we provide additional analysis and
+visual insight into why FiLM conditioning in the prior might
+be beneficial (Section 6.3). Finally, we present an ablation
+that compares more variations of conditioning for predictor
+and prior (Section 6.4). For each experiment, we also list the
+exact hyperparameters in Appendix D and implementation
+details in Appendix C. Additionally, we analyse sensitivity
+to hyperparameters in Appendix E.
+6.1. Evaluation on the Gym Domain
+Setup. We evaluate our method on all available datasets for
+the HalfCheetah, Walker2d and Hopper tasks in the Gym do-
+main of the D4RL benchmark. For ensemble-free baselines,
+we chose CQL (Kumar et al., 2020), IQL (Kostrikov et al.,
+2021), TD3+BC (Fujimoto & Gu, 2021), which show good
+results and are widely used in practice. For ensemble-based
+baselines, we chose SAC-N & EDAC (An et al., 2021) and
+the more recent RORL (Yang et al., 2022), which currently
+achieve state-of-the-art scores in this domain. We follow the
+An et al. (2021) and train for 3M gradient steps, evaluating
+on 10 episodes.
+Results. The resulting scores are presented in Table 2. We
+see that SAC-RND stands out from the ensemble-free meth-
+ods and outperforms them by a wide margin, achieving a
+mean score comparable to EDAC and only slightly behind
+RORL. Note that we do not use ensembles, whereas SAC-N
+can require up to 500 critics, EDAC up to 50 and RORL up
+to 20. In addition, we compare our proposed changes with
+the naive predictor and prior, confirming that our modifi-
+cations are essential for achieving good performance (see
+Figure 1).
+6.2. Evaluation on the AntMaze Domain
+Setup. We evaluate our method on all datasets available for
+the AntMaze domain of the D4RL benchmark. Ensemble-
+free baselines are the same as in Section 6.1. For ensemble-
+based baselines, we chose RORL (Yang et al., 2022) and
+MSG (Ghasemipour et al., 2022), the latter of which, to
+our knowledge, currently has the best mean score for this
+domain. We do not include SAC-N and EDAC, as there are
+no public results for them on this domain, and we were also
+unable to obtain a non-zero result. We follow the An et al.
+(2021) and train for 3M gradient steps, evaluating on 100
+episodes.
+Results. The resulting scores are presented in Table 3.
+Kostrikov et al. (2021) has shown that many offline RL
+methods that perform well on the Gym domain fail on the
+AntMaze domain. It can be seen that, on the AntMaze do-
+main, SAC-RND shows good results that are on par with
+ensembles, and outperforms ensemble-free methods. This
+also shows that our choice of predictor and prior generalises
+well to new domains. Note that, in addition to ensembles,
+both MSG and RORL require pre-training or supervision
+with behavioural cloning in order to achieve reported results,
+
+Anti-Exploration by Random Network Distillation
+Table 2. SAC-RND evaluation on the Gym domain. We report the final normalized score averaged over 4 random seeds on v2 datasets.
+TD3 + BC and IQL scores are taken from Lyu et al. (2022). CQL, SAC-N and EDAC scores are taken from An et al. (2021). RORL
+scores are taken from Yang et al. (2022).
+Ensemble-free
+Ensemble-based
+Task Name
+TD3+BC
+IQL
+CQL
+SAC-N
+EDAC
+RORL
+SAC-RND
+halfcheetah-random
+11.0 ± 1.1
+13.1 ± 1.3
+31.1 ± 3.5
+28.0 ± 0.9
+28.4 ± 1.0
+28.5 ± 0.8
+29.0 ± 1.5
+halfcheetah-medium
+48.3 ± 0.3
+47.4 ± 0.2
+46.9 ± 0.4
+67.5 ± 1.2
+65.9 ± 0.6
+66.8 ± 0.7
+66.6 ± 1.6
+halfcheetah-expert
+96.7 ± 1.1
+95.0 ± 0.5
+97.3 ± 1.1
+105.2 ± 2.6
+106.8 ± 3.4
+105.2 ± 0.7
+105.8 ± 1.9
+halfcheetah-medium-expert
+90.7 ± 4.3
+86.7 ± 5.3
+95.0 ± 1.4
+107.1 ± 2.0
+106.3 ± 1.9
+107.8 ± 1.1
+107.6 ± 2.8
+halfcheetah-medium-replay
+44.6 ± 0.5
+44.2 ± 1.2
+45.3 ± 0.3
+63.9 ± 0.8
+61.3 ± 1.9
+61.9 ± 1.5
+54.9 ± 0.6
+halfcheetah-full-replay
+-
+-
+76.9 ± 0.9
+84.5 ± 1.2
+84.6 ± 0.9
+-
+82.7 ± 0.9
+hopper-random
+8.5 ± 0.6
+7.9 ± 0.2
+5.3 ± 0.6
+31.3 ± 0.0
+25.3 ± 10.4
+31.4 ± 0.1
+31.3 ± 0.1
+hopper-medium
+59.3 ± 4.2
+66.2 ± 5.7
+61.9 ± 6.4
+100.3 ± 0.3
+101.6 ± 0.6
+104.8 ± 0.1
+97.8 ± 2.3
+hopper-expert
+107.8 ± 7.0
+109.4 ± 0.5
+106.5 ± 9.1
+110.3 ± 0.3
+110.1 ± 0.1
+112.8 ± 0.2
+109.7 ± 0.3
+hopper-medium-expert
+98.0 ± 9.4
+91.5 ± 14.3
+96.9 ± 15.1
+110.1 ± 0.3
+110.7 ± 0.1
+112.7 ± 0.2
+109.8 ± 0.6
+hopper-medium-replay
+60.9 ± 18.8
+94.7 ± 8.6
+86.3 ± 7.3
+101.8 ± 0.5
+101.0 ± 0.5
+102.8 ± 0.5
+100.5 ± 1.0
+hopper-full-replay
+-
+-
+101.9 ± 0.6
+102.9 ± 0.3
+105.4 ± 0.7
+-
+107.3 ± 0.1
+walker2d-random
+1.6 ± 1.7
+5.4 ± 1.2
+5.1 ± 1.7
+21.7 ± 0.0
+16.6 ± 7.0
+21.4 ± 0.2
+21.5 ± 0.1
+walker2d-medium
+83.7 ± 2.1
+78.3 ± 8.7
+79.5 ± 3.2
+87.9 ± 0.2
+92.5 ± 0.8
+102.4 ± 1.4
+91.6 ± 2.8
+walker2d-expert
+110.2 ± 0.3
+109.9 ± 1.2
+109.3 ± 0.1
+107.4 ± 2.4
+115.1 ± 1.9
+115.4 ± 0.5
+114.3 ± 0.6
+walker2d-medium-expert
+110.1 ± 0.5
+109.6 ± 1.0
+109.1 ± 0.2
+116.7 ± 0.4
+114.7 ± 0.9
+121.2 ± 1.5
+105.0 ± 7.9
+walker2d-medium-replay
+81.8 ± 5.5
+73.8 ± 7.1
+76.8 ± 10.0
+78.7 ± 0.7
+87.1 ± 2.4
+90.4 ± 0.5
+88.7 ± 7.7
+walker2d-full-replay
+-
+-
+94.2 ± 1.9
+94.6 ± 0.5
+99.8 ± 0.7
+-
+109.2 ± 1.8
+Average
+67.5
+68.9
+73.6
+84.4
+85.2
+85.7
+85.2
+while our method does not require any additional modifica-
+tions.
+Table 3. SAC-RND evaluation on AntMaze domain. We report
+the final normalized score averaged over 4 random seeds on v1
+datasets. IQL, CQL, MSG scores are taken from Ghasemipour
+et al. (2022). TD3+BC, RORL scores are taken from Yang et al.
+(2022).
+Ensemble-free
+Ensemble-based
+Task Name
+TD3+BC
+IQL
+CQL
+RORL
+MSG
+SAC-RND
+antmaze-umaze
+78.6
+87.5
+74.0
+97.7 ± 1.9
+97.8 ± 1.2
+97.2 ± 1.2
+antmaze-umaze-diverse
+71.4
+62.2
+84.0
+90.7 ± 2.9
+81.8 ± 3.0
+83.5 ± 7.7
+antmaze-medium-play
+10.6
+71.2
+61.2
+76.3 ± 2.5
+89.6 ± 2.2
+65.5 ± 35.7
+antmaze-medium-diverse
+3.0
+70.0
+53.7
+69.3 ± 3.3
+88.6 ± 2.6
+88.5 ± 9.2
+antmaze-large-play
+0.2
+39.6
+15.8
+16.3 ± 11.1
+72.6 ± 7.0
+67.2 ± 6.1
+antmaze-large-diverse
+0.0
+47.5
+14.9
+41.0 ± 10.7
+71.4 ± 12.2
+57.6 ± 22.7
+Average
+27.3
+63.0
+50.6
+65.2
+83.6
+76.6
+6.3. Why is FiLM Conditioning Beneficial for Bonus
+Minimization?
+In Section 4, we showed that FiLM conditioning in the RND
+prior significantly improved the actors’ ability to minimize
+the anti-exploration bonus. Since the issue occurred during
+actor training, we hypothesize that this may be related to
+the anti-exploration bonus optimization landscape. In this
+section, we analyze the anti-gradient fields for conditioning
+with concatenation or FiLM for the prior network.
+For the purpose of analysis, we design a toy dataset with
+only four categorical states and two-dimensional actions
+sampled uniformly in each corner of the grid (see Ap-
+pendix B for dataset visualization and generation details).
+We fix the hyperparameters and pretrain two RNDs that
+differ only in the type of prior conditioning. The predictor
+uses simple concatenation. Next, in Figure 4, we plot the
+two-dimensional anti-gradient field for the anti-exploration
+bonus conditioned on each state. The effect of FiLM be-
+comes more apparent in these plots. While the resulting
+anti-gradients for concatenation are noisy and only point
+in the direction of the minimum in a small neighbourhood,
+the directions for FiLM are smooth over the entire available
+action space and point to the correct global minimum for
+each state. While we cannot draw general conclusions from
+such a demonstration, based on the results of Section 4,
+we hypothesize that a similar phenomenon might exist in
+high-dimensional problems as well.
+6.4. Exploring More Conditioning Pairs
+One might wonder (1) how different types of conditioning
+for predictor and prior interact with each other and (2) where
+to introduce conditioning in terms of depth for it to be most
+beneficial.
+To answer these questions, we return to the experiment from
+Section 4 and generate more variations for each type (where
+it is possible): conditioning on the first layer, on the last
+layer, and on all layers. We also look at two variations
+of the bilinear layer: full, as presented in Jayakumar et al.
+(2020), and simplified, which is used by default in PyTorch.
+In Figure 5 we plot the final MSE between the resulting
+policy and the behavioural one on the training data. Two
+interesting observations can be made from these findings.
+
+Anti-Exploration by Random Network Distillation
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.0025
+0.0050
+0.0075
+0.0100
+0.0125
+0.0150
+0.0175
+0.0200
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.005
+0.010
+0.015
+0.020
+0.025
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.005
+0.010
+0.015
+0.020
+0.025
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.1
+0.2
+0.3
+0.4
+(a) State 0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+(b) State 1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+(c) State 2
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+(d) State 3
+Figure 4. Actions’ anti-gradient field for the anti-exploration bonus conditioned on four categorical states at each corner for the toy problem
+introduced in Section 6.3. We visualize the dataset in Figure 7 in the appendix. The top row corresponds to RND with concatenation
+conditioning in the prior, while the bottom row corresponds to FiLM conditioning. As can be seen, the resulting anti-gradients for
+concatenation are noisy, while the directions for FiLM are smooth over the entire available action space.
+First, FiLM may not be the only architecture with the right
+inductive biases for the prior, and both bilinear types with
+conditioning on all layers can also achieve similar results.
+However, compared to FiLM, inner bilinear layers are much
+more computationally expensive, as they involve at least
+one 3D weight tensor and two additional 2D weight tensors,
+and the hidden dimensions are usually much higher than the
+input dimensions.
+Second, it appears that conditioning on the last layer is most
+beneficial for the predictor, while conditioning on all layers
+is beneficial for the prior. In spite of that, it is difficult to
+draw broad conclusions, as different types may work well
+for new problems and domains.
+7. Related Work
+Model-free offline RL. Most offline RL approaches focus
+on the distribution shift problem and overestimation bias
+of Q-values for OOD actions. Some researchers address
+this by imposing strict constraints for policy updates, pe-
+nalizing the divergence from the behavioral policy with KL
+divergence, maximum mean discrepancy (MMD) distance
+(Kumar et al., 2019; Wu et al., 2019), simple mean squared
+error (MSE) (Fujimoto & Gu, 2021), or by re-weighting
+behavioral policy actions with the estimated advantages
+(Nair et al., 2020). Others directly regularize Q-values
+by lowering return estimates for OOD actions, preventing
+gated
+concat_first
+concat_last
+concat_full
+bilinear_first
+bilinear_last
+bilinear_full
+torch_bilinear_first
+torch_bilinear_last
+torch_bilinear_full
+film_full
+film_first
+film_last
+prior
+gated
+concat_first
+concat_last
+concat_full
+bilinear_first
+bilinear_last
+bilinear_full
+torch_bilinear_first
+torch_bilinear_last
+torch_bilinear_full
+film_full
+film_first
+film_last
+predictor
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Figure 5. Mean squared error between actions of the actor trained
+with different conditioning for the predictor & prior and actions
+of the behavioral policy. We use the halfcheetah, walker2d and
+hopper medium datasets, with 3 seeds each. It can be seen that
+conditioning on each layer is beneficial for the priors, while for the
+predictors, it is better to condition on the last layer. Note that this
+experiment is in the setting of Section 4, that is, without a critic.
+overestimation for unseen actions. For instance, Kumar
+et al. (2020), Ghasemipour et al. (2022) and Rezaeifar et al.
+(2022) explicitly introduce an optimization term that lowers
+Q-values for OOD actions, while An et al. (2021) penalizes
+implicitly by utilizing the lower-confidence bound (LCB)
+of Q-values. Alternatively, the evaluation of OOD actions
+can be avoided altogether by using the upper expectile value
+
+Anti-Exploration by Random Network Distillation
+function (Kostrikov et al., 2021) or by policy optimization
+within a latent action space (Chen et al., 2022; Zhou et al.,
+2021; Akimov et al., 2022).
+In our work, we follow the anti-exploration approach
+(Rezaeifar et al., 2022). In contrast to An et al. (2021);
+Ghasemipour et al. (2022); Yang et al. (2022), we com-
+pletely eliminate ensembles for uncertainty estimation, thus
+reducing computational overhead without sacrificing perfor-
+mance. Moreover, unlike Rezaeifar et al. (2022), we have
+succeeded in using an RND for novelty detection in offline
+RL for continuous action spaces.
+Estimation bias in Q-learning. In both offline and on-
+line reinforcement learning, off-policy Q-learning methods
+suffer from an overestimation bias in the temporal differ-
+ence learning target (Van Hasselt et al., 2016; Fujimoto
+et al., 2018). This phenomenon is orthogonal to overes-
+timation due to unseen actions in offline RL, as it occurs
+even in the presence of strong conservatism constraints. It
+is mainly caused by target prediction errors for seen transi-
+tions and their propagation due to the maximum operation
+maxa′∈AQ(s′, a′). To address this problem, Fujimoto et al.
+(2018) introduced clipped double Q learning (Van Hasselt
+et al., 2016) in TD3, which uses a minimum of two critics.
+This approach was later used by Haarnoja et al. (2018) in
+SAC to improve stability and accelerate convergence.
+In our work, we use clipped double Q-learning (Fujimoto
+et al., 2018), since SAC-RND is based on SAC (Haarnoja
+et al., 2018), and found it beneficial for stability. However,
+to ensure that it does not introduce additional conservatism,
+which can be a confounding factor for the impact of RND,
+we always set the number of critics to two.
+Uncertainty estimation in offline RL. Uncertainty estima-
+tion is a popular technique in reinforcement learning and is
+used for a variety of purposes such as exploration, planning,
+and robustness. In offline RL, its use is mostly limited to
+modeling epistemic uncertainty (Clements et al., 2019), in-
+cluding measuring the prediction confidence of dynamics
+models (Yu et al., 2020; Kidambi et al., 2020) or critics (An
+et al., 2021; Rezaeifar et al., 2022). This approach can be
+further used to induce uncertainty-aware penalization during
+training.
+Alternatively, uncertainty can help overcome suboptimal
+conservatism by designing more flexible offline approaches,
+e.g., conditioning on different levels of confidence to dy-
+namically adjust the level of conservatism during evaluation
+(Hong et al., 2022) or using Bayesian perspective to design
+an optimal adaptive offline RL policy (Ghosh et al., 2022).
+In our work, we estimate epistemic uncertainty and use it as
+an anti-exploration bonus to induce conservatism. Unlike
+previous approaches, we do not use ensembles to estimate
+epistemic uncertainty.
+Efficient ensembles Ensembles are a powerful and sim-
+ple non-Bayesian baseline for uncertainty estimation that
+outperform Bayesian neural networks in practice (Lakshmi-
+narayanan et al., 2017). However, training deep ensembles
+can be both memory intensive and computationally demand-
+ing, making the design of efficient ensembles an attractive
+research direction for which numerous methods have been
+developed. For example, Gal & Ghahramani (2016) pro-
+posed to use dropout to approximate Bayesian inference in
+deep Gaussian processes, and Durasov et al. (2021) derived
+a method to interpolate between dropout and full ensembles
+with fixed masks and controllable overlap between them.
+Meanwhile, Wen et al. (2020) significantly reduced the cost
+by defining each weight matrix as the Hadamard product
+of a shared weight among all ensemble members and a
+rank-one matrix per member.
+Recently, Ghasemipour et al. (2022) showed that, in offline
+RL, none of the most popular approaches for efficient en-
+sembles are capable of delivering performance that is com-
+parable to naive ensembles, and that more work is needed in
+this research direction. In our work, we chose an alternative
+path for uncertainty estimation with RND, which was shown
+to a fast and competitive counterpart to ensembles (Ciosek
+et al., 2019).
+8. Conclusion
+In this work, we revisited the results from previous research
+(Rezaeifar et al., 2022), showing that with a naive choice
+of conditioning for the RND prior, it becomes infeasible
+for the actor to effectively minimize the anti-exploration
+bonus and discriminativity is not an issue. To solve this,
+we proposed conditioning based on FiLM, which led us
+to a new ensemble-free method called SAC-RND. We em-
+pirically validated that it achieves results comparable to
+ensemble-based methods and outperforms its ensemble-free
+counterparts. As such, we believe that our work is a valuable
+contribution to anti-exploration and uncertainty estimation
+in offline RL.
+References
+Akimov, D., Kurenkov, V., Nikulin, A., Tarasov, D., and
+Kolesnikov, S. Let offline rl flow: Training conserva-
+tive agents in the latent space of normalizing flows. In
+3rd Offline RL Workshop: Offline RL as a”Launchpad”,
+2022.
+An, G., Moon, S., Kim, J.-H., and Song, H. O. Uncertainty-
+based offline reinforcement learning with diversified q-
+ensemble. Advances in neural information processing
+systems, 34:7436–7447, 2021.
+Ba, J. L., Kiros, J. R., and Hinton, G. E. Layer normalization.
+
+Anti-Exploration by Random Network Distillation
+arXiv preprint arXiv:1607.06450, 2016.
+Badia, A. P., Piot, B., Kapturowski, S., Sprechmann, P.,
+Vitvitskyi, A., Guo, Z. D., and Blundell, C. Agent57:
+Outperforming the atari human benchmark. In Interna-
+tional Conference on Machine Learning, pp. 507–517.
+PMLR, 2020.
+Baker, B., Akkaya, I., Zhokhov, P., Huizinga, J., Tang, J.,
+Ecoffet, A., Houghton, B., Sampedro, R., and Clune, J.
+Video pretraining (vpt): Learning to act by watching un-
+labeled online videos. arXiv preprint arXiv:2206.11795,
+2022.
+Berner, C., Brockman, G., Chan, B., Cheung, V., D˛ebiak, P.,
+Dennison, C., Farhi, D., Fischer, Q., Hashme, S., Hesse,
+C., et al. Dota 2 with large scale deep reinforcement
+learning. arXiv preprint arXiv:1912.06680, 2019.
+Bradbury, J., Frostig, R., Hawkins, P., Johnson, M. J., Leary,
+C., Maclaurin, D., Necula, G., Paszke, A., VanderPlas, J.,
+Wanderman-Milne, S., and Zhang, Q. JAX: composable
+transformations of Python+NumPy programs, 2018. URL
+http://github.com/google/jax.
+Burda, Y., Edwards, H., Storkey, A., and Klimov, O. Ex-
+ploration by random network distillation. arXiv preprint
+arXiv:1810.12894, 2018.
+Chen, X., Ghadirzadeh, A., Yu, T., Gao, Y., Wang, J., Li,
+W., Liang, B., Finn, C., and Zhang, C. Latent-variable
+advantage-weighted policy optimization for offline rl.
+arXiv preprint arXiv:2203.08949, 2022.
+Ciosek, K., Fortuin, V., Tomioka, R., Hofmann, K., and
+Turner, R. Conservative uncertainty estimation by fitting
+prior networks. In International Conference on Learning
+Representations, 2019.
+Clements, W. R., Van Delft, B., Robaglia, B.-M., Slaoui,
+R. B., and Toth, S. Estimating risk and uncertainty in deep
+reinforcement learning. arXiv preprint arXiv:1905.09638,
+2019.
+Dumoulin, V., Perez, E., Schucher, N., Strub, F., Vries,
+H. d., Courville, A., and Bengio, Y. Feature-wise trans-
+formations. Distill, 2018. doi: 10.23915/distill.00011.
+https://distill.pub/2018/feature-wise-transformations.
+Durasov, N., Bagautdinov, T., Baque, P., and Fua, P.
+Masksembles for uncertainty estimation. In Proceed-
+ings of the IEEE/CVF Conference on Computer Vision
+and Pattern Recognition, pp. 13539–13548, 2021.
+Fu, J., Kumar, A., Nachum, O., Tucker, G., and Levine,
+S. D4rl: Datasets for deep data-driven reinforcement
+learning. arXiv preprint arXiv:2004.07219, 2020.
+Fujimoto, S. and Gu, S. S. A minimalist approach to offline
+reinforcement learning. Advances in neural information
+processing systems, 34:20132–20145, 2021.
+Fujimoto, S., Hoof, H., and Meger, D. Addressing function
+approximation error in actor-critic methods. In Interna-
+tional conference on machine learning, pp. 1587–1596.
+PMLR, 2018.
+Fujimoto, S., Meger, D., and Precup, D. Off-policy deep
+reinforcement learning without exploration. In Interna-
+tional conference on machine learning, pp. 2052–2062.
+PMLR, 2019.
+Gal, Y. and Ghahramani, Z. Dropout as a bayesian approx-
+imation: Representing model uncertainty in deep learn-
+ing. In international conference on machine learning, pp.
+1050–1059. PMLR, 2016.
+Ghasemipour, S. K. S., Gu, S. S., and Nachum, O. Why so
+pessimistic? estimating uncertainties for offline rl through
+ensembles, and why their independence matters. arXiv
+preprint arXiv:2205.13703, 2022.
+Ghosh, D., Ajay, A., Agrawal, P., and Levine, S. Offline
+rl policies should be trained to be adaptive. In Interna-
+tional Conference on Machine Learning, pp. 7513–7530.
+PMLR, 2022.
+Haarnoja, T., Zhou, A., Abbeel, P., and Levine, S. Soft
+actor-critic: Off-policy maximum entropy deep reinforce-
+ment learning with a stochastic actor. In International
+conference on machine learning, pp. 1861–1870. PMLR,
+2018.
+Hong, J., Kumar, A., and Levine, S. Confidence-conditioned
+value functions for offline reinforcement learning. arXiv
+preprint arXiv:2212.04607, 2022.
+Jarrett, D., Tallec, C., Altché, F., Mesnard, T., Munos, R.,
+and Valko, M. Curiosity in hindsight. arXiv preprint
+arXiv:2211.10515, 2022.
+Jayakumar, S. M., Czarnecki, W. M., Menick, J., Schwarz,
+J., Rae, J., Osindero, S., Teh, Y. W., Harley, T., and
+Pascanu, R. Multiplicative interactions and where to find
+them. 2020.
+Kidambi, R., Rajeswaran, A., Netrapalli, P., and Joachims,
+T. Morel: Model-based offline reinforcement learning.
+Advances in neural information processing systems, 33:
+21810–21823, 2020.
+Kingma, D. P. and Ba, J. Adam: A method for stochastic
+optimization. arXiv preprint arXiv:1412.6980, 2014.
+Kostrikov, I., Nair, A., and Levine, S. Offline reinforce-
+ment learning with implicit q-learning. arXiv preprint
+arXiv:2110.06169, 2021.
+
+Anti-Exploration by Random Network Distillation
+Kumar, A., Fu, J., Soh, M., Tucker, G., and Levine, S.
+Stabilizing off-policy q-learning via bootstrapping error
+reduction. Advances in Neural Information Processing
+Systems, 32, 2019.
+Kumar, A., Zhou, A., Tucker, G., and Levine, S. Con-
+servative q-learning for offline reinforcement learning.
+Advances in Neural Information Processing Systems, 33:
+1179–1191, 2020.
+Kumar, A., Agarwal, R., Geng, X., Tucker, G., and Levine,
+S. Offline q-learning on diverse multi-task data both
+scales and generalizes. arXiv preprint arXiv:2211.15144,
+2022.
+Kurenkov, V. and Kolesnikov, S. Showing your offline
+reinforcement learning work: Online evaluation budget
+matters. In International Conference on Machine Learn-
+ing, pp. 11729–11752. PMLR, 2022.
+Lakshminarayanan, B., Pritzel, A., and Blundell, C. Simple
+and scalable predictive uncertainty estimation using deep
+ensembles. Advances in neural information processing
+systems, 30, 2017.
+Lee, K.-H., Nachum, O., Yang, M., Lee, L., Freeman,
+D., Xu, W., Guadarrama, S., Fischer, I., Jang, E.,
+Michalewski, H., et al. Multi-game decision transformers.
+arXiv preprint arXiv:2205.15241, 2022.
+Levine, S., Kumar, A., Tucker, G., and Fu, J. Offline rein-
+forcement learning: Tutorial, review, and perspectives on
+open problems. arXiv preprint arXiv:2005.01643, 2020.
+Lillicrap, T. P., Hunt, J. J., Pritzel, A., Heess, N., Erez,
+T., Tassa, Y., Silver, D., and Wierstra, D. Continuous
+control with deep reinforcement learning. arXiv preprint
+arXiv:1509.02971, 2015.
+Lyu, J., Ma, X., Li, X., and Lu, Z. Mildly conservative q-
+learning for offline reinforcement learning. arXiv preprint
+arXiv:2206.04745, 2022.
+Nair, A., Gupta, A., Dalal, M., and Levine, S. Awac: Accel-
+erating online reinforcement learning with offline datasets.
+arXiv preprint arXiv:2006.09359, 2020.
+Nikulin, A., Kurenkov, V., Tarasov, D., Akimov, D., and
+Kolesnikov, S. Q-ensemble for offline rl: Don’t scale
+the ensemble, scale the batch size.
+arXiv preprint
+arXiv:2211.11092, 2022.
+Perez, E., Strub, F., De Vries, H., Dumoulin, V., and
+Courville, A. Film: Visual reasoning with a general con-
+ditioning layer. In Proceedings of the AAAI Conference
+on Artificial Intelligence, volume 32, 2018.
+Reed, S., Zolna, K., Parisotto, E., Colmenarejo, S. G.,
+Novikov, A., Barth-Maron, G., Gimenez, M., Sulsky,
+Y., Kay, J., Springenberg, J. T., et al. A generalist agent.
+arXiv preprint arXiv:2205.06175, 2022.
+Rezaeifar, S., Dadashi, R., Vieillard, N., Hussenot, L.,
+Bachem, O., Pietquin, O., and Geist, M. Offline rein-
+forcement learning as anti-exploration. In Proceedings
+of the AAAI Conference on Artificial Intelligence, vol-
+ume 36, pp. 8106–8114, 2022.
+Schrittwieser, J., Antonoglou, I., Hubert, T., Simonyan, K.,
+Sifre, L., Schmitt, S., Guez, A., Lockhart, E., Hassabis,
+D., Graepel, T., et al. Mastering atari, go, chess and shogi
+by planning with a learned model. Nature, 588(7839):
+604–609, 2020.
+Smith, L., Kostrikov, I., and Levine, S.
+A walk in the
+park: Learning to walk in 20 minutes with model-free
+reinforcement learning. arXiv preprint arXiv:2208.07860,
+2022.
+Srivastava, R. K., Shyam, P., Mutz, F., Ja´skowski, W., and
+Schmidhuber, J. Training agents using upside-down re-
+inforcement learning. arXiv preprint arXiv:1912.02877,
+2019.
+Tarasov, D., Nikulin, A., Akimov, D., Kurenkov, V., and
+Kolesnikov, S. CORL: Research-oriented deep offline
+reinforcement learning library. In 3rd Offline RL Work-
+shop: Offline RL as a ”Launchpad”, 2022. URL https:
+//openreview.net/forum?id=SyAS49bBcv.
+Van Hasselt, H., Guez, A., and Silver, D. Deep reinforce-
+ment learning with double q-learning. In Proceedings of
+the AAAI conference on artificial intelligence, volume 30,
+2016.
+Wen, Y., Tran, D., and Ba, J. Batchensemble: an alterna-
+tive approach to efficient ensemble and lifelong learning.
+arXiv preprint arXiv:2002.06715, 2020.
+Wu, Y., Tucker, G., and Nachum, O.
+Behavior regu-
+larized offline reinforcement learning. arXiv preprint
+arXiv:1911.11361, 2019.
+Yang, R., Bai, C., Ma, X., Wang, Z., Zhang, C., and Han,
+L. Rorl: Robust offline reinforcement learning via con-
+servative smoothing. arXiv preprint arXiv:2206.02829,
+2022.
+Yu, T., Thomas, G., Yu, L., Ermon, S., Zou, J. Y., Levine, S.,
+Finn, C., and Ma, T. Mopo: Model-based offline policy
+optimization. Advances in Neural Information Processing
+Systems, 33:14129–14142, 2020.
+Zhou, W., Bajracharya, S., and Held, D. Plas: Latent action
+space for offline reinforcement learning. In Conference
+on Robot Learning, pp. 1719–1735. PMLR, 2021.
+
+Anti-Exploration by Random Network Distillation
+A. Previous Research Results
+Figure 6. Anti-exploration bonus on walker2d-medium dataset for RND and CVAE. Note that figure taken from Rezaeifar et al. (2022) for
+a convenient comparison with our results in Figure 2.
+B. Toy Dataset
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+State
+0
+1
+2
+3
+Figure 7. Toy dataset visualization introduced in Section 6.3. This toy dataset consists of four categorical states for each corner of the
+limited 2D actions grid. For each state, we uniformly sample 4096 two-dimensional actions within a limited square. We use one-hot
+encoding for the states during RND training.
+C. Implementation Details
+In our experiments, we use hyperparameters from Table 4 where possible and sweep over α to pick the best value for each
+dataset. We implement all of our models using the Jax (Bradbury et al., 2018) framework. For the exact implementation
+of conditioning variants for predictor and prior networks, refer to our code at https://github.com/tinkoff-ai/
+sac-rnd. Similarly to Nikulin et al. (2022); Kumar et al. (2022); Smith et al. (2022), we add Layer Normalization (Ba
+et al., 2016) to the critic after each layer as it greatly improves stability and convergence speed. For SAC-N in Section 4 we
+use the implementation from the CORL library (Tarasov et al., 2022). All experiments were performed on V100 and A100
+GPUs. With our implementation, each training for 3 million training steps usually takes ∼ 40 minutes to run (∼ 15 minutes
+for the typical 1 million steps).
+Gym Domain.
+We use the v2 version of each dataset.
+We follow the An et al. (2021) approach and run our
+algorithms for 3 million training steps and report the final normalized average score over 10 evaluation episodes.
+For the final experiments, we use 4 seeds, while using less for hyperparameter tuning.
+We tune the α co-
+efficient over the {1.0, 3.0, 4.0, 5.0, 8.0, 9.0, 10.0, 13.0, 15.0, 20.0, 25.0} range for the walker and hopper datasets.
+We found that the halfcheetah datasets require a lower level of conservatism, which is why we tune over the
+{0.001, 0.1, 0.3, 0.5, 0.8, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0} range for these datasets while keeping the same number of candidates.
+We follow the Ghasemipour et al. (2022) approach and choose the best α for each dataset (see Table 5).
+AntMaze Domain. We use the v1 version of each dataset due to the fact that the v0 version has major problems and
+bugs during generation (e.g., some trajectories have discontinuities where the agent teleports from one part of the maze to
+
+CVAE
+RND
+300000-
+400000
+Dataset actions
+Shuffled actions
+Random actions
+300000
+Dataset actions + Gaussian noise (std = 0.25)
+200000
+Dataset actions + Gaussian noise (std = O.5)
+200000
+100000
+100000
+0
+0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+1e-7Anti-Exploration by Random Network Distillation
+another 3). We follow the An et al. (2021) approach and run our algorithms for 3 million training steps and report the final
+normalized average score over 100 evaluation episodes. Same as Chen et al. (2022), we scale the reward by 100.0. We found
+that actor and critic require different levels of conservatism in these tasks, which is why we chose to decouple α and use
+separate values (the same approach was used in Rezaeifar et al. (2022)). We tune the α for the actor in the {0.5, 1.0, 1.5}
+range, and α for the critic in the {0.001, 0.01, 0.1} range. We follow the Ghasemipour et al. (2022) approach and choose
+the best α for each dataset (see Table 6).
+D. Hyperparameters
+Table 4. SAC-RND general hyperparameters.
+Parameter
+Value
+optimizer
+Adam (Kingma & Ba, 2014)
+batch size
+1024 (256 on antmaze-*)
+learning rate (all networks)
+1e-3 (3e-4 on antmaze-*)
+tau (τ)
+5e-3
+hidden dim (all networks)
+256
+num layers (all networks)
+4
+RND embedding dim (all tasks)
+32
+target entropy
+-action_dim
+gamma (γ)
+0.99 (0.999 on antmaze-*)
+nonlinearity
+ReLU
+Table 5. SAC-RND best hyperparameters used in D4RL Gym domain.
+Task Name
+α
+halfcheetah-random
+0.1
+halfcheetah-medium
+0.3
+halfcheetah-expert
+6.0
+halfcheetah-medium-expert
+0.1
+halfcheetah-medium-replay
+0.1
+halfcheetah-full-replay
+3.0
+hopper-random
+5.0
+hopper-medium
+25.0
+hopper-expert
+20.0
+hopper-medium-expert
+15.0
+hopper-medium-replay
+8.0
+hopper-full-replay
+3.0
+walker2d-random
+1.0
+walker2d-medium
+8.0
+walker2d-expert
+4.0
+walker2d-medium-expert
+25.0
+walker2d-medium-replay
+8.0
+walker2d-full-replay
+3.0
+Table 6. SAC-RND best hyperparameters used in D4RL AntMaze domain.
+Task Name
+α (actor)
+α (critic)
+antmaze-umaze
+1.0
+0.1
+antmaze-umaze-diverse
+1.0
+0.1
+antmaze-medium-play
+0.5
+0.001
+antmaze-medium-diverse
+1.0
+0.01
+antmaze-large-play
+1.0
+0.01
+antmaze-large-diverse
+0.5
+0.01
+3https://github.com/Farama-Foundation/D4RL/issues/77
+
+Anti-Exploration by Random Network Distillation
+E. Sensitivty to Hyperparameters
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Policies Evaluated Online
+30
+40
+50
+60
+70
+80
+90
+Expected Online Performance
+HalfCheetah
+Hopper
+Walker2D
+AntMaze
+Figure 8. Expected Online Performance (Kurenkov & Kolesnikov, 2022) under uniform offline policy selection. It can be seen, that for
+satisfactory results in all domains a budget of at least five policies for online evaluations is needed.
+F. Pseudocode
+Algorithm 1 Soft Actor-Critic with Random Network Distillation (SAC-RND)
+Initialize policy parameters θ, Double Q-function parameters {φ1, φ2}, RND predictor and prior parameters {ψ, ψ′}, and
+offline replay buffer D
+for desired number of pretraining steps do
+Sample a mini-batch B = {(s, a)} from D
+Update RND predictor weights ψ with gradient descent using
+∇ψ
+1
+|B|
+�
+s∈B
+�
+∥fψ(s, a) − ¯f ¯
+ψ(s, a)∥2
+2
+�
+end for
+for desired number of training steps do
+Sample a mini-batch B = {(s, a, r, s′)} from D
+Compute target Q-values (shared by all Q-functions):
+y(r, s′) = r + γ
+�
+min
+j=1,2 Q ¯φi(s′, a′) − β log πθ(a′|s′) − αb(s′, a′)
+�
+where a′ ∼ πθ(·|s′) and b(s′, a′) is an anti-exploration bonus defined by Eq. (3).
+Update each Q-function Qφi with gradient descent using
+∇φi
+1
+|B|
+�
+(s,a,r,s′)∈B
+�
+Qφi(s, a) − y(r, s′)
+�2
+Update policy with gradient ascent using
+∇θ
+1
+|B|
+�
+s∈B
+�
+min
+j=1,2 Qφi(s, ˜aθ(s)) − β log π(˜aθ(s)|s) − αb(s, ˜aθ(s))
+�
+where ˜aθ(s) is a sample from π(·|s) which is differentiable w.r.t. θ via the reparametrization trick.
+Update target networks with ¯φi ← (1 − ρ) ¯φi + ρφi
+end for
+
+Anti-Exploration by Random Network Distillation
+Algorithm 2 Simplified SAC-RND (without a critic) used in experiments for Section 4 and Section 6.4.
+Initialize policy parameters θ, RND predictor and prior parameters {ψ, ψ′}, and offline replay buffer D
+for desired number of pretraining steps do
+Sample a mini-batch B = {(s, a)} from D
+Update RND predictor weights ψ with gradient descent using
+∇ψ
+1
+|B|
+�
+s∈B
+�
+∥fψ(s, a) − ¯f ¯
+ψ(s, a)∥2
+2
+�
+end for
+for desired number of training steps do
+Sample a mini-batch B = {(s, a, r, s′)} from D
+Update policy with gradient descent using
+∇θ
+1
+|B|
+�
+s∈B
+�
+β log π(˜aθ(s)|s) + b(s, ˜aθ(s))
+�
+where ˜aθ(s) is a sample from π(·|s) which is differentiable w.r.t. θ via the reparametrization trick and b(s′, a′) is an
+anti-exploration bonus defined by Eq. (3).
+end for
+
diff --git a/9NFRT4oBgHgl3EQfqTc6/content/tmp_files/load_file.txt b/9NFRT4oBgHgl3EQfqTc6/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7b5bb5bee2966cc832e7c6c672e54d0082f95ed1
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@@ -0,0 +1,1605 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf,len=1604
+page_content='Anti-Exploration by Random Network Distillation  Alexander Nikulin 1 Vladislav Kurenkov 1 Denis Tarasov 1 Sergey Kolesnikov 1  Abstract  Despite the success of Random Network Distilla-  tion (RND) in various domains, it was shown as  not discriminative enough to be used as an uncer-  tainty estimator for penalizing out-of-distribution  actions in offline reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In this  paper, we revisit these results and show that, with  a naive choice of conditioning for the RND prior,  it becomes infeasible for the actor to effectively  minimize the anti-exploration bonus and discrim-  inativity is not an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We show that this lim-  itation can be avoided with conditioning based  on Feature-wise Linear Modulation (FiLM), re-  sulting in a simple and efficient ensemble-free  algorithm based on Soft Actor-Critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We eval-  uate it on the D4RL benchmark, showing that it  is capable of achieving performance comparable  to ensemble-based methods and outperforming  ensemble-free approaches by a wide margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' 1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Introduction  In recent years, significant success has been achieved in ap-  plying Reinforcement Learning (RL) to challenging and  large-scale tasks such as Atari (Badia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020), Go  (Schrittwieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020), Dota 2 (Berner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2019),  and Minecraft (Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' However, the online na-  ture of such RL algorithms makes it difficult to apply them  in the real world, where online collection of large amounts  of exploratory data may not be feasible for safety or fi-  nancial reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Offline Reinforcement Learning (Levine  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020) promises a more controllable and data-driven  approach, focusing on algorithms that can learn from a fixed,  pre-recorded dataset without requiring additional environ-  ment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  The use of ensembles for uncertainty-based penalization has  proven to be one of the most effective approaches for offline  RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Ensemble-based algorithms, such as SAC-N, EDAC  (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2021), and MSG (Ghasemipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022)  1Tinkoff, Moscow, Russia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Correspondence to: Alexander  Nikulin <a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='nikulin@tinkoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='ai>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  1Our implementation is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  com/tinkoff-ai/sac-rnd  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  training steps  1e6  0  20  40  60  80  100  average normalized score  CQL (ensemble-free)  SAC-N (ensemble-based)  SAC-RND (Naive)  SAC-RND (Ours)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Mean performance of SAC-RND variants on walker and  hopper medium-* datasets, each averaged over 3 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We plot  performance for the naive version, which uses concatenation con-  ditioning, and our final version, which is described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  We also plot the final scores for the ensemble-free CQL (Kumar  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020) and the ensemble-based SAC-N (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' It  can be seen that our version is a significant improvement over the  naive version, achieving performance comparable to ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  currently achieve state-of-the-art results on most D4RL (Fu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020) datasets, outperforming ensemble-free methods  by a wide margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Unfortunately, in order to achieve the best  performance, these algorithms may require tens or hundreds  of ensemble members, leading to significant computational  and memory overhead, as well as extended training duration  (Nikulin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Recent research (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022) has successfully reduced  the ensemble size to tens of Q-networks in the worst-case  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' However, given the general trend for model scal-  ing in offline RL (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Reed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Lee  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022), efficiently training even ten Q-networks with  80 million parameters each is not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Furthermore,  Ghasemipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022) showed that methods for efficient  ensemble training found in supervised learning literature  do not deliver performance comparable to naive ensembles  and can even worsen the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Thus, further research  on efficient uncertainty estimation for offline RL is needed,  with the goal of reducing the size of the ensemble as much  as possible or even fully removing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  In this work, we move away from ensembles and take an  alternative approach to uncertainty estimation, proposing an  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='13616v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='LG]  31 Jan 2023  Anti-Exploration by Random Network Distillation  efficient offline RL method with ensemble-free uncertainty  estimation via Random Network Distillation (RND) (Burda  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' RND, a simple and fast ensemble competitor  for epistemic uncertainty estimation (Ciosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2019),  is an attractive choice for offline RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' However, previous  research (Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022) found RND to be insuffi-  ciently discriminative for good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  In our preliminary experiment (Section 3), we show that  RND is discriminative enough to detect OOD actions, which  contradicts the previous study (Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Nev-  ertheless, our results show that the naive application of RND  does indeed not lead to good results (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Building  upon these findings, we further simplify the problem and  analyze the reasons for this issue (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We discover  that a naive choice of conditioning for the RND prior can  hinder the minimization of the anti-exploration bonus by  the actor, and that conditioning based on Feature-wise Lin-  ear Modulation (FiLM) (Perez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2018) is particularly  effective in solving this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Based on our findings, we propose a new ensemble-free of-  fline RL algorithm called SAC-RND (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We eval-  uate our method on the D4RL (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020) benchmark  (Section 6), and show that SAC-RND achieves performance  comparable to ensemble-based methods while outperform-  ing ensemble-free approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Background  Offline Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Reinforcement learning  problem can be described as a Markov Decision Process  (MDP) defined by the {S, A, P, R, γ} tuple with state space  S ⊂ RN, action space A ⊂ RM, transition dynamics P :  S × A → S, reward function R : S × A → R, and a  discount factor γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' The goal of reinforcement learning in  an infinite horizon setting is to produce a policy π(a|s)  that maximizes the expected cumulative discounted return  Eπ[�∞  t=0 γtr(st, at)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  In offline reinforcement learning, a policy must be learned  from a fixed dataset D collected under a different policy or  mixture of policies, without any environment interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  This setting poses unique fundamental challenges (Levine  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020), since the learning policy is unable to explore  and has to deal with distributional shift and extrapolation  errors (Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2019) for actions not represented in  the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Offline RL as Anti-Exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' There are numerous ap-  proaches for offline RL, a substantial part of which constrain  the learned policy to stay within the support of the train-  ing dataset, thus reducing (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020) or avoiding  (Kostrikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2021) extrapolation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For our work,  it is essential to understand how such a constraint can be  framed as anti-exploration (Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Similarly to online RL, where novelty bonuses are used as  additive intrinsic rewards for effective exploration, in offline  RL, novelty bonuses can induce conservatism, reducing the  reward in unseen state-action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Hence the name anti-  exploration, since the same approaches from exploration  can be used, but a bonus is subtracted from the extrinsic  reward instead of being added to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  However, unlike online RL, subtracting a bonus from the  raw reward would not be as useful, since the novelty bonus  is, by design, close to zero for in-dataset state-action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Therefore, it is more effective to apply it where the overesti-  mation for OOD actions emerges — the temporal difference  learning target:  r + γEa′∼π(·|s′)[Q(s′, a′) − b(s′, a′)]  (1)  where the actor is trained to maximize the expected Q-value,  as is usually done in off-policy actor-critic algorithms (Lil-  licrap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' It can be shown  that, theoretically, these approaches are equivalent, but the  latter is more suited for use in offline RL (Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  An illustrative example of how such framing can be effective  are ensemble-based approaches such as SAC-N & EDAC  (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2021) and MSG (Ghasemipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022),  which currently outperform their ensemble-free counterparts  by a large margin on most D4RL (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020) benchmark  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For the anti-exploration bonus, these methods use  ensemble disagreement as a proxy for epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  However, a large number of ensemble members is usually  required for a competitive result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Random Network Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Random network distilla-  tion (RND) was first proposed in online RL (Burda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  2018) as a simple and effective exploration bonus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' To this  day, RND is still considered a strong baseline for explo-  ration that can work well even in stochastic environments,  contrary to some more modern approaches (Jarrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  RND consists of two neural networks: a fixed and randomly  initialized prior network ¯f ¯  ψ, and a predictor network fψ  which learns to predict the prior outputs on the training data:  ∥fψ(s) − ¯f ¯  ψ(s)∥2  2  (2)  Both networks map states to embeddings in RK, and the  gradient through prior is disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' The interpretation of  the novelty is straightforward: with the sufficiently diverse  prior, the predictor must learn to match embeddings on data  points similar to the training dataset, while failing to predict  on new examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' A bonus in such a case may simply be a  prediction error, as in Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Anti-Exploration by Random Network Distillation  In a subsequent work, Ciosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2019) analyses the  success of RND in a supervised setting, and shows that  fitting random priors can be a competitive alternative to  ensembles for estimating epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Note that in practice, the choice of predictor and prior having  the same architecture and the estimation of novelty from  states only are very common, but arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Moreover, for  offline RL, we are interested in estimating the novelty of an  action conditioned on the state, which is why in our work  RND depends on both: fψ(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Multiplicative Interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' The most common way to  fuse two different streams of information is feature con-  catenation, which is straightforward but can be suboptimal  (Dumoulin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2020) shows that  multiplicative interactions provide a powerful inductive bias  for fusing or conditioning from multiple streams and are  superior in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We provide a brief review of those used  in our work (excluding concatenation): gating, bilinear, and  feature-wise linear modulation (FiLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Simple conditioning with two linear layers and  pointwise multiplication of the resulting features (Srivastava  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  f(a, s) = tanh(W1a + b1) ⊙ σ(W2s + b2)  Bilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Bilinear layer in its most general form, as pro-  posed by Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  f(a, s) = sT Wa + sT U + Va + b  where W is a 3D tensor, U, V are regular matrices and b  is a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' However, in our work, we also use the imple-  mentation as in PyTorch, which does not learn U, V by  default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  FiLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Special case of a bilinear layer with low-rank weight  matrices (Perez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  f(h, s) = γ(s) ⊙ h + β(s)  Usually, FiLM operates on hidden activations h before non-  linearity between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Thus, the main network takes a as  an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Random Network Distillation is  Discriminative Enough  To better understand the possible difficulties of applying  RND to offline RL, we first reproduce the main experiment  from Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022), which showed that RND is not  discriminative enough to be used as a novelty bonus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For  convenience, we provide the original figure from Rezaeifar  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022) in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We also compare RND with  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  standard deviation  0  20000  40000  Q-Ensemble  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  prediction error  RND  dataset  uniform  dataset + noise (std .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='25)  dataset + noise (std .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5)  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Anti-exploration bonus (Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022) on the  walker2d-medium dataset for trained SAC-N (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2021),  Q-ensemble (N = 25) and RND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Bonus is computed for state-  action pairs from the original dataset and different perturbations of  actions: random actions, dataset actions to which Gaussian noise  is added with different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Both RND networks use simple  state-action concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' The result is strikingly different from a  similar figure in the Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022) (we provide the original  figure in the Appendix A for convenience).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Contrary to previous  research, it can be seen that RND is capable of distinguishing ID  from OOD actions and is comparable to a trained Q-ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  a trained Q-ensemble (N = 25) from the SAC-N algorithm  (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Similarly to Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022), we  use simple state-action concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Predictor and prior  share the identical architecture of 4-layer MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  The goal of the experiment (see Figure 2) is to visually  plot the anti-exploration bonus for ID state-action pairs  and different perturbations of actions to model OOD data:  random actions sampled from a uniform distribution and  dataset actions to which Gaussian noise with different scales  is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  To our surprise, the result on Figure 2 is strikingly different  from previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' It shows that RND is able to discrim-  inate between ID and OOD actions with varying degrees  of distributional shift and is comparable to a trained Q-  ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In contrast, Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022) hypothesizes  that RND can only work well out of the box for discrete ac-  tion spaces and visual features, and concludes that extending  it to continuous action spaces is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  After further investigation of the open-sourced codebase2 in  search of discrepancies with our implementation, we found  that the only difference is that, contrary to the advice of  Ciosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2019), Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022) sets the predictor  smaller than prior by two layers during RND pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' It  is important to make the predictor larger or comparable in  capacity to the prior so that it can minimize the loss to zero  on the training dataset (Ciosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' However, the  actual RND hyperparameters used in the final publication  were not listed, so we cannot draw a definitive conclusion  about the reason for such different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='com/shidilrzf/Anti-exploration-RL  Anti-Exploration by Random Network Distillation  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Concatenation Prior Hinders Bonus  Minimization  A well-behaved anti-exploration bonus for continuous action  spaces, be it RND or any other, should satisfy at least two  criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' First, it should be discriminative enough to detect  novel actions and downweight their value estimates (see  Equation (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Ideally, the bonus should be close to zero for  ID data so that we do not bias the Q-function, as this can  be detrimental to training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Second, it should allow the actor  to easily minimize the bonus with gradient descent during  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  In Section 3, we showed that RND can detect OOD ac-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Nevertheless, naive use of RND as an anti-exploration  bonus on top of the Soft Actor Critic algorithm (Haarnoja  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2018) still does not provide satisfactory performance  (see Figure 1) with scores lower than CQL (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  2020) and SAC-N (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' This gives us an hint  that the problem may not be the discriminative power of  RND, but that the actor cannot effectively minimize the  anti-exploration bonus during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  To test our hypothesis that the actor cannot effectively min-  imize the anti-exploration bonus, we further simplify the  problem by removing the critic from the SAC algorithm  but keeping the entropy bonus (see Algorithm 2 in the Ap-  pendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We expect that, in such a setting, the actor will be  able to successfully minimize the anti-exploration bonus to  the possible minimum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' comparable to the bonus for the  ground truth data at the end of the RND pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' As a  consequence, since dataset actions provide the minimum  bonus by design, we also expect that the distance from the  agent to dataset actions should be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  We set predictor architecture to state-action concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Additionally, we explore different conditioning schemes for  the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We use the halfcheetah, walker2d and hopper  medium datasets, with 3 seeds each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Figure 3 compares  the anti-exploration bonus for dataset actions during RND  pretraining (see Figure 3a) and for agent actions during  training (see Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  As one can see for all prior architectures except one, the  anti-exploration bonus during actor training is much higher  than it should be according to the values on the dataset  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' These results confirm our hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Furthermore,  we can note from Figure 3c that the actor cannot clone the  behavioral policy, since the distance to the dataset actions  can even increase during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  However, RND with the FiLM prior architecture allows the  actor to effectively minimize the anti-exploration bonus and  successfully clone the behavioral policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' This suggests that,  with the right inductive bias for the prior, we can solve the  problems of naive RND and possibly achieve better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Comparison of different RND predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Prior uses FiLM  conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Predictor uses conditioning in the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' All  scores are averaged over 3 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Halfcheetah tasks are  ommited, as we found them non-representative of the final perfor-  mance on harder tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Task Name  Concat  Gating  Bilinear  FiLM  hopper-medium-v2  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  hopper-medium-expert-v2  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7  hopper-medium-replay-v2  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  walker2d-medium-v2  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  walker2d-medium-expert-v2  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9  110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  walker2d-medium-replay-v2  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7  Average  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Anti-Exploration by Random Network  Distillation  We are now ready to present SAC-RND: a new offline RL  method for continuous action spaces, based on our findings  in Section 3 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' It is simple, ensemble-free and  achieves state-of-the-art results comparable to ensemble-  based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  We have chosen the Soft Actor-Critic  (Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2018) algorithm as the backbone of the  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In this section, we will explain how the RND is  trained and how we define the anti-exploration bonus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Random Network Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We pretrain RND with  MSE loss between prior and predictor embeddings, stop-  ping gradient through prior and freezing both networks after-  wards during SAC training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We keep both networks similar  in size to the agent and critic, which are 4 layer MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Con-  trary to Burda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Ciosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2019), we do not  add additional layers to the predictor to prevent undesirable  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' This is because, when the predictor size is bigger  than prior on state-based tasks (not image-based as in orig-  inal work by Burda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2018)), we observe that it can  sometimes overgeneralize to OOD prior embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  According to Section 4, for the prior, we use FiLM condi-  tioning on penultimate layer before nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In prin-  ciple, the predictor can be arbitrary (Ciosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2019),  but in practice, its architecture and conditioning type can  also affect performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We conduct a preliminary study  on a small subset of the D4RL Gym tasks to select the best-  performing conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Based on the results in Table 1,  we chose a predictor with bilinear conditioning in the first  layer, as it showed the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Anti-Exploration Bonus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We define the anti-exploration  bonus similarly to RND loss as  b(s, a) = ∥fψ(s, a) − ¯f ¯  ψ(s, a)∥2  2  (3)  and additionally divide it by RND loss running standard  deviation (which is tracked during pretraining phase) to  increase its scale uniformly among environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Such  Anti-Exploration by Random Network Distillation  0  50000  100000  150000  200000  250000  300000  350000  training steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7  RND bonus  Concat  FiLM  Bilinear  Gated  (a) RND bonus for dataset actions  0  20000  40000  60000  80000  100000  training steps  0  1  2  3  4  5  6  7  RND bonus  Concat  FiLM  Bilinear  Gated  (b) RND bonus for actor actions  0  20000  40000  60000  80000  100000  training steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  mean squared error  Concat  FiLM  Bilinear  Gating  (c) Distance to dataset actions  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Effect of different state-action conditioning in the prior of RND on actor training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We use the halfcheetah, walker2d and hopper  medium datasets, with 3 seeds each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For training procedure, see Algorithm 2 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (a) Anti-exploration bonus for in-dataset  actions during RND pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We additionally divide the bonus by the RND loss running standard deviation to increase its scale  (see Section 5) so the anti-exploration bonus increases slightly over time as standard deviation decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' However, this does not affect  minimization by the actor and is needed to highlight the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (b) Anti-exploration bonus for actor actions during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Ideally,  it should converge to values close to the final values in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (c) Distance of actor actions to true in-dataset actions during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Ideally,  it should decrease, as actions closer to the behavioral policy have the lowest bonus by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  scaling simplifies hyperparameter search, shrinking the pos-  sible range of useful α coefficients that control the level of  conservatism during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  For detailed training procedure and full SAC losses, we  refer to Algorithm 1 in the Appendix (differences with the  original SAC algorithm are highlighted in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Experiments  In this section, we present an empirical evaluation of our  method using the D4RL benchmark on the Gym domain  (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1) and the more challenging AntMaze domain  (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Next, we provide additional analysis and  visual insight into why FiLM conditioning in the prior might  be beneficial (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Finally, we present an ablation  that compares more variations of conditioning for predictor  and prior (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For each experiment, we also list the  exact hyperparameters in Appendix D and implementation  details in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Additionally, we analyse sensitivity  to hyperparameters in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Evaluation on the Gym Domain  Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We evaluate our method on all available datasets for  the HalfCheetah, Walker2d and Hopper tasks in the Gym do-  main of the D4RL benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For ensemble-free baselines,  we chose CQL (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020), IQL (Kostrikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  2021), TD3+BC (Fujimoto & Gu, 2021), which show good  results and are widely used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For ensemble-based  baselines, we chose SAC-N & EDAC (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2021) and  the more recent RORL (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022), which currently  achieve state-of-the-art scores in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We follow the  An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2021) and train for 3M gradient steps, evaluating  on 10 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' The resulting scores are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We  see that SAC-RND stands out from the ensemble-free meth-  ods and outperforms them by a wide margin, achieving a  mean score comparable to EDAC and only slightly behind  RORL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Note that we do not use ensembles, whereas SAC-N  can require up to 500 critics, EDAC up to 50 and RORL up  to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In addition, we compare our proposed changes with  the naive predictor and prior, confirming that our modifi-  cations are essential for achieving good performance (see  Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Evaluation on the AntMaze Domain  Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We evaluate our method on all datasets available for  the AntMaze domain of the D4RL benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Ensemble-  free baselines are the same as in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For ensemble-  based baselines, we chose RORL (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022) and  MSG (Ghasemipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022), the latter of which, to  our knowledge, currently has the best mean score for this  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We do not include SAC-N and EDAC, as there are  no public results for them on this domain, and we were also  unable to obtain a non-zero result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We follow the An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  (2021) and train for 3M gradient steps, evaluating on 100  episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' The resulting scores are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Kostrikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2021) has shown that many offline RL  methods that perform well on the Gym domain fail on the  AntMaze domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' It can be seen that, on the AntMaze do-  main, SAC-RND shows good results that are on par with  ensembles, and outperforms ensemble-free methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' This  also shows that our choice of predictor and prior generalises  well to new domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Note that, in addition to ensembles,  both MSG and RORL require pre-training or supervision  with behavioural cloning in order to achieve reported results,  Anti-Exploration by Random Network Distillation  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' SAC-RND evaluation on the Gym domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We report the final normalized score averaged over 4 random seeds on v2 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  TD3 + BC and IQL scores are taken from Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' CQL, SAC-N and EDAC scores are taken from An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' RORL  scores are taken from Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Ensemble-free  Ensemble-based  Task Name  TD3+BC  IQL  CQL  SAC-N  EDAC  RORL  SAC-RND  halfcheetah-random  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  halfcheetah-medium  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
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+page_content='2  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
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+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  hopper-full-replay 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  walker2d-random  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  walker2d-medium  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8  102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8  walker2d-expert  110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4  115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9  115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  walker2d-medium-expert  110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4  114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9  121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9  walker2d-medium-replay  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7  walker2d-full-replay 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8  Average  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  while our method does not require any additional modifica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' SAC-RND evaluation on AntMaze domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We report  the final normalized score averaged over 4 random seeds on v1  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' IQL, CQL, MSG scores are taken from Ghasemipour  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' TD3+BC, RORL scores are taken from Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Ensemble-free  Ensemble-based  Task Name  TD3+BC  IQL  CQL  RORL  MSG  SAC-RND  antmaze-umaze  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  antmaze-umaze-diverse  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7  antmaze-medium-play  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
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+page_content='5  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
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+page_content='2  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
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+page_content='7  antmaze-medium-diverse  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
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+page_content='0  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
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+page_content='5  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='9  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
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+page_content='7  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
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+page_content='2  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6 ± 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='7  Average  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Why is FiLM Conditioning Beneficial for Bonus  Minimization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  In Section 4, we showed that FiLM conditioning in the RND  prior significantly improved the actors’ ability to minimize  the anti-exploration bonus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Since the issue occurred during  actor training, we hypothesize that this may be related to  the anti-exploration bonus optimization landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In this  section, we analyze the anti-gradient fields for conditioning  with concatenation or FiLM for the prior network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  For the purpose of analysis, we design a toy dataset with  only four categorical states and two-dimensional actions  sampled uniformly in each corner of the grid (see Ap-  pendix B for dataset visualization and generation details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  We fix the hyperparameters and pretrain two RNDs that  differ only in the type of prior conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' The predictor  uses simple concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Next, in Figure 4, we plot the  two-dimensional anti-gradient field for the anti-exploration  bonus conditioned on each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' The effect of FiLM be-  comes more apparent in these plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' While the resulting  anti-gradients for concatenation are noisy and only point  in the direction of the minimum in a small neighbourhood,  the directions for FiLM are smooth over the entire available  action space and point to the correct global minimum for  each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' While we cannot draw general conclusions from  such a demonstration, based on the results of Section 4,  we hypothesize that a similar phenomenon might exist in  high-dimensional problems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Exploring More Conditioning Pairs  One might wonder (1) how different types of conditioning  for predictor and prior interact with each other and (2) where  to introduce conditioning in terms of depth for it to be most  beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  To answer these questions, we return to the experiment from  Section 4 and generate more variations for each type (where  it is possible): conditioning on the first layer, on the last  layer, and on all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We also look at two variations  of the bilinear layer: full, as presented in Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  (2020), and simplified, which is used by default in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  In Figure 5 we plot the final MSE between the resulting  policy and the behavioural one on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Two  interesting observations can be made from these findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Anti-Exploration by Random Network Distillation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
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+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  a1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='40  (d) State 3  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Actions’ anti-gradient field for the anti-exploration bonus conditioned on four categorical states at each corner for the toy problem  introduced in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We visualize the dataset in Figure 7 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' The top row corresponds to RND with concatenation  conditioning in the prior, while the bottom row corresponds to FiLM conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' As can be seen, the resulting anti-gradients for  concatenation are noisy, while the directions for FiLM are smooth over the entire available action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  First, FiLM may not be the only architecture with the right  inductive biases for the prior, and both bilinear types with  conditioning on all layers can also achieve similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  However, compared to FiLM, inner bilinear layers are much  more computationally expensive, as they involve at least  one 3D weight tensor and two additional 2D weight tensors,  and the hidden dimensions are usually much higher than the  input dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Second, it appears that conditioning on the last layer is most  beneficial for the predictor, while conditioning on all layers  is beneficial for the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In spite of that, it is difficult to  draw broad conclusions, as different types may work well  for new problems and domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Related Work  Model-free offline RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Most offline RL approaches focus  on the distribution shift problem and overestimation bias  of Q-values for OOD actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Some researchers address  this by imposing strict constraints for policy updates, pe-  nalizing the divergence from the behavioral policy with KL  divergence, maximum mean discrepancy (MMD) distance  (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2019), simple mean squared  error (MSE) (Fujimoto & Gu, 2021), or by re-weighting  behavioral policy actions with the estimated advantages  (Nair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Others directly regularize Q-values  by lowering return estimates for OOD actions, preventing  gated  concat_first  concat_last  concat_full  bilinear_first  bilinear_last  bilinear_full  torch_bilinear_first  torch_bilinear_last  torch_bilinear_full  film_full  film_first  film_last  prior  gated  concat_first  concat_last  concat_full  bilinear_first  bilinear_last  bilinear_full  torch_bilinear_first  torch_bilinear_last  torch_bilinear_full  film_full  film_first  film_last  predictor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Mean squared error between actions of the actor trained  with different conditioning for the predictor & prior and actions  of the behavioral policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We use the halfcheetah, walker2d and  hopper medium datasets, with 3 seeds each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' It can be seen that  conditioning on each layer is beneficial for the priors, while for the  predictors, it is better to condition on the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Note that this  experiment is in the setting of Section 4, that is, without a critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  overestimation for unseen actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For instance, Kumar  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2020), Ghasemipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022) and Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  (2022) explicitly introduce an optimization term that lowers  Q-values for OOD actions, while An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2021) penalizes  implicitly by utilizing the lower-confidence bound (LCB)  of Q-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Alternatively, the evaluation of OOD actions  can be avoided altogether by using the upper expectile value  Anti-Exploration by Random Network Distillation  function (Kostrikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2021) or by policy optimization  within a latent action space (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Akimov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  In our work, we follow the anti-exploration approach  (Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In contrast to An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Ghasemipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022), we com-  pletely eliminate ensembles for uncertainty estimation, thus  reducing computational overhead without sacrificing perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Moreover, unlike Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022), we have  succeeded in using an RND for novelty detection in offline  RL for continuous action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Estimation bias in Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In both offline and on-  line reinforcement learning, off-policy Q-learning methods  suffer from an overestimation bias in the temporal differ-  ence learning target (Van Hasselt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Fujimoto  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' This phenomenon is orthogonal to overes-  timation due to unseen actions in offline RL, as it occurs  even in the presence of strong conservatism constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' It  is mainly caused by target prediction errors for seen transi-  tions and their propagation due to the maximum operation  maxa′∈AQ(s′, a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' To address this problem, Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  (2018) introduced clipped double Q learning (Van Hasselt  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2016) in TD3, which uses a minimum of two critics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  This approach was later used by Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2018) in  SAC to improve stability and accelerate convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  In our work, we use clipped double Q-learning (Fujimoto  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2018), since SAC-RND is based on SAC (Haarnoja  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2018), and found it beneficial for stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' However,  to ensure that it does not introduce additional conservatism,  which can be a confounding factor for the impact of RND,  we always set the number of critics to two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Uncertainty estimation in offline RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Uncertainty estima-  tion is a popular technique in reinforcement learning and is  used for a variety of purposes such as exploration, planning,  and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In offline RL, its use is mostly limited to  modeling epistemic uncertainty (Clements et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2019), in-  cluding measuring the prediction confidence of dynamics  models (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Kidambi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2020) or critics (An  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' This approach can be  further used to induce uncertainty-aware penalization during  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Alternatively, uncertainty can help overcome suboptimal  conservatism by designing more flexible offline approaches,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', conditioning on different levels of confidence to dy-  namically adjust the level of conservatism during evaluation  (Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022) or using Bayesian perspective to design  an optimal adaptive offline RL policy (Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  In our work, we estimate epistemic uncertainty and use it as  an anti-exploration bonus to induce conservatism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Unlike  previous approaches, we do not use ensembles to estimate  epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Efficient ensembles Ensembles are a powerful and sim-  ple non-Bayesian baseline for uncertainty estimation that  outperform Bayesian neural networks in practice (Lakshmi-  narayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' However, training deep ensembles  can be both memory intensive and computationally demand-  ing, making the design of efficient ensembles an attractive  research direction for which numerous methods have been  developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For example, Gal & Ghahramani (2016) pro-  posed to use dropout to approximate Bayesian inference in  deep Gaussian processes, and Durasov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2021) derived  a method to interpolate between dropout and full ensembles  with fixed masks and controllable overlap between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Meanwhile, Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2020) significantly reduced the cost  by defining each weight matrix as the Hadamard product  of a shared weight among all ensemble members and a  rank-one matrix per member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Recently, Ghasemipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022) showed that, in offline  RL, none of the most popular approaches for efficient en-  sembles are capable of delivering performance that is com-  parable to naive ensembles, and that more work is needed in  this research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In our work, we chose an alternative  path for uncertainty estimation with RND, which was shown  to a fast and competitive counterpart to ensembles (Ciosek  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Conclusion  In this work, we revisited the results from previous research  (Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022), showing that with a naive choice  of conditioning for the RND prior, it becomes infeasible  for the actor to effectively minimize the anti-exploration  bonus and discriminativity is not an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' To solve this,  we proposed conditioning based on FiLM, which led us  to a new ensemble-free method called SAC-RND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We em-  pirically validated that it achieves results comparable to  ensemble-based methods and outperforms its ensemble-free  counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' As such, we believe that our work is a valuable  contribution to anti-exploration and uncertainty estimation  in offline RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  References  Akimov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Kurenkov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Nikulin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tarasov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and  Kolesnikov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Let offline rl flow: Training conserva-  tive agents in the latent space of normalizing flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In  3rd Offline RL Workshop: Offline RL as a”Launchpad”,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  An, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Moon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Song, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Uncertainty-  based offline reinforcement learning with diversified q-  ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Advances in neural information processing  systems, 34:7436–7447, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Ba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Kiros, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Layer normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Anti-Exploration by Random Network Distillation  arXiv preprint arXiv:1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='06450, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Badia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Piot, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Kapturowski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Sprechmann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  Vitvitskyi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Guo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Blundell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Agent57:  Outperforming the atari human benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In Interna-  tional Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' 507–517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Baker, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Akkaya, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Zhokhov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Huizinga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  Ecoffet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Houghton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Sampedro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Clune, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Video pretraining (vpt): Learning to act by watching un-  labeled online videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='11795,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Berner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Brockman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Chan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Cheung, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', D˛ebiak, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  Dennison, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Farhi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Fischer, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Hashme, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Hesse,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Dota 2 with large scale deep reinforcement  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='06680, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Bradbury, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Frostig, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Hawkins, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Johnson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Leary,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Maclaurin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Necula, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Paszke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', VanderPlas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  Wanderman-Milne, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' JAX: composable  transformations of Python+NumPy programs, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' URL  http://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='com/google/jax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Burda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Edwards, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Storkey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Klimov, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Ex-  ploration by random network distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint  arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='12894, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Ghadirzadeh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Yu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Li,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Liang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Finn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Latent-variable  advantage-weighted policy optimization for offline rl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='08949, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Ciosek, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Fortuin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tomioka, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Hofmann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and  Turner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Conservative uncertainty estimation by fitting  prior networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In International Conference on Learning  Representations, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Clements, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Van Delft, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Robaglia, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Slaoui,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Toth, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Estimating risk and uncertainty in deep  reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='09638,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Dumoulin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Perez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Schucher, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Strub, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Vries,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Courville, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Bengio, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Feature-wise trans-  formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Distill, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='23915/distill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='00011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  https://distill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='pub/2018/feature-wise-transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Durasov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Bagautdinov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Baque, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Fua, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Masksembles for uncertainty estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In Proceed-  ings of the IEEE/CVF Conference on Computer Vision  and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' 13539–13548, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Fu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Nachum, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tucker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Levine,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' D4rl: Datasets for deep data-driven reinforcement  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='07219, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Fujimoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' and Gu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' A minimalist approach to offline  reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Advances in neural information  processing systems, 34:20132–20145, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Fujimoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Hoof, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Meger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Addressing function  approximation error in actor-critic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In Interna-  tional conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' 1587–1596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Fujimoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Meger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Precup, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Off-policy deep  reinforcement learning without exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In Interna-  tional conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' 2052–2062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Gal, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' and Ghahramani, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Dropout as a bayesian approx-  imation: Representing model uncertainty in deep learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In international conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  1050–1059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' PMLR, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Ghasemipour, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Gu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Nachum, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Why so  pessimistic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' estimating uncertainties for offline rl through  ensembles, and why their independence matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv  preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='13703, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Ghosh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Ajay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Agrawal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Offline  rl policies should be trained to be adaptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In Interna-  tional Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' 7513–7530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Haarnoja, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Zhou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Abbeel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Soft  actor-critic: Off-policy maximum entropy deep reinforce-  ment learning with a stochastic actor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In International  conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' 1861–1870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' PMLR,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Hong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Confidence-conditioned  value functions for offline reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv  preprint arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='04607, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Jarrett, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tallec, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Altché, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Mesnard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Munos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  and Valko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Curiosity in hindsight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint  arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='10515, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Jayakumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Czarnecki, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Menick, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Schwarz,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Rae, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Osindero, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Teh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Harley, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and  Pascanu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Multiplicative interactions and where to find  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Kidambi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Rajeswaran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Netrapalli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Joachims,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Morel: Model-based offline reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Advances in neural information processing systems, 33:  21810–21823, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Kingma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' and Ba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Adam: A method for stochastic  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6980, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Kostrikov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Nair, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Offline reinforce-  ment learning with implicit q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint  arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='06169, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Anti-Exploration by Random Network Distillation  Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Fu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Soh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tucker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Stabilizing off-policy q-learning via bootstrapping error  reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Advances in Neural Information Processing  Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Zhou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tucker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Con-  servative q-learning for offline reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Advances in Neural Information Processing Systems, 33:  1179–1191, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Agarwal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Geng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tucker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Levine,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Offline q-learning on diverse multi-task data both  scales and generalizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='15144,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Kurenkov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' and Kolesnikov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Showing your offline  reinforcement learning work: Online evaluation budget  matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In International Conference on Machine Learn-  ing, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' 11729–11752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Lakshminarayanan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Pritzel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Blundell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Simple  and scalable predictive uncertainty estimation using deep  ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Advances in neural information processing  systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Nachum, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Lee, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Freeman,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Guadarrama, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Fischer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Jang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  Michalewski, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Multi-game decision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='15241, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tucker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Fu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Offline rein-  forcement learning: Tutorial, review, and perspectives on  open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='01643, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Lillicrap, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Hunt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Pritzel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Heess, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Erez,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tassa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Silver, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Wierstra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Continuous  control with deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint  arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='02971, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Lyu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Lu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Mildly conservative q-  learning for offline reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint  arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='04745, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Nair, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Dalal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Awac: Accel-  erating online reinforcement learning with offline datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='09359, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Nikulin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Kurenkov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tarasov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Akimov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and  Kolesnikov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Q-ensemble for offline rl: Don’t scale  the ensemble, scale the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  arXiv preprint  arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='11092, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Perez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Strub, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', De Vries, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Dumoulin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and  Courville, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Film: Visual reasoning with a general con-  ditioning layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In Proceedings of the AAAI Conference  on Artificial Intelligence, volume 32, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Reed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Zolna, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Parisotto, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Colmenarejo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  Novikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Barth-Maron, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Gimenez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Sulsky,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Kay, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Springenberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' A generalist agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='06175, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Rezaeifar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Dadashi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Vieillard, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Hussenot, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  Bachem, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Pietquin, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Geist, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Offline rein-  forcement learning as anti-exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In Proceedings  of the AAAI Conference on Artificial Intelligence, vol-  ume 36, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' 8106–8114, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Schrittwieser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Antonoglou, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Hubert, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Simonyan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  Sifre, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Schmitt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Guez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Lockhart, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Hassabis,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Graepel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Mastering atari, go, chess and shogi  by planning with a learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Nature, 588(7839):  604–609, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Smith, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Kostrikov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  A walk in the  park: Learning to walk in 20 minutes with model-free  reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='07860,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Srivastava, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Shyam, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Mutz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Ja´skowski, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and  Schmidhuber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Training agents using upside-down re-  inforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='02877,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Tarasov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Nikulin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Akimov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Kurenkov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and  Kolesnikov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' CORL: Research-oriented deep offline  reinforcement learning library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In 3rd Offline RL Work-  shop: Offline RL as a ”Launchpad”, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' URL https:  //openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='net/forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='id=SyAS49bBcv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Van Hasselt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Guez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Silver, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Deep reinforce-  ment learning with double q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In Proceedings of  the AAAI conference on artificial intelligence, volume 30,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Wen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tran, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Ba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Batchensemble: an alterna-  tive approach to efficient ensemble and lifelong learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  arXiv preprint arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='06715, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Tucker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Nachum, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Behavior regu-  larized offline reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint  arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='11361, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Yang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Bai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Han,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Rorl: Robust offline reinforcement learning via con-  servative smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='02829,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Yu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Thomas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Yu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Ermon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Zou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=',  Finn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Ma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Mopo: Model-based offline policy  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Advances in Neural Information Processing  Systems, 33:14129–14142, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', Bajracharya, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', and Held, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Plas: Latent action  space for offline reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' In Conference  on Robot Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' 1719–1735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Anti-Exploration by Random Network Distillation  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Previous Research Results  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Anti-exploration bonus on walker2d-medium dataset for RND and CVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Note that figure taken from Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022) for  a convenient comparison with our results in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Toy Dataset  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
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+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  a1  State  0  1  2  3  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Toy dataset visualization introduced in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' This toy dataset consists of four categorical states for each corner of the  limited 2D actions grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For each state, we uniformly sample 4096 two-dimensional actions within a limited square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We use one-hot  encoding for the states during RND training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Implementation Details  In our experiments, we use hyperparameters from Table 4 where possible and sweep over α to pick the best value for each  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We implement all of our models using the Jax (Bradbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2018) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For the exact implementation  of conditioning variants for predictor and prior networks, refer to our code at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='com/tinkoff-ai/  sac-rnd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Similarly to Nikulin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022), we add Layer Normalization (Ba  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2016) to the critic after each layer as it greatly improves stability and convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' For SAC-N in Section 4 we  use the implementation from the CORL library (Tarasov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' All experiments were performed on V100 and A100  GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' With our implementation, each training for 3 million training steps usually takes ∼ 40 minutes to run (∼ 15 minutes  for the typical 1 million steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Gym Domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  We use the v2 version of each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  We follow the An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2021) approach and run our  algorithms for 3 million training steps and report the final normalized average score over 10 evaluation episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  For the final experiments, we use 4 seeds, while using less for hyperparameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  We tune the α co-  efficient over the {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0} range for the walker and hopper datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  We found that the halfcheetah datasets require a lower level of conservatism, which is why we tune over the  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0} range for these datasets while keeping the same number of candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  We follow the Ghasemipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022) approach and choose the best α for each dataset (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  AntMaze Domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We use the v1 version of each dataset due to the fact that the v0 version has major problems and  bugs during generation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=', some trajectories have discontinuities where the agent teleports from one part of the maze to  CVAE  RND  300000-  400000  Dataset actions  Shuffled actions  Random actions  300000  Dataset actions + Gaussian noise (std = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='25)  200000  Dataset actions + Gaussian noise (std = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5)  200000  100000  100000  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  1e-7Anti-Exploration by Random Network Distillation  another 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We follow the An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2021) approach and run our algorithms for 3 million training steps and report the final  normalized average score over 100 evaluation episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Same as Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022), we scale the reward by 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We found  that actor and critic require different levels of conservatism in these tasks, which is why we chose to decouple α and use  separate values (the same approach was used in Rezaeifar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We tune the α for the actor in the {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5}  range, and α for the critic in the {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1} range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' We follow the Ghasemipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (2022) approach and choose  the best α for each dataset (see Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Hyperparameters  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' SAC-RND general hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Parameter  Value  optimizer  Adam (Kingma & Ba, 2014)  batch size  1024 (256 on antmaze-*)  learning rate (all networks)  1e-3 (3e-4 on antmaze-*)  tau (τ)  5e-3  hidden dim (all networks)  256  num layers (all networks)  4  RND embedding dim (all tasks)  32  target entropy  action_dim  gamma (γ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='99 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='999 on antmaze-*)  nonlinearity  ReLU  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' SAC-RND best hyperparameters used in D4RL Gym domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Task Name  α  halfcheetah-random  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  halfcheetah-medium  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='3  halfcheetah-expert  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  halfcheetah-medium-expert  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  halfcheetah-medium-replay  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  halfcheetah-full-replay  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  hopper-random  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  hopper-medium  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  hopper-expert  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  hopper-medium-expert  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  hopper-medium-replay  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  hopper-full-replay  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  walker2d-random  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  walker2d-medium  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  walker2d-expert  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  walker2d-medium-expert  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  walker2d-medium-replay  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  walker2d-full-replay  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' SAC-RND best hyperparameters used in D4RL AntMaze domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Task Name  α (actor)  α (critic)  antmaze-umaze  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  antmaze-umaze-diverse  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='1  antmaze-medium-play  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='001  antmaze-medium-diverse  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='01  antmaze-large-play  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='01  antmaze-large-diverse  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='01  3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='com/Farama-Foundation/D4RL/issues/77  Anti-Exploration by Random Network Distillation  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Sensitivty to Hyperparameters  1  2  3  4  5  6  7  8  9  10  Policies Evaluated Online  30  40  50  60  70  80  90  Expected Online Performance  HalfCheetah  Hopper  Walker2D  AntMaze  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Expected Online Performance (Kurenkov & Kolesnikov, 2022) under uniform offline policy selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' It can be seen, that for  satisfactory results in all domains a budget of at least five policies for online evaluations is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Pseudocode  Algorithm 1 Soft Actor-Critic with Random Network Distillation (SAC-RND)  Initialize policy parameters θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' Double Q-function parameters {φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' φ2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' RND predictor and prior parameters {ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' ψ′},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' and  offline replay buffer D  for desired number of pretraining steps do  Sample a mini-batch B = {(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' a)} from D  Update RND predictor weights ψ with gradient descent using  ∇ψ  1  |B|  �  s∈B  �  ∥fψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' a) − ¯f ¯  ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' a)∥2  2  �  end for  for desired number of training steps do  Sample a mini-batch B = {(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' s′)} from D  Compute target Q-values (shared by all Q-functions):  y(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' s′) = r + γ  �  min  j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='2 Q ¯φi(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' a′) − β log πθ(a′|s′) − αb(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' a′)  �  where a′ ∼ πθ(·|s′) and b(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' a′) is an anti-exploration bonus defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Update each Q-function Qφi with gradient descent using  ∇φi  1  |B|  �  (s,a,r,s′)∈B  �  Qφi(s, a) − y(r, s′)  �2  Update policy with gradient ascent using  ∇θ  1  |B|  �  s∈B  �  min  j=1,2 Qφi(s, ˜aθ(s)) − β log π(˜aθ(s)|s) − αb(s, ˜aθ(s))  �  where ˜aθ(s) is a sample from π(·|s) which is differentiable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' θ via the reparametrization trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Update target networks with ¯φi ← (1 − ρ) ¯φi + ρφi  end for  Anti-Exploration by Random Network Distillation  Algorithm 2 Simplified SAC-RND (without a critic) used in experiments for Section 4 and Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  Initialize policy parameters θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' RND predictor and prior parameters {ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' ψ′},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' and offline replay buffer D  for desired number of pretraining steps do  Sample a mini-batch B = {(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' a)} from D  Update RND predictor weights ψ with gradient descent using  ∇ψ  1  |B|  �  s∈B  �  ∥fψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' a) − ¯f ¯  ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' a)∥2  2  �  end for  for desired number of training steps do  Sample a mini-batch B = {(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' s′)} from D  Update policy with gradient descent using  ∇θ  1  |B|  �  s∈B  �  β log π(˜aθ(s)|s) + b(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' ˜aθ(s))  �  where ˜aθ(s) is a sample from π(·|s) which is differentiable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' θ via the reparametrization trick and b(s′, a′) is an  anti-exploration bonus defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
+page_content='  end for' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf'}
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf,len=372
+page_content='FEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE TOMAS CABEZON PEDROSO1 and JINMO RHEE2 and DARAGH BYRNE3 1,2,3Carnegie Mellon University, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 1tcabezon@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='edu, 0000-0002-5483-2676 2jinmor@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='edu, 0000-0003-4710-7385 3daraghb@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='edu, 0000-0001-7193-006X  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' This paper compares the parametric design space with a feature space generated by the extraction of design features using deep learning (DL) as an alternative way for design space exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' In this comparison, the parametric design space is constructed by creating a synthetic dataset of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000 elements using a parametric algorithm and reducing its dimensions for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The feature space — reduced-dimensionality vector space of embedded data features — is constructed by training a DL model on the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' We analyze and compare the extracted design features by reducing their dimension and visualizing the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' We demonstrate that parametric design space is narrow in how it describes the design solutions because it is based on the combination of individual parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' In comparison, we observed that the feature design space can intuitively represent design solutions according to complex parameter relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Based on our results, we discuss the potential of translating the features learned by DL models to provide a mechanism for intuitive design exploration space and visualization of possible design solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Deep Learning, VAE, Design Space, Feature Design Space, Parametric Design Space, Design Exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Introduction  Parametric modeling has acquired widespread acceptance among creative practitioners as it allows the synthesis of various design options and solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Changing the parameters in this modeling process, either manually or randomly, can rapidly create a vast set of design variations (Toulkeridou, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Navigating the resulting parametric design space — where the design variants are topologically placed by their parameters — is part of the design exploration process — a crucial step in the development of new alternatives and design solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Exploration of the parametric design space allows creative practitioners  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' CABEZON PEDROSO, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' RHEE AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' BYRNE  many benefits: to reach satisfying solutions, better define design problems, and understand the opportunities and limitations of the possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Despite these benefits, design exploration is laborious within the parametric space and challenged along two fronts: comparison and selection (Fuchkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Parametric design exploration is an iterative process that focuses on the variation of these individual parameters, rather than on the relationship among them (Yamamoto and Nakakoji, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Hence, comparing one design solution with others by their parameters alone does not always result in a superior solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' for example, the variants generated by the local combination of parameters might not match the design requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Moreover, infinite alternative design solutions can be generated by inputting new parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Thus, the parametric design space consists of a huge amount of design variants that cannot be fully or sufficiently explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  We propose an alternative way to construct and examine the design space, by extracting features from a DL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' By comparing and analyzing how the DL feature design space differs from the parametric design space, we illustrate the potential of feature design space for design practitioners during the design exploration process and provide a new way to compare, examine and select the design alternatives based on the exploration of a properly constrained design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  No previous approach to compare the parametric design space and feature design space as design exploration tools has been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' To demonstrate how the feature space compares to the parametric space, we designed an experiment to construct both a parametric design space and a feature design space using the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The dataset consists of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000 synthetic 3D models produced by a parametric algorithm with five parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' This parametric design space consists of five axes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' each axis corresponds to each of the parameters that are used as inputs of the parametric algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Subsequently, this same dataset is used to train a DL model to compress the data into a feature vector of 128 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Both the parametric space (five-axes) and the feature space (128 axes) are not directly visualizable due to their high dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Nevertheless, as visual feedback plays an important role in design exploration (Bradner, Iorio and Davis, 2014), we employ a dimensionality reduction algorithm (t-SNE) to the design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' We are able to illustrate the design exploration space, showing how the data is distributed across both the parametric and feature design spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  In the next section, we describe the generation of the dataset, as well as the construction of parametric design space and its visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', we illustrate how training a DL model resulted in a feature space for design exploration and comparison with the parametric approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Then, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', we will compare, contrast, and discuss the characteristics of the DL feature space and the parametric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=')  FEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The overall process of comparing parametric design space and feature space from deep learning  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Constructing Parametric Design Space 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' DATASET GENERATION  To conduct a design space comparison, a simple parametric modeling system was designed: a parametric algorithm for generating different styles of vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' As with handcraft of pottery wheel throwing, a simple Bezier curve with three control points was turned around an axis to generate each 3D digital vessels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' the form of each vessel is specified by the five parameters that were used as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' These parameters, as can be seen in Figure 2, are: the height of the vessel, the width of the base, the width of the top opening, and the horizontal and vertical coordinates of the central control point of the Bezier curve that are used to create the curve of the form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The five parameters are represented as a vector, and each vector corresponds to a specific 3D model of a vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Using this system, we created a 3D vessel dataset by randomly generating a total of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000 different vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The total shape of the parametric representation of the vessel dataset is [15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000, 5], however, as it will be explained in the next section,  Parametric 5 algorithm Parametric Data parameters Design Space Comparison Feature VAE DesignSpace ENCODER DECODERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' CABEZON PEDROSO, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' RHEE AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' BYRNE  only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000 vessels were used for the space exploration and visualization, so this will be a design space of size [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Upper: An illustration of the dataset parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Lower: Three illustrative examples from the dataset with the parameters and the resulting 3D form side-by-side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' DIMENSIONALITY REDUCTION  As a five-dimensional space makes it hard to compare models and to visualize and compare the characteristics, we employed a dimensionality reduction process to reduce the space to two-dimensions and enable the objects to be plotted and compared to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' shows the overall process of visualizing the space using t-Distributed Stochastic Neighbour Embedding (t-SNE) algorithm (van der Maaten and Hinton, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' t-SNE is a popular dimensionality-reduction algorithm for visualizing high-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The hyper-parameters used for this reduction are: perplexity: 30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' learning rate: 200;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and iterations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Illustration of the dimensional reduction process for the 3D vessel dataset, and the construction of a parametric design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  After dimensional  reduction, each point in the plot represents the corresponding embedding of a vessel in the parametric design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Each point is expressed as  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Pot top width 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Pot bottom width 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Control point horizontal coordinate 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Control point vertical coordinate 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Pot height 2TopLevelApp (400 x 400) Parameters: bottomWidth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
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+page_content='755 Figure1 Press: d, to display the mesh bezier s, to save the,sti flle 75 50 25 0 -25 50 75 100 75 50 25 -100 0 -75 _50 -25 -25 0 50 25 75 50 75 100 100[3k,5] [3k,2] dimension 1 PLOT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' t-SNE x Parametric Dimensional dimension 0 Design Space ReductionFEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE a 2D image of the profile cut section of the corresponding vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' represents the reduced parametric design space of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' A 2D visualization of the parametric design space of the vessel dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Inset image: a detailed section for a subset of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Constructing the Feature Space  To construct the design space based on the features and not the parameters, we used a Variational Autoencoder (VAE) as a tool for extracting the morphological features of the vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' VAEs (Kingma and Welling, 2013) are a type of generative deep neural network used for statistical inference problems as they generalize a probabilistic distribution of the given dataset and synthesize new data samples from that distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' VAEs are composed of two modules: encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The encoder abstracts the input data into smaller dimensional vectors, latent vectors, and the decoder reconstructs the latent vector back into a 3D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' During the encoding process, the network captures and extracts the features of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' These features can be topologically placed in the data space, namely, latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' In the latent space, the distance between two data points represents the degree of resemblance of data: the closer points, the more resembled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' We translate this latent space as the feature space for an alternative way to explore the design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' DATA PRE-PROCESSING  Different representations of 3D data have been used in DL research, like point clouds (Achlioptas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', 2018), meshes (Ranjan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', 2018), or voxels (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' As resolutions of the data is not key for our purpose rather than the extracted features of it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and because we will implement a VAE for this experiment that needs fixed space inputs for the Convolutional Neural Networks (CNNs), we will be representing our 3D data with voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Voxels are discretized three- dimensional grids containing a binary value of volumetric occupancy of an object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' they distinguish between the elements on the grid that are filled with material and those that are empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The size of the voxel will determine the number of divisions 1  4 I  -- I 1 1 1 1 1  I !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 1 I 1 1 1 I 1 1 1 I 1 I =T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' CABEZON PEDROSO, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' RHEE AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' BYRNE  of the grid, consequently, the resolution at which we represent our 3D models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' the larger size, the more detailed 3D models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' In this experiment, we used 32-sized voxels so that a 3D vessel model is represented by 32x32x32 grid, the shape of the entire dataset is [15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000, 32, 32, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Finally, the dataset was then divided into two groups: 80% of the dataset (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000 vessels) was used for training the DL model, and the remaining 20% (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000 vessels) was used for testing the model and the parametric and feature space analysis and comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' TRAINING  For training the model, we adopted the VAE architecture implemented in ‘Adversarial Generation of Continuous Implicit Shape Representations’ (Kleineberg, Fey and Weichert, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The encoder consists of four residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Each residual block is composed of a 3D convolution layer, followed by a batch normalization and a Leaky ReLu activation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The decoder, on the contrary, comprises four residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Each block starts with a batch normalization, followed by a Leaky ReLu activation layer, and finally a 3D transposed convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The following hyper-parameters are used for training the VAE with the voxelized vessel dataset: batch size 32,  Adam optimizer (Kingma and Ba, 2015), learning rate 5e-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The model was trained in Google Colab Pro using the Nvidia Tesla T4 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Training process losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  The model was trained for a total of 240 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' We early stopped the model before the model started to overfit, Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The loss function used during training was a combination of two losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The first one, is the Kullback–Leibler divergence (KLD) loss (Kullback and Leibler, 1951), with a weight in the total loss formula of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' This function is a measurement of the difference between two statistical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The second loss is the Minimum Square Distance (MSE) loss (Sammut and Webb, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' It is used as the reconstruction loss and measures the error between the input voxels and the reconstructed output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' shows the reasonable quality of the reconstruction of the training result after 240 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' To ensure the performance of the model, it was evaluated using the test set and showed that the model maintained the accuracy with the new dataset, which shows  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='07 IMSE Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='06 KLD Loss Total Loss Total Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='01 0 50 100 150 200 250 EpochsFEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE that the model generalizes well to new data and is able to encode never seen before 3D vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Two examples of reconstructions from the trained VAE: the section slides and 3D voxels of the ground truth (the top row of each example) and the reconstruction (bottom of each example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' DIMENSIONALITY REDUCTION  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Feature space generation and visualization diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Once the VAE is trained, the encoder is used to extract the features of each vessel in the test dataset from 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='768 dimensions, the size of each voxelized vessel, into 128-dimensional vectors, the latent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Consequently, the entire test dataset of the vessels is represented into vectors whose total shape is [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000, 128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Like in the parametric case, 128 dimensions are non-visualizable so the same process as in Section 2 is followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' t-SNE algorithm is used to reduce the dimensionality of each vector and plot the resulting two dimensions in an image with the section of each of the vessels (Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The hyper-parameters used for this reduction are: perplexity: 50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' learning rate: 700;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and iterations: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' shows the results of distributed feature vectors in the reduced dimensional space, the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  1- Input (Ground truth) 10 10 10 20 30 05 20 20 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 Output 10 to 10 O1 1o 20 20 20 20 20 20 30 20 0 20 20 20 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 2- Input (Ground truth) 10 10 10 10 10 20 20 20 20 30 20 20 20 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 Output 10 OT 10 10 10 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='6 20 20 20 20 20 20 20 30 20 20 20 20 0 20 o 20 0 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0[3k,128] [3k,2] dimension ENCODER PLOT voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='npy t-SNE : dimension 0 Latent Dimensional object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='stl Space ReductionT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' CABEZON PEDROSO, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' RHEE AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' BYRNE  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' A 2D visualization of the feature design space of the vessel dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Inset image: a detailed section for a subset of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Comparison Between the spaces  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' shows that similar vessels have been clustered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Thinner vessels are located at the top right of the image, in contrast to the opposite lower bottom corner with the bigger vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The figure illustrates how the VAE model is able to understand the relationship between the parameters and their influence on the output morphological shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  On the contrary, in the parametric space (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' ), we can see how concave vessels were gathered at the bottom of the image, however, if the height of the vessels is considered, we can see that this parameter was not considered when clustering the vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Parametric space is based on each parameter independently, and not on the relationship among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Therefore, we observe that parametric design space insufficiently expresses the final form characteristics of the vessels by the combinations of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' On the contrary, in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', the feature space, a gradual change in the shape or concavity as well as height or width is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  To further examine and compare the characteristics of both design spaces, we used a clustering, algorithm: a Density-Based Spatial Clustering of Applications with Noise (DBSCAN) (Ester M et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' It is one of the most common clustering algorithms that finds core samples of high density and expands clusters with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' shows the results of this clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  The parametric design space has a total of seven clusters: three of them large, and four of them small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' It shows how the parametric design space doesn’t provide enough information to intuitively compare the design variants locally, this space shows extreme changes in vessel forms even in the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  "FEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE The feature design space, on the contrary, has a total of nine clusters: six main big clusters, and three smaller ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' In the feature design space, we can trace smooth changes in the forms as we move through the different clusters (local changes) and along the whole image (global changes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Shorter vessels are located on the top, while taller ones are on the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' If we move on the horizontal axis, the curve that generates the vessels goes from a concave shape on the right to a convex shape on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' This gives the designer the ability to locally compare similar design alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Parametric Design Space:                        Feature Design Space:  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Final visualization and clusters of the parametric and feature design spaces with representative vessels of each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Conclusion and Future work  We constructed the parametric and the feature design spaces using a custom synthetic dataset and a VAE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' By comparing the parametric and feature design spaces, we observed improved distributions of design alternatives in the later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' When the multi-dimensional parametric design space is projected into a 2D space (Figures 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' left), the clusters are insufficiently relevant to the morphological characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' On the other hand, when the multi-dimensional feature space is projected into a 2D space (Figures 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' right), the clusters show sufficient relevance to the features of the data they represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Based on this comparison, we conclude that combination of individual parameters in the parametric design space is limited in representing the morphological characteristics of the shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' However, we showed that DL models can be used to extract design features from 3D models and that the extracted features are more complex than the combinations of individual parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Hence, we conclude that the extracted features, that include information of the relationships between the parameters, can construct a well-distributed design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' For that reason, we propose feature design space as a tool for design space exploration that creative practitioners can use as a new way for looking at objects beyond the parametric design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' CABEZON PEDROSO, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' RHEE AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' BYRNE  Our results and implications are limited to a single dataset and DL model, however the results seem promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Future work will expand on this study with more diverse datasets generated by more complex parametric algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Accordingly, to perform the feature extraction, we would like to train other types of DL models to investigate different potentials of DL in design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' References  Achlioptas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (2018) ‘Learning Representations and Generative Models for 3D Point Clouds’, In International conference on machine learning (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 40-49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Bradner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', Iorio, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and Davis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (2014) ‘Parameters Tell the Design Story: Ideation and Abstraction in Design Optimization’, In Proceedings of the symposium on simulation for architecture & urban design (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 26) Ester M et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (1996) ‘A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise.’, KDD’96 Proceedings of the Second International Conference on Knowledge Discovery and Data Mining, 96, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 226–231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Fuchkina, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (2018) ‘Design Space Exploration Framework’, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Kingma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and Ba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (2015) ‘Adam: A method for stochastic optimization’, in 3rd International Conference on Learning Representations, ICLR 2015 - Conference Track Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' International Conference on Learning Representations, ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Available at: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='org/abs/1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='6980v9 (Accessed: 30 May 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Kingma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and Welling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (2013) ‘Auto-Encoding Variational Bayes’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Available at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='6114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Kleineberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', Fey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and Weichert, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (2020) ‘Adversarial Generation of Continuous Implicit Shape Representations’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Available at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='00349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Kullback, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and Leibler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (1951) ‘On Information and Sufficiency’, The Annals of Mathematical Statistics, 22(1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 79–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Available at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='1214/aoms/1177729694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  van der Maaten, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (2008) ‘Viualizing data using t-SNE’, Journal of Machine Learning Research, 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 2579–2605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Ranjan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (2018) ‘Generating 3D Faces using Convolutional Mesh Autoencoders’, in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Proceedings of the European Conference on Computer Vision (ECCV), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 704–720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Available at: https://openaccess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='thecvf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='com/content_ECCV_2018/html/Anurag_Ranjan_Generating_3 D_Faces_ECCV_2018_paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='html (Accessed: 7 December 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Sammut, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and Webb, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (eds) (2010) ‘Mean Squared Error’, in Encyclopedia of Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Boston, MA: Springer US, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 653–653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Available at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='1007/978-0-387-30164-8_528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Toulkeridou, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (2019) ‘Steps towards AI augmented parametric modeling systems for supporting design exploration’, in Blucher Design Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 37 Education and Research in Computer Aided Architectural Design in Europe and XXIII Iberoamerican Society of Digital Graphics, Joint Conference (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 1), Porto, Portugal: Editora Blucher, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 81–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Available at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='5151/proceedings-ecaadesigradi2019_602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (2017) ‘Learning a Probabilistic Latent Space of Object Shapes via 3D Generative-Adversarial Modeling’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Available at: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='org/abs/1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='07584 (Accessed: 7 December 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Yamamoto, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and Nakakoji, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (2005) ‘Interaction design of tools for fostering creativity in the early stages of information design’, International Journal of Human-Computer Studies, 63(4–5), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 513–535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Available at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='ijhcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
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+1
+A Fundamental Tradeoff Among Storage,
+Computation, and Communication for
+Distributed Computing over Star Network
+Qifa Yan Member, IEEE, Xiaohu Tang Senior Member, IEEE,
+Meixia Tao Fellow, IEEE, and Qin Huang Senior Member, IEEE
+Abstract
+Coded distributed computing can alleviate the communication load by leveraging the redundant
+storage and computation resources with coding techniques in distributed computing. In this paper, we
+study a MapReduce-type distributed computing framework over star topological network, where all
+the workers exchange information through a common access point. The optimal tradeoff among the
+normalized number of the stored files (storage load), computed intermediate values (computation load)
+and transmitted bits in the uplink and downlink (communication loads) are characterized. A coded
+computing scheme is proposed to achieve the Pareto-optimal tradeoff surface, in which the access point
+only needs to perform simple chain coding between the signals it receives, and information-theretical
+bound matching the surface is also provided.
+Index Terms
+Storage, coded computing, communication, MapReduce, star network
+I. INTRODUCTION
+The rapid growth of computationally intensive applications on mobile devices has attracted
+much research interest in designing efficient distributed computing frameworks. One of the most
+important programing models for distributed computing is MapReduce [1], [2], which has been
+utilized to deal with computation tasks with data sizes as large as tens of terabytes.
+MapReduce framework allows to assign multiple computation tasks to distributed nodes, where
+each node only stores a subset of files. This is done by decomposing each function to be computed
+into a set of “map” functions and a “reduce” function, where each map function can be computed
+from a batch of data, with the output called intermediate values (IVs), while the computation of
+a “reduce” function needs to collect the IVs from all the data as inputs. The whole procedure is
+composed of three phases, i.e., map, shuffle and reduce. In the map phase, each distributed node
+computes the map functions on its local file batch assigned by the server and generates output
+IVs; in the shuffle phase, the nodes exchange their computed IVs to facitate each node to obtain
+the IVs needed by its assigned reduce functions; in the reduce phase, each node computes its
+assigned reduce functions by decoding all the corresponding IVs.
+Q. Yan and X. Tang are with the Information Coding & Transmission Key Lab of Sichuan Province, CSNMT Int. Coop. Res.
+Centre (MoST), Southwest Jiaotong University, Chengdu 611756, China(email: qifayan@swjtu.edu.cn, xhutang@swjtu.edu.cn).
+M.
+Tao
+is
+with
+the
+Department
+of
+Electronic
+Engineering,
+Shanghai
+Jiao
+Tong
+University,
+Shanghai
+200240,
+China(email:mxtao@sjtu.edu.cn).
+Q. Huang is with the School of Electronic and Information Engineering, Beihang University, Beijing 100191, China
+(email:qinhuang@buaa.edu.cn).
+arXiv:2301.03788v1  [cs.IT]  10 Jan 2023
+
+2
+Recently, a coded distributed computing (CDC) scheme was proposed by Li et al. [3], where
+the files are stored multiple times across the distributed nodes in the map phase. The IVs are also
+computed multiple times accordingly, such that multicast opportunities are created for the shuffle
+phase. As a result, the communication load was reduced significantly compared to traditional
+uncoded scheme. It was proved in [3] that the scheme achieves the optimal communication load
+for a given total storage requirements. Interestingly, the normalized number of files stored across
+the nodes was termed computation load by Li et al, because each node calculates all the IVs that
+can be obtained from the data stored at that node in the model therein, no matter if these IVs
+are used or not in the subsequent phases. Subsequently, Ezzeldin [4] and Yan et al [5], [6] found
+that some IVs are computed but not used in the model. For this reason, Yan et al reformulated
+the problem as a tradeoff between storage, computation, and communication loads in [7], which
+allows each node to choose any subset of IVs to compute from its stored files.
+Some interesting works that extend CDC have been proposed, for example, the technique was
+combined with maximum distance separable (MDS) code in matrix-vector multiplication tasks to
+resist stragglers in [8]; stragglers with general functions are considered in [9], [10]; the optimal
+resource allocations are considered in [11]; [12]–[14] investigated the iterative procedures of data
+computing and shuffling; [15] studied the case when each node has been randomly allocated
+files; [16] investigated the case with random connectivity between nodes.
+The coded distributed computing technique is extended to wireless distributed computing
+[17], [18], where the computation is typically carried out by the wireless devices. Due to the
+decentralized natural of the wireless networks, the nodes in wireless networks normally need a
+central Access Point (AP) to exchange data, which leads to uplink and downlink communications.
+For example, smart-phone end users typically communicate with each other through a base station
+in cellular networks, which operates in a star network. In [19] and [20], Li et al. investigated
+distributed computing in a wireless network where the nodes performs data shuffling through an
+AP. The optimal storage-communication tradeoff was characterized for both uplink and downlink
+transmissions.
+In this paper, following the conventions of Ezzeldin [4] and Yan et al [7], we investigate
+a distributed computing system with star network, where all nodes exchange IVs through an
+AP, but each node is allowed to choose any arbitrary subset of IVs to compute from its stored
+files. In particular, in addition to the storage and computation loads as considered in [7], the
+communication load includes both upload and download. The main contribution of this paper is
+the characterization of the Pareto-optimal surface in the storage-computation-upload-download
+space for distributed computing over star network. The idea is to form the same multicast
+signals as in CDC scheme but compute less IVs by ignoring the un-used IVs in the map phase
+in the uplink, and combine them through a simple chain coding to form the downlink signals at
+the AP. It turns out that, for any given storage-computation pair, both the optimal upload and
+download communication costs can be simultaneously achieved by a coded computing scheme
+that oriented from CDC. The information-theoretical bound matching the Pareto-optimal surface
+is also presented.
+Paper Organization: Section II presents the system model. Section III summarizes the main
+results. Section IV presents the coded computing scheme that achieves the optimal surface, and
+Section V provides information-theoretical bound. Finally, Section VI concludes the paper.
+Notations: Let N+ be the set of positive integers, and F2 be the binary field. For m, n ∈ N+,
+denote the n-dimensional vector space over F2 by Fn
+2, and the integer set {1, . . . , n} by [n]. If
+m < n, we use [m : n] to denote the set {m, m + 1, . . . , n}. We also use interval notations, e.g.,
+
+3
+Fig. 1: A Distributed Computing System with Star Network
+[a, b] ≜ {x : a ≤ x ≤ b} and [a, b) ≜ {x : a ≤ x < b} for real numbers a, b such that a < b. The
+bitwise exclusive OR (XOR) operation is denoted by ⊕. For sets we use upper case calligraphic
+font, e.g., A, and for collections (sets of sets) we use upper case Greek letters with bold font,
+e.g., Ω. We denote a point in two or three dimensional Euclidean space by an upper case letter.
+A line segment with end points A1, A2 or a line through the points A1, A2 is denoted by A1A2.
+A triangle with vertices A1, A2, A3 is denoted by △A1A2A3. A trapezoid with the four edges
+A1A2, A2A3, A3A4, and A4A1, where A1A2 is parallel to A3A4, is denoted by ⊟A1A2A3A4.
+Let F be a set of facets, if the facets in F form a continuous surface, then we refer to this
+surface simply as F.
+II. SYSTEM MODEL
+Let K, N, W, U, V be given positive integers. Consider a star network consisting of K dis-
+tributed computing nodes {1, . . . , K} that can communicate with each other through a common
+AP, as illustrated in Fig. 1. Each of the K nodes can transmit signals to the AP through an
+uplink channel, while the AP can broadcast signals to all the K nodes via a downlink channel.
+Each of the K nodes aims to compute an individual function from a set of N files,
+W = {w1, . . . , wN},
+wn ∈ FW
+2 , ∀ n ∈ [N],
+each of size W bits. Node k aims to compute an output function
+φk : FNW
+2
+→ FU
+2 ,
+which maps all the files to a bit stream
+uk = φk(w1, . . . , wN) ∈ FU
+2
+of length U. Assume that each output function φk decomposes as:
+φk(w1, . . . , wN) = hk(fk,1(w1), . . . , fk,N(wN)),
+(1)
+where
+
+Files 
+Map
+Node 3
+IVs
+Reduce
+The set of files
+X(X1, X2, X3)
+X1
+X2
+个
+Reduce
+Reduce
+Files 
+Files 
+Map
+Map
+IVs
+IVs
+Node 1
+Node 24
+• Each “map” function fk,n is of the form
+fk,n : FW
+2 → FV
+2 ,
+and maps the file wn into the IV
+vk,n ≜ fk,n(wn) ∈ FV
+2 .
+• The “reduce” function hk is of the form
+hk : FNV
+2
+→ FU
+2 ,
+and maps the IVs
+Vk ≜ {vk,n : n ∈ [N]}
+into the output stream
+uk = hk(vk,1, . . . , vk,N).
+Notice that one trivial decompositon is that, the map functions are identity functions and the
+reduce functions are the output functions, i.e., gk,n(wn) = wn, and hk = φk, ∀ n ∈ [N], k ∈ [K].
+But in practice, many output functions can be decomposed such that the main computation load is
+dominated by the map functions. For example, in federated learning, it typically needs to collect
+the sum of the gradients over all data blocks, where the map functions are used to compute the
+gradients of the loss functions over a data block, while the reduce function is the sum operation.
+The described structure of the output functions φ1, . . . , φK, allows the nodes to perform their
+computation in the following three-phase procedure.
+1) Map Phase: Each node k ∈ [K] chooses to store a subset of files Mk ⊆ W. For each file
+wn ∈ Mk, node k computes a subset of IVs
+Ck,n = {vq,n : q ∈ Zk,n},
+where Zk,n ⊆ [K]. Denote the set of IVs computed at node k by Ck, i.e.,
+Ck ≜
+�
+n:wn∈Mk
+Ck,n.
+(2)
+2) Shuffle Phase: The K nodes exchange some of their computed IVs through the AP via
+upload and download sub-phases:
+In the upload sub-phase, each node k generates a coded signal
+Xk = ϕk (Ck)
+of some length lk ∈ N and sends it to the AP, using a function
+ϕk : F|Ck|V
+2
+→ Flk
+2 .
+In the download sub-phase, receiving all the signals {X1, . . . , XK}, the AP generates a signal
+X = χ(X1, X2, . . . , XK)
+(3)
+of length l ∈ N, and broadcasts it to all nodes, where the encoding function is
+χ : Fl1+l2+...+lK
+2
+→ Fl
+2.
+
+5
+3) Reduce Phase: Using the received signal X broadcast from the AP in the shuffle phase
+and its own IVs Ck computed locally in the map phase, each node k now computes the IVs
+(vk,1, . . . , vk,N) = ψk (X, Ck) ,
+(4)
+for some function
+ψk : Fl
+2 × F|Ck|V
+2
+→ FNV
+2
+.
+Finally, it computes
+uk = hk(vk,1, . . . , vk,N).
+(5)
+To measure the storage, computation, and communication costs of the described procedure,
+following the convention in [7], we introduce the following definitions.
+Definition 1 (Storage load). Storage load r is defined as the total number of files stored across
+the K nodes normalized by the total number of files N:
+r ≜
+�K
+k=1 |Mk|
+N
+.
+(6)
+Definition 2 (Computation load). Computation load c is defined as the total number of map
+functions computed across the K nodes, normalized by the total number of map functions NK:
+c ≜
+�K
+k=1 |Ck|
+NK
+.
+(7)
+Definition 3 (Communication Load). The communication load is characterized by the tuple
+(L, D), where L (resp. D) is the upload (resp. download) defined as the total number of the bits
+sent by the K nodes (resp. AP) during the upload (resp. download) sub-phase, normalized by
+the total length of all intermediate values NKV :
+L ≜
+�K
+k=1 lk
+NKV ,
+D ≜
+l
+NKV .
+Remark 1 (Nontrivial Regime). In general, the non-trivial regime in our setup is
+1 ≤ c ≤ r ≤ K,
+(8a)
+0 ≤ D ≤ L ≤ 1 − r
+K .
+(8b)
+For completeness, we justify them by the following observations.
+• Justification of (8a): Since each IV needs to be computed at least once somewhere, we
+have c ≥ 1. Moreover, the definition of Ck in (2) implies that |Ck| ≤ |Mk|K, and thus by
+(6) and (7), c ≤ r. Finally, the regime r > K is not interesting, because in this case each
+node stores all the files, and can thus locally compute all the IVs required to compute its
+output function. In this case, c ≥ 1, D ≥ 0 and L ≥ 0, can be arbitrary.
+• Justification of (8b): D ≥ 0 is trivial. By (3), as the down-link signal X is created from the
+upload signals X1, . . . , XK, D = L is sufficient to communicate all the received information.
+Finally, each node k can trivially compute |Mk| of its desired IVs locally and thus only
+needs to receive N − |Mk| IVs from other nodes. Thus, such an uncoded manner requires
+an upload of L =
+�K
+k=1(N−|Mk|)V
+NKV
+= 1 − r
+K.
+
+6
+In the trivial case that the AP simply forwards all the receiving signals, i.e., X = (X1, . . . , XK),
+then D = L, and the model degrades to the distributed model without the AP as in [7], where
+the non-trivial region on the triple (r, c, L) was 1 ≤ c ≤ r ≤ K, 0 ≤ L ≤ 1 − r
+K.
+Definition 4 (Fundamental SCC Region). A Storage-Computation-Communication1 (SCC) quadru-
+ple (r, c, L, D) satisfying (8) is achievable if for any ϵ > 0 and sufficiently large N, W, V , there
+exist map, shuffle, and reduce procedures with storage load, computation load, upload and
+download less than r + ϵ, c + ϵ, L + ϵ and D + ϵ, respectively. The fundamental SCC region is
+defined as the set of all feasible SCC quadruple:
+R = {(r, c, L, D) : (r, c, L, D) is feasible}.
+Definition 5 (Optimal Tradeoff Surface). An SCC quadruple (r, c, L, D) is called Pareto-optimal
+if it is feasible and if no feasible SCC quadruple (r′, c′, L′, D′) exists so that r′ ≤ r, c′ ≤ c, L′ ≤ L
+and D ≤ D′ with one or more of the inequalities being strict. The set of all Pareto-optimal SCC
+quadruples is defined as the optimal tradeoff surface:
+O ≜ {(r, c, L, D) : (r, c, L, D) is Pareto-optimal}.
+The goal of this paper is to characterize the fundamental SCC region R and the optimal
+tradeoff surface O in our setup.
+III. MAIN RESULTS
+Before we present the main theorem, let us provide a toy example to illustrate the key idea
+of the proposed achievable scheme.
+A. An Toy Example for Achievable Scheme
+Consider the case, where there aorre K = 3 nodes and N = 6 files. Each node wants to
+compute an individual function from the N = 6 files as in (1). Fig. 2 illustrates the strategy
+achieving the Pareto-optimal point (r, c, L, D) = (2, 4
+3, 1
+6, 1
+9), where the uplink and downlink
+transmissions are illustrated in Fig.2(a) and 2(b), respectively.
+In Fig. 2, the three nodes are denoted by three boxes with red, green and blue edges re-
+spectively. The top-most lines in each of the three boxes indicate the files stored at the node.
+The rectangle below this line indicates the map functions at the node. The computed IVs
+are depicted below the rectangle, where red circles, green squares, and blue triangles indi-
+cate IVs {v1,1, · · · , v1,6}, {v2,1, · · · , v2,6}, and {v3,1, · · · , v3,6}, respectively. The dashed cir-
+cles/squares/triangles stand for the IVs that are not computed from the stored files. The last
+line of each box indicates the IVs that the node needs to learn during the shuffle phase.
+The N = 6 files
+W = {w1, w2, w3, w4, w5, w6}
+are partitioned into
+�K
+r
+�
+= 3 batches, i.e., {w1, w2}, {w3, w4}, {w5, w6}. In the map phase, the
+files {w1, w2} are simultaneously stored at nodes 1 and 3; the files {w3, w4} at nodes 1 and 2;
+and the files {w5, w6} at nodes 2 and 3. For each node, the computed IVs can be classified into
+two types: the IVs that will be used by its own reduce function (the first line below the “map”
+1The communication load includes both upload and download.
+
+7
+rectangle) and the IVs that will be used for transmission or decoding (the second and third lines
+below the “map” rectangle).
+In the shuffle phase, during the upload sub-phase, each node creates a coded signal by XORing
+two IVs and sends it to the AP as illustrated in Fig. 2(a), i.e., Nodes 1, 2 and 3 sends coded
+IVs v1,1 ⊕ v3,3, v3,4 ⊕ v2,5 and v6,2 ⊕ v2,1, respectively; during the download sub-phase, the AP
+combines the three received signals by a simple chain coding, i.e., the two downlink signals are
+formed by XORing the signals from nodes 1 and 2, and the signals from 2 and 3, respectively.
+The combined signals are sent to all the three nodes.
+In the reduce phase, for each node, since the two chain coded signals involve a coded signal
+transmitted by itself, the node can decode the two coded signals from the other two nodes.
+Moreover, from each of the coded singals, the node can further decode an IVs it needs, by
+XORing the coding signal with one of its computed IV. For example, Node 1 first decodes the
+two signals v3,4 ⊕ v2,5 and v2,6 ⊕ v1,2, then it can further decodes the IVs v2,5 and v2,6, since the
+IVs v3,4 and v1,2 have been computed locally. Finally, each node collects all IVs for its assigned
+reduce function, and computes the final output.
+B. Fundamental SCC Region and Optimal Tradeoff Surface
+For each i ∈ [K], define two SCC quadrules
+Pi ≜
+�
+i, i
+�
+1 − i − 1
+K
+�
+, 1
+i
+�
+1 − i
+K
+�
+,
+1
+i + 1
+�
+1 − i
+K
+��
+,
+Qi ≜
+�
+i, i, 1
+i
+�
+1 − i
+K
+�
+,
+1
+i + 1
+�
+1 − i
+K
+��
+.
+In the following, we will use P u
+i , Qu
+i , P d
+i , Qd
+i to denote the projections of Pi, Qi into the uplink
+and downlink SCC subspaces2, i.e.,
+P u
+i ≜
+�
+i, i
+�
+1 − i − 1
+K
+�
+, 1
+i
+�
+1 − i
+K
+��
+,
+Qu
+i ≜
+�
+i, i, 1
+i
+�
+1 − i
+K
+��
+,
+P d
+i ≜
+�
+i, i
+�
+1 − i − 1
+K
+�
+,
+1
+i + 1
+�
+1 − i
+K
+��
+,
+(9)
+Qd
+i ≜
+�
+i, i,
+1
+i + 1
+�
+1 − i
+K
+��
+.
+The main result of this paper is summarized in the following theorem, where the proofs are
+provided in the following sections.
+Theorem 1. The fundamental SCC region R is given by
+R =
+�
+(r, c, L, D) : 1 ≤ c ≤ r ≤ K, L∗(r, c) ≤ L ≤ 1 − r
+K , D∗(r, c) ≤ D ≤ L
+�
+,
+where L∗(r, c) is a function such that {(r, c, L∗(r, c)) : 1 ≤ c ≤ r ≤ K} forms the surface
+Fu ≜ △P u
+1 P u
+2 Qu
+2 ∪
+K−1
+∪
+i=2 △P u
+i−1P u
+i P u
+K ∪
+K−1
+∪
+i=2 ⊟P u
+i Qu
+i Qu
+i+1P u
+i+1
+2In this paper, we will refer r-c-L subspace as the uplink SCC subspace, and the r-c-D subspace the downlink SCC subspace.
+The superscripts “u” and “d” indicate “uplink” and “downlink”, respectively.
+
+8
+(a)
+(b)
+Fig. 2: Illustration of the CDC for star network: (a) Uplink (b) Downlink
+in the uplink SCC subspace, and D∗(r, c) is a function such that {(r, c, D∗(r, c)) : 1 ≤ c ≤ r ≤
+K} forms the surface
+Fd ≜ △P d
+1 P d
+2 Qd
+2 ∪
+K−1
+∪
+i=2 △P d
+i−1P d
+i P d
+K ∪
+K−1
+∪
+i=2 ⊟P d
+i Qd
+i Qd
+i+1P d
+i+1
+in the downlink SCC subspace. The optimal tradeoff surface is given by
+O =
+K−1
+∪
+i=2 {θ1Pi−1 + θ2Pi + θ3PK : θ1, θ2, θ3 ∈ [0, 1], θ1 + θ2 + θ3 = 1}.
+(10)
+In Fig. 3, the functions L∗(r, c) and D∗(r, c) are ploted for K = 10 nodes. Notice that, by
+setting r = c, we recover the optimal upload and download as investigated in [20]3, i.e.,
+3The measurement of communication load is up to a scalar “K” in [20] compared Definition 3, and a slightly difference in
+assumption in [20] is that each node has a fixed storage load.
+
+Files
+25
+6
+Map
+Node 3
+Computes
+12
+56
+1:256
+Needs
+3
+Files
+3
+2
+4
+3
+4
+Files
+T9
+1
+Map
+1④3
+Map
+3456
+Computes
+Computes 3 4 .5..6
+14
+3:4:60
+Needs
+Needs
+12
+Node 1
+Node 2Files
+25
+6
+Map
+Node 3
+Computes
+12
+1256
+Needs
+3
+2
+Files
+2
+3
+4
+Files
+3
+4
+5
+T9
+Map
+Map
+3451
+16
+Computes
+Computes 3 4 .5..6
+3:4:60
+Needs
+Needs
+12
+Node 1
+Node 29
+1) the optimal upload for given storage is given by
+L∗(r) ≜ Conv
+�1
+r
+�
+1 − r
+K
+��
+,
+which corresponds the curve formed by the line segments Qu
+1Qu
+2, Qu
+2Qu
+3, . . . , Qu
+K−1Qu
+K.
+2) the optimal download for given storage is given by
+D∗(r) ≜ Conv
+�
+1
+r + 1
+�
+1 − r
+K
+��
+,
+which corresponds to the curve formed by the line segments Qd
+1Qd
+2, Qd
+2Qd
+3, . . . , Qd
+K−1Qd
+K.
+Observe that, the line segments Qu
+i P u
+i in the uplink SCC space and Qd
+i P d
+i in the downlink
+space (i = 2, 3, . . . , K) are parellel to the c-axis, which indicate that the computation load can be
+saved to achieve L∗(r). The length of the line segments indicates the amount of the computation
+load that can be saved. Thus, with larger storage load r, the saving of computation load to
+achieve L∗(r) and D∗(r) is larger. It will be clear later that the saving on the computation load
+is due to the fact that, under the assumption that each not computes all IVs it can computes,
+some of the IVs computed are not used in neither generating the signal, nor in the decoding
+process.
+The projections of the Pareto-optimal surface O into the uplink and downlink SCC space
+correspond to the surfaces
+Ou ≜ {P u
+i−1P u
+i P u
+K : i ∈ [2 : K − 1]}
+and
+Od ≜ {P d
+i−1P d
+i P d
+K : i ∈ [2 : K − 1]},
+respectively. Observe that, for a given feasible (r, c) pair, the optimal upload is strictly larger
+than the optimal download. We will see that this is achieved by performing some simple chain
+coding at the AP to combine the signals from different nodes. Interestingly, both the upload and
+download can be simultaneously achieved for a fixed (r, c) pair.
+Remark 2 (Relation to Results in [7]). One can observe that, the surfaces composing L∗(r, c) and
+Ou concide with the optimal communication load and the Pareto-optimal SCC tradeoff surface
+in the setup where the nodes directly connect to each other through a shared link (c.f. [7, Fig.
+2]), respectively. It was showed in [7], by dropping the computations of the IVs that are not
+used in the CDC scheme [3] as in [4], the resultant coded computing scheme can achieve the
+corner points of the Pareto-optimal SCC tradeoff surface (which is same as Ou). In fact, in
+our proposed scheme, each node performs the same map procedures as in [4], but the signals
+are sent to the AP. The AP performs a simple chain coding on the received signals to further
+compress the length of the signals, which leads to a further decrease of the download compared
+to the upload. We will present the whole process in Section IV.
+IV. ACHIEVABILITY
+Since the set O is exactly all the Pareto-optimal points of the set R (Appendix A), we
+only need to prove the achievability of the hypersurface O. We will derive a coded computing
+scheme that achieves the SCC quadruple Pi. Moreover, for any fixed θ1, θ2, θ3 ∈ [0, 1] such that
+
+10
+Fig. 3: The functions L∗(r, c) and D∗(r, c).
+θ1+θ2+θ3 = 1, divide the N files into three groups of sizes4 θ1N, θ2N and θ3N. By applying the
+scheme achieving the points Pi−1, Pi and PK on the three groups of files, the resultant scheme
+achieves the point P = θ1Pi−1 + θ2Pi + θPK. Thus, we only need to prove the achievability of
+Pi, i ∈ [K].
+A. Coded Distributed Computing for Star Network
+We now describe the scheme achieving Pi for a fixed i ∈ [K].
+Define
+Ωi ≜ {T ⊆ [K] : |T | = i} ,
+∀ i ∈ [K].
+For i = K, PK = (K, 1, 0, 0) is trivial, since each node can simply store all the files and
+computes their IVs as well as their reduce functions locally, with no communication loads.
+Consider a fixed i ∈ [K − 1], the N files are partitioned into
+�K
+i
+�
+batches, each containing
+ηi = N
+�K
+i
+�
+(11)
+files. Each batch is then associated with a subset T of [K] of cardinality i, i.e., an element in
+Ωi. Let WT denote the batch of the ηi files associated with set T . Then,
+W = {w1, . . . , wN} =
+�
+T ∈Ωi
+WT .
+4This requires that θ1, θ2, θ3 have to be rational. If any one is irrational, one can replace it by a rational number arbitrarily
+close to it.
+
+Optimal Upload and Download, K = 10
+pu
+0.9 -
+..The Function L*(r, c)
+(/T)
+0.8 ~
+-- The Function D*(r,c)
+0.7
+-Plane r = c
+Communication load (
+0.6 ~
+Pf,
+0.5 
+Q
+0.4 ~
+0.3 ~
+Qu
+0.2 -
+0.1 
+0.
+Storage load (r)s
+Pl0/ P10
+2
+3
+5
+4
+10 Q1 /Qil 0
+6
+6
+8
+>
+Computation load (c)11
+Further let UT ,k be the set of IVs for output function φk that can be computed from the files in
+WT :
+UT ,k ≜ {vk,n : wn ∈ WT }.
+We now describe the map, shuffle, and reduce procedures.
+1) Map Phase: Each node k stores
+Mk =
+�
+T ∈Ωi:k∈T
+WT ,
+and computes the IVs
+Ck = C1
+k ∪ C2
+k,
+(12)
+where
+C1
+k =
+�
+T ∈Ωi:k∈T
+UT ,k,
+(13a)
+C2
+k =
+�
+T ∈Ωi:k∈T
+�
+q∈K\T
+UT ,q.
+(13b)
+In other words, for each batch T , each node k computes all the IVs for its own function
+k, and all the IVs for the function q if node q does not have the batch T .
+2) Shuffle Phase: For each element T ∈ Ωi and each index j ∈ K\T , we partition the set
+UT ,j into i smaller subsets
+UT ,j =
+�
+U k
+T ,j : k ∈ T
+�
+(14)
+of equal size.
+In the upload sub-phase, for each S ∈ Ωi+1 and k ∈ S, by (13b), node k can compute
+the signal
+Xk
+S ≜
+�
+l∈S\{k}
+U k
+S\{l},l
+from the IVs calculated during the map phase. Node k thus sends the multicast signal
+Xk =
+�
+Xk
+S : S ∈ Ωi+1 such that k ∈ S
+�
+to the AP R. Thus, the AP R receives the signals X1, . . . , XK.
+In the download sub-phase, for each S = {k1, . . . , ki+1} ∈ Ωi+1, the AP R creates a
+signal,
+XS ≜ (Xk1
+S ⊕ Xk2
+S , Xk2
+S ⊕ Xk3
+S , . . . , Xki
+S ⊕ Xki+1
+S
+).
+(15)
+Then the AP broadcast the signal
+X ≜ {XS : S ∈ Ωi+1}.
+(16)
+3) Reduce Phase: Notice that C2
+k only contains the IVs vq,n where q ̸= k. Thus, by (12) and
+
+12
+(13a), during the shuffle phase each node k needs to learn all the IVs in
+�
+T ∈Ωi : k/∈T
+UT ,k.
+Fix an arbitrary T ∈ Ωi such that k /∈ T . From the received multicast message XT ∪{k},
+since the signal Xk
+T ∪{k} is generated by node k, by (15), node k can decode Xj
+T ∪{k} for
+all j ∈ T , where the signal
+Xj
+T ∪{k} =
+�
+l∈T ∪{k}\{j}
+U j
+T ∪{k}\{l},l
+is sent by node j during the shuffle phase. For any fixed j ∈ T , node k can recover the
+missing IV U j
+T ,k through a simple XOR operation:
+U j
+T ,k = Xj
+T ∪{k} ⊕
+�
+l∈T \{j}
+U j
+T ∪{k}\{l},l,
+(17)
+where U j
+T ∪{k}\{l},l is calculated at node k by (13b) and (14) for all l ∈ T \{j}. Moreover,
+node k can decode UT ,k from
+�
+Xj
+T ∪{k} : j ∈ T
+�
+.
+by (14) and (17). After collecting all the missing IVs, node k can proceed to compute the
+reduce function (5).
+Remark 3 (Comparison with [20]). Compared to the coded computing scheme in [20], two
+differences of the above scheme are:
+1) In the map phase, each node only needs to compute the IVs described in (12) and (13),
+because only those IVs are useful for creating or decoding the coded signals, while in
+[20], all the IVs pertaining to the files in Mk are computed, i.e., node k computes
+�C ≜
+�
+T ∈Ωi,k∈T
+�
+q∈[K]
+UT ,q.
+(18)
+This scheme in fact achieves the point Qi, which is inferior to Pi for i > 0 in computation
+load. The idea of removing the redundancy has been proposed in the setup where the nodes
+connects to each other directly through a bus link by Ezzeldin [4] and Yan et al [7].
+2) In (15), for any node set S of size i + 1, we used a simple chain coding on the signals to
+form i signals, while in [20], it uses random coding on the signals {Xk
+S : k ∈ S} to form
+i coded signals. The advantage of chain coding in (15) is obvious:
+a) It has smaller encoding and decoding complexities;
+b) It can be operated on the binary field F2;
+c) The order of nodes in the chain can be arbitrary. It makes sense in some scenarios:
+the signals {Xk
+S : k ∈ S} may arrive at different time points. Consider the case that
+the signals arrive in the ordder Xk1
+S , Xk2
+S , . . . , Xki+1
+S
+, to perform the encoding (15), at
+any time the AP only needs to keep one signal in its buffer, because each coordinate
+in (15) only depends on two consecutive signals. While with random linear coding,
+the AP typically have to wait for all signals {Xk
+S : k ∈ S}. Thus, the chain coding
+can reduce the buffer size at the AP and the node to node delay.
+
+13
+Remark 4 (PDA framework). In [7], [21], [22], a coded computing scheme was derived based on
+placement delivery array (PDA), which was proposed in [23] to explore coded caching schemes
+with uncoded placement [24]. In particular, it turns out that the Maddah-Ali and Niesen’s coded
+caching scheme corresponds to a special structure of PDA (referred to as MAN-PDA). It was
+showed in [7] that, with any given PDA belonging to a special class (defined as PDA for
+distributed computing (Comp-PDA)), one can always obtain a coded computing scheme. The class
+of PDAs achieving the Pareto-optimal tradeoff surface was characterized in [7]. The advantage of
+establishing the PDA framework is, various known PDA structure, e.g., the constructions in [23],
+[25], [26] can be directly utilized to obtain coded computing schemes with low file complexity5.
+In our setup, similar connections between coded computing schemes and Comp-PDA can be
+established, by following the same steps as in [7] for upload singals, and incoporating the chain
+coding (15) on all multicast signals from the Comp-PDA for the downlink signals. For example,
+the scheme described in Fig. 2 can be derived from the PDA
+�
+�
+∗
+1
+∗
+1
+∗
+∗
+∗
+∗
+1
+�
+� ,
+for details of forming the upload signals in Fig. 2(a), one can refer to [7, Example 4].
+B. Performance Analysis
+We analyze the performance of the scheme.
+1) Storage Load: The number of batches in Mk is
+�K−1
+i−1
+�
+, each consisting of ηi files. Thus,
+the storage load is
+r = 1
+N · K ·
+�K − 1
+i − 1
+�
+· ηi = i.
+(19)
+2) Computation Load: Since C1
+k ∩ C2
+k = ∅, we have |Ck| = |C1
+k| + |C2
+k|. From (11), (13a), and
+(13b), we have
+|C1
+k| =
+�K − 1
+i − 1
+�
+· ηi = iN
+K ,
+|C2
+k| =
+�K − 1
+i − 1
+�
+· (K − i) · ηi
+=
+�
+1 − i
+K
+�
+· i · N.
+Thus, the computation load is
+c =
+�K
+k=1 |Ck|
+NK
+= i
+�
+1 − i − 1
+K
+�
+.
+(20)
+3) Communication Load: The number of signals that each node k transmits is
+�K−1
+i
+�
+, each
+of size ηi·V
+i
+bits. Thus, the length of the signal Xk is lk =
+�K−1
+i
+� ηi·V
+i
+bits. Therefore, the
+5The file complexity of a coded computing scheme is defined as the smallest number of files required to implement the
+scheme, e.g., the file complexity of the proposed scheme achiving Pi is
+�K
+i
+�
+.
+
+14
+upload is
+L =
+�K
+k=1 lk
+NKV
+= 1
+i ·
+�
+1 − i
+K
+�
+.
+(21)
+By (15) and (16), the AP R transmits
+� K
+i+1
+�
+· i signals, each of size
+ηi·V
+i
+bits, thus the
+download is
+D =
+1
+NKV ·
+� K
+i + 1
+�
+· i · ηi · V
+i
+=
+1
+i + 1
+�
+1 − i
+K
+�
+.
+(22)
+From (19), (20), (21) and (22), we show the achievability of the SCC quadruple Pi.
+V. CONVERSE
+We need to prove that for any achievable (r, c, L, D) satisfying (8),
+L ≥ L∗(r, c),
+(23a)
+D ≥ D∗(r, c).
+(23b)
+Consider a coded distributed computing scheme achieving (r, c, L, D), with file allocations M[K],
+IV allocations C[K], uplink signals X[K] and downlink signal X. By the decoding condition (4),
+H(Vk|X, Ck) = 0,
+∀ k ∈ [K].
+Thus for any k ∈ [K],
+H(Vk|X1, . . . , XK, Ck)
+(a)
+= H(Vk|X1, . . . , XK, X, Ck)
+≤ H(Vk|X, Ck)
+= 0,
+where (a) follows since the downlink signal X is determined by the uplink signals X[K] by (3).
+That is, with the signals X1, . . . , XK and the locally computed IVs Ck, node k can decode
+all the IVs it needs. As a result, the file allocations M[K], IV allocations C[K] and the uplink
+singals X[K] consisitute an valid scheme for the distributed computing system where the nodes
+are connected through a bus shared link directly, as investigated in [7]. Therefore, by the results
+in [7, Theorem 2], we have proved (23a).
+We proceed to prove the (23b). For any k ∈ [K] and nonempty S ⊆ [K]\{k}, define
+Bk,S ≜ {vk,n : vk,n is exclusively computed by the nodes in S},
+�Bk ≜ {vk,n : vk,n is the computed by node k}.
+Let bk,S be the cardinality of the set Bk,S and ˜bk be the cardinality of �Bk. Obviously, the subsets
+{Bk,S : S ⊆ [K]\{k}, S ̸= ∅} and �Bk form a partition of the IVs Vk, thus
+˜bk +
+�
+S⊆[K],S̸=∅
+bk,S = N.
+
+15
+For each j ∈ [K − 1], the set of IVs not computed locally but exclusively computed by j other
+nodes are
+Bj =
+�
+k∈[K]
+�
+S⊆[K]\{k},|S|=j
+Bk,S.
+Then the cardinality of set Bj is given by
+bj ≜
+�
+k∈[K]
+�
+S⊆[K]\{k},|S|=j
+bk,S,
+∀j ∈ [K − 1].
+(24)
+To prove the lower bound in (23b), we need the following two lemmas.
+Lemma 1. The entropy of the download signal X satisfy
+H(X) ≥ V
+K−1
+�
+j=1
+bj
+j + 1.
+Proof: Assume that the AP holds all IVs Vk, then the access point can create the signal X.
+Consider the data exchange problem6 formed by the AP and the K nodes, where only the AP
+sends the signal X to all the K nodes. Notice that, in this system, each bits in Bk,S is cached
+at the AP and the nodes in S, but only demanded by node k. Thus, by the lower bound in [27,
+Theorem 1],
+H(X) ≥ V
+�
+k∈[K]
+�
+S⊆[K]\{k}
+1
+(|S| + 1) + 1 − 1 · bk,S
+= V
+K
+�
+k=1
+K−1
+�
+j=1
+�
+S⊆[K]\{k},|S|=j
+1
+j + 1 · bk,S
+(a)
+= V
+K−1
+�
+j=1
+1
+j + 1
+K
+�
+k=1
+�
+S⊆[K]\{k},|S|=j
+bk,S
+= V
+K−1
+�
+j=1
+bj
+j + 1,
+where in (a), we utilized (24).
+The following lemma was proved in [7, Lemma 2].
+Lemma 2. The parameters b1, . . . , bK−1 defined in (24) satisfy
+K−1
+�
+j=1
+bj ≥ N(K − r),
+K−1
+�
+j=1
+(j − 1)bj ≤ (c − 1)NK.
+6Data exchange problem was defined in [27], where each of the nodes holds a subset of the information bits, and request
+another subset of information bits.
+
+16
+For a fixed r ∈ [1 : K], and each i ∈ [K], define
+ci ≜ 1 +
+�
+1 − r
+K
+�
+(i − 1).
+(25)
+Let λi, µi ∈ R+ such that
+λix + µi|x=ci−1 =
+1
+ci−1 + 1 − 2r/K ·
+�
+1 − r
+K
+�2
+= 1
+i
+�
+1 − r
+K
+�
+,
+(26a)
+λix + µi|x=ci =
+1
+ci + 1 − 2r/K ·
+�
+1 − r
+K
+�2
+=
+1
+i + 1
+�
+1 − r
+K
+�
+.
+(26b)
+From (26a) and (26b), the following relationships hold:
+λi = −
+1
+i(i + 1) < 0,
+(27a)
+µi = 2i − 1
+i(i + 1)
+�
+1 − r
+K
+�
++
+1
+i(i + 1) > 0,
+(27b)
+λi + µi = 2i − 1
+i(i + 1)
+�
+1 − r
+K
+�
+> 0.
+(27c)
+Moreover, by its convexity over x ∈ [1, ∞), the function
+1
+x + 1 − 2r/K
+�
+1 − r
+K
+�2
+− (λix + µi)
+must be nonnegative outside the interval formed by the two zero points, i.e.,
+1
+x + 1 − 2r/K
+�
+1 − r
+K
+�2
+≥ λix + µi,
+∀ x ∈ [1, ci−1] ∪ [ci, ∞).
+Therefore,
+1
+cj + 1 − 2r/K
+�
+1 − r
+K
+�2
+≥ λicj + µi,
+∀ j ∈ [K − 1].
+(28)
+Now, we are ready to derive the lower bound for the download D:
+D ≥ H(X)
+NKV
+(a)
+≥
+K−1
+�
+j=1
+bj
+NK ·
+1
+j + 1
+(b)
+=
+1
+N(K − r)
+K−1
+�
+j=1
+bj ·
+1
+cj + 1 − 2r/K
+�
+1 − r
+K
+�2
+(c)
+≥
+1
+N(K − r)
+K−1
+�
+j=1
+bj(λicj + µi)
+
+17
+(d)
+=
+1
+N(K − r)
+K−1
+�
+j=1
+bj
+�
+λi
+�
+1 +
+�
+1 − r
+K
+�
+(j − 1)
+�
++ µi
+�
+=
+λi
+NK ·
+K−1
+�
+j=1
+(j − 1)bj +
+λi + µi
+N(K − r) ·
+K−1
+�
+j=1
+bj
+(e)
+≥
+λi
+NK · (c − 1)NK +
+λi + µi
+N(K − r) · N(K − r)
+= λic + µi
+= −
+2i − 1
+Ki(i + 1)r −
+1
+i(i + 1)c +
+2
+i + 1.
+(29)
+where (a) follows from Lemma 1; (b) and (d) follow from the definition of ci in (25); (c) follows
+from (28); and (e) follows from Lemma 2 and the signs of λi and λi + µi in (27).
+Notice that the three points P d
+i−1, P d
+i and P d
+K defined in (9) satisfy (29) with equality. Thus, the
+inequalities above indicate that all the feasible points (r, c, L, D) must satisfy that the projection
+into the download SCC space (r, c, D) must lie above the plane containing △P d
+i−1P d
+i P d
+K.
+Furthmore, D should be lower bounded by the optimal download even if each node computes
+all the IVs that can be computed locally from their stored file, i.e., a similar setup as in [20].
+The converse in [20] indicates that L is lower bounded as follows in the r-D plane7:
+D ≥ Conv
+�1
+r
+�
+1 − r
+K
+��
+,
+r ∈ {1, 2, . . . , K}.
+(30)
+Finally, by the lower bounds in (29) for i = 2, 3, . . . , K − 1 and (30), D is lower bounded by
+D∗(r, c), i.e., the lower bound (23b) is proved.
+VI. CONCLUSION
+In this paper, the Pareto-optimal storage-computation-upload-download tradeoff surface is
+characterized for the MapReduce distributed computing system, where the nodes have to exchage
+intermediate values through an access point that can broadcast signals to all nodes. It turns
+out that, for a given storage-computation pair (r, c), the optimal upload and download can be
+simultaneously achieved. Information-theoretical bounds matching the achievable communication
+load are provided for both uplink and downlink.
+APPENDIX A
+THE RELATION OF HYPERSURFACE O AND REGION R
+We now prove that O is the Pareto-optimal surface of the region R. Obviously, all Pareto-
+optimal points must lie on the surface
+F = {(r, c, L∗(r, c), D∗(r, c)) : 1 ≤ c ≤ r ≤ K}.
+7Although the setup in [20] assumes a fixed storage capacity at each node, the proof the following inequality do not rely on
+this assumption.
+
+18
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Fig. 4: The projections of Pi and Qi (i ∈ [K]) to the storage-computation subspace (r-c plane).
+Let the projections of points Pi and Qi into the r-c plane be P ′
+i and Q′
+i (i ∈ [K]), respectively8,
+i.e.,
+P ′
+i =
+�
+i, i
+�
+1 − i − 1
+K
+��
+,
+Q′
+i = (i, i).
+Let the projection of the surface F to the r-c plane be
+F′ ≜ {(r, c) : 1 ≤ c ≤ r ≤ K} = △P ′
+1P ′
+KQ′
+K.
+Notice that, here the “projection” map is one-to-one. Moreover, F′ can be decomposed into (see
+Fig. 4)
+F′ = △P ′
+1P ′
+2Q′
+2 ∪
+K−1
+∪
+i=2 △P ′
+i−1P ′
+iP ′
+K ∪
+K−1
+∪
+i=2 ⊟P ′
+iQ′
+iQ′
+i+1P ′
+i+1.
+Since the triangle △P u
+1 P u
+2 Qu
+2 and the trapezoids ⊟P u
+i Qu
+i Qu
+i+1P u
+i+1 in the uplink SCC space
+(i ∈ [2 : K − 1]) are parallel to c-axis, and so as the triangle △P d
+1 P d
+2 Qd
+2 and the trapezoids
+⊟P d
+i Qd
+i Qd
+i+1P d
+i+1 in the downlink SCC space, all the points (r, c, L∗(r, c), D∗(r, c)) ∈ F such
+that
+(r, c) ∈ △P ′
+1P ′
+2Q′
+2 ∪
+K−1
+∪
+i=2 ⊟P ′
+iQ′
+iQ′
+i+1P ′
+i+1\
+K−1
+∪
+i=2 △P ′
+i−1PiP ′
+K
+(31)
+cannot be Pareto-optimal. In the following, we prove that, all the points (r, c, L∗(r, c), D∗(r, c))
+such that
+(r, c) ∈
+K−1
+∪
+i=2 △P ′
+i−1P ′
+iP ′
+K
+(32)
+are Pareto-optimal.
+Now fix a quadruple (r1, c1, L∗(r1, c1), D∗(r1, c1)) ∈ F that satisfies (32). We show that it is
+Pareto-optimal. To this end, consider any other triple (r2, c2, L2, D2) ∈ R that satisfies
+r2 ≤ r1,
+c2 ≤ c1,
+(33a)
+L2 ≤ L∗(r1, c1),
+D2 ≤ D∗(r1, c1).
+(33b)
+We show by contradiction that all four inequalities must hold with equality. Notice that, (r2, c2)
+either satisfies (31) or (32).
+8Notice that the projections of P u
+i , Qu
+i and P d
+i , Qd
+i into the r-c plane are the same as the ones of the points Pi and Qi. As a
+result, the projections of △P u
+1 P u
+2 Qu
+2/△P d
+1 P d
+2 Qd
+2, △P u
+i−1P u
+i P u
+K/△P u
+i−1P u
+i P u
+K, and ⊟P u
+i Qu
+i Qu
+i+1P u
+i+1/⊟P d
+i Qd
+i Qd
+i+1P d
+i+1 into
+the r-c plane are △P ′
+1P ′
+2Q′
+2, △P ′
+i−1P ′
+iP ′
+K and ⊟P ′
+iQ′
+iQ′
+i+1P ′
+i+1, respectively.
+
+19
+1) Assume that (r2, c2) satisfies (32). If r2 < r1 or c2 < c1, then consider the uplink SCC
+subspace, one can verify that the points P u
+i−1, P u
+i and P u
+K are on the surface
+L = −
+1
+i(i − 1)c − 2
+Kir + 2i − 1
+i(i − 1).
+(34)
+Therefore, it must hold that
+L∗(r2, c2) > L∗(r1, c1),
+(35)
+because all the surfaces containing △P u
+i−1P u
+i P u
+K (i ∈ [2 : K − 1]) have positive directional
+derIVtives along (r2 − r1, c2 − c1) by (34). Since (r2, c2, L2, D2) ∈ R, we have L2 ≥
+L∗(r2, c2) and thus by (35), L2 > L∗(r1, c1), which contradicts (33). Therefore, it must
+hold that r2 = r1 and c2 = c1. Then obviously, L2 ≥ L∗(r2, c2) = L∗(r1, c1) and D2 ≥
+D∗(r2, c2) = D∗(r1, c1), thus all equalities in (33) hold.
+2) Assume now that (r2, c2) satisfies (31). Then, (r2, c2) must lie on at least one of the K −1
+facets
+△P ′
+1P ′
+2Q′
+2 or ⊟ P ′
+iQ′
+iQ′
+i+1P ′
+i+1,
+i ∈ [2 : K − 1],
+and it must not lie on the line segments P ′
+i−1P ′
+i, i ∈ [2 : K]. As the facets △P u
+1 P u
+2 Qu
+2,
+⊟P u
+i Qu
+i Qu
+i+1P u
+i+1 (i ∈ [2 : K −1]) in the uplink SCC subspace are all parellel to the c-axis,
+and so as the facets △P d
+1 P d
+2 Qd
+2, ⊟P d
+i Qd
+i Qd
+i+1P d
+i+1 (i ∈ [2 : K − 1]) in the downlink SCC
+facets, there exists c3 < c2 ≤ c1 such that (r2, c3) satisfies (32), and
+L∗(r2, c3) = L∗(r2, c2), D∗(r2, c3) = D∗(r2, c2).
+Therefore,
+L2 ≥ L∗(r2, c2) = L∗(r2, c3)
+(a)
+> L∗(r1, c1),
+(36)
+where (a) follows by proof step 1). But (36) contradicts with (33).
+From the above analysis, we conclude that, the set of all Pareto-optimal points of R is exactly
+all the quadruples (r, c, L∗(r, c), D∗(r, c)) ∈ F satisfying (32). Notice that those points are exactly
+the surface O defined in (10).
+REFERENCES
+[1] J. Dean and S. Ghemawat, “MapReduce: Simplified data processing on large clusters,” Sixth USENIX OSDI, Dec. 2004.
+[2] M. Isard, M. Budiu, Y. Yu, A. Birrell, and D. Fetterly, “Dryad: distributed data-parallel programs from sequential building
+blocks,” in Proc. the 2nd ACM SIGOPS/EuroSys’07, Mar. 2007.
+[3] S. Li, M. A. Maddah-Ali, Q. Yu, and A. S. Avestimehr, “A fundamental tradeoff between computation and communication
+in distributed computing,” IEEE Trans. Inf. Theory, vol. 64, no. 1, pp. 109–128, Jan. 2018.
+[4] Y. H. Ezzeldin, M. Karmoose, and C. Fragouli, “Communication vs distributed computation: An alternative trade-off curve,”
+in Proc. IEEE Inf. Theory Workshop (ITW), Kaohsiung, Taiwan, pp. 279–283, Nov. 2017.
+[5] Q. Yan, S. Yang, and M. Wigger, “A storage-computation-communication tradeoff for distributed computing,” in Proc. 2018
+15th Int. Symp. Wireless Commun. Sys. (ISWCS), Lisbon, Portugal, Aug. 28-31, 2018.
+[6] Q. Yan, S. Yang, and M. Wigger, “Storage, computation and communication: A fundamental tradeoff in distributed
+computing,” in Proc. IEEE Inf. Thoery Workshop (ITW), Guangzhou, China, Nov. 2018.
+[7] Q. Yan, S. Yang, and M. Wigger, “Storage-computation-communication tradeoff in distributed computing: Fundamental
+limits and complexity,” IEEE Trans. Inf. Theory, vol. 68, no. 8, pp. 5496-5512, Aug. 2022.
+[8] S. Li, M. A. Maddah-Ali, and A. S. Avestimehr, “A unified coding framework for distributed computing with straggling
+servers,” in Proc. IEEE Globecom Works (GC Wkshps), Washington, DC, USA, Dec. 2016.
+[9] Q. Yan, M. Wigger, S. Yang, and X. Tang, “A fundamental storage-communication tradeoff for distriubted computing with
+straggling nodes,” IEEE Trans. Commun. , vol. 68., no. 12, Dec. 2020.
+
+20
+[10] Q. Yan, M. Wigger, S. Yang, and X. Tang, “A fundamental storage-communication tradeoff for distriubted computing with
+straggling nodes,” In Proc. IEEE Int. Symp. Inf. Theory (ISIT), Paris, France, Jul. 7-12, 2019.
+[11] Q. Yu, S. Li, M. A. Maddah-Ali, and A. S. Avestimehr, “How to optimally allocate resources for coded distributed
+computing,” in Proc. IEEE Int. Conf. Commun. (ICC), 2017, Paris, France, 21–25, May 2017.
+[12] M. A. Attia and R. Tandon, “On the worst-case communication overhead for distributed data shuffling,” in Proc. 54th
+Allerton Conf. Commun., Control, Comput., Monticello, IL, USA, pp. 961–968, Sep. 2016.
+[13] M. A. Attia and R. Tandon, “Information theoretic limits of data shuffling for distributed learning,” in Proc. IEEE Glob.
+Commun. Conf. (Globlcom), Washington, DC, USA, Dec. 2016.
+[14] A. Elmahdy and S. Mohajer, “On the fundamental limits of coded data shuffling,” in Proc. IEEE Int. Symp. Inf. Theory,
+Vail, CO, USA, pp. 716–720, Jun. 2018.
+[15] L. Song, S. R. SrinIVsavaradhan, and C. Fragouli, “The benefit of being flexible in distributed computation,” in Proc.
+IEEE Inf. Theory Workshop (ITW), Kaohsiung, Taiwan, pp. 289–293, Nov. 2017.
+[16] S. R. SrinIVsavaradhan, L. Song, and C. Fragouli, “Distributed computing trade-offs with random connectivity,” in Proc.
+IEEE Int. Symp. Inf. Theory, Vail, CO, USA, pp. 1281–1285, Jun. 2018.
+[17] F. Li, J. Chen, and Z. Wang, “Wireless Map-Reduce distributed computing,” in Proc. IEEE Int. Symp. Inf. Theory, Vail,
+CO, USA, pp. 1286–1290, Jun. 2018.
+[18] E. Parrinello, E. Lampiris, and P. Elia, “Coded distributed computing with node cooperation substantially increases speedup
+factors,” in Proc. IEEE Int. Symp. Inf. Theory, Vail, CO, USA, pp. 1291–1295, Jun. 2018.
+[19] S. Li, Q. Yu, M. A. Maddah-Ali, and A. S. Avestimehr, “Edge-facilitated wireless distributed computing,” in Proc. IEEE
+Glob. Commun. Conf. (Globlcom), Washington, DC, USA, Dec. 2016.
+[20] S. Li, Q. Yu, M. A. Maddah-Ali, and A. S. Avestimehr, “A scalable framework for wireless distributed computing,”
+IEEE/ACM Trans. Netw., vol. 25, no. 5, pp. 2643–2653, Oct. 2017.
+[21] Q. Yan, X. Tang, and Q. Chen, “Placement delivery array and its applications,” in Proc. IEEE Inf. Theory Workshop (ITW),
+Guangzhou, China, Nov. 2018.
+[22] V. Ramkumar and P. V. Kumar, “Coded mapreduce schemes based on placement delivery array,” in Proc. IEEE Int. Symp.
+Inf. Theory, Paris, France, pp. 3087–3091, Jul. 2019.
+[23] Q. Yan, M. Cheng, X. Tang, and Q. Chen, “On the placement delivery array design for centralized coded caching scheme,”
+IEEE Trans. Inf. Theory, vol. 63, no. 9, pp. 5821–5833, Sep. 2017.
+[24] M. A. Maddah-Ali and U. Niesen, “Fundamental limits of caching,” IEEE Trans. Inf. Theory, vol. 60, no. 5, pp. 2856–2867,
+May 2014.
+[25] C. Shangguan, Y. Zhang, and G. Ge, “Centralized coded caching schemes: A hypergraph theoretical approach.” IEEE
+Trans. Inf. Theory, vol. 64, no. 8, pp. 5755-5766, Aug. 2018.
+[26] Q. Yan, X. Tang, Q. Chen, and M. Cheng, “Placement delivery array design through strong edge coloring of bipartite
+graphs,” IEEE Commun. Lett., vol. 22, no. 2, pp. 236–239, Feb. 2018.
+[27] P. Krishnan, L. Natarajan, and V. Lalitha, “An umbrella for data exchange: Applied to caching, computing, shuffling &
+rebalancing,” 2020 IEEE Inf. Theory Workshop (ITW), RIV del Garda, Italy, Apr., 2021.
+
diff --git a/C9E2T4oBgHgl3EQfSAeL/content/tmp_files/load_file.txt b/C9E2T4oBgHgl3EQfSAeL/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..39d84da13a20c4448aa2e83ddd64ad7a6eb84855
--- /dev/null
+++ b/C9E2T4oBgHgl3EQfSAeL/content/tmp_files/load_file.txt
@@ -0,0 +1,818 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf,len=817
+page_content='1  A Fundamental Tradeoff Among Storage,  Computation, and Communication for  Distributed Computing over Star Network  Qifa Yan Member, IEEE, Xiaohu Tang Senior Member, IEEE,  Meixia Tao Fellow, IEEE, and Qin Huang Senior Member, IEEE  Abstract  Coded distributed computing can alleviate the communication load by leveraging the redundant  storage and computation resources with coding techniques in distributed computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' In this paper, we  study a MapReduce-type distributed computing framework over star topological network, where all  the workers exchange information through a common access point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The optimal tradeoff among the  normalized number of the stored files (storage load), computed intermediate values (computation load)  and transmitted bits in the uplink and downlink (communication loads) are characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' A coded  computing scheme is proposed to achieve the Pareto-optimal tradeoff surface, in which the access point  only needs to perform simple chain coding between the signals it receives, and information-theretical  bound matching the surface is also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Index Terms  Storage, coded computing, communication, MapReduce, star network  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' INTRODUCTION  The rapid growth of computationally intensive applications on mobile devices has attracted  much research interest in designing efficient distributed computing frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' One of the most  important programing models for distributed computing is MapReduce [1], [2], which has been  utilized to deal with computation tasks with data sizes as large as tens of terabytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  MapReduce framework allows to assign multiple computation tasks to distributed nodes, where  each node only stores a subset of files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' This is done by decomposing each function to be computed  into a set of “map” functions and a “reduce” function, where each map function can be computed  from a batch of data, with the output called intermediate values (IVs), while the computation of  a “reduce” function needs to collect the IVs from all the data as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The whole procedure is  composed of three phases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', map, shuffle and reduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' In the map phase, each distributed node  computes the map functions on its local file batch assigned by the server and generates output  IVs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' in the shuffle phase, the nodes exchange their computed IVs to facitate each node to obtain  the IVs needed by its assigned reduce functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' in the reduce phase, each node computes its  assigned reduce functions by decoding all the corresponding IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yan and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Tang are with the Information Coding & Transmission Key Lab of Sichuan Province, CSNMT Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Coop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Centre (MoST), Southwest Jiaotong University, Chengdu 611756, China(email: qifayan@swjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='cn, xhutang@swjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Tao  is  with  the  Department  of  Electronic  Engineering,  Shanghai  Jiao  Tong  University,  Shanghai  200240,  China(email:mxtao@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Huang is with the School of Electronic and Information Engineering, Beihang University, Beijing 100191, China  (email:qinhuang@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='03788v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='IT]  10 Jan 2023  2  Recently, a coded distributed computing (CDC) scheme was proposed by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' [3], where  the files are stored multiple times across the distributed nodes in the map phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The IVs are also  computed multiple times accordingly, such that multicast opportunities are created for the shuffle  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' As a result, the communication load was reduced significantly compared to traditional  uncoded scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' It was proved in [3] that the scheme achieves the optimal communication load  for a given total storage requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Interestingly, the normalized number of files stored across  the nodes was termed computation load by Li et al, because each node calculates all the IVs that  can be obtained from the data stored at that node in the model therein, no matter if these IVs  are used or not in the subsequent phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Subsequently, Ezzeldin [4] and Yan et al [5], [6] found  that some IVs are computed but not used in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' For this reason, Yan et al reformulated  the problem as a tradeoff between storage, computation, and communication loads in [7], which  allows each node to choose any subset of IVs to compute from its stored files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Some interesting works that extend CDC have been proposed, for example, the technique was  combined with maximum distance separable (MDS) code in matrix-vector multiplication tasks to  resist stragglers in [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' stragglers with general functions are considered in [9], [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' the optimal  resource allocations are considered in [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' [12]–[14] investigated the iterative procedures of data  computing and shuffling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' [15] studied the case when each node has been randomly allocated  files;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' [16] investigated the case with random connectivity between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  The coded distributed computing technique is extended to wireless distributed computing  [17], [18], where the computation is typically carried out by the wireless devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Due to the  decentralized natural of the wireless networks, the nodes in wireless networks normally need a  central Access Point (AP) to exchange data, which leads to uplink and downlink communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  For example, smart-phone end users typically communicate with each other through a base station  in cellular networks, which operates in a star network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' In [19] and [20], Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' investigated  distributed computing in a wireless network where the nodes performs data shuffling through an  AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The optimal storage-communication tradeoff was characterized for both uplink and downlink  transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  In this paper, following the conventions of Ezzeldin [4] and Yan et al [7], we investigate  a distributed computing system with star network, where all nodes exchange IVs through an  AP, but each node is allowed to choose any arbitrary subset of IVs to compute from its stored  files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' In particular, in addition to the storage and computation loads as considered in [7], the  communication load includes both upload and download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The main contribution of this paper is  the characterization of the Pareto-optimal surface in the storage-computation-upload-download  space for distributed computing over star network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The idea is to form the same multicast  signals as in CDC scheme but compute less IVs by ignoring the un-used IVs in the map phase  in the uplink, and combine them through a simple chain coding to form the downlink signals at  the AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' It turns out that, for any given storage-computation pair, both the optimal upload and  download communication costs can be simultaneously achieved by a coded computing scheme  that oriented from CDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The information-theoretical bound matching the Pareto-optimal surface  is also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Paper Organization: Section II presents the system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Section III summarizes the main  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Section IV presents the coded computing scheme that achieves the optimal surface, and  Section V provides information-theoretical bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Finally, Section VI concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Notations: Let N+ be the set of positive integers, and F2 be the binary field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' For m, n ∈ N+,  denote the n-dimensional vector space over F2 by Fn  2, and the integer set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , n} by [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' If  m < n, we use [m : n] to denote the set {m, m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' We also use interval notations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=',  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 1: A Distributed Computing System with Star Network  [a, b] ≜ {x : a ≤ x ≤ b} and [a, b) ≜ {x : a ≤ x < b} for real numbers a, b such that a < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The  bitwise exclusive OR (XOR) operation is denoted by ⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' For sets we use upper case calligraphic  font, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', A, and for collections (sets of sets) we use upper case Greek letters with bold font,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' We denote a point in two or three dimensional Euclidean space by an upper case letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  A line segment with end points A1, A2 or a line through the points A1, A2 is denoted by A1A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  A triangle with vertices A1, A2, A3 is denoted by △A1A2A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' A trapezoid with the four edges  A1A2, A2A3, A3A4, and A4A1, where A1A2 is parallel to A3A4, is denoted by ⊟A1A2A3A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Let F be a set of facets, if the facets in F form a continuous surface, then we refer to this  surface simply as F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' SYSTEM MODEL  Let K, N, W, U, V be given positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Consider a star network consisting of K dis-  tributed computing nodes {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , K} that can communicate with each other through a common  AP, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Each of the K nodes can transmit signals to the AP through an  uplink channel, while the AP can broadcast signals to all the K nodes via a downlink channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Each of the K nodes aims to compute an individual function from a set of N files,  W = {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , wN},  wn ∈ FW  2 , ∀ n ∈ [N],  each of size W bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Node k aims to compute an output function  φk : FNW  2  → FU  2 ,  which maps all the files to a bit stream  uk = φk(w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , wN) ∈ FU  2  of length U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Assume that each output function φk decomposes as:  φk(w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , wN) = hk(fk,1(w1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , fk,N(wN)),  (1)  where  Files  Map  Node 3  IVs  Reduce  The set of files  X(X1, X2, X3)  X1  X2  个  Reduce  Reduce  Files  Files  Map  Map  IVs  IVs  Node 1  Node 24  Each “map” function fk,n is of the form  fk,n : FW  2 → FV  2 ,  and maps the file wn into the IV  vk,n ≜ fk,n(wn) ∈ FV  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  The “reduce” function hk is of the form  hk : FNV  2  → FU  2 ,  and maps the IVs  Vk ≜ {vk,n : n ∈ [N]}  into the output stream  uk = hk(vk,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , vk,N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Notice that one trivial decompositon is that, the map functions are identity functions and the  reduce functions are the output functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', gk,n(wn) = wn, and hk = φk, ∀ n ∈ [N], k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  But in practice, many output functions can be decomposed such that the main computation load is  dominated by the map functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' For example, in federated learning, it typically needs to collect  the sum of the gradients over all data blocks, where the map functions are used to compute the  gradients of the loss functions over a data block, while the reduce function is the sum operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  The described structure of the output functions φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , φK, allows the nodes to perform their  computation in the following three-phase procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  1) Map Phase: Each node k ∈ [K] chooses to store a subset of files Mk ⊆ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' For each file  wn ∈ Mk, node k computes a subset of IVs  Ck,n = {vq,n : q ∈ Zk,n},  where Zk,n ⊆ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Denote the set of IVs computed at node k by Ck, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=',  Ck ≜  �  n:wn∈Mk  Ck,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (2)  2) Shuffle Phase: The K nodes exchange some of their computed IVs through the AP via  upload and download sub-phases:  In the upload sub-phase, each node k generates a coded signal  Xk = ϕk (Ck)  of some length lk ∈ N and sends it to the AP, using a function  ϕk : F|Ck|V  2  → Flk  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  In the download sub-phase, receiving all the signals {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , XK}, the AP generates a signal  X = χ(X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , XK)  (3)  of length l ∈ N, and broadcasts it to all nodes, where the encoding function is  χ : Fl1+l2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='+lK  2  → Fl  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  5  3) Reduce Phase: Using the received signal X broadcast from the AP in the shuffle phase  and its own IVs Ck computed locally in the map phase, each node k now computes the IVs  (vk,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , vk,N) = ψk (X, Ck) ,  (4)  for some function  ψk : Fl  2 × F|Ck|V  2  → FNV  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Finally, it computes  uk = hk(vk,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , vk,N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (5)  To measure the storage, computation, and communication costs of the described procedure,  following the convention in [7], we introduce the following definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Definition 1 (Storage load).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Storage load r is defined as the total number of files stored across  the K nodes normalized by the total number of files N:  r ≜  �K  k=1 |Mk|  N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (6)  Definition 2 (Computation load).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Computation load c is defined as the total number of map  functions computed across the K nodes, normalized by the total number of map functions NK:  c ≜  �K  k=1 |Ck|  NK  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (7)  Definition 3 (Communication Load).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The communication load is characterized by the tuple  (L, D), where L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' D) is the upload (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' download) defined as the total number of the bits  sent by the K nodes (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' AP) during the upload (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' download) sub-phase, normalized by  the total length of all intermediate values NKV :  L ≜  �K  k=1 lk  NKV ,  D ≜  l  NKV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Remark 1 (Nontrivial Regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' In general, the non-trivial regime in our setup is  1 ≤ c ≤ r ≤ K,  (8a)  0 ≤ D ≤ L ≤ 1 − r  K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (8b)  For completeness, we justify them by the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Justification of (8a): Since each IV needs to be computed at least once somewhere, we  have c ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Moreover, the definition of Ck in (2) implies that |Ck| ≤ |Mk|K, and thus by  (6) and (7), c ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Finally, the regime r > K is not interesting, because in this case each  node stores all the files, and can thus locally compute all the IVs required to compute its  output function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' In this case, c ≥ 1, D ≥ 0 and L ≥ 0, can be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Justification of (8b): D ≥ 0 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' By (3), as the down-link signal X is created from the  upload signals X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , XK, D = L is sufficient to communicate all the received information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Finally, each node k can trivially compute |Mk| of its desired IVs locally and thus only  needs to receive N − |Mk| IVs from other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Thus, such an uncoded manner requires  an upload of L =  �K  k=1(N−|Mk|)V  NKV  = 1 − r  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  6  In the trivial case that the AP simply forwards all the receiving signals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , XK),  then D = L, and the model degrades to the distributed model without the AP as in [7], where  the non-trivial region on the triple (r, c, L) was 1 ≤ c ≤ r ≤ K, 0 ≤ L ≤ 1 − r  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Definition 4 (Fundamental SCC Region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' A Storage-Computation-Communication1 (SCC) quadru-  ple (r, c, L, D) satisfying (8) is achievable if for any ϵ > 0 and sufficiently large N, W, V , there  exist map, shuffle, and reduce procedures with storage load, computation load, upload and  download less than r + ϵ, c + ϵ, L + ϵ and D + ϵ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The fundamental SCC region is  defined as the set of all feasible SCC quadruple:  R = {(r, c, L, D) : (r, c, L, D) is feasible}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Definition 5 (Optimal Tradeoff Surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' An SCC quadruple (r, c, L, D) is called Pareto-optimal  if it is feasible and if no feasible SCC quadruple (r′, c′, L′, D′) exists so that r′ ≤ r, c′ ≤ c, L′ ≤ L  and D ≤ D′ with one or more of the inequalities being strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The set of all Pareto-optimal SCC  quadruples is defined as the optimal tradeoff surface:  O ≜ {(r, c, L, D) : (r, c, L, D) is Pareto-optimal}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  The goal of this paper is to characterize the fundamental SCC region R and the optimal  tradeoff surface O in our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' MAIN RESULTS  Before we present the main theorem, let us provide a toy example to illustrate the key idea  of the proposed achievable scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' An Toy Example for Achievable Scheme  Consider the case, where there aorre K = 3 nodes and N = 6 files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Each node wants to  compute an individual function from the N = 6 files as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2 illustrates the strategy  achieving the Pareto-optimal point (r, c, L, D) = (2, 4  3, 1  6, 1  9), where the uplink and downlink  transmissions are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='2(a) and 2(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2, the three nodes are denoted by three boxes with red, green and blue edges re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The top-most lines in each of the three boxes indicate the files stored at the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  The rectangle below this line indicates the map functions at the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The computed IVs  are depicted below the rectangle, where red circles, green squares, and blue triangles indi-  cate IVs {v1,1, · · · , v1,6}, {v2,1, · · · , v2,6}, and {v3,1, · · · , v3,6}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The dashed cir-  cles/squares/triangles stand for the IVs that are not computed from the stored files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The last  line of each box indicates the IVs that the node needs to learn during the shuffle phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  The N = 6 files  W = {w1, w2, w3, w4, w5, w6}  are partitioned into  �K  r  �  = 3 batches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', {w1, w2}, {w3, w4}, {w5, w6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' In the map phase, the  files {w1, w2} are simultaneously stored at nodes 1 and 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' the files {w3, w4} at nodes 1 and 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  and the files {w5, w6} at nodes 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' For each node, the computed IVs can be classified into  two types: the IVs that will be used by its own reduce function (the first line below the “map”  1The communication load includes both upload and download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  7  rectangle) and the IVs that will be used for transmission or decoding (the second and third lines  below the “map” rectangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  In the shuffle phase, during the upload sub-phase, each node creates a coded signal by XORing  two IVs and sends it to the AP as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2(a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', Nodes 1, 2 and 3 sends coded  IVs v1,1 ⊕ v3,3, v3,4 ⊕ v2,5 and v6,2 ⊕ v2,1, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' during the download sub-phase, the AP  combines the three received signals by a simple chain coding, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', the two downlink signals are  formed by XORing the signals from nodes 1 and 2, and the signals from 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  The combined signals are sent to all the three nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  In the reduce phase, for each node, since the two chain coded signals involve a coded signal  transmitted by itself, the node can decode the two coded signals from the other two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Moreover, from each of the coded singals, the node can further decode an IVs it needs, by  XORing the coding signal with one of its computed IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' For example, Node 1 first decodes the  two signals v3,4 ⊕ v2,5 and v2,6 ⊕ v1,2, then it can further decodes the IVs v2,5 and v2,6, since the  IVs v3,4 and v1,2 have been computed locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Finally, each node collects all IVs for its assigned  reduce function, and computes the final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Fundamental SCC Region and Optimal Tradeoff Surface  For each i ∈ [K], define two SCC quadrules  Pi ≜  �  i, i  �  1 − i − 1  K  �  , 1  i  �  1 − i  K  �  ,  1  i + 1  �  1 − i  K  ��  ,  Qi ≜  �  i, i, 1  i  �  1 − i  K  �  ,  1  i + 1  �  1 − i  K  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  In the following, we will use P u  i , Qu  i , P d  i , Qd  i to denote the projections of Pi, Qi into the uplink  and downlink SCC subspaces2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=',  P u  i ≜  �  i, i  �  1 − i − 1  K  �  , 1  i  �  1 − i  K  ��  ,  Qu  i ≜  �  i, i, 1  i  �  1 − i  K  ��  ,  P d  i ≜  �  i, i  �  1 − i − 1  K  �  ,  1  i + 1  �  1 − i  K  ��  ,  (9)  Qd  i ≜  �  i, i,  1  i + 1  �  1 − i  K  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  The main result of this paper is summarized in the following theorem, where the proofs are  provided in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The fundamental SCC region R is given by  R =  �  (r, c, L, D) : 1 ≤ c ≤ r ≤ K, L∗(r, c) ≤ L ≤ 1 − r  K , D∗(r, c) ≤ D ≤ L  �  ,  where L∗(r, c) is a function such that {(r, c, L∗(r, c)) : 1 ≤ c ≤ r ≤ K} forms the surface  Fu ≜ △P u  1 P u  2 Qu  2 ∪  K−1  ∪  i=2 △P u  i−1P u  i P u  K ∪  K−1  ∪  i=2 ⊟P u  i Qu  i Qu  i+1P u  i+1  2In this paper, we will refer r-c-L subspace as the uplink SCC subspace, and the r-c-D subspace the downlink SCC subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  The superscripts “u” and “d” indicate “uplink” and “downlink”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  8  (a)  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2: Illustration of the CDC for star network: (a) Uplink (b) Downlink  in the uplink SCC subspace, and D∗(r, c) is a function such that {(r, c, D∗(r, c)) : 1 ≤ c ≤ r ≤  K} forms the surface  Fd ≜ △P d  1 P d  2 Qd  2 ∪  K−1  ∪  i=2 △P d  i−1P d  i P d  K ∪  K−1  ∪  i=2 ⊟P d  i Qd  i Qd  i+1P d  i+1  in the downlink SCC subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The optimal tradeoff surface is given by  O =  K−1  ∪  i=2 {θ1Pi−1 + θ2Pi + θ3PK : θ1, θ2, θ3 ∈ [0, 1], θ1 + θ2 + θ3 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (10)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 3, the functions L∗(r, c) and D∗(r, c) are ploted for K = 10 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Notice that, by  setting r = c, we recover the optimal upload and download as investigated in [20]3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=',  3The measurement of communication load is up to a scalar “K” in [20] compared Definition 3, and a slightly difference in  assumption in [20] is that each node has a fixed storage load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Files  25  6  Map  Node 3  Computes  12  56  1:256  Needs  3  Files  3  2  4  3  4  Files  T9  1  Map  1④3  Map  3456  Computes  Computes 3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='.6  14  3:4:60  Needs  Needs  12  Node 1  Node 2Files  25  6  Map  Node 3  Computes  12  1256  Needs  3  2  Files  2  3  4  Files  3  4  5  T9  Map  Map  3451  16  Computes  Computes 3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='.6  3:4:60  Needs  Needs  12  Node 1  Node 29  1) the optimal upload for given storage is given by  L∗(r) ≜ Conv  �1  r  �  1 − r  K  ��  ,  which corresponds the curve formed by the line segments Qu  1Qu  2, Qu  2Qu  3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , Qu  K−1Qu  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  2) the optimal download for given storage is given by  D∗(r) ≜ Conv  �  1  r + 1  �  1 − r  K  ��  ,  which corresponds to the curve formed by the line segments Qd  1Qd  2, Qd  2Qd  3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , Qd  K−1Qd  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Observe that, the line segments Qu  i P u  i in the uplink SCC space and Qd  i P d  i in the downlink  space (i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , K) are parellel to the c-axis, which indicate that the computation load can be  saved to achieve L∗(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The length of the line segments indicates the amount of the computation  load that can be saved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Thus, with larger storage load r, the saving of computation load to  achieve L∗(r) and D∗(r) is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' It will be clear later that the saving on the computation load  is due to the fact that, under the assumption that each not computes all IVs it can computes,  some of the IVs computed are not used in neither generating the signal, nor in the decoding  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  The projections of the Pareto-optimal surface O into the uplink and downlink SCC space  correspond to the surfaces  Ou ≜ {P u  i−1P u  i P u  K : i ∈ [2 : K − 1]}  and  Od ≜ {P d  i−1P d  i P d  K : i ∈ [2 : K − 1]},  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Observe that, for a given feasible (r, c) pair, the optimal upload is strictly larger  than the optimal download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' We will see that this is achieved by performing some simple chain  coding at the AP to combine the signals from different nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Interestingly, both the upload and  download can be simultaneously achieved for a fixed (r, c) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Remark 2 (Relation to Results in [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' One can observe that, the surfaces composing L∗(r, c) and  Ou concide with the optimal communication load and the Pareto-optimal SCC tradeoff surface  in the setup where the nodes directly connect to each other through a shared link (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' [7, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  2]), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' It was showed in [7], by dropping the computations of the IVs that are not  used in the CDC scheme [3] as in [4], the resultant coded computing scheme can achieve the  corner points of the Pareto-optimal SCC tradeoff surface (which is same as Ou).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' In fact, in  our proposed scheme, each node performs the same map procedures as in [4], but the signals  are sent to the AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The AP performs a simple chain coding on the received signals to further  compress the length of the signals, which leads to a further decrease of the download compared  to the upload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' We will present the whole process in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' ACHIEVABILITY  Since the set O is exactly all the Pareto-optimal points of the set R (Appendix A), we  only need to prove the achievability of the hypersurface O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' We will derive a coded computing  scheme that achieves the SCC quadruple Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Moreover, for any fixed θ1, θ2, θ3 ∈ [0, 1] such that  10  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 3: The functions L∗(r, c) and D∗(r, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  θ1+θ2+θ3 = 1, divide the N files into three groups of sizes4 θ1N, θ2N and θ3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' By applying the  scheme achieving the points Pi−1, Pi and PK on the three groups of files, the resultant scheme  achieves the point P = θ1Pi−1 + θ2Pi + θPK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Thus, we only need to prove the achievability of  Pi, i ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Coded Distributed Computing for Star Network  We now describe the scheme achieving Pi for a fixed i ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Define  Ωi ≜ {T ⊆ [K] : |T | = i} ,  ∀ i ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  For i = K, PK = (K, 1, 0, 0) is trivial, since each node can simply store all the files and  computes their IVs as well as their reduce functions locally, with no communication loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Consider a fixed i ∈ [K − 1], the N files are partitioned into  �K  i  �  batches, each containing  ηi = N  �K  i  �  (11)  files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Each batch is then associated with a subset T of [K] of cardinality i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', an element in  Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Let WT denote the batch of the ηi files associated with set T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Then,  W = {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , wN} =  �  T ∈Ωi  WT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  4This requires that θ1, θ2, θ3 have to be rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' If any one is irrational, one can replace it by a rational number arbitrarily  close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Optimal Upload and Download, K = 10  pu  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='9 -  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='.The Function L*(r, c)  (/T)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='8 ~  -- The Function D*(r,c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='7  Plane r = c  Communication load (  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='6 ~  Pf,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='5  Q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='4 ~  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='3 ~  Qu  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='2 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Storage load (r)s  Pl0/ P10  2  3  5  4  10 Q1 /Qil 0  6  6  8  >  Computation load (c)11  Further let UT ,k be the set of IVs for output function φk that can be computed from the files in  WT :  UT ,k ≜ {vk,n : wn ∈ WT }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  We now describe the map, shuffle, and reduce procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  1) Map Phase: Each node k stores  Mk =  �  T ∈Ωi:k∈T  WT ,  and computes the IVs  Ck = C1  k ∪ C2  k,  (12)  where  C1  k =  �  T ∈Ωi:k∈T  UT ,k,  (13a)  C2  k =  �  T ∈Ωi:k∈T  �  q∈K\\T  UT ,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (13b)  In other words, for each batch T , each node k computes all the IVs for its own function  k, and all the IVs for the function q if node q does not have the batch T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  2) Shuffle Phase: For each element T ∈ Ωi and each index j ∈ K\\T , we partition the set  UT ,j into i smaller subsets  UT ,j =  �  U k  T ,j : k ∈ T  �  (14)  of equal size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  In the upload sub-phase, for each S ∈ Ωi+1 and k ∈ S, by (13b), node k can compute  the signal  Xk  S ≜  �  l∈S\\{k}  U k  S\\{l},l  from the IVs calculated during the map phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Node k thus sends the multicast signal  Xk =  �  Xk  S : S ∈ Ωi+1 such that k ∈ S  �  to the AP R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Thus, the AP R receives the signals X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , XK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  In the download sub-phase, for each S = {k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , ki+1} ∈ Ωi+1, the AP R creates a  signal,  XS ≜ (Xk1  S ⊕ Xk2  S , Xk2  S ⊕ Xk3  S , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , Xki  S ⊕ Xki+1  S  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (15)  Then the AP broadcast the signal  X ≜ {XS : S ∈ Ωi+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (16)  3) Reduce Phase: Notice that C2  k only contains the IVs vq,n where q ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Thus, by (12) and  12  (13a), during the shuffle phase each node k needs to learn all the IVs in  �  T ∈Ωi : k/∈T  UT ,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Fix an arbitrary T ∈ Ωi such that k /∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' From the received multicast message XT ∪{k},  since the signal Xk  T ∪{k} is generated by node k, by (15), node k can decode Xj  T ∪{k} for  all j ∈ T , where the signal  Xj  T ∪{k} =  �  l∈T ∪{k}\\{j}  U j  T ∪{k}\\{l},l  is sent by node j during the shuffle phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' For any fixed j ∈ T , node k can recover the  missing IV U j  T ,k through a simple XOR operation:  U j  T ,k = Xj  T ∪{k} ⊕  �  l∈T \\{j}  U j  T ∪{k}\\{l},l,  (17)  where U j  T ∪{k}\\{l},l is calculated at node k by (13b) and (14) for all l ∈ T \\{j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Moreover,  node k can decode UT ,k from  �  Xj  T ∪{k} : j ∈ T  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  by (14) and (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' After collecting all the missing IVs, node k can proceed to compute the  reduce function (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Remark 3 (Comparison with [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Compared to the coded computing scheme in [20], two  differences of the above scheme are:  1) In the map phase, each node only needs to compute the IVs described in (12) and (13),  because only those IVs are useful for creating or decoding the coded signals, while in  [20], all the IVs pertaining to the files in Mk are computed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', node k computes  �C ≜  �  T ∈Ωi,k∈T  �  q∈[K]  UT ,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (18)  This scheme in fact achieves the point Qi, which is inferior to Pi for i > 0 in computation  load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The idea of removing the redundancy has been proposed in the setup where the nodes  connects to each other directly through a bus link by Ezzeldin [4] and Yan et al [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  2) In (15), for any node set S of size i + 1, we used a simple chain coding on the signals to  form i signals, while in [20], it uses random coding on the signals {Xk  S : k ∈ S} to form  i coded signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The advantage of chain coding in (15) is obvious:  a) It has smaller encoding and decoding complexities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  b) It can be operated on the binary field F2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  c) The order of nodes in the chain can be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' It makes sense in some scenarios:  the signals {Xk  S : k ∈ S} may arrive at different time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Consider the case that  the signals arrive in the ordder Xk1  S , Xk2  S , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , Xki+1  S  , to perform the encoding (15), at  any time the AP only needs to keep one signal in its buffer, because each coordinate  in (15) only depends on two consecutive signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' While with random linear coding,  the AP typically have to wait for all signals {Xk  S : k ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Thus, the chain coding  can reduce the buffer size at the AP and the node to node delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  13  Remark 4 (PDA framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' In [7], [21], [22], a coded computing scheme was derived based on  placement delivery array (PDA), which was proposed in [23] to explore coded caching schemes  with uncoded placement [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' In particular, it turns out that the Maddah-Ali and Niesen’s coded  caching scheme corresponds to a special structure of PDA (referred to as MAN-PDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' It was  showed in [7] that, with any given PDA belonging to a special class (defined as PDA for  distributed computing (Comp-PDA)), one can always obtain a coded computing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The class  of PDAs achieving the Pareto-optimal tradeoff surface was characterized in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The advantage of  establishing the PDA framework is, various known PDA structure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', the constructions in [23],  [25], [26] can be directly utilized to obtain coded computing schemes with low file complexity5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  In our setup, similar connections between coded computing schemes and Comp-PDA can be  established, by following the same steps as in [7] for upload singals, and incoporating the chain  coding (15) on all multicast signals from the Comp-PDA for the downlink signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' For example,  the scheme described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2 can be derived from the PDA  �  �  ∗  1  ∗  1  ∗  ∗  ∗  ∗  1  �  � ,  for details of forming the upload signals in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2(a), one can refer to [7, Example 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Performance Analysis  We analyze the performance of the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  1) Storage Load: The number of batches in Mk is  �K−1  i−1  �  , each consisting of ηi files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Thus,  the storage load is  r = 1  N · K ·  �K − 1  i − 1  �  ηi = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (19)  2) Computation Load: Since C1  k ∩ C2  k = ∅, we have |Ck| = |C1  k| + |C2  k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' From (11), (13a), and  (13b), we have  |C1  k| =  �K − 1  i − 1  �  ηi = iN  K ,  |C2  k| =  �K − 1  i − 1  �  (K − i) · ηi  =  �  1 − i  K  �  i · N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Thus, the computation load is  c =  �K  k=1 |Ck|  NK  = i  �  1 − i − 1  K  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (20)  3) Communication Load: The number of signals that each node k transmits is  �K−1  i  �  , each  of size ηi·V  i  bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Thus, the length of the signal Xk is lk =  �K−1  i  � ηi·V  i  bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Therefore, the  5The file complexity of a coded computing scheme is defined as the smallest number of files required to implement the  scheme, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', the file complexity of the proposed scheme achiving Pi is  �K  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  14  upload is  L =  �K  k=1 lk  NKV  = 1  i ·  �  1 − i  K  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (21)  By (15) and (16), the AP R transmits  � K  i+1  �  i signals, each of size  ηi·V  i  bits, thus the  download is  D =  1  NKV ·  � K  i + 1  �  i · ηi · V  i  =  1  i + 1  �  1 − i  K  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (22)  From (19), (20), (21) and (22), we show the achievability of the SCC quadruple Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' CONVERSE  We need to prove that for any achievable (r, c, L, D) satisfying (8),  L ≥ L∗(r, c),  (23a)  D ≥ D∗(r, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (23b)  Consider a coded distributed computing scheme achieving (r, c, L, D), with file allocations M[K],  IV allocations C[K], uplink signals X[K] and downlink signal X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' By the decoding condition (4),  H(Vk|X, Ck) = 0,  ∀ k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Thus for any k ∈ [K],  H(Vk|X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , XK, Ck)  (a)  = H(Vk|X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , XK, X, Ck)  ≤ H(Vk|X, Ck)  = 0,  where (a) follows since the downlink signal X is determined by the uplink signals X[K] by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  That is, with the signals X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , XK and the locally computed IVs Ck, node k can decode  all the IVs it needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' As a result, the file allocations M[K], IV allocations C[K] and the uplink  singals X[K] consisitute an valid scheme for the distributed computing system where the nodes  are connected through a bus shared link directly, as investigated in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Therefore, by the results  in [7, Theorem 2], we have proved (23a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  We proceed to prove the (23b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' For any k ∈ [K] and nonempty S ⊆ [K]\\{k}, define  Bk,S ≜ {vk,n : vk,n is exclusively computed by the nodes in S},  �Bk ≜ {vk,n : vk,n is the computed by node k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Let bk,S be the cardinality of the set Bk,S and ˜bk be the cardinality of �Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Obviously, the subsets  {Bk,S : S ⊆ [K]\\{k}, S ̸= ∅} and �Bk form a partition of the IVs Vk, thus  ˜bk +  �  S⊆[K],S̸=∅  bk,S = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  15  For each j ∈ [K − 1], the set of IVs not computed locally but exclusively computed by j other  nodes are  Bj =  �  k∈[K]  �  S⊆[K]\\{k},|S|=j  Bk,S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Then the cardinality of set Bj is given by  bj ≜  �  k∈[K]  �  S⊆[K]\\{k},|S|=j  bk,S,  ∀j ∈ [K − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (24)  To prove the lower bound in (23b), we need the following two lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The entropy of the download signal X satisfy  H(X) ≥ V  K−1  �  j=1  bj  j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Proof: Assume that the AP holds all IVs Vk, then the access point can create the signal X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Consider the data exchange problem6 formed by the AP and the K nodes, where only the AP  sends the signal X to all the K nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Notice that, in this system, each bits in Bk,S is cached  at the AP and the nodes in S, but only demanded by node k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Thus, by the lower bound in [27,  Theorem 1],  H(X) ≥ V  �  k∈[K]  �  S⊆[K]\\{k}  1  (|S| + 1) + 1 − 1 · bk,S  = V  K  �  k=1  K−1  �  j=1  �  S⊆[K]\\{k},|S|=j  1  j + 1 · bk,S  (a)  = V  K−1  �  j=1  1  j + 1  K  �  k=1  �  S⊆[K]\\{k},|S|=j  bk,S  = V  K−1  �  j=1  bj  j + 1,  where in (a), we utilized (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  The following lemma was proved in [7, Lemma 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' The parameters b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , bK−1 defined in (24) satisfy  K−1  �  j=1  bj ≥ N(K − r),  K−1  �  j=1  (j − 1)bj ≤ (c − 1)NK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  6Data exchange problem was defined in [27], where each of the nodes holds a subset of the information bits, and request  another subset of information bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  16  For a fixed r ∈ [1 : K], and each i ∈ [K], define  ci ≜ 1 +  �  1 − r  K  �  (i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (25)  Let λi, µi ∈ R+ such that  λix + µi|x=ci−1 =  1  ci−1 + 1 − 2r/K ·  �  1 − r  K  �2  = 1  i  �  1 − r  K  �  ,  (26a)  λix + µi|x=ci =  1  ci + 1 − 2r/K ·  �  1 − r  K  �2  =  1  i + 1  �  1 − r  K  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (26b)  From (26a) and (26b), the following relationships hold:  λi = −  1  i(i + 1) < 0,  (27a)  µi = 2i − 1  i(i + 1)  �  1 − r  K  �  +  1  i(i + 1) > 0,  (27b)  λi + µi = 2i − 1  i(i + 1)  �  1 − r  K  �  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (27c)  Moreover, by its convexity over x ∈ [1, ∞), the function  1  x + 1 − 2r/K  �  1 − r  K  �2  − (λix + µi)  must be nonnegative outside the interval formed by the two zero points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=',  1  x + 1 − 2r/K  �  1 − r  K  �2  ≥ λix + µi,  ∀ x ∈ [1, ci−1] ∪ [ci, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Therefore,  1  cj + 1 − 2r/K  �  1 − r  K  �2  ≥ λicj + µi,  ∀ j ∈ [K − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (28)  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' we are ready to derive the lower bound for the download D:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='D ≥ H(X)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='NKV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='(a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='≥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='K−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='bj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='NK ·  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='j + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='(b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='N(K − r)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='K−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='bj ·  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='cj + 1 − 2r/K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='1 − r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='(c)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='≥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='N(K − r)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='K−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='bj(λicj + µi)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='(d)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='N(K − r)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='K−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='bj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='λi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='1 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='1 − r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='(j − 1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='+ µi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='λi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='NK ·  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='K−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='(j − 1)bj +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='λi + µi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='N(K − r) ·  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='K−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='bj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='(e)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='≥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='λi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='NK · (c − 1)NK +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='λi + µi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='N(K − r) · N(K − r)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='= λic + µi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='= −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='2i − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='Ki(i + 1)r −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='i(i + 1)c +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (29)  where (a) follows from Lemma 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' (b) and (d) follow from the definition of ci in (25);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' (c) follows  from (28);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' and (e) follows from Lemma 2 and the signs of λi and λi + µi in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Notice that the three points P d  i−1, P d  i and P d  K defined in (9) satisfy (29) with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Thus, the  inequalities above indicate that all the feasible points (r, c, L, D) must satisfy that the projection  into the download SCC space (r, c, D) must lie above the plane containing △P d  i−1P d  i P d  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Furthmore, D should be lower bounded by the optimal download even if each node computes  all the IVs that can be computed locally from their stored file, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', a similar setup as in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  The converse in [20] indicates that L is lower bounded as follows in the r-D plane7:  D ≥ Conv  �1  r  �  1 − r  K  ��  ,  r ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (30)  Finally, by the lower bounds in (29) for i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , K − 1 and (30), D is lower bounded by  D∗(r, c), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', the lower bound (23b) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' CONCLUSION  In this paper, the Pareto-optimal storage-computation-upload-download tradeoff surface is  characterized for the MapReduce distributed computing system, where the nodes have to exchage  intermediate values through an access point that can broadcast signals to all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' It turns  out that, for a given storage-computation pair (r, c), the optimal upload and download can be  simultaneously achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Information-theoretical bounds matching the achievable communication  load are provided for both uplink and downlink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  APPENDIX A  THE RELATION OF HYPERSURFACE O AND REGION R  We now prove that O is the Pareto-optimal surface of the region R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Obviously, all Pareto-  optimal points must lie on the surface  F = {(r, c, L∗(r, c), D∗(r, c)) : 1 ≤ c ≤ r ≤ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  7Although the setup in [20] assumes a fixed storage capacity at each node, the proof the following inequality do not rely on  this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  18  1  2  3  4  5  6  7  8  9  10  1  2  3  4  5  6  7  8  9  10  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 4: The projections of Pi and Qi (i ∈ [K]) to the storage-computation subspace (r-c plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Let the projections of points Pi and Qi into the r-c plane be P ′  i and Q′  i (i ∈ [K]), respectively8,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=',  P ′  i =  �  i, i  �  1 − i − 1  K  ��  ,  Q′  i = (i, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Let the projection of the surface F to the r-c plane be  F′ ≜ {(r, c) : 1 ≤ c ≤ r ≤ K} = △P ′  1P ′  KQ′  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Notice that, here the “projection” map is one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Moreover, F′ can be decomposed into (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 4)  F′ = △P ′  1P ′  2Q′  2 ∪  K−1  ∪  i=2 △P ′  i−1P ′  iP ′  K ∪  K−1  ∪  i=2 ⊟P ′  iQ′  iQ′  i+1P ′  i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Since the triangle △P u  1 P u  2 Qu  2 and the trapezoids ⊟P u  i Qu  i Qu  i+1P u  i+1 in the uplink SCC space  (i ∈ [2 : K − 1]) are parallel to c-axis, and so as the triangle △P d  1 P d  2 Qd  2 and the trapezoids  ⊟P d  i Qd  i Qd  i+1P d  i+1 in the downlink SCC space, all the points (r, c, L∗(r, c), D∗(r, c)) ∈ F such  that  (r, c) ∈ △P ′  1P ′  2Q′  2 ∪  K−1  ∪  i=2 ⊟P ′  iQ′  iQ′  i+1P ′  i+1\\  K−1  ∪  i=2 △P ′  i−1PiP ′  K  (31)  cannot be Pareto-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' In the following, we prove that, all the points (r, c, L∗(r, c), D∗(r, c))  such that  (r, c) ∈  K−1  ∪  i=2 △P ′  i−1P ′  iP ′  K  (32)  are Pareto-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Now fix a quadruple (r1, c1, L∗(r1, c1), D∗(r1, c1)) ∈ F that satisfies (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' We show that it is  Pareto-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' To this end, consider any other triple (r2, c2, L2, D2) ∈ R that satisfies  r2 ≤ r1,  c2 ≤ c1,  (33a)  L2 ≤ L∗(r1, c1),  D2 ≤ D∗(r1, c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (33b)  We show by contradiction that all four inequalities must hold with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Notice that, (r2, c2)  either satisfies (31) or (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  8Notice that the projections of P u  i , Qu  i and P d  i , Qd  i into the r-c plane are the same as the ones of the points Pi and Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' As a  result, the projections of △P u  1 P u  2 Qu  2/△P d  1 P d  2 Qd  2, △P u  i−1P u  i P u  K/△P u  i−1P u  i P u  K, and ⊟P u  i Qu  i Qu  i+1P u  i+1/⊟P d  i Qd  i Qd  i+1P d  i+1 into  the r-c plane are △P ′  1P ′  2Q′  2, △P ′  i−1P ′  iP ′  K and ⊟P ′  iQ′  iQ′  i+1P ′  i+1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  19  1) Assume that (r2, c2) satisfies (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' If r2 < r1 or c2 < c1, then consider the uplink SCC  subspace, one can verify that the points P u  i−1, P u  i and P u  K are on the surface  L = −  1  i(i − 1)c − 2  Kir + 2i − 1  i(i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  (34)  Therefore, it must hold that  L∗(r2, c2) > L∗(r1, c1),  (35)  because all the surfaces containing △P u  i−1P u  i P u  K (i ∈ [2 : K − 1]) have positive directional  derIVtives along (r2 − r1, c2 − c1) by (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Since (r2, c2, L2, D2) ∈ R, we have L2 ≥  L∗(r2, c2) and thus by (35), L2 > L∗(r1, c1), which contradicts (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Therefore, it must  hold that r2 = r1 and c2 = c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Then obviously, L2 ≥ L∗(r2, c2) = L∗(r1, c1) and D2 ≥  D∗(r2, c2) = D∗(r1, c1), thus all equalities in (33) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  2) Assume now that (r2, c2) satisfies (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Then, (r2, c2) must lie on at least one of the K −1  facets  △P ′  1P ′  2Q′  2 or ⊟ P ′  iQ′  iQ′  i+1P ′  i+1,  i ∈ [2 : K − 1],  and it must not lie on the line segments P ′  i−1P ′  i, i ∈ [2 : K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' As the facets △P u  1 P u  2 Qu  2,  ⊟P u  i Qu  i Qu  i+1P u  i+1 (i ∈ [2 : K −1]) in the uplink SCC subspace are all parellel to the c-axis,  and so as the facets △P d  1 P d  2 Qd  2, ⊟P d  i Qd  i Qd  i+1P d  i+1 (i ∈ [2 : K − 1]) in the downlink SCC  facets, there exists c3 < c2 ≤ c1 such that (r2, c3) satisfies (32), and  L∗(r2, c3) = L∗(r2, c2), D∗(r2, c3) = D∗(r2, c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Therefore,  L2 ≥ L∗(r2, c2) = L∗(r2, c3)  (a)  > L∗(r1, c1),  (36)  where (a) follows by proof step 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' But (36) contradicts with (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  From the above analysis, we conclude that, the set of all Pareto-optimal points of R is exactly  all the quadruples (r, c, L∗(r, c), D∗(r, c)) ∈ F satisfying (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Notice that those points are exactly  the surface O defined in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  REFERENCES  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Dean and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Ghemawat, “MapReduce: Simplified data processing on large clusters,” Sixth USENIX OSDI, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Isard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Budiu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Birrell, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Fetterly, “Dryad: distributed data-parallel programs from sequential building  blocks,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' the 2nd ACM SIGOPS/EuroSys’07, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Maddah-Ali, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Avestimehr, “A fundamental tradeoff between computation and communication  in distributed computing,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 64, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 109–128, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Ezzeldin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Karmoose, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content=' Yan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content=' Wigger, “Storage, computation and communication: A fundamental tradeoff in distributed  computing,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content='  [7] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content=' 5496-5512, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Maddah-Ali, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Avestimehr, “A unified coding framework for distributed computing with straggling  servers,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' IEEE Globecom Works (GC Wkshps), Washington, DC, USA, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [9] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Wigger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Tang, “A fundamental storage-communication tradeoff for distriubted computing with  straggling nodes,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' , vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content=' 12, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content='  20  [10] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Wigger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Tang, “A fundamental storage-communication tradeoff for distriubted computing with  straggling nodes,” In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content='  [11] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Maddah-Ali, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Avestimehr, “How to optimally allocate resources for coded distributed  computing,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content='  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Attia and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Tandon, “On the worst-case communication overhead for distributed data shuffling,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 54th  Allerton Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content=', Control, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', Monticello, IL, USA, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 961–968, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Attia and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Tandon, “Information theoretic limits of data shuffling for distributed learning,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' IEEE Glob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content='  [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Elmahdy and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Mohajer, “On the fundamental limits of coded data shuffling,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content=' Theory,  Vail, CO, USA, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 716–720, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' SrinIVsavaradhan, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Fragouli, “The benefit of being flexible in distributed computation,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content=' Theory Workshop (ITW), Kaohsiung, Taiwan, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 289–293, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content='  [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' SrinIVsavaradhan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Song, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Fragouli, “Distributed computing trade-offs with random connectivity,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+page_content=' Theory, Vail, CO, USA, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 1281–1285, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [17] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Chen, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Wang, “Wireless Map-Reduce distributed computing,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Theory, Vail,  CO, USA, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 1286–1290, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [18] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Parrinello, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Lampiris, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Elia, “Coded distributed computing with node cooperation substantially increases speedup  factors,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Theory, Vail, CO, USA, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 1291–1295, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Maddah-Ali, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Avestimehr, “Edge-facilitated wireless distributed computing,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' IEEE  Glob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' (Globlcom), Washington, DC, USA, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Maddah-Ali, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Avestimehr, “A scalable framework for wireless distributed computing,”  IEEE/ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2643–2653, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [21] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Tang, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Chen, “Placement delivery array and its applications,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' IEEE Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Theory Workshop (ITW),  Guangzhou, China, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [22] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Ramkumar and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Kumar, “Coded mapreduce schemes based on placement delivery array,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Theory, Paris, France, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 3087–3091, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [23] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Cheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Tang, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Chen, “On the placement delivery array design for centralized coded caching scheme,”  IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 63, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 5821–5833, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Maddah-Ali and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Niesen, “Fundamental limits of caching,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 60, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2856–2867,  May 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Shangguan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Zhang, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Ge, “Centralized coded caching schemes: A hypergraph theoretical approach.” IEEE  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 64, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 5755-5766, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [26] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Yan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Tang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Chen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Cheng, “Placement delivery array design through strong edge coloring of bipartite  graphs,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 236–239, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content='  [27] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Krishnan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Natarajan, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Lalitha, “An umbrella for data exchange: Applied to caching, computing, shuffling &  rebalancing,” 2020 IEEE Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=' Theory Workshop (ITW), RIV del Garda, Italy, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
+page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQfSAeL/content/2301.03788v1.pdf'}
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+arXiv:2301.00793v1  [stat.ML]  2 Jan 2023
+Causal Inference (C-inf) — closed form worst case
+typical phase transitions
+Agostino Capponi ∗ Mihailo Stojnic †
+Department of Industrial Engineering and Operations Research
+Columbia University, New York, NY 10027, USA
+Abstract
+In this paper we establish a mathematically rigorous connection between Causal inference (C-inf)
+and the low-rank recovery (LRR). Using Random Duality Theory (RDT) concepts developed in [46,48,50]
+and novel mathematical strategies related to free probability theory, we obtain the exact explicit typical
+(and achievable) worst case phase transitions (PT). These PT precisely separate scenarios where causal
+inference via LRR is possible from those where it is not. We supplement our mathematical analysis with
+numerical experiments that confirm the theoretical predictions of PT phenomena, and further show that
+the two closely match for fairly small sample sizes. We obtain simple closed form representations for the
+resulting PTs, which highlight direct relations between the low rankness of the target C-inf matrix and the
+time of the treatment. Hence, our results can be used to determine the range of C-inf’s typical applicability.
+Index Terms: Causal inference; Random duality theory; Algorithms; Matrix completion; Spar-
+sity.
+1
+Introduction
+The Causal inference (C-inf) discipline deals with design of methods for estimating causal effects in
+panel data settings, where a subset of the units are exposed to a treatment during some time periods. The
+goal is estimating counterfactual outcomes, i.e., that outcome for those units were the treatment not been
+applied for that period of time.
+Casual inference plays a key role in decision making, and is essential in business decisions, network design,
+medical sciences, and many others. It allows answering questions of the form: What would happen to a data
+center’s latency if a new congestion control algorithm were not used? What would have been the systolic
+blood pressure of a patient if the new drug were not given to her?
+This problem of estimating the counterfactual appears in many disciplines, including economics, finance,
+health, and social sciences (see, e.g. [2, 16, 18, 19, 37, 38, 58]), and computer science (see, e.g. [30–33]). The
+increasing availability of big data through digital services and smart sensors calls makes it possible to design
+efficient algorithmic techniques to address these fundamental questions in counterfactual estimation.
+The econometrics and social sciences communities have proposed three main approaches to causal in-
+ference: 1) the unconfoundedness (see, e.g. [19, 37]); 2) the synthetic control (see, e.g. [1, 2, 16]); and 3)
+the matrix completion (see, e.g. [3, 4, 20]. Matrix completion methods build upon the foundation works
+of [9, 11, 34]). Perhaps unexpectedly, all three methods heavily rely on mathematical, statistical, and ulti-
+mately algorithmic concepts with very deep roots in information theory. Our work is positioned within the
+third line of work that mathematically resembles the matrix completion (MC) problem.
+Our main contribution is to develop a rigorous connection between Causal inference (C-inf) from
+observational data and the low-rank recovery (LRR) problem in compressed sensing. Methodologically,
+we integrate Random Duality Theory (RDT) concepts developed in [46, 48, 50] with novel mathematical
+∗e-mail: ac3827@columbia.edu
+†e-mail: flatoyer@gmail.com
+1
+
+strategies related to free probability theory, and manage to obtain the exact explicit typical (and achievable)
+worst case phase transitions (PT). These PT provide a precise separation between regions of the parameter
+space where the counterfactual can be perfectly estimated via LRR from those regions where this is not
+possible for large sample sizes. Our numerical study complements our theoretical predictions by showing
+that theory and numerical simulation closely match even if the sample size is fairly small.
+2
+Mathematics of causal inference
+To put our contribution into a proper context, we begin by presenting the explicit causal inference (C-inf)
+↔ matrix completion (MC) connection.
+To introduce the most basic mathematical description of the main C-inf concepts we adopt the standard
+matrix completion (MC) terminology. As is well known the matrix completion problems belong to a class
+of the so-called low-rank recovery problems (LRR) which themselves belong to a broader class of the so-
+called structured recovery (SR) problems. In its most general way the description of the structured recovery
+problems starts by assuming that one has access to the observation vector y ∈ Rm given by the following
+yi = fi(Ai,:xsol),
+(1)
+where A ∈ Rm×n is a known system matrix (with rows Ai,:, 1 ≤ i ≤ m), xsol ∈ Rn is the unknown vector,
+and fi(·) : R −→ R are known real functions. In the treatments that will be of our interest here the functions
+fi(·) will be assumed to be identically linear, i.e. it will be assumed that fi(x) = x. Then the structured
+recovery assumes utilizing the xsol ’s a priori known structure to eventually recover it. Most often, that
+boils down practically to finding efficient algorithmic designs that take as inputs the observation vector y
+and the system matrix A and successfully output xsol or its a sufficiently close approximation. Moreover,
+under efficient algorithms one typically views those that run preferably provably in polynomial time. What
+typically differentiates between various forms of the SR problems are the structures that xsol possesses as
+well as the structure of the system matrix A. Along the same lines, what typically differentiates the level
+of success (or usefulness) that some of these forms might achieve is the ultimate theoretical and practical
+capabilities of the algorithms employed for their solving.
+The structured recovery problems have been the subject of an extensive research for a better half of the
+last century and many beautiful results appeared regarding their various theoretical and practical aspects.
+However, it is the emergence of compressed sensing (CS) about 20 years ago that almost singlehandedly
+made a key transformational change in how these problems and the surrounding research are perceived. As a
+result of such a perceptional change, the interest in the structured recovery, its importance, popularity, and
+ultimate usefulness skyrocketed to the heights unseen ever before. It is, of course, hard to pinpoint exactly
+what could be the secret behind the compressed sensing meteoric rise to success, but the overall conceptual
+simplicity probably contributed to some degree. We refer for more on the CS invention, simplicity, and
+further developments to e.g. [7,8,13,14,44,45,47–50,52]. At the same time, since (mathematically speaking)
+the CS is not precisely the main topic of our interest here, we, before proceeding further, just briefly mention
+that it basically assumes the above mentioned structured recovery setup with the sparsity of xsol being the
+underlying structuring.
+2.1
+Low rank recovery (LRR)
+While the mathematical problems typically seen in compressed sensing will not exactly match the key
+mathematical problems of this paper, the above mentioned low-rank recovery (LRR) ones will. To introduce
+the LRR problems, we first note that almost everything mentioned above for the generic structured recovery
+remains in place in the LRR scenarios. In fact, there will be only one key difference compared to the standard
+CS setup. The xsol (which is a sparse vector in the CS context) will within the LRR considerations be viewed
+as a vectorized unknown matrix Xsol and consequently the imposed a priori known structure will be the
+low-rankness of Xsol. This, of course, is nothing but the sparsity of the vector of the singular values of
+Xsol. In other words, in compressed sensing the unknown vector xsol itself is sparse, whereas in the LRR
+the vector of the singular values of the unknown matrix Xsol is sparse.
+2
+
+In more mathematical terms, the LRR can then be described in the following way. One first starts with
+the above mentioned vectorizing of matrix Xsol
+xsol = vec(Xsol),
+(2)
+with vec(·) stacking its matrix argument columns one after another (starting from the very first one) into
+a column vector. Assuming Xsol ∈ Rn×n, trivial dimensional adjustments then give A ∈ Rm×n2.
+The
+low-rankness (rank(Xsol) = k ≤ n) and the corresponding sparsity are imposed via the singular value
+decomposition (SVD)
+X = UΣV T ,
+(3)
+where
+σ(X) ≜ diag(Σ)
+and
+U T U = In×n
+and
+V T V = In×n,
+(4)
+with In×n being the identity matrix of size n × n and diag(·) being the operator that extracts the diagonal
+from its matrix argument and puts it into a column vector. When clear from the context, we will abbreviate
+and write just σ instead of σ(X). Moreover, σi(X) will stand for the i-th component of σ(X) (with σi
+often being its shorter version). As is of course well known, the elements of σ ∈ Rn are precisely the above
+mentioned singular values of X and the number of nonzero such elements is precisely the rank of Xsol. It
+is probably obvious, but we state it to ensure the overall clarity, that in typical SVD treatments in the
+mathematical literature one will often find in (4) Ik×k as a replacement for In×n. In other words, one will
+often find that the underlying identity matrix is of size k × k instead of size n × n. The reason for our choice
+is to ensure a complete parallelism between the LRR and the compressed sensing. Basically, this choice
+makes the underlying vector of the singular values visibly sparse (by this definition it will automatically have
+n − k zeros) and as such more in parallel with the corresponding sparse vector structure one typically finds
+in the compressed sensing setup. To further maintain the parallelism with compressed sensing, we also find
+it useful to introduce ℓ∗
+p(X) as the so-called ℓ∗
+p quasi-norm of X
+ℓ∗
+p(X) ≜ ℓp(σ(X)), p ∈ R+,
+(5)
+where, ℓp(·) is the standard vector ℓp (quasi-) norm, and to note that the following useful matrix-vector
+limiting ℓp(·) connections also hold
+ℓ∗
+0(Xsol) ≜ ℓ0(σ(Xsol)) = ∥σ(Xsol)∥0 = lim
+p−→0 ∥σ(Xsol)∥p = lim
+p−→0 ℓp(σ(Xsol)) = lim
+p−→0 ℓ∗
+p(Xsol).
+(6)
+One can then restate (1) adapted to fit the LRR context as
+yi = Ai,:xsol = Ai,:vec(Xsol)
+where
+ℓ∗
+0(Xsol) = ℓ0(σ(Xsol)) = ∥σ(Xsol)∥0 = k, k ∈ N,
+(7)
+and Ai,: being the i-th row of A. Finally one can summarize the LRR mechanism as:
+Generic LRR
+Observations – forming y from Xsol
+Given A and Xsol create y as
+y = Avec(Xsol), ℓ∗
+0(Xsol) = k, k ∈ N.
+(8)
+Observations – forming y from Xsol
+Given y and A from (8) can one efficiently
+recover Xsol back?
+It is not that difficult to see that ℓ∗
+0(Xsol) basically serves as a counting function that counts the number
+of the nonzero singular values of Xsol. Its analogy with the function ℓ0(xsol), that appears in the compressed
+sensing setup and counts the number of the nonzero elements of the unknown sparse vector, is rather obvious.
+Moreover, both such an analogy and the underlying sparsity that enables it suggest an algorithmic way that
+3
+
+one can try to employ to ultimately recover Xsol. By the above definition, the LRR is the inverse problem
+for (8) and can be posed as the so-called ℓ∗
+0-minimization optimization problem. Since such a minimization
+is a highly non-convex problem it is not known to be solvable in polynomial time. To design polynomially
+solvable heuristics one then, following into the compressed sensing footsteps, introduces the tightest convex
+norm relaxation concept and replaces the ℓ∗
+0- with the ℓ∗
+1-minimization.
+ℓ∗
+0-minimization (LRR/MC/C-inf – VMT)
+min
+X
+ℓ∗
+0(X)
+subject to
+y = Avec(X).
+(9)
+ℓ∗
+1-minimization (LRR/MC/C-inf – VMT)
+min
+X
+ℓ∗
+1(X)
+subject to
+y = Avec(X).
+(10)
+More on the history, usefulness, and applicability of the above introduced generic low rank recovery (LRR)
+via the tightest convex norm relaxation can be found in the introductory papers [35,41]. We also point out
+that we might on occasion refer to the above LRR description as the “vectorized matrix terminology” (VMT).
+Below, we wll also find it useful to introduce the very same LRR via the corresponding, so-called, “masking
+matrix terminology” (MMT).
+2.2
+Matrix completion (MC) – a special case of LRR
+The above is of course a generic LRR description. The matrix completion is a particular type of the LRR
+or, in technical terms, a special case of the above described LRR mathematical framework. In the matrix
+completion scenario one deals with a very particular type of system matrix A. Instead of (7) one then has
+yi = Avec(Xsol)
+where
+ℓ∗
+0(Xsol) = ℓ0(σ(Xsol)) = ∥σ(Xsol)∥0 = k, k ∈ N
+and
+∀p ∈ R+, ∥Ai,:∥p = 1.
+(11)
+This practically means that each row of system matrix A has exactly one element equal to one and all other
+elements equal to zero. In a way, one can think of A as being a cardinality m subset of rows of In2×n2.
+The reason for the appearance of such a matrix A is of course the origin of the matrix completion itself.
+Basically, the matrix A essentially emulates a “mask” that one puts on matrix X which allows reading out
+only m of its n2 elements. More on the origin of the matrix completion, its importance, and different related
+algorithmic considerations can be found in the introductory papers [9,41] as well as in many further studies
+that followed later on (see, e.g. [10,21–27,36]).
+To ensure the completeness of the overall presentation and to be in alignment with the standard rep-
+resentation of the matrix completion problem usually seen throughout the literature we reformulate (11)
+through the above mentioned masking matrices terminology. Let M ∈ Rn×n be a masking matrix such that
+Mi,j =
+�
+1,
+(i, j)-th element of Xsol is observed
+0,
+otherwise.
+(12)
+One can then describe creating the linear observations in the matrix completion as the following
+Y = M ◦ Xsol,
+(13)
+where ◦ stands for the component-wise multiplication. Clearly, ones in matrix M allow reading out cor-
+responding elements of Xsol while zeros block (mask) them. To fit in the above linear description in (11)
+one would then create A by removing all the zero rows from diag−1(vec(M))In2×n2 (diag−1(·) creates the
+diagonal matrix with the elements on the main diagonal equal to its vector argument and, in a way, is the
+inverse of the earlier introduced diag(·)). Consequently, y would be obtained as y = AXsol.
+2.3
+Causal inference (C-inf) – a special case of MC
+Finally, the causal inference, which will be the main topic of our interest, is yet another special case of the
+above low rank recovery framework. In fact, to be a bit more precise, the matrix completion is a special case
+4
+
+of the above LRR and the causal inference is a special case of the matrix completion itself. The connection
+between the matrix completion (MC) and the causal inference (C-inf) was established in [4]. Moreover, a
+very nice additional connections to the unconfoundedness and the synthetic control were established in [4]
+as well. Here, we will also work within the context of the matrix completion-causal inference connection.
+To that end we start by making this connection mathematically more precise. Namely, if one thinks of
+the matrix completion as a way of “masking” X and reading out the unmasked elements, then the causal
+inference does exactly the same thing while additionally imposing a particular structure on the mask itself.
+Let, as above, M ∈ Rn×n be a masking matrix. Then for a fixed l ≤ n one, in a causal inference context,
+has
+M ≜ M (l)
+and
+Mi,j = M (l)
+i,j =
+�
+1,
+if min(i, j) ≤ l
+0,
+otherwise.
+(14)
+The above is the so-called block causal inference setup. Figures 1 and 2 showcase the key difference
+between the generic matrix completion and the causal inference. In the generic matrix completion the mask
+matrix M can have zeros and ones located within the matrix in a basically arbitrary way. On the other
+hand, in the causal inference setup the structure of M is somewhat particular. In general, in a row of the
+matrix M the first zero can appear not later than the first zeros appeared in any of the previous rows. After
+a zero all other remaining elements in the same row must also be zero. For the block case of our interest
+here it is as shown in Figure 2.
+0
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+0
+1
+0
+0
+1
+0
+0
+1
+M =
+Matrix M – general matrix completion
+0 and 1 randomly scattered
+Figure 1: Matrix M – general matrix completion (MC) setup
+The rationale for the use of the block causal inference is the most easily understood if one views things in
+the time domain. Namely, if the columns of the masking matrix M represent time axis then the observations
+related to ceratin rows of the matrix will not be available after a fixed point in time. In the block scenario
+this point is fixed across the affected rows. However, it does not necessarily need to be fixed (for more in this
+direction we refer to [2] (in particular, the California tobacco example), [57] (in particular, the latent factor
+modeling in the context of the simultaneous/staggered treatment adoption), and to [5,6,39] (in particular,
+the health care applications) as excellent references for understanding the need of various C-inf scenarios).
+As this is the introductory paper, where we present the overall methodology, we selected the block causal
+inference scenario as probably the most representative and well-known one.
+In some of our companion
+papers we will show how the methodology that we are introducing here can be utilized to handle other C-inf
+scenarios as well.
+Under this casual inference assumption one then has the following for the collection of the observation Y
+5
+
+M =
+1
+Matrix M – block causal inference (C-inf)
+1
+0
+1
+0 and 1 grouped in blocks
+l × l block of all 1s
+l × (n − l) block of all 1s
+(n − l) × (n − l) block of all 0s
+(n − l) × l block of all 1s
+Figure 2: Matrix M ≜ M (l) – block causal inference (C-inf) setup
+in the matrix completion
+Y = M ◦ X = M (l) ◦ X.
+(15)
+We will more often than not avoid specifying superscript to make writing easier. From the context it will
+be clear if it should be there and, if so, what its value should be. To fit in the linear description of (11)
+one would create A in the following way: 1) start by choosing the first ln rows of In2×n2; and 2) then for
+any next set of n rows of In2×n2 continue by choosing its subset of first l rows while skipping the remaining
+n − l. Similarly, y would be obtained by stacking the columns of Y and skipping the elements Yi,j where
+min(i, j) > l. As mentioned above, we will find it useful later on to have (9) and (10) rewritten in the
+“masking matrix terminology” (MMT) as well;
+ℓ∗
+0-minimization (C-inf – MMT)
+min
+X
+ℓ∗
+0(X)
+subject to
+Y = M ◦ X.
+(16)
+ℓ∗
+1-minimization (C-inf – MMT)
+min
+X
+ℓ∗
+1(X)
+subject to
+Y = M ◦ X.
+(17)
+2.4
+C-inf ↔ MC connection via counterfactuals
+To understand where such a structure may come from, it is useful to connect it to the context of treat-
+ment/units/times following the terminology in [4]. In such context, one assumes that the matrix X contains
+observations about a certain set of, say, n units (e.g. individuals, subpopulations, and geographic regions)
+over a period of say, n, time instances. Assuming that the rows of X correspond to the units and the columns
+to the time instances, one wants to estimate the effects that a certain treatment may have on the treated
+units. A subset of the units (say those that correspond to the rows i > l) is then at time l exposed to an
+irreversible treatment (once the treatment starts its effects can not be reversed). Examples of treatments
+include health therapies, socio-economic policies, and taxes. To be able to appropriately assess the resulting
+treatment effects, in addition to having the values of X after the treatment, one would need to have the
+access to the so-called counterfactuals – the values of the treated units – had the treatment not been
+applied. Switching back to the matrix completion terminology, one would need to estimate (a presumably
+low rank) X while not having access to its portion covered by the block-mask M = M (l). In other words,
+one would need to solve (16) with M = M (l).
+6
+
+Of course, as the change in the structure of A or M does not prevent the utilization of the above ℓ∗
+0 −→ ℓ∗
+1
+relaxation concept, one typically employs it as a provably polynomial heuristic strategy to solve the matrix
+completion/causal inference problems approximately. While it is somewhat intuitive that, as the closest
+convex norm relaxation, the above ℓ∗
+1-minimization might produce a matrix similar to the unknown Xsol it
+is perhaps quite surprising that in certain scenarios it actually perfectly recovers the exact Xsol. Potential
+existence of such an ℓ∗
+0 − ℓ∗
+1 equivalence is rather remarkable phenomenon and determining when or how
+often it happens is pretty much the key task in the mathematical analysis of the convex norm relaxation
+based LRR algorithms. Along the same lines, answering this very same question will be precisely the main
+mathematical contribution of this paper.
+A lot of work will need to be performed, however, before we get to the point where we can say a few
+more concrete words about the ultimate mathematical contributions. We will utilize to a large degree some
+of the Random Duality Theory (RDT) concepts presented and discussed in details in a long line of
+work [42–46,48,50,51]). On top of that, quite a few additional mathematical concepts will be needed as well.
+While we will try to explain all the needed mathematical tools in sufficient detail, a solid level of familiarity
+with the RDT might be helpful.
+Before moving to a more thorough discussion particularly related to the mathematical analysis of the
+causal inference we first briefly digress to address a seemingly paradoxical situation regarding the level of
+simplicity/difficulty of the above LRR on the one side and the MC/C-inf, as its special cases, on the other.
+2.5
+Special cases are not necessarily simpler
+The above introduction of the matrix completion and the causal inference concepts through the generic LRR
+mechanism might portrait them as subproblems of a more general class of recovery problems. Moreover, one
+then may be tempted to believe that as such they can be both solved and analyzed in the very same way as
+the generic LRR problems. That would basically mean that all the results that one could conceivably create
+for the LRR would automatically translate to hold in a similar form in the MC and the C-inf scenarios. The
+part related to the solving of these problems is indeed true. The same algorithm (say ℓ∗
+1-minimization) that
+can be used as a heuristic for the generic LRR can be used for the MC and the C-inf as well. On the other
+hand, the part related to the analysis could not be further from the truth. Not only would not the results
+obtained for the LRR directly translate to the MC and the C-inf but they actually often might need to be
+proven in a completely different way.
+The key to fully understanding this paradox is in distinguishing the additional structuring of the unknown
+vector Xsol from the additional structuring of the system matrix A. When the unknown vectors/matrices
+are additionally structured it is quite likely that the corresponding performance analyses of the underlying
+algorithms are translatable (see, e.g.
+the companion paper [12]).
+On the other hand when the system
+matrices A are additionally structured then not only that the corresponding analyses might be difficult to
+translate but they also might actually need to be completely replaced. Along the same lines, since the MC
+and the C-inf are the special cases of the LRR with regard to the additional structuring of A, it is not a
+priori clear that the ability to analytically handle the corresponding generic LRR in any way guarantees the
+existence of such an ability when it comes to analytically handling the MC and the C-inf. We will see some
+aspects of this reasoning already in one of the sections that will follow later on.
+3
+Causal inference – ℓ∗
+0 − ℓ∗
+1 relaxation equivalence
+As mentioned earlier, solving the generic LRR (and consequently the C-inf as its a special case) might be
+difficult due to a highly non-convex objective function in (16). Various heuristics can be employed depending
+on the practical scenarios that one can face. In the mathematically most challenging so-called linear regime,
+the above mentioned ℓ∗
+1-minimization relaxation heuristic is typically viewed as the best known provably
+polynomial one.
+We adopt the same view in what follows and take it as a current benchmark for the
+algorithmic handling of the C-inf.
+As mentioned above, a rather remarkable feature of this heuristic is
+that sometimes it can actually solve the underlying problems exactly. When that happens we say that the
+following ℓ∗
+0 − ℓ∗
+1-equivalence phenomenon occurs.
+7
+
+ℓ∗
+0 − ℓ∗
+1-equivalence (C-inf):
+Let Xsol be the solution of (16) or (9) and let ˆX be a solution of (17) or (10) and set
+RMSE ≜ ∥vec( ˆX) − vec(Xsol)∥2.
+If and only if ( ˆX = Xsol and RMSE = 0)
+then
+(ℓ∗
+0 − minimization ⇐⇒ ℓ∗
+1 − minimization).
+(18)
+The above basically means that when the ℓ∗
+0 − ℓ∗
+1-equivalence happens the optimization problems in (16)
+and (17) are equivalent and as such replaceable by each other. That would, of course, be an ideal scenario
+where it would be basically possible to replace the non-convex optimization problem with the convex one
+without losing anything in terms of the accuracy of the obtained solutions. Since the mere existence of such a
+scenario is already a remarkable phenomenon we will in this paper be interested in uncovering the underlying
+intricacies that enable for it ro happen. Moreover, as it will turn out that its occurrence is not an anomaly
+but rather a consequence of a generic property, we will then raise the bar accordingly and attempt to provide
+not only the proof of the existence but also a complete analytical characterization of this property. This
+will include a full characterization as to how often and in what scenarios it might happen. To do so we will
+combine the Random Duality Theory (RDT) tools from [42–46,48,50,51] and several advanced sophisticated
+probabilistic concepts that we will introduce along the way in the sections that follow below.
+In the rest of this section we will focus on some algebraic ℓ∗
+0 − ℓ∗
+1-equivalence preliminaries conceptually
+borrowed from the RDT. We start things off with a generic LRR ℓ∗
+0 − ℓ∗
+1-equivalence result (the result is
+basically an adaptation of the general CS equivalence condition result from [44–46] to the corresponding one
+for the ℓ1 norm of the singular/eigenvalues (similar adaptation can also be found in [29])).
+Theorem 1. (ℓ∗
+0 − ℓ∗
+1-equivalence condition (LRR) – general X) Consider a ¯U ∈ Rn×k such that
+¯U T ¯U = Ik×k and a ¯V ∈ Rn×k such that ¯V T ¯V = Ik×k and a rank− k matrix Xsol = X ∈ Rn×n with all of its
+columns belonging to the span of ¯U and all of its rows belonging to the span of ¯V T . Also, let the orthogonal
+spans ¯U ⊥ ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) be such that U ≜
+� ¯U
+¯U ⊥�
+and V ≜
+� ¯V
+¯V ⊥�
+and
+U T U ≜
+� ¯U
+¯U ⊥�T � ¯U
+¯U ⊥�
+= In×n
+and
+V T V ≜
+� ¯V
+¯V ⊥�T � ¯V
+¯V ⊥�
+= In×n.
+(19)
+For a given matrix A ∈ Rm×n2 (m ≤ n2) assume that y = Avec(X) = Avec(Xsol) ∈ Rm and let ˆX be the
+solution of (17). If
+(∀W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n)
+− tr ( ¯U T W ¯V ) < ℓ∗
+1(( ¯U ⊥)T W ¯V ⊥),
+(20)
+then
+ℓ∗
+0 ⇐⇒ ℓ∗
+1
+and
+RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0,
+(21)
+and the solutions of (16) (or (9)) and (17) (or (10)) coincide. Moreover, if
+(∃W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n)
+− tr ( ¯U T W ¯V ) ≥ ℓ∗
+1(( ¯U ⊥)T W ¯V ⊥),
+(22)
+then there is an X from the above set of matrices with columns belonging to the span of ¯U and rows belonging
+to the span of ¯V such that the solutions of (16) (or (9)) and (17) (or (10)) are different.
+Proof. The proof is a trivial adaptation of the proof for symmetric matrices given in Appendix A.
+The condition in the theorem relates matrix W to the null-space of matrix A and as such is VMT based.
+In the MC and C-inf cases that are of our interest here, it is more convenient to deal with its an MMT
+analogue. Recalling on the proof of Theorem 1 from Appendix A and the origin and role of matrix W within
+that proof, one has that stating that W belongs to the null-space of A is basically equivalent to stating that
+M ◦ W = 0n×n. In other words, one has the equivalence between the following two sets
+(W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n) ⇐⇒ (W ∈ Rn×n|M ◦ W = 0n×n, W ̸= 0n×n).
+(23)
+8
+
+Continuing further in the spirit of the RDT the following corollary of the above theorem can be established
+as well.
+Corollary 1. (ℓ∗
+0−ℓ∗
+1-equivalence condition via masking matrix (MC/C-inf) – general X) Assume
+the setup of Theorem 1 with Xsol being the unique solution of (16) (or (9)). Let the masking matrix M ∈
+Rn×n have m ones and (n2 − m) zeros and let A be generated via M, i.e. let A be the matrix obtained after
+removing all the zero rows from diag−1(vec(M))In2×n2. If and only if
+min
+W,W T W=1,M◦W=0n×n
+tr ( ¯U T W ¯V ) + ℓ∗
+1(( ¯U ⊥)T W ¯V ⊥) ≥ 0,
+(24)
+then
+ℓ∗
+0 ⇐⇒ ℓ∗
+1
+and
+RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0,
+(25)
+and the solutions of (16) (or (9)) and (17) (or (10)) coincide.
+Proof. Follows immediately as a combination of (20), (22), and (23).
+Remark: Carefully comparing the conditions in (20) and (24) one can observe that a strict inequality is
+loosened up a bit at the expense of the uniqueness assumption. With a little bit of extra effort one may
+avoid this. However, to make writings below substantially easier we will work with a non-strict inequality.
+To analyze the optimization problem in (24) we follow into the footsteps of [45] where similar optimization
+problems were handled on multiple occasions through a very generic Lagrangian mechanism. The first step
+of such a procedure is writing down explicitly the optimization from (24)
+fpr(M; U, V ) ≜ min
+W
+tr ( ¯U T W ¯V ) + ℓ∗
+1(( ¯U ⊥)T W ¯V ⊥)
+subject to
+M ◦ W = 0n×n
+W T W = 1.
+(26)
+One can then write the corresponding Lagrangian and the Lagrange dual function to obtain
+L(W, Λ, Θ, γ)
+≜
+tr ( ¯U T W ¯V ) + ℓ∗
+1(( ¯U ⊥)T W ¯V ⊥) + Θ(M ◦ W) + γ
+�
+tr (W T W) − 1
+�
+=
+max
+Λ,ΛT Λ≤I
+�
+tr ( ¯U T W ¯V ) + tr (Λ(( ¯U ⊥)T W ¯V ⊥)) + Θ(M ◦ W) + γ
+�
+tr (W T W) − 1
+��
+=
+max
+Λ,ΛT Λ≤I
+�
+tr
+�
+( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)W
+�
++ γtr (W T W) − γ
+�
+,
+(27)
+and
+g(Θ, γ)
+≜
+min
+W L(W, Λ, Θ, γ)
+=
+min
+W
+max
+Λ,ΛT Λ≤I
+�
+tr
+�
+( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)W
+�
++ γtr (W T W) − γ
+�
+.
+(28)
+Utilizing the Lagrangian duality one then further has
+g(Θ, γ)
+=
+min
+W
+max
+Λ,ΛT Λ≤I
+�
+tr
+�
+( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)W
+�
++ γtr (W T W) − γ
+�
+≥
+max
+Λ,ΛT Λ≤I min
+W
+�
+tr
+�
+( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)W
+�
++ γtr (W T W) − γ
+�
+=
+max
+Λ,ΛT Λ≤I
+�
+− 1
+4γ tr
+�
+( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)T �
+− γ
+�
+.
+(29)
+Moreover,
+fpr(M; U, V )
+≥
+max
+Θ,γ g(Θ, γ)
+=
+max
+Λ,ΛT Λ≤I,Θ,γ
+�
+− 1
+4γ tr
+�
+( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)T �
+− γ
+�
+.
+9
+
+(30)
+We proceed by further optimizing over γ.
+fpr(M; U, V )
+≥
+max
+Λ,ΛT Λ≤I,Θ,γ
+�
+− 1
+4γ tr
+�
+( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)T �
+− γ
+�
+=
+max
+Λ,ΛT Λ≤I,Θ −
+�
+tr
+�
+( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)T �
+=
+−
+min
+Λ,ΛT Λ≤I,Θ
+�
+tr
+�
+( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)T �
+.
+(31)
+We now particularize the above to the block causal inference scenario. In the block C-inf scenario the
+matrix M is as defined in (14). For the concreteness and to facilitate writings later on we will introduce
+matrix I(l) and also keep in mind the following characterization of matrix M
+M matrix in causal inference (C-inf):
+M ≜ M (l) ≜ 1n×11T
+n×1 − I(l)(I(l))T 1n×11T
+n×1I(l)(I(l))T
+and
+I(l) ≜
+�
+0l×(n−l)
+I(n−l)×(n−l)
+�
+.
+(32)
+In the above definition/representation of M, 1/0 stand for vectors and matrices of all ones/zeros with the
+dimensions specified in their subscripts.
+To avoid overwhelming the notation, we may on occasion skip
+specifying the underlying dimensions of these vectors. However, they should be easy to infer from the overall
+context. Also, l is, of course, adjusted so that M has m nonzero elements, i.e. m elements equal to one
+which basically amounts to having the following identity to hold
+m
+=
+n2 − (n − l)2.
+(33)
+Using the above C-inf M in (31) the only terms that will be left in
+� ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T �
+after the
+optimization are those that correspond to the zero elements of M, i..e. the only ones where the presence of
+(and consequently the optimization over) Θ can not have an effect. This basically implies
+fpr(M; U, V ) ≥ −
+min
+Λ,ΛT Λ≤I
+�
+tr
+��
+(I(l))T � ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T �
+I(l)� �
+(I(l))T � ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T �
+I(l)�T �
+.
+(34)
+For the overall success of the whole machinery one would need that Λ in the above optimization can be chosen
+so that the overall optimum is nonnegative. This is then sufficient to establish the following alternative to
+Corollary 1.
+Corollary 2. (ℓ∗
+0 − ℓ∗
+1-equivalence condition via masking matrix (C-inf) – general X) Assume the
+setup of Theorem 1 and Corollary 1. Let I(l) be as in (32). Then
+C-inf perfectly succeeds: ℓ∗
+0 ⇐⇒ ℓ∗
+1
+and
+RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0
+If and only if
+∃Λ|ΛTΛ ≤ I
+and
+(I(l))T � ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T �
+I(l) = 0
+(35)
+Proof. The “if” part follows from Corollary 1, (34), and the above discussion. The “only if” part follows
+after noting that all the above inequalities in (29)-(34) are written for generic instructional purposes. Due to
+the underlying convexity and the strong duality they all actually can be replaced with equalities as well.
+10
+
+For the time being we will assume k ≤ l (later on this assumption will be rigorously justified). From (35)
+one then easily has
+Λ = ((I(l))T ¯V ⊥)−1(I(l))T ¯V ¯U T I(l)(( ¯U ⊥)T I(l))−1
+=⇒
+(I(l))T � ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T �
+I(l) = 0,
+(36)
+where (·)−1 stands for the pseudo-inverse. To make writing a bit easier we can set
+Λopt
+≜
+ΛV ΛT
+U
+ΛV
+≜
+((I(l))T ¯V ⊥)−1(I(l))T ¯V
+ΛU
+≜
+((I(l))T ¯U ⊥)−1(I(l))T ¯U.
+(37)
+Let λmax(·) be the maximum eigenvalue of its symmetric matrix argument. After combining (35)-(37) we
+conclude that if
+λmax(ΛT
+optΛopt) ≤ 1,
+(38)
+then (35) will be satisfied. After basic algebraic transformations (38) can also be rewritten as
+λmax(ΛT
+optΛopt) = λmax(ΛoptΛT
+opt) = λmax(ΛV ΛT
+UΛUΛT
+V ) = λmax(ΛT
+V ΛV ΛT
+UΛU) ≤ 1.
+(39)
+From (39) it is rather clear that the spectrum of ΛT
+V ΛV ΛT
+UΛU as well as the spectra of ΛT
+V ΛV and ΛT
+UΛU
+play an important role in the ℓ∗
+0 − ℓ∗
+1-equivalence. We first observe a worst case bound. Namely, since
+λmax(ΛT
+optΛopt) = λmax(ΛT
+V ΛV ΛT
+UΛU) ≤ λmax(ΛT
+V ΛV )λmax(ΛT
+UΛU)),
+(40)
+one has that if the individual spectra of ΛT
+V ΛV and ΛT
+UΛU do not exceed one then the ℓ∗
+0 − ℓ∗
+1-equivalence
+holds. Given the obvious importance of these spectra we will below look at them in more detail. Clearly,
+due to symmetry we need to focus on only one of them. To that end we start by observing
+((I(l))T ¯V ⊥)−1
+=
+( ¯V ⊥)T I(l) �
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1
+,
+(41)
+From (41) one quickly finds
+�
+((I(l))T ¯V ⊥)−1�T
+((I(l))T ¯V ⊥)−1
+=
+�
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1
+.
+(42)
+We can then write
+(I(l))T ¯V ¯V T I(l)
+=
+(I(l))T �
+I − ¯V ⊥( ¯V ⊥)T �
+I(l).
+(43)
+From (37) we have
+Q1 ≜ ΛT
+V ΛV
+=
+��
+((I(l))T ¯V ⊥)−1(I(l))T ¯V
+�T �
+((I(l))T ¯V ⊥)−1(I(l))T ¯V
+��
+=
+¯V T I(l) �
+((I(l))T ¯V ⊥)−1�T
+((I(l))T ¯V ⊥)−1 �
+(I(l))T ¯V
+�
+.
+(44)
+Now, we will find it more convenient to work with a slightly change version of matrix Q1. Namely, after
+combining (37), (42), and (43) we obtain
+Q
+≜
+��
+((I(l))T ¯V ⊥)−1�T
+((I(l))T ¯V ⊥)−1 �
+(I(l))T ¯V ¯V T I(l)��
+=
+��
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1 �
+(I(l))T �
+I − ¯V ⊥( ¯V ⊥)T �
+I(l)��
+=
+�
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1
+− I.
+(45)
+11
+
+Clearly, all the nonzero eigenvalues of Q1 and Q are identical. When k ≤ n−l then Q has all the eigenvalues
+of Q1 plus n − l − k extra zeros. On the other hand, when k ≥ n − l then Q1 has all the eigenvalues of Q
+plus k − (n − l) extra zeros. Since adding or removing zeros from the spectra will not change any of their
+features of our interests here, instead of working directly with Q1, we can work with Q. In particular, we
+have
+λmax(Q1) = λmax(ΛT
+V ΛV )
+=
+λmax
+��
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1�
+− 1 = λmax(Q).
+(46)
+We are now in position to establish a spectral alternative to Corollary 2.
+Corollary 3. (ℓ∗
+0 − ℓ∗
+1-equivalence condition via mask-modified bases spectra (C-inf) – general
+X) Assume the setup of Theorem 1 and Corollaries 1 and 2 with k ≤ l. Let λV and λU be defined as in
+(37). Then
+C-inf perfectly succeeds: ℓ∗
+0 ⇐⇒ ℓ∗
+1
+and
+RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0
+If and only if
+λmax(ΛT
+V ΛV ΛT
+UΛU) ≤ 1.
+(47)
+Moreover, if
+�
+λmax
+��
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1�
+− 1
+� �
+λmax
+��
+(I(l))T ¯U ⊥( ¯U ⊥)T I(l)�−1�
+− 1
+�
+≤ 1,
+(48)
+then again ℓ∗
+0 ⇐⇒ ℓ∗
+1 and RMSE = ∥vec( ˆX − vec(Xsol)∥2 = 0 and the C-inf perfectly succeeds as well.
+Proof. The first part follows from Corollaries 1 and 2, (36), (37), (39), the above discussion and some
+additional considerations while the second part follows by additional taking into account (40) and (46). We
+below present all the details split into three parts: the first two relate to the equivalence condition (equation
+(47)) and third one to (48).
+1) =⇒ – The “if part” of condition (47): Choosing Λ = Λopt
+Λopt
+≜
+ΛV ΛT
+U = −((I(l))T ¯V ⊥)−1(I(l))T ¯V ¯U T I(l)(( ¯U ⊥)T I(l))−1,
+(49)
+(where (·)−1 stands for the pseudo-inverse) one ensures
+(I(l))T � ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T �
+I(l) = 0.
+(50)
+Let λmax(·) be the maximum eigenvalue of its symmetric matrix argument. A combination of (35)-(50)
+ensures that if
+λmax(ΛT
+optΛopt) ≤ 1,
+(51)
+then Λopt satisfies (35). (51) is implied by (47) since
+λmax(ΛT
+optΛopt) = λmax(ΛUΛT
+V ΛV ΛT
+U) = λmax(ΛT
+V ΛV ΛT
+UΛU) ≤ 1,
+which suffices to complete the proof of the “if part”.
+2) ⇐= – The “only if part” of condition (47): Consider SVDs
+B ≜ (I(l))T V ⊥ = UBΣBV T
+B ,
+C ≜ (I(l))T U ⊥ = UCΣCV T
+C
+(52)
+with unitary UB, VB, UC, VC and diagonal (with no zeros on the main diagonal) ΣB, ΣC. Any Λ can be
+parameterized as
+Λ = VBHT + V ⊥
+B DT ,
+H ≜ VCE + V ⊥
+C F
+(53)
+12
+
+for some E, F, D and unitary V ⊥
+B and V ⊥
+C such that V T
+B V ⊥
+B = V T
+C V ⊥
+C = 0. Also, one can set Λ∗ and write
+the SVD of E
+Λ∗ ≜ VBET V T
+C ,
+E = UEΣEV T
+E ,
+(54)
+where UE, VE are unitary and ΣE is diagonal with entries on the main diagonal being nonzero and in
+ascending order. Let ue be the last column of UE (i.e. the eigenvector of EET that corresponds to its
+largest eigenvalue). Since ∥VCue∥2 = 1,
+λmax(ΛT Λ)
+≥
+uT
+e V T
+C ΛT ΛVCue
+=
+uT
+e V T
+C HHT VCue + uT
+e V T
+C DDT VCue
+≥
+uT
+e V T
+C (VCE + V ⊥
+C F)(VCE + V ⊥
+C F)T VCue
+=
+uT
+e EET ue = λmax(EET )
+=
+λmax(VCEET V T
+C ) = λmax(ΛT
+∗ Λ∗).
+(55)
+If Λ satisfies the condition of (35) then a combination of (35) and (52)-(54) gives
+(I(l))T ¯V ¯UI(l) + BΛ∗CT = 0,
+(56)
+and a combination of (37), (52), and (56) gives
+ΛV ΛT
+U = −B−1(I(l))T ¯V ¯UI(l)(CT )−1 = Λ∗.
+(57)
+Finally, for Λ that fits (35), from (55) and (57) one has
+1
+≥
+λmax(ΛT Λ) > λmax(ΛT
+∗ Λ∗) = λmax(Λ∗ΛT
+∗ )
+=
+λmax(ΛV ΛT
+UΛUΛT
+V ) = λmax(ΛT
+V ΛV ΛT
+UΛU),
+(58)
+which completes the proof of the “only if part”.
+3) Suffciency of the condition (48): Since
+λmax(ΛT
+V ΛV ΛT
+UΛU) ≤ λmax(ΛT
+V ΛV )λmax(ΛT
+UΛU),
+(59)
+one has that if the individual spectra of ΛT
+V ΛV and ΛT
+UΛU do not exceed one then the ℓ∗
+0 − ℓ∗
+1-equivalence
+holds. Due to symmetry we focus only on one of them. First we observe
+((I(l))T ¯V ⊥)−1 = ( ¯V ⊥)T I(l) �
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1
+,
+and quickly find
+�
+((I(l))T ¯V ⊥)−1�T
+((I(l))T ¯V ⊥)−1 =
+�
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1
+.
+(60)
+We also note
+(I(l))T ¯V ¯V T I(l)
+=
+(I(l))T �
+I − ¯V ⊥( ¯V ⊥)T �
+I(l),
+(61)
+and
+Q1
+≜
+ΛT
+V ΛV
+=
+��
+((I(l))T ¯V ⊥)−1(I(l))T ¯V
+�T �
+((I(l))T ¯V ⊥)−1(I(l))T ¯V
+��
+=
+¯V T I(l) �
+((I(l))T ¯V ⊥)−1�T
+((I(l))T ¯V ⊥)−1 �
+(I(l))T ¯V
+�
+.
+(62)
+13
+
+A combination of (60), (61), and (62) produces
+Q
+≜
+��
+((I(l))T ¯V ⊥)−1�T
+((I(l))T ¯V ⊥)−1 �
+(I(l))T ¯V ¯V T I(l)��
+=
+��
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1 �
+(I(l))T �
+I − ¯V ⊥( ¯V ⊥)T �
+I(l)��
+=
+�
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1
+− I.
+(63)
+Since all the nonzero eigenvalues of Q1 and Q are identical
+λmax(ΛT
+V ΛV ) = λmax(Q1) = λmax(Q).
+(64)
+Repeating the above with V replaced by U, Q1 by Q⊥
+1 , and Q by Q⊥ one arrives at the following analogue
+of (64)
+λmax(ΛT
+UΛU) = λmax(Q⊥
+1 ) = λmax(Q⊥).
+(65)
+A combination of (59) and (62) - (65) completes the proof of the condition (48)’s sufficiency for the ℓ∗
+0 − ℓ∗
+1-
+equivalence.
+All the three above corollaries provide useful characterization of the ℓ∗
+0 − ℓ∗
+1-equivalence. Depending on
+what kind of scenario one faces and what kind of numerical/computational/statistical resources might be
+available each of them could be used. In the following sections we will focus on a particular type of analysis
+that will primarily relate to the spectral characterizations given in Corollary 3.
+4
+Typical worst case analysis of the ℓ∗
+0 − ℓ∗
+1-equivalence
+In this section we provide an analysis that sheds a bit more light on when the conditions from Corollary 3 are
+indeed met. We will work in a typical statistical scenario. We will first assume that V and U are statistical
+objects and under such an assumption we will try to see if there are regimes where the ℓ∗
+0 − ℓ∗
+1-equivalence
+generically holds. There are of course many valid candidates for the statistics of V and U. We will assume
+the most generic typical uniformly random scenario from the spectral theory. That means that both ¯V and ¯U
+will be Haar distributed. The experts in sparse recovery will quickly recognize that this is in a way analogous
+to assuming that the locations of the nonzero components of a k-sparse vector in the standard compressed
+sensing are uniformly randomly chosen. Apart from the RDT considerations from [42,43,45,46,48,50] and
+the high-dimensional geometry considerations from [13,15] we are unaware of any other techniques that can
+avoid assumptions of this type and still achieve the ultimate exact phase transition (PT) characterizations
+for the conditions of the type similar to the one appearing in Theorem 1 and Corollaries 1-3.
+Conducting the analysis assuming the statistical nature of V and U we will uncover a very interesting
+and rather remarkable connection, among three a priori not necessarily related fields: 1) the compressed
+sensing (CS), 2) the causal inference (C-inf), and 3) the free probability theory (FPT). As the connection
+between the former two exists even outside a statistical context we were able to partially incorporate it in
+our earlier discussion presented in the previous sections. On the other hand, in the sections that follow, we
+will deepen our understanding of such a connection while relating it to the free probability theory and a
+collection of very generic concepts from the modern spectral theory of random matrices.
+4.1
+Free probability theory (FPT) – preliminaries
+Since this is the introductory paper where we are establishing the connection between the causal inference
+and the free probability theory (FPT) we will find it useful to first, in this subsection, recall on some
+FPT basics. As the FPT theory is mathematically very deep and involved we will restrict ourselves to the
+introduction of the basic FPT definitions, the explanations of the key technical results, and finally to a brief
+description related to the practical utilization of these results. After that, in the sections that follow, we will
+see how some of the introduced FPT concepts can be incorporated to strengthen our understanding of the
+causal inference itself.
+14
+
+The FPT is, of course, a very generic and abstract concept. Below we focus on some of its key implications
+related to the spectral theory of random matrices and start by sketching a bit of main motivation behind the
+FPT. That effectively means that we start with the simplest possible matrices which are of course scalars.
+4.1.1
+Basics of FPT – random scalar variables
+It is well known that if one has two independent random variables A and B with respective pdfs fA(·)
+and fB(·), then the standard way of determining the distribution of their sum or product goes through the
+characteristic functions and the corresponding inverse Fourier transform considerations. To be a bit more
+concrete, one first recognizes that the individual characteristic functions for both variables are given as
+FA(jw)
+≜
+EejwA =
+�
+ejwafA(a)da ≜ F(fA(a))
+FB(jw)
+≜
+EejwB =
+�
+ejwbfB(b)db ≜ F(fB(b)).
+(66)
+Assuming that
+C = A + B,
+(67)
+we analogously to (66) also have
+FC(jw) = EejwC =
+�
+ejwcfC(c)dc.
+(68)
+Moreover,
+FC(jw) = EejwC = Eejw(A+B) = EejwAEejwB = FA(jw)FB(jw) =
+�
+ejwafA(a)da
+�
+ejwbfB(b)db,
+(69)
+and finally
+fC(c) = F−1(FC(jw)).
+(70)
+It is then easy to see that (69) and (70) are sufficient to determine the pdf of C = A + B starting from the
+individual pdfs of A and B. The key that leads to the success of the above mechanism is the introduction of
+the Fourier transform and the so-called characteristic function. It turns out that in the transform’s domain
+the sum of random variables in a way corresponds to their product and, as a consequence, one can successfully
+separate them and then rely on their individual pdfs. While this methodology is fairly simple and has been
+known for almost two centuries, the existence of a similar one for matrices was not discovered until only a
+couple of decades ago. Moreover, the path to its discovery turned out to be more thornier and unpredictable
+than one could have ever imagined.
+The work od Dan Voiculescu on group theories (see, e.g. [54–56])
+uncovered it in an almost by-product type of way. Of course, due to an enormous practical importance it
+immediately drew a substantial interest and in the years that followed immediately after its discovery a few
+nice results appeared that helped make it presentable in a relatively simple and easily understandable way.
+We below follow into the same footsteps, leave all the abstractions out, and focus on presenting how the
+main FPT mechanism actually works (more details can be found in e.g. [17,28,40,53–56]).
+4.1.2
+Basics of FPT – random matrix variables
+As was the case above for scalars, we here also start with two random variables, A and B. This time though,
+these two variables are symmetric matrices, i.e. A = AT ∈ Rn×n and B = BT ∈ Rn×n. We will also assume
+large n regime and that the eigenspaces of these matrices are Haar distributed. Moreover, we will assume
+that their individual respective spectral laws are fA(·) and fB(·). Similarly to what we showed above in
+the scalar case, we will here also rely on introducing distributional transform. However, differently from
+the scalar case, here we will introduce not one but three different transforms. We start with the so-called
+15
+
+Stieltjes transform (or as we will often call it G-transform) of a pdf f(·)
+G(z)
+≜
+�
+If
+f(x)
+z − xdx,
+z ∈ C \ If,
+(71)
+where If is the domain of f(·). One then also has the inverse relation (somewhat analogous to the above
+relation between the inverse Fourier and the underlying pdf of the sum of random variables)
+f(x) = lim
+ǫ→0+
+G(x − iǫ) − G(x + iǫ)
+2iπ
+or
+f(x) = − lim
+ǫ→0+
+imag(G(x + iǫ))
+π
+.
+(72)
+For the above to hold it makes things easier to implicitly assume that f(x) is continuous. We will, however,
+utilize it even in discrete (or semi-discrete) scenarios since the obvious asymptotic translation to continuity
+would make it fully rigorous. A bit later though, when we see some concrete examples where things of this
+nature may appear, we will say a few more words and explain more thoroughly what exactly can be discrete
+and how one can deal with such a discreteness. In the meantime we proceed with general principles not
+necessarily worrying about all the underlying technicalities that may appear in scenarios deviating from the
+typically seen ones and potentially requiring additional separate addressing. To that end we continue by
+considering the R(·)- and S(·)-transforms that satisfy the following
+R(G(z)) +
+1
+G(z) = z,
+(73)
+and
+S(z) =
+1
+R(zS(z))
+and
+R(z) =
+1
+S(zR(z)).
+(74)
+Let fA(·) and fB(·) be the spectral distributions of A and B and let RA(z)/SA(z) and RB(z)/SB(z) be their
+associated R(·)-/S(·)-transforms. One then has the following
+Key Voiculescu’s FPT concepts [54, 55]:
+C
+=
+A + B
+=⇒
+RC(z)
+=
+RA(z) + RB(z)
+C
+=
+AB
+=⇒
+SC(z)
+=
+SA(z)SB(z).
+(75)
+Now it is relatively easy to see that (71)-(75) are sufficient to determine the spectral distribution of the sum
+or the product of two independent matrices with given spectral densities and the Haar distributed bases of
+eigenspaces. The above is of course generic principle. It can be applied pretty much always as long as one
+has access to the statistics of the underlying matrices A and B. In the following section we will raise the bar
+a bit higher and show that in the case of the causal inference one can use all of the above in such a manner
+that eventually all the quantities of interest are explicitly determined. Moreover, although the methodology
+may, on occasion, seem a bit involved the final results will turn out to be presentable in fairly neat and
+elegant closed forms.
+4.2
+Uncovering the C-inf ←→ FPT connection
+We are now in position to finally move to one of the key aspects of this paper, namely the uncovering of a
+rather unexpected connection between the C-inf and the FPT. The first part of the connection is in a way
+implicit and includes what we presented in in the previous section. Namely, utilizing the key compressed
+sensing concepts and the Random duality theory (RDT) we connected the success of the causal inference to
+behavior of ceratin algebraic structures. In particular, we have established the so-called ℓ∗
+0 − ℓ∗
+1-equivalence
+as the key concept in determining the ultimate level of success of C-inf. The second part of the connection
+builds on the first and proceeds by analyzing the ℓ∗
+0 − ℓ∗
+1-equivalence via the FPT machinery. In the sections
+that follow we provide such a very detailed and self-contained analysis.
+16
+
+4.2.1
+The spectral approach to the analysis of the ℓ∗
+0 − ℓ∗
+1-equivalence
+As mentioned earlier, we below rely on the spectral characterization of the ℓ∗
+0 − ℓ∗
+1-equivalence provided in
+Corollary 3. Clearly, determining the spectrum of λT
+V λV λT
+UλU would be sufficient to determine when the
+ℓ∗
+0 − ℓ∗
+1-equivalence occurs (in fact, determining the edges of the spectrum is sufficient as well). While we
+will in the sections that follow below indeed determine the spectrum of λT
+V λV λT
+UλU, here we note that in
+the worst case that may not be necessary. Namely, in the worst case one has
+λmax(λT
+V λV λT
+UλU) ≤ λmax(λT
+V λV )λmax(λT
+UλU).
+(76)
+In the large n limit due to the concentrations and identically Haar distributed V and U one also has
+λmax(λT
+V λV λT
+UλU) ≤ λmax(λT
+V λV )λmax(λT
+UλU) −→
+�
+λmax(λT
+V λV )
+�2 .
+(77)
+Moreover, in the worst case, U = V , the equality is actually achieved since
+λmax(λT
+V λV λT
+V λV ) = λmax((λT
+V λV )2) =
+�
+λmax(λT
+V λV )
+�2 .
+(78)
+This basically means that in the worst case it is sufficient to consider only the spectrum of Q with
+Q ≜ λT
+V λV .
+(79)
+4.2.2
+The FPT analysis of the spectrum of Q
+We start by recalling from (44) and (45)
+Q1
+≜
+ΛT
+V ΛV
+Q
+≜
+�
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1
+− I
+Sp(Q1)
+⇐⇒\0
+Sp(Q),
+(80)
+where Sp(·) stands for the spectrum of the matrix argument and ⇐⇒\0 means the equivalence of the parts
+of the spectra outside the zero eigenvalues. It is rather obvious that it will then be sufficient to handle the
+spectrum of
+D
+≜
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l).
+(81)
+Consider Haar distributed ¯U ⊥
+D ∈ Rn×(n−l) with ( ¯U ⊥
+D)T ¯U ⊥
+D = I(n−l)×(n−l) and let
+UD
+=
+� ¯UD
+¯U ⊥
+D
+�
+with
+U T
+DUD = In×n.
+(82)
+Also, we assume that ¯U ⊥
+D (and UD) are independent of ¯V ⊥. After setting
+¯D
+≜
+(I(l))T U T
+D ¯V ⊥( ¯V ⊥)T UDI(l),
+(83)
+we have that the spectra of D and ¯D are statistically identical, i.e.
+Sp(D) ≜ Sp((I(l))T ¯V ⊥( ¯V ⊥)T I(l)) ⇐⇒P Sp((I(l))T U T
+D ¯V ⊥( ¯V ⊥)T UDI(l)) ≜ Sp( ¯D),
+(84)
+where ⇐⇒P stands for the statistical/probabilistic equivalence. Two facts enable the above statistical iden-
+tity: 1) the spectrum of the projector ¯V ⊥( ¯V ⊥)T does not change under pre- and post-unitary multiplications
+on both sides; and 2) the Haar structure of ¯V ⊥ remains preserved. Modulo zero eigenvalues, we then further
+have
+Sp((I(l))T U T
+D ¯V ⊥( ¯V ⊥)T UDI(l)) ⇐⇒P\0 Sp( ¯V ⊥( ¯V ⊥)T UDI(l)(I(l))T U T
+D) ⇐⇒ Sp( ¯V ⊥( ¯V ⊥)T ¯U ⊥
+D( ¯U ⊥
+D)T ), (85)
+17
+
+where, similarly as above, ⇐⇒P\0 stands for the statistical/probabilistic equivalence in the part of the
+spectrum outside the zero eignevalues (introduced due to the non-square underlying matrices). Clearly, the
+key object of our interest below will be
+˜D
+≜
+¯V ⊥( ¯V ⊥)T ¯U ⊥
+D( ¯U ⊥
+D)T ,
+(86)
+where both ¯V ⊥ and ¯U ⊥
+D are Haar distributed and independent of each other. After setting
+V
+≜
+¯V ⊥( ¯V ⊥)T
+U
+≜
+¯U ⊥
+D( ¯U ⊥
+D)T ,
+(87)
+we easily have from (86)
+˜D
+≜
+VU,
+(88)
+and below first focus on handling the spectrum and the corresponding relevant transforms of V. Since we
+will be working in the mathematically most challenging large n linear regime, we find it useful to introduce
+the following large dimensional scalings
+β ≜ lim
+n→∞
+k
+n
+and
+η ≜ lim
+n→∞
+l
+n
+and
+α ≜ lim
+n→∞
+m
+n2 = lim
+n→∞
+n2 − (n − l)2
+n2
+= 1 − (1 − η)2.
+(89)
+We start with a trivial observation. Let fV(·) be the spectral distribution of V. Then
+fV(x) = (1 − β)δ(1 − x) + βδ(x),
+(90)
+where δ(·) stands for the standard delta function with nonzero value only when its argument takes value
+zero. Using the definition of the G-transform from (71) we can find
+GV(z) =
+� fV(x)
+z − x dx =
+� (1 − β)δ(1 − x) + βδ(x)
+z − x
+dx = 1 − β
+z − 1 + β
+z = z − β
+z2 − z .
+(91)
+Also, from (85) we have
+RV(y) = z − 1
+y
+with
+y = GV(z)
+and
+z = G−1
+V (y).
+(92)
+From (91) and (92) we further find
+GV(z) = y
+⇐⇒
+z − β
+z2 − z = y
+⇐⇒
+z2y − z(y + 1) + β = 0.
+(93)
+Solving for z gives
+z = y + 1 ±
+�
+(y + 1)2 − 4βy
+2y
+.
+(94)
+Combining (92) and (94) we obtain for the R-transform
+RV(y) = z − 1
+y = y − 1 ±
+�
+(y + 1)2 − 4βy
+2y
+,
+(95)
+where we for the completeness adopt the strategy to keep both ± signs. To determine the S-transform we
+start by combining (74) and (95)
+SV(z) =
+1
+RV(zSV(z)) =
+1
+zSV(z)−1±√
+(zSV(z)+1)2−4βzSV(z)
+2zSV(z)
+.
+(96)
+18
+
+After a bit of algebraic transformations we have
+zSV(z) − 1 − 2z
+=
+∓
+�
+(zSV(z) + 1)2 − 4βzSV(z)
+⇐⇒
+(zSV(z) − 1 − 2z)2
+=
+(zSV(z) + 1)2 − 4βzSV(z)
+⇐⇒
+(zSV(z))2 − 2(2z + 1)zSV(z) + (2z + 1)2
+=
+(zSV(z))2 + 2zSV + 1 − 4βzSV(z)
+⇐⇒
+−2(2z + 1)zSV(z) + 4z2 + 4z
+=
+2zSV(z) − 4βzSV(z)
+⇐⇒
+4z2 + 4z
+=
+(4z2 + 4z)SV(z) − 4βzSV(z)
+⇐⇒
+z + 1
+=
+SV(z)(z + 1 − β).
+(97)
+From (97) we finally have
+SV(z) =
+z + 1
+z + 1 − β .
+(98)
+As this is a very generic result it is useful to have it formalized in the following lemma.
+Lemma 1. Let ¯V ⊥ ∈ Rn×(n−k) be Haar distributed unitary basis of an (n − k)-dimensional subspace of Rn.
+Let V be as in (87), i.e.
+V ≜ ¯V ⊥( ¯V ⊥)T .
+(99)
+In the large n linear regime, with β ≜ limn→∞ k
+n, the S-transform of the spectral density of V, fV(·), is
+SV(z) =
+z + 1
+z + 1 − β .
+(100)
+Proof. Follows from the above discussion.
+Since V and U are structurally identical (with the only difference being one of their dimensions) we easily
+have
+SU(z) =
+z + 1
+z + 1 − η .
+(101)
+A combination of (75), (88), (98), and (101) gives
+S ˜
+D(z) =
+(z + 1)2
+(z + 1 − β)(z + 1 − η).
+(102)
+From (74) we also have
+R ˜
+D(z) =
+1
+S ˜
+D(zR ˜
+D(z)) =
+1
+(zR ˜
+D(z)+1)2
+(zR ˜
+D(z)+1−β)(zR ˜
+D(z)+1−η)
+= (zR ˜
+D(z) + 1 − β)(zR ˜
+D(z) + 1 − η)
+(zR ˜
+D(z) + 1)2
+.
+(103)
+Moreover, (72) gives
+R ˜
+D(G ˜
+D(z)) +
+1
+G ˜
+D(z) = z,
+(104)
+and
+G ˜
+D(z)R ˜
+D(G ˜
+D(z)) = zG ˜
+D(z) − 1,
+(105)
+From (103) one further finds
+R ˜
+D(G ˜
+D(z)) = (G ˜
+D(z)R ˜
+D(G ˜
+D(z)) + 1 − β)(G ˜
+D(z)R ˜
+D(G ˜
+D(z)) + 1 − η)
+(G ˜
+D(z)R ˜
+D(G ˜
+D(z)) + 1)2
+.
+(106)
+19
+
+After plugging (105) in (106) we have
+R ˜
+D(G ˜
+D(z)) = (zG ˜
+D(z) − 1 + 1 − β)(zG ˜
+D(z) − 1 + 1 − η)
+(zG ˜
+D(z) − 1 + 1)2
+= (zG ˜
+D(z) − β)(zG ˜
+D(z) − η)
+(zG ˜
+D(z))2
+.
+(107)
+A combination of (104) and (107) further gives
+z −
+1
+G ˜
+D(z) = (zG ˜
+D(z) − β)(zG ˜
+D(z) − η)
+(zG ˜
+D(z))2
+.
+(108)
+From (108) we quickly find
+z3(G ˜
+D(z))2 − z2G ˜
+D(z) = z2(G ˜
+D(z))2 − (β + η)zG ˜
+D(z) + βη,
+(109)
+and
+(G ˜
+D(z))2(z3 − z2) − G ˜
+D(z)(z2 − z(β + η)) − βη = 0.
+(110)
+Solving for G ˜
+D(z) finally gives
+G±
+˜
+D(z) = z2 − z(β + η) ±
+�
+(z2 − z(β + η))2 + 4βη(z3 − z2)
+2(z3 − z2)
+,
+(111)
+or
+G±
+˜
+D(z) = z − (β + η) ±
+�
+(z − (β + η))2 + 4βη(z − 1)
+2(z2 − z)
+.
+(112)
+The above is sufficient to establish the following lemma.
+Lemma 2. Let ¯V ⊥ ∈ Rn×(n−k) and ¯U ⊥
+D ∈ Rn×(n−k) be Haar distributed unitary bases of (n−k)-dimensional
+subspaces of Rn. Let V and U be as in (87) and ˜D as in (88), i.e.
+V
+≜
+¯V ⊥( ¯V ⊥)T
+U
+≜
+¯U ⊥
+D( ¯U ⊥
+D)T
+˜D
+≜
+VU.
+(113)
+In the large n linear regime, with β ≜ limn→∞ k
+n, the G-transform of the spectral density of ˜D, f ˜
+D(·), is
+G±
+˜
+D(z) = z − (β + η) ±
+�
+(z − (β + η))2 + 4βη(z − 1)
+2(z2 − z)
+.
+(114)
+Proof. Follows from the above discussion. The “+/−” signs are taken for negative/positive imaginary part
+under the root.
+One then relies on (72) to determine f ˜
+D(x) as
+f ˜
+D(x) = − lim
+ǫ→0+
+imag(G ˜
+D(x + iǫ))
+π
+.
+(115)
+The above is a generic procedure and we in Figure 3 show the results that one can get for two concrete values
+β = 0.2 and η = 0.6. One should note that it is not clear a priori which of the two ± signs should be used.
+As Figure 3 indicates one most definitely has to be fairly careful and account for both signs. From Figure 3
+one further observes that there are four critical points in the spectrum itself: the locations of the two delta
+functions, zero and one, and two edges of the spectrum’s bulk, xl and xu. The values of these points are
+shown in the plots on the right hand side. In general one can actually determine their closed forms as well.
+20
+
+Moreover, it turns out that one can determine the closed form of the entire spectral function. The section
+that follows analyzes the spectrum of ˜D in more details and eventually provides the closed form expressions
+for all the relevant spectral features.
+x
+0
+0.2
+0.4
+0.6
+0.8
+1
+f ˜D(x)
+-0.5
+0
+0.5
+1
+1.5
+f ˜D(x) obtained using G+
+˜D(z); β = 0.2, η = 0.6
+simulated
+theory
+xl
+xc
+xu
+f ˜D(x) = −limǫ→0+ imag(G+
+˜D(x+iǫ))
+π
+x
+0
+0.2
+0.4
+0.6
+0.8
+1
+f ˜D(x)
+-0.5
+0
+0.5
+1
+1.5
+f ˜D(x) obtained using G ˜D(z); β = 0.2, η = 0.6
+simulated
+theory
+xu = 0.9519
+xl = 0.1681
+xl = β + η − 2βη = 0.56
+G ˜D = G−
+˜D
+G ˜D = G+
+˜D
+f ˜D(x) = −limǫ→0+ imag(G ˜D(x+iǫ))
+π
+Figure 3: Both G+
+˜
+D(z) and G−
+˜
+D(z) need to be taken into account
+4.2.2.1
+The spectrum of ˜D – closed form expressions
+As one of our main concerns in this paper is the utilization of the final results that we will get in this
+section and not necessarily the presentation of the tiny details needed to get them, we will sketch all the key
+arguments and leave out all the unnecessary minute details. However, we do emphasize that the sketch will
+contain all the key pointers so that with a little bit of effort one, if in a need, can fill in all the missing pieces
+of the overall mosaic.
+As mentioned above, looking at the denominator of (112) and keeping in mind the f ←→ G connection
+from (115) one observes that the pdf of interest, f ˜
+D(x), potentially has two delta functions, one at zero and
+the other one at one. Moreover, the bulk of the spectrum will be in the range where the real part under
+the root is negative. It also goes almost without saying that the entire spectrum will be located between
+zero and one. Finally, the breaking point, xc, where one needs to switch from G+
+˜
+D(z) to G−
+˜
+D(z) in (115) is
+determined as the value where the imaginary part under the root changes its sign. Equipped with these
+observations one can then proceed to actually concretely determine some of the relevant quantities.
+Based on what we have just observed above, we first express f ˜
+D(x) as the sum of its three key constitutive
+parts (two delta functions and the bulk)
+f ˜
+D(x) = f0δ(x − 0) + f (b)
+˜
+D (x) + f1δ(x − 1).
+(116)
+From (116) one has that f ˜
+D(x) will be fully specified if one can determine the delta multipliers f0 and f1,
+and the bulk pdf f (b)
+˜
+D (·).
+1) Finding f0: To determine f0 we start by observing from (115) for x = 0
+f ˜
+D(0) = − lim
+ǫ→0+
+imag(G ˜
+D(iǫ))
+π
+.
+(117)
+Utilizing (112) we further have
+f ˜
+D(0)
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+iǫ − (β + η) ±
+�
+(iǫ − (β + η))2 + 4βη(iǫ − 1)
+2((iǫ)2 − iǫ)
+�
+21
+
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+−(β + η) ±
+�
+−ǫ2 + (β + η)2 − 4βη − 2iǫ(β + η − 2βη)
+2(−ǫ2 − iǫ)
+�
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+−(β + η) ±
+�
+(β − η)2
+2(−iǫ)
+�
+=
+− 1
+π lim
+ǫ→0+ imag
+�−(β + η) − |β − η|
+−2iǫ
+�
+=
+(β + η + |β − η|)
+�
+− 1
+π lim
+ǫ→0+ imag
+� 1
+iǫ
+��
+=
+max(β, η)δ(0),
+(118)
+where the fourth equality (the choice of the “−” sign in ±) follows since 0 ≤ xc (the spectrum belongs to
+the interval [0, 1] and xc must be in the spectrum) and the last equality follows since by convention
+δ(0) =
+�
+− 1
+π lim
+ǫ→0+ imag
+� 1
+iǫ
+��
+.
+(119)
+To see the rationale behind (119) we briefly digress and start with
+g(x) = δ(x).
+(120)
+Then from (71)
+G(z) =
+�
+x
+δ(x)dx
+z − x = 1
+z ,
+(121)
+and from (72)
+δ(x) = − lim
+ǫ→0+
+imag(G(x + iǫ))
+π
+.
+(122)
+For x = 0 then
+δ(0) = − lim
+ǫ→0+
+imag(G(iǫ))
+π
+= − lim
+ǫ→0+ imag
+� 1
+πiǫ
+�
+= − 1
+π lim
+ǫ→0+ imag
+� 1
+iǫ
+�
+,
+(123)
+which is identical to (119). The above description of the delta function may not necessarily be the most
+adequate one. However, for what we need here it is conceptually sufficient. Namely, we are here interested
+in determining the proportionality constants that multiply the delta functions rather than the functions’
+expressions themselves. One way to make everything more adequate would be to translate everything into
+the continuous domain by choosing a continuous function as an asymptotic replacement for δ(x).
+For
+example, one can use the Gaussian continual approximation
+δ(x) = lim
+σ→0+
+e− x2
+2σ2
+√
+2πσ2 .
+(124)
+Then all the above holds for small σ = ǫ√π/2 and
+δ(x) → lim
+σ→0+
+e− x2
+2σ2
+√
+2πσ2 → lim
+σ→0+
+e− x2
+πǫ2
+πǫ
+and
+δ(0) → lim
+σ→0+
+1
+πǫ.
+(125)
+The difference though would be that when computing and maneuvering with all the above transforms one
+would need to account for the resulting/induced ǫ-differences. These are of course practically and concep-
+tually negligible and all the results that we presented would continue to hold in the limit of small σ or ǫ.
+However, the writing would be substantially more tedious and a tone of additional minute details would
+need to be added to express all the ǫ-type of modifications and to show that their contributions are indeed
+marginal. These things are conceptually highly trivial but require a tedious detail-oriented work. Since, on
+22
+
+the other hand, they contribute exactly nothing to the essence of the arguments and final results we chose to
+operate in a semi-discrete domain with the delta functions. As a consequence one has the expressions given
+in (119) and (123). We believe that a little bit of conventional inadequacy is better than to overwhelm the
+presentation with a tone of details which would avoid it but at the same time make the overall content less
+accessible and potentially even less understandable.
+2) Finding f1: To determine f1 we follow the above methodology and start by observing from (115)
+for x = 1
+f ˜
+D(1) = − lim
+ǫ→0+
+imag(G ˜
+D(1 + iǫ))
+π
+.
+(126)
+Further utilization of (112) gives
+f ˜
+D(1)
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+1 + iǫ − (β + η) ±
+�
+(1 + iǫ − (β + η))2 + 4βηiǫ
+2((1 + iǫ)2 − 1 − iǫ)
+�
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+1 − (β + η) ±
+�
+−ǫ2 + (1 − (β + η))2 − 2iǫ(−1 + β + η − 2βη)
+2(−ǫ2 + iǫ)
+�
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+1 − (β + η) ±
+�
+(1 − (β − η))2
+2(iǫ)
+�
+=
+− 1
+π lim
+ǫ→0+ imag
+�1 − (β + η) + |1 − (β + η)|
+2iǫ
+�
+=
+(1 − (β + η) + |1 − (β + η)|)
+�
+− 1
+π lim
+ǫ→0+ imag
+� 1
+iǫ
+��
+=
+max(1 − (β + η), 0)δ(0),
+(127)
+where the fourth equality (the choice of the “+” sign in ±) follows since now xc ≤ 1 and the last equality
+follows by the above discussed δ(0) convention.
+3) Finding f (b)
+˜
+D (x): To determine f (b)
+˜
+D (x) for x /∈ {0, 1} we again start with (115) and wrte the following
+for a general x from the bulk of the spectrum
+f (b)
+˜
+D (x) = − lim
+ǫ→0+
+imag(G ˜
+D(x + iǫ))
+π
+.
+(128)
+Relying once again on (112) we, for x /∈ {0, 1}, have
+f (b)
+˜
+D (x)
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+x + iǫ − (β + η) ±
+�
+(x + iǫ − (β + η))2 + 4βη(x + iǫ − 1
+2((x + iǫ)2 − x − iǫ)
+�
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+x − (β + η) ±
+�
+−ǫ2 + (x − (β + η))2 + 4βη(x − 1) − 2iǫ(−x + β + η − 2βη)
+2(x2 − x − ǫ2 + iǫ(2x − 1))
+�
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+x − (β + η) ±
+�
+(x − (β + η))2 + 4βη(x − 1) − 2iǫ(−x + β + η − 2βη)
+2(x2 − x)
+�
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+±
+�
+(x − (β + η))2 + 4βη(x − 1) − 2iǫ(−x + β + η − 2βη)
+2(x2 − x)
+�
+.
+(129)
+Now, since one is interested in the imaginary part of interest is the region of x where the real part under the
+root is negative (outside that region, i.e in the region of x where the real part under the root is nonnegative
+f (b)
+˜
+D (x) is zero). To determine the region of interest we start by setting
+T ˜
+D ≜ {x ∈ R|(x − (β + η))2 + 4βη(x − 1) ≤ 0}
+and
+xc ≜ β + η − 2βη.
+(130)
+23
+
+To explicitly characterize T ˜
+D we look at the following
+(x − (β + η))2 + 4βη(x − 1)
+=
+0
+⇐⇒
+x2 − 2x(β + η − 2βη) + (β + η)2 − 4βη
+=
+0
+⇐⇒
+x2 − 2x(β + η − 2βη) + (β − η)2
+=
+0.
+(131)
+Solving for x one finds
+x = 2(β + η − 2βη) ±
+�
+(2(β + η − 2βη))2 − 4(β − η)2
+2
+= β +η −2βη ±
+�
+(β + η − 2βη)2 − (β − η)2. (132)
+Setting
+xl
+≜
+β + η − 2βη −
+�
+(β + η − 2βη)2 − (β − η)2
+xu
+≜
+β + η − 2βη +
+�
+(β + η − 2βη)2 − (β − η)2,
+(133)
+one has
+T ˜
+D = {x ∈ R|x ∈ [xl, xu]}.
+(134)
+Moreover, from (133), one also has
+0 ≤ xl ≤ xu ≤ 1,
+(135)
+with
+xl = 0
+if
+β = η
+and
+xu = 1
+if
+β = η = 0.5.
+(136)
+The first two inequalities in (133) are trivial, whereas the third one follows after noting
+β + η − 2βη ≤ max(β, 1 − β) ≤ 1,
+(137)
+and observing the following sequence
+β + η − 2βη +
+�
+(β + η − 2βη)2 − (β − η)2
+≤
+1
+⇐⇒
+(β + η − 2βη)2 − (β − η)2
+≤
+(1 − (β + η − 2βη))2
+⇐⇒
+−(β − η)2
+≤
+1 − 2(β + η − 2βη)
+⇐⇒
+−(β − η)2 + 2(β + η − 2βη) − 1
+≤
+0
+⇐⇒
+−(β + η)2 + 2(β + η) − 1
+≤
+0
+⇐⇒
+−(1 − (β + η))2
+≤
+0.
+(138)
+Returning to (129) we further have for x ∈ T ˜
+D = [xl, xu]
+f (b)
+˜
+D (x)
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+±
+�
+(x − (β + η))2 + 4βη(x − 1) − 2iǫ(−x + β + η − 2βη)
+2(x2 − x)
+�
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+±
+�
+(x − (β + η))2 + 4βη(x − 1) − 2iǫ(xc − x)
+2(x2 − x)
+�
+=
+− 1
+π lim
+ǫ→0+ imag
+�
+i
+�
+−(x − (β + η))2 − 4βη(x − 1)
+2(x2 − x)
+�
+,
+(139)
+where the “+” plus sign is chosen if xc ≤ x ≤ xu and the “−” sign is chosen if xl ≤ x ≤ xc. Finally, from
+(139) one easily finds
+f (b)
+˜
+D (x)
+=
+�
+−(x − (β + η))2 − 4βη(x − 1)
+2π(x − x2)
+if
+xl ≤ x ≤ xu.
+(140)
+24
+
+The above is then sufficient to completely characterize the spectral distribution f ˜
+D(x). We summarize
+the results in the following lemma.
+Lemma 3. Assume large n linear regime with
+β ≜ lim
+n→∞
+k
+n
+and
+η ≜ lim
+n→∞
+l
+n.
+(141)
+Let ¯V ⊥ ∈ Rn×(n−k) be a Haar distributed basis of an n − k-dimensional subspace of Rn.
+Analogously,
+let ¯U ⊥
+D ∈ Rn×(n−k) be a Haar distributed basis of an n − l-dimensional subspace of Rn.
+Moreover, let
+¯V ⊥ ∈ Rn×(n−k) and ¯U ⊥
+D ∈ Rn×(n−k) be independent of each other. Also, let V, U, and ˜D be as defined in
+(87) and (88), i.e. let
+V
+≜
+¯V ⊥( ¯V ⊥)T
+U
+≜
+¯U ⊥
+D( ¯U ⊥
+D)T
+˜D
+≜
+VU.
+(142)
+Set xl and xu as in (133), i.e.
+xl
+≜
+β + η − 2βη −
+�
+(β + η − 2βη)2 − (β − η)2
+xu
+≜
+β + η − 2βη +
+�
+(β + η − 2βη)2 − (β − η)2,
+(143)
+Then the limiting spectral distribution of ˜D, f ˜
+D(x), is
+f ˜
+D(x) = f0δ(x) + f (b)
+˜
+D (x) + f1f0δ(x − 1) = max(β, η)δ(x) + f (b)
+˜
+D (x) + max(1 − (β + η), 0)δ(x − 1),
+(144)
+with
+f (b)
+˜
+D (x)
+=
+�√
+−(x−(β+η))2−4βη(x−1)
+2π(x−x2)
+,
+if xl ≤ x ≤ xu.
+0,
+otherwise.
+(145)
+Proof. Follows through a combination of (116), (118), (127), (140), and the above discussion.
+In Figure 4 we show the spectral function obtained based on the above lemma for β = 0.1 and η = 0.8.
+We observe a very strong agreement between the simulated results and the above theoretical predictions.
+Simulation results were obtained using moderately large n = 4000.
+In Figure 5 we show the spectral function obtained based on the above lemma for β = 0.2 and η = 0.9.
+Due to a remarkable property of the underlying functions the spectrum is identical as in Figure 4 apart
+from the fact that the multiplier of the delta function at zero is increased from 0.8 to 0.9 at the expense of
+removing the delta function at one. We also again observe a very strong agreement between the simulated
+results and the theoretical predictions. As in Figure 4, Simulation results were again obtained for n = 4000.
+4.2.2.2
+The spectrum of ¯D/D – closed form expressions
+We recall on (82) and (83) to set
+¯D
+≜
+(I(l))T U T
+D ¯V ⊥( ¯V ⊥)T UDI(l) = ( ¯U (⊥)
+D )T ¯V ⊥( ¯V ⊥)T ¯U (⊥)
+D ,
+(146)
+Comparing (86) and (146) we observe that their spectra are modulo scalings basically identical. To be a bit
+more precise, ¯D has all eigenvalues that ˜D has with ηn zeros less. That basically means that one needs to
+adjust the multiplier of δ(x) in f ˜
+D(x) and to scale everything by (1 − η). The following lemma summarizes
+the final results of such a procedure.
+Lemma 4. Assume the setup of Lemma 3. Let ¯D be as in (146), i.e.
+¯D
+≜
+( ¯U (⊥)
+D )T ¯V ⊥( ¯V ⊥)T ¯U (⊥)
+D ,
+(147)
+25
+
+x
+0
+0.2
+0.4
+0.6
+0.8
+1
+f ˜D(x)
+-0.5
+0
+0.5
+1
+1.5
+f ˜
+D(x); β = 0.1, η = 0.8
+simulated
+theory
+xu = 0.98 (with “+” sign)
+xl = 0.5 (with “−” sign)
+Bulk
+xl/u = β + η − 2βη ± �(β + η − 2βη)2 − (β − η)2
+f ˜
+D(0) = max(β, η)δ(0) = 0.8δ(0)
+f ˜
+D(1) = max(1 − (β + η), 0)δ(0) = 0.1δ(0)
+f (b)
+˜
+D (x) =
+√
+−(x−(β+η))2−4βη(x−1)
+2π(x−x2)
+Figure 4: f ˜
+D(x) – spectral function of ˜D; β = 0.1 and η = 0.8
+and let xl and xu be as in (143). Then the limiting spectral distribution of ¯D, f ¯
+D(x), is
+f ¯
+D(x) = (max(β, η) − η)
+1 − η
+δ(x) + f (b)
+¯
+D (x) + max(1 − (β + η), 0)
+1 − η
+δ(x − 1),
+(148)
+with
+f (b)
+¯
+D (x)
+=
+�√
+−(x−(β+η))2−4βη(x−1)
+2π(x−x2)(1−η)
+,
+if xl ≤ x ≤ xu.
+0,
+otherwise.
+(149)
+Moreover, let D be as in (81), i.e.
+D
+≜
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l).
+(150)
+The limiting spectral distribution of D, fD(x), is
+fD(x) = f ¯
+D(x).
+(151)
+Proof. The part that relates to the spectral distribution of ¯D follows by removing l = ηn zeros from the
+spectrum of ˜D and appropriately scaling the residual pdf by (1 − η). The part that relates to the spectral
+distribution of D follows from (84) which itself is a consequence of the spectral invariance under unitary
+multiplications.
+In Figure 6 we show the spectral function obtained based on the above lemma for β = 0.1 and η = 0.8.
+We observe a very strong agreement between the simulated results and the above theoretical predictions.
+Simulation results were obtained using moderately large n = 4000.
+4.2.2.3
+The spectrum of Q – closed form expressions
+We start by recalling on (81)
+Q = D−1 − I.
+(152)
+26
+
+x
+0
+0.2
+0.4
+0.6
+0.8
+1
+f ˜D(x)
+-0.5
+0
+0.5
+1
+1.5
+f ˜
+D(x); β = 0.2, η = 0.9
+simulated
+theory
+xu = 0.98 (with “+” sign)
+xl = 0.5 (with “−” sign)
+Bulk
+xl/u = β + η − 2βη ± �(β + η − 2βη)2 − (β − η)2
+f ˜
+D(0) = max(β, η)δ(0) = 0.9δ(0)
+f (b)
+˜
+D (x) =
+√
+−(x−(β+η))2−4βη(x−1)
+2π(x−x2)
+Figure 5: f ˜
+D(x) – spectral function of ˜D; β = 0.2 and η = 0.9
+Almost all of the above holds without explicitly assuming k ≤ l. From this point on such an assumption is
+needed. Since k ≤ l means β ≤ η one has that D has no zeros in its spectrum and can be inverted. Since
+the portion of the spectrum at one remains the same after the inversion, one then basically has that the
+spectrum of Q is the same as the spectrum of D modulo the inversion of the bulk of D. After a change of
+variables y = 1
+x one has for the spectral distribution of the inverted bulk
+f (b)
+¯
+D−1(y)
+=
+
+
+
+−
+�
+−( 1
+y −(β+η))2−4βη( 1
+y −1)
+2π( 1
+y −( 1
+y )2)(1−η)y2
+,
+if
+1
+xu ≤ y ≤
+1
+xl .
+0,
+otherwise,
+(153)
+which after elementary algebraic transformations becomes
+f (b)
+¯
+D−1(x)
+=
+�√
+−(1−x(β+η))2−4βηx(1−x)
+2π(1−x)(1−η)x
+,
+if
+1
+xu ≤ x ≤
+1
+xl .
+0,
+otherwise,
+(154)
+Noting that β ≤ η implies max(β, η) − η = 0, one can combine (154 together with (148) to obtain
+f ¯
+D−1(x) = f (b)
+¯
+D−1(x) + max(1 − (β + η), 0)
+1 − η
+δ(x − 1).
+(155)
+Finally adjusting for a subtracted identity matrix corresponds to subtracting one from any point in the
+spectrum (or basically to shifting the entire spectral distribution to the left by one). In other words
+fQ(x) = f ¯
+D−1−I(x) = f (b)
+¯
+D−1−I(x) + max(1 − (β + η), 0)
+1 − η
+δ(x),
+(156)
+with
+f (b)
+Q (x) ≜ f (b)
+¯
+D−1−I(x)
+=
+�√
+−(1−(x+1)(β+η))2−4βηx(x+1)
+2πx(x+1)(1−η)
+,
+if
+1
+xu − 1 ≤ x ≤
+1
+xl − 1.
+0,
+otherwise,
+(157)
+The following lemma summarizes the key results of this section.
+27
+
+x
+0
+0.2
+0.4
+0.6
+0.8
+1
+fD(x)
+-2
+-1
+0
+1
+2
+3
+4
+5
+6
+fD(x); β = 0.1, η = 0.8
+simulated
+theory
+xu = 0.98 (with “+” sign)
+xl = 0.5 (with “−” sign)
+Bulk
+xl/u = β + η − 2βη ± �(β + η − 2βη)2 − (β − η)2
+f (b)
+D (x) =
+√
+−(x−(β+η))2−4βη(x−1)
+2π(x−x2)(1−η)
+fD(1) = max(1−(β+η),0)
+1−η
+δ(0) = 0.5δ(0)
+Figure 6: fD(x) – spectral function of D; β = 0.1 and η = 0.8
+Lemma 5. Assume the setup of Lemma 4. Let Q be as in (81), i.e.
+Q ≜ D−1 − I =
+�
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1
+− I,
+(158)
+and let xl and xu be as in (143). Then the limiting spectral distribution of Q, fQ(x), is
+fQ(x) = f (b)
+Q (x) + max(1 − (β + η), 0)
+1 − η
+δ(x),
+(159)
+with
+f (b)
+Q (x)
+=
+�√
+−(1−(x+1)(β+η))2−4βηx(x+1)
+2πx(x+1)(1−η)
+,
+if
+1
+xu − 1 ≤ x ≤
+1
+xl − 1,
+0,
+otherwise.
+(160)
+Proof. Follows from the above discussion.
+In Figure 7 we show the spectral function, fQ(x), obtained based on the above lemma for β = 0.1 and
+η = 0.8. One observes a very strong agreement between what the theory predicts and what the simulations
+produce. As in all earlier experiments n = 4000 was used here again.
+4.2.3
+ℓ∗
+0 − ℓ∗
+1-equivalence via the spectral limit
+From Corollary 3, (47), (48), and (77) one has in the worst case
+ℓ∗
+0 − ℓ∗
+1 − equivalence
+⇐⇒
+λmax(Q) ≤ 1.
+(161)
+From Lemma 5 we have
+λmax(Q) = 1
+xl
+− 1,
+(162)
+28
+
+x
+0
+0.2
+0.4
+0.6
+0.8
+1
+fQ(x)
+-2
+-1
+0
+1
+2
+3
+4
+5
+6
+fQ(x); β = 0.1, η = 0.8
+simulated
+theory
+xl/u = β + η − 2βη ± �(β + η − 2βη)2 − (β − η)2
+Bulk
+fQ(0) = max(1−(β+η),0)
+1−η
+δ(0) = 0.5δ(0)
+x∗ =
+1
+xu − 1 ≈ 0.02
+x∗ =
+1
+xl − 1 = 1
+f (b)
+Q (x) =
+√
+−(1−(x+1)(β+η))2−4βηx(x+1)
+2πx(x+1))(1−η)
+Figure 7: fQ(x) – spectral function of Q; β = 0.1 and η = 0.8
+with xl as in (143). Moreover, from (161) and (162), one arrives at the following necessary and sufficient
+condition to achieve the ℓ∗
+0 − ℓ∗
+1-equivalence
+ℓ∗
+0 − ℓ∗
+1 − equivalence
+⇐⇒
+1
+xl
+− 1 ≤ 1,
+(163)
+or
+ℓ∗
+0 − ℓ∗
+1 − equivalence
+⇐⇒
+1
+2 ≤ xl.
+(164)
+Recalling on (143)
+xl
+≜
+β + η − 2βη −
+�
+(β + η − 2βη)2 − (β − η)2,
+(165)
+and combining further with (164) we have
+1
+2
+≤
+xl
+⇐⇒
+1
+2
+=
+β + η − 2βη −
+�
+(β + η − 2βη)2 − (β − η)2
+⇐⇒
+(β + η − 2βη)2 − (β − η)2
+≤
+�
+β + η − 2βη − 1
+2
+�2
+⇐⇒
+(β + η − 2βη)2 − (β − η)2
+≤
+(β + η − 2βη)2 − (β + η − 2βη) + 1
+4
+⇐⇒
+−(β − η)2
+≤
+− (β + η − 2βη) + 1
+4
+⇐⇒
+0
+≤
+β2 + η2 − (β + η) + 1
+4
+⇐⇒
+η − η2
+≤
+� 1
+2 − β
+�2
+⇐⇒
+β
+≤
+1
+2 −
+�
+η − η2.
+(166)
+From (164) and (166) we finally have
+ℓ∗
+0 − ℓ∗
+1 − equivalence
+⇐⇒
+β ≤ 1
+2 −
+�
+η − η2.
+(167)
+We are now in position to formalize the key causal inference results that establish the so-called phase-
+transition phenomenon as well as its a precise worst case location in a typical statistical scenario.
+Theorem 2. (ℓ∗
+1 – phase transition – C-inf (typical worst case)) Consider a rank-k matrix Xsol =
+29
+
+X ∈ Rn×n with the Haar distributed ( not necessarily independent) bases of its orthogonal row and column
+spans ¯U ⊥ ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) (XT
+sol ¯U ⊥ = Xsol ¯V ⊥ = 0n×(n−k)). Let M ≜ M (l) ∈ Rn×n be as
+defined in (15) or (32). Assume a large n linear regime with β ≜ limn→∞ k
+n and η ≜ limn→∞ l
+n and let βwc
+and η satisfy the following
+C-inf ℓ∗
+1 worst case phase transition (PT) characterization
+ξ(wc)
+η
+(β) ≜ β − 1
+2 +
+�
+η − η2 = 0.
+(168)
+If and only if β ≤ βwc
+lim
+n→∞ P(ℓ∗
+0 ⇐⇒ ℓ∗
+1) =
+lim
+n→∞ P(RMSE = 0) = 1,
+(169)
+and the solutions of (16) and (17) coincide with overwhelming probability.
+Proof. The “if” part follows through a combination of Theorem 1, Corollary 3, Lemma 5, and the above
+discussion from (161) to (167). The “only if” part additionally assumes ¯U ⊥ = ¯V ⊥ and then relying on (77)
+and (78) ensures that the results are in the worst case achievable.
+The results obtained based on the above theorem are shown in Figure 9. As can be seen from the figure,
+the phase transition curve splits the entire (β, η) region into two subregions. The first of the subregions is
+below (or to the right of) the curve and in that region the ℓ∗
+0 − ℓ∗
+1-equivalence phenomenon occurs. This
+means that one can recover Xsol masked by M as in (16) via the ℓ∗
+1 heuristic from (17) with the residual
+mean square error (RMSE) equal to zero. In other words, for the system parameters (β, η) that belong to
+the subregion below the curve one has a perfect recovery with Xsol and ˆX (the respective solutions of (16)
+and (17)) being equal to each other and consequently with RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0. On the
+other hand, in the subregion above the curve, the ℓ∗
+1 heuristic fails and one can even find an Xsol for which
+RMSE → ∞.
+η
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+β
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+0.5
+(η, β) region of success/failure — C-inf ℓ∗1 PT
+RMSE −→ ∞, ℓ∗1
+fails
+RMSE = 0, ℓ∗1 succeeds
+ℓ∗1’s PT: ξ(wc)
+η
+(β) = β − 1
+2 + �η − η2 = 0
+Figure 8: Causal inference (C-inf) – typical worst case ℓ∗
+1 phase transition
+We make a few remarks that relate to some general features of matrix completion, and their differences
+with respect with standard compressed sensing.
+Even in the so-called random-masking scenario (where
+elements of M take values 0/1 with probability one half), one may have troubles recovering low-rank matrices.
+30
+
+For example, in such a scenario, it is statistically unlikely that one can guarantee universal recovery even
+of rank-1 matrices. To see this, one can choose rank-1 Xsol with one in the upper left corner and all other
+elements equal to zero. Then such a matrix will be recoverable from M ◦ Xsol only if the element in the
+upper left corner of M is one. Since that happens with probability 1/2, one can not have a reliable recovery
+with probability going to one as n → ∞. It is the discreteness in the process of acquiring observations Y that
+enables scenarios like this, and makes the matrix completion (MC) substantially different from its compressed
+sensing vector analogue. In that light, it is somewhat surprising that any form of the phase-transition can
+be established in the linear regime. The key behind the success of the above machinery is the focus on the
+typical worst case and the causal inference internal structure. In the vector compressed sensing setup, the
+above described scenario cannot happen and one consequently does not need to resort to the typicality and
+instead can formulate more generic phase transition concepts (more on the non-typical compressed sensing
+approaches can be found in, e.g. [13,45,48] and on the corresponding typical ones in, e.g. [7,14]).
+Corollary 4. (ℓ∗
+1 – phase transition – C-inf (typical worst case; standard (α, β) representation))
+Assume the setup of Theorem 2. Let m be the total number of ones in matrix M and let α ≜ limn→∞ m
+n2 .
+Let β and αw satisfy the
+C-inf ℓ∗
+1 worst case PT (standard (α, β) representation)
+ξ(wc,s)
+β
+(α) ≜ β − 1
+2 +
+�√
+1 − α − 1 + α = 0.
+(170)
+If and only if α ≥ αw
+lim
+n→∞ P(ℓ∗
+0 ⇐⇒ ℓ∗
+1) =
+lim
+n→∞ P(RMSE = 0) = 1,
+(171)
+and the solutions of (16) and (17) coincide with overwhelming probability.
+Proof. Follows immediately from Theorem 2 after observing that m = n2 − (n − l)2 and consequently
+α = 1 − (1 − η)2.
+Figure 9 shows the results obtained based on the above corollary in the standard (α, β) region format.
+As earlier, in the subregion to the right of (or below) the curve RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0. In
+the subregion to the left (or above) the curve, the ℓ∗
+1 heuristic generally fails and RMSE → ∞ can even be
+achieved.
+4.3
+Numerical results
+We conducted a set of numerical experiments to complement the above theoretical findings and see how well
+the entire theoretical machinery characterizes the utilization of the ℓ∗
+1-minimization heuristic in the causal
+inference problems. Figure 10 shows the performance obtained through the numerical experiments as well
+as the corresponding theoretical worst case predictions discussed above. We clearly observe the existence
+of the phase transition and a solid agreement between the theoretical predictions and the results obtained
+through the numerical experiments.
+We should also add that we conducted the numerical experiments for fairly small matrix sizes. On the
+other hand, the theoretical predictions assume large n (basically an n → ∞). In particular, we chose n = 80
+and η in the range [0.6, 0.95]. Such a choice shows that even though the theory is predicated on the large n
+assumption, its conclusions may be applicable for smaller values of n as well. Viewed a bit alternatively, the
+large n regime, needed for the theory to properly operate, practically may start ro kick in already for not
+necessarily super large values of n. This also means that the presented theory may actually be of a practical
+use as well. We do, however, mention that for larger values of n an even better fit between the theoretical
+and simulated results might be expected.
+Finally, we should emphasize that in the numerical experiments here (as well as in a significant portion of
+the paper) we considered the so-called typical behavior. Also, it should be mentioned that we followed into
+the footsteps of our theoretical analyses from the previous sections and presented the simulations results for
+the square matrices. With a little bit of extra effort, all of our theoretical considerations can be repeated
+31
+
+α
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+β/α
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+0.5
+(α, β) region of success/failure — C-inf ℓ∗1 PT
+RMSE = 0, ℓ∗1 succeeds
+ℓ∗1’s PT: ξ(wc,s)
+β
+(α) = β − 1
+2 +
+�√1 − α − 1 + α = 0
+RMSE −→ ∞, ℓ∗1
+fails
+Figure 9: Causal inference (C-inf) – typical worst case ℓ∗
+1 phase transition ((α, β) region)
+in the non-square scenarios as well. Since the writings would be a bit more involved we found it useful
+to preserve the clarity of the presentation at the expense of rather trivial generalizations. Finally, in all
+numerical experiments that we present we chose the unknown matrices with the singular values equal to
+one. While a thorough discussion regarding this choice goes well beyond the scope of this paper, we just
+briefly recall that these types of structures typically serve as the worst case examples in establishing the
+reversal ℓ0 − ℓ1-equivalence conditions. In other words, they are usually the examples that make many of
+our key results/theorems hold as equivalences rather than just as implications. We also conducted numerical
+experiments where singular values were randomly chosen with results being identical to the ones shown in
+Figure 10 or better.
+5
+Conclusion
+In this paper, we have explored the Causal inference (C-inf) ↔ low-rank recovery (LRR) connection.
+We have analyzed how the best known convex type of heuristic, called ℓ∗
+1-minimization, fairs when used for
+solving the C-inf. We have shown both theoretically and numerically that in a typical statistical context,
+causal inference exhibits the so-called phase transition (PT) phenomenon. This ensures that for certain
+range of system parameters ℓ∗
+1 succeeds in recovering the unobserved potential outcomes, and outside such
+a range it fails. Moreover, we have obtained the exact explicit functional characterization for the location
+of the worst case phase transition (PT) curve. As a byproduct of our analysis, we have obtained (somewhat
+surprisingly) that the underlying functional characterization admits a fairly simple form, which elegantly
+pins down the relation between the low rankness of the target C-inf matrix and the time when the treatment
+is applied. We also emphasize that, while establishing the mathematical methodology to provide the C-
+inf PT characterizations, we uncovered a rather interesting connection between C-inf via LRR and the
+free probability theory (FPT) from modern spectral theory of random matrices. After establishing the
+connection between C-inf and the compressed sensing via the Random duality theory (RDT), we have
+proceeded to recognize the role that FPT plays in the overall mosaic that ultimately enables handling the
+C-inf problem.
+On a path to achieving complete handling of the causal inference we have created quite a few mathematical
+results that are of independent interest. To ensure a completeness of the overall treatment, we for all of
+them also ran the corresponding numerical experiments and again observed a rather overwhelming agreement
+between the theoretical predictions and simulations.
+32
+
+η
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+β
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+0.5
+(η, β) region of success/failure — ℓ∗1’s PT; simulated/theory
+ℓ∗
+1’s PT – simulated
+ℓ∗
+1’s PT – theory
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Failure
+Success
+Figure 10: C-inf ℓ∗
+1’s phase transition (PT)
+While there exists different approaches that can be explored to attack the problems considered here
+(with some of them being even conceptually simpler), our choice is partly motivated by the ability to handle
+more complicated problems in future extensions of this work. Being this the introductory paper, we stopped
+short of showcasing how the introduced theory fairs when applied to many such specific, more complex
+instances (the simplest among them would be the noisy and approximately low-rank corresponding ones).
+In our companion papers, we will establish results along these directions, which all rely on the mathematical
+framework presented in this paper.
+References
+[1] A. Abadie. Using synthetic controls: Feasibility, data requirements, and methodological aspects. Journal
+of Economic Literature, 2019.
+[2] A. Abadie, A. Diamond, and J. Hainmueller. Synthetic control methods for comparative case stud-
+ies: Estimating the effect of california’s tobacco control program. Journal of the American Statistical
+Association, 105:493–505, 2010.
+[3] A. Agarwal, M. Dahel, D. Shah, and D. Shen. Causal matrix completion. available online at arxiv.
+[4] S. Athey, M. Bayati, N. Doudchenko, G. Imbens, and K. Khosravi. Matrix completion methods for
+causal panel data models. Journal of the American Statistical Association, 116(536):1716–1730, 2021.
+[5] S. Athey and G. W. Imbens. Design-based analysis in differencein- differences settings with staggered
+adoption. Technical Report, National Bureau of Economic Research, 2018.
+[6] S. Athey and S. Stren. The impact of information technology on emergency health care outcomes. The
+RAND Journal of Economics, 33:399–432, 2002.
+[7] M. Bayati and A. Montanari. The dynamics of message passing on dense graphs, with applications to
+compressed sensing. available online at http://arxiv.org/abs/1001.3448.
+33
+
+[8] E. Candes, J. Romberg, and T. Tao. Robust uncertainty principles: exact signal reconstruction from
+highly incomplete frequency information. IEEE Trans. on Information Theory, 52:489–509, December
+2006.
+[9] E. J. Candes and B. Recht. Exact matrix completion via convex optimization. Foundations of Compu-
+tational Mathematics, (9):717, 2009.
+[10] E. J. Candes and T. Tao. The power of convex relaxation: Near-optimalmatrix completion. IEEE
+Transactions on Information Theory, 56:2053–2080, 2010.
+[11] E. J. Cand`es and Y. Plan. Matrix completion with noise. Proceedings of the IEEE, 98:925–936, 2010.
+[12] A. Capponi and M. Stojnic. Causal inference (C-inf) — asymmetric scenario of typical phase transitions.
+available online at arxiv.
+[13] D. Donoho. High-dimensional centrally symmetric polytopes with neighborlines proportional to dimen-
+sion. Disc. Comput. Geometry, 35(4):617–652, 2006.
+[14] D. Donoho, A. Maleki, and A. Montanari. Message-passing algorithms for compressed sensing. Proc.
+National Academy of Sciences, 106(45):18914–18919, Nov. 2009.
+[15] D. Donoho and J. Tanner. Sparse nonnegative solutions of underdetermined linear equations by linear
+programming. Proc. National Academy of Sciences, 102(27):9446–9451, 2005.
+[16] N. Doudchenko and G. W. Imbens. Balancing, regression, difference-in-differences and synthetic control
+methods: A synthesis. Technical Report, National Bureau of Economic Research, 2016.
+[17] U. Haagerup. On Voiculescu’s R- and S-transforms for free non-commuting random variables. In: Free
+Probability Theory, Fields Institute Communications, 12:127–148, 1997.
+[18] M. A. Hernan and J. M. Robins. Causal Inference. Boca Raton, FL: CRC Press, 2010.
+[19] G. W. Imbens and D. B. Rubin. Causal Inference in Statistics, Social, and Biomedical Sciences. New
+York: Cambridge University Press, 2015.
+[20] N. Kallus, X. Mao, and M. Udell.
+Causal inference with noisy and missing covariates via matrix
+factorization. Proceedings of the 32rd International Conference on Neural Information Processing, pages
+6921—-6932, December 2018. available online.
+[21] R. H. Keshavan, A. Montanari, and S. Oh. Matrix completion from a few entries. IEEE Transactions
+on Information Theory, 56:2980–2998, 2010.
+[22] R. H. Keshavan, A. Montanari, and S. Oh. Matrix completion from noisy entries. Journal of Machine
+Learning Research, 11:2057–2078, 2010.
+[23] O. Klopp. Noisy low-rank matrix completion with general sampling distribution. Bernoulli, 20:282–303,
+2014.
+[24] V. Koltchinskii, K. Lounici, and A. B. Tsybakov. Nuclear-norm penalization and optimal rates for noisy
+low-rank matrix completion. The Annals of Statistics, 39:2302–2329, 2011.
+[25] R. Mazumder, T. Hastie, and R. Tibshirani.
+Spectral regularization algorithms for learning large
+incomplete matrices. Journal of Machine Learning Research, 11:2287–2322, 2010.
+[26] S. N. Negahban and M. J. Wainwright. Estimation of (near) low-rank matriceswith noise and high-
+dimensional scaling. The Annals of Statistics, 39:1069–1097, 2011.
+[27] S. N. Negahban and M. J. Wainwright. Restricted strong convexity and weighted matrix completion:
+Optimal bounds with noise. Journal of Machine Learning Research, 13:1685–1697, 2012.
+[28] A. Nica and R. Speicher.
+London Mathematical Society Lecture Note Series, vol. 335.
+Cambridge
+University Press, 2006.
+34
+
+[29] S. Oymak and B. Hassibi. New null space results and recovery thresholds for matrix rank minimization.
+Nov 2010. available online at http://arxiv.org/abs/1011.6326.
+[30] J. Pearl. Causal inference in statistics: An overview. Statistics Surveys, 3:96–146, 2009.
+[31] J. Pearl. Causality: Models, Reasoning, and Inference. 2nd. Cambridge University Press, New York,
+2009.
+[32] J. Pearl and E. Bareinboim. A note on ‘generalizability of study results’. Journal of Epidemiology,
+30:186–188, 2019.
+[33] J. Pearl and D. Mackenzie. The Book of Why. Basic Books, New York, 2018.
+[34] B. Recht. A simpler approach to matrix completion. Journal of Machine Learning Research, 12:3413–
+3430, 2011.
+[35] B. Recht, M. Fazel, and P. A. Parrilo. Guaranteed minimum-rank solution of linear matrix equations
+via nuclear norm minimization. 2007. available online at http://www.dsp.ece.rice.edu/cs/.
+[36] A. Rohde and A. B. Tsybakov. Estimation of high-dimensional low-rank matrices.
+The Annals of
+Statistics, 39:887–930, 2011.
+[37] P. R. Rosenbaum and D. B. Rubin. Thecentral role of the propensity score in observational studies for
+causal effects. Biometrika, 70, 1983.
+[38] D. B. Rubin. Matched Sampling for Causal Effects. Cambridge: Cambridge University Press, 2006.
+[39] A. Shaikh and P. Toulis.
+Randomization tests in observational studies with staggered adoption of
+treatment.
+University of Chicago, Becker Friedman Institute for Economics Working Paper, (144),
+2019.
+[40] R. Speicher. Free probability and random matrices. In: Proc. ICM, III:477–501, 2014.
+[41] N. Srebro, N. Alon, and T. S. Jaakkola. Generalization error bounds for collaborative prediction with
+low-rank matrices. in Advances in Neural Information Processing Systems, 17:1321–1328, 2005. eds. L.
+K. Saul, Y.Weiss, and L. Bottou.
+[42] M. Stojnic. A framework for perfromance characterization of LASSO algortihms. available online at
+http://arxiv.org/abs/1303.7291.
+[43] M. Stojnic. Regularly random duality. available online at http://arxiv.org/abs/1303.7295.
+[44] M. Stojnic. Upper-bounding ℓ1-optimization weak thresholds. available online at http://arxiv.org/
+abs/1303.7289.
+[45] M. Stojnic. Various thresholds for ℓ1-optimization in compressed sensing. available online at http://
+arxiv.org/abs/0907.3666.
+[46] M. Stojnic. A simple performance analysis of ℓ1-optimization in compressed sensing. ICASSP, Inter-
+national Conference on Acoustics, Signal and Speech Processing, pages 3021–3024, 15-19 April 2009.
+Taipei, Taiwan.
+[47] M. Stojnic. Block-length dependent thresholds for ℓ2/ℓ1-optimization in block-sparse compressed sens-
+ing. ICASSP, IEEE International Conference on Acoustics, Signal and Speech Processing, pages 3918–
+3921, 14-19 March 2010. Dallas, TX.
+[48] M. Stojnic. ℓ1 optimization and its various thresholds in compressed sensing. ICASSP, IEEE Inter-
+national Conference on Acoustics, Signal and Speech Processing, pages 3910–3913, 14-19 March 2010.
+Dallas, TX.
+[49] M. Stojnic. ℓ2/ℓ1-optimization in block-sparse compressed sensing and its strong thresholds. IEEE
+Journal of Selected Topics in Signal Processing, 4(2):350–357, 2010.
+35
+
+[50] M. Stojnic. Recovery thresholds for ℓ1 optimization in binary compressed sensing. ISIT, IEEE Inter-
+national Symposium on Information Theory, pages 1593 – 1597, 13-18 June 2010. Austin, TX.
+[51] M. Stojnic. Towards improving ℓ1 optimization in compressed sensing. ICASSP, IEEE International
+Conference on Acoustics, Signal and Speech Processing, pages 3938–3941, 14-19 March 2010. Dallas,
+TX.
+[52] M. Stojnic, F. Parvaresh, and B. Hassibi. On the reconstruction of block-sparse signals with an optimal
+number of measurements. IEEE Trans. on Signal Processing, 57(8):3075–3085, August 2009.
+[53] A. M. Tulino and S. Verd´u. Random Matrix Theory and Wireless Communications. Foundations and
+Trends® in Communications and Information Theory: Vol. 1. Now Publishers, Hanover, 2004.
+[54] D. Voiculescu. Addition of certain non-commuting random variables. J. Funct. Anal., 66(3):323–346,
+1986.
+[55] D. Voiculescu. Multiplication of certain noncommuting random variables. J. Operator Theory, 18:2223–
+2235, 1987.
+[56] D. Voiculescu. Limit laws for random matrices and free products. Invent. Math., 104(1):201–220, 1991.
+[57] R. Xiong and M. Pelger. Large dimensional latent factor modeling with missing observations and appli-
+cations to causal inference. 2018. Electronic copy available at: https://ssrn.com/abstract=3465357.
+[58] Y. Xu. Generalized synthetic control method: Causal inference with interactive fixed effects models.
+Political Analysis, 25:57–76, 2017.
+A
+Proof of Theorem 1
+As mentioned earlier, the proof of Theorem 1 is conceptually identical to the corresponding proof when
+matrix X is symmetric. A detailed proof for the symmetric matrices is given below. Before being able to
+present the proof we need a couple of technical lemmas.
+Lemma 6. Let C = CT ∈ Rn×n. Also let all eigenvalues of C belong to the interval [−1, 1]. Finally, let the
+first k entries on the main diagonal, Ci,i, 1 ≤ i ≤ k, be larger than or equal to 1. Then the upper k × k left
+block of C, C1:k,1:k, is an identity matrix, i.e.
+C1:k,1:k
+=
+Ik×k.
+(172)
+Proof. Let λmax(C) be the maximum eigenvalue of C. Then
+λmax(C)
+≜
+max
+∥c∥2=1 cT Cc.
+(173)
+Since by assumptions 1 ≤ Ci,i, 1 ≤ i ≤ k and λmax(C) ≤ 1 we also have for any 1 ≤ i ≤ k
+1 ≤ Ci,i ≤ max
+∥c∥2=1 cT Cc ≜ λmax(C) ≤ 1,
+(174)
+which implies C(i, i) = 1, 1 ≤ i ≤ k. The proof that all other elements of C1:k,1:k are equal to zero proceeds
+inductively.
+1) Induction move from l = 1 to l = 2: First we look at the upper block of size 2 × 2, i.e. at C1:2,1:2.
+We then have
+1 ≥ max
+∥c∥2=1 cT Cc ≥
+max
+∥c1:2∥2=1 cT
+1:2C1:2,1:2c1:2
+≥
+max
+∥c1:2∥2=1 (∥c1:2∥2 + 2|c1c2C1,2|)
+≥
+max
+∥c1:2∥2=1 (1 + 2|c1c2C1,2|) ≥ 1,
+(175)
+36
+
+which implies C1,2 = 0.
+2) Induction move from l to l + 1: Now we look at the upper block of size (l + 1) × (l + 1), i.e. at
+C1:l+1,1:l+1 while assuming that C1:l,1:l = Il×l. We then have
+1
+≥
+max
+∥c∥2=1 cT Cc
+≥
+max
+∥c1:l+1∥2=1 cT
+1:l+1C1:l+1,1:l+1c1:l+1
+≥
+max
+∥c1:l+1∥2=1
+�
+∥c1:l+1∥2 + 2|cT
+1:lC1:l,l+1cl+1|
+�
+≥
+max
+∥c1:l+1∥2=1
+�
+1 + 2|cT
+1:lC1:l,l+1cl+1|
+�
+≥
+1,
+(176)
+which implies C1:l,l+1 = 0l×1 and completes the proof.
+Lemma 7. Assume the setup of Lemma 6. Then the upper k × k left block of C, C1:k,1:k, is an identity
+matrix and the upper k × (n − k) right block of C, C1:k,n−k+1:n is a zero matrix, i.e.
+C1:k,1:k
+=
+Ik×k
+C1:k,n−k+1:n
+=
+0k×(n−k).
+(177)
+Proof. The first part follows by Lemma 6. We now focus on the second part. Consider the following partition
+of matrix C
+C
+=
+�
+C1:k,1:k
+C1:k,n−k+1:n
+Cn−k+1:n,1:k
+Cn−k+1:n,n−k+1:n
+�
+=
+�
+Ik×k
+C1:k,n−k+1:n
+Cn−k+1:n,1:k
+Cn−k+1:n,n−k+1:n
+�
+.
+(178)
+Then assuming that the largest nonzero singular value of C1:k,n−k+1:n is equal to b > 0, we have
+1
+≥
+max
+∥c∥2=1 cT Cc
+≥
+max
+∥c1:k∥2=a,cn−k+1:n
+�
+cT
+1:kC1:k,1:kc1:k + 2|cT
+1:kC1:k,n−k+1:ncn−k+1:n| + cT
+n−k+1:nCn−k+1:n,n−k+1:ncn−k+1:n
+�
+≥
+max
+∥c1:k∥2=a,cn−k+1:n
+�
+a2 + 2|cT
+1:kC1:k,n−k+1:ncn−k+1:n| + cT
+n−k+1:nCn−k+1:n,n−k+1:ncn−k+1:n
+�
+≥
+max
+∥c1:k∥2=a,cn−k+1:n
+�
+a2 + 2|cT
+1:kC1:k,n−k+1:ncn−k+1:n| − cT
+n−k+1:ncn−k+1:n
+�
+≥
+max
+a∈[0,1]
+�
+a2 + 2ba
+�
+1 − a2 − (1 − a2)
+�
+=
+max
+a∈[0,1]
+�
+2a2 − 1 + 2ba
+�
+1 − a2
+�
+,
+(179)
+where the fourth inequality follows since the minimum eigenvalue of Cn−k+1:n,n−k+1:n is larger than or equal
+to the minimum eigenvalue of C which is by the lemma’s assumption larger than or equal to -1. Now, we
+further have
+c ≜ 2a
+�
+1 − a2
+and
+2a2 − 1 + 2ba
+�
+1 − a2 =
+�
+1 − c2 + bc,
+(180)
+and
+d(
+√
+1 − c2 + bc)
+dc
+=
+−c
+√
+1 − c2 + b = 0.
+(181)
+37
+
+From (181) we then easily obtain
+c =
+b
+√
+1 + b2 .
+(182)
+A combination of (179), (180), and (182) gives
+1 ≥ max
+∥c∥2=1 cT Cc ≥ max
+a∈[0,1]
+�
+2a2 − 1 + 2ba
+�
+1 − a2
+�
+=
+√
+1 + b2,
+(183)
+which implies b = 0 and automatically C1:k,n−k+1:n = 0k×1. This completes the proof.
+Now we can consider the above mentioned theorem that adapts the general ℓ1 equivalence condition
+result from [44–46] to the corresponding one for the ℓ1 norm of the singular/eigenvalues (similar adaptation
+can also be found in [29]).
+Theorem 3. (ℓ∗
+0 − ℓ∗
+1-equivalence condition (LRR) – symmetric X) Consider a ¯U ∈ Rn×k such that
+¯U T ¯U = Ik×k and a rank − k a priori known to be symmetric matrix Xsol = X ∈ Rn×n with all of its
+columns belonging to the span of ¯U. For concreteness, and without loss of generality, assume that X has only
+positive nonzero eigenvalues. For a given matrix A ∈ Rm×n2 (m ≤ n2) assume that y = Avec(X) ∈ Rm. If
+(∀W ∈ Rn×n|Avec(W) = 0m×1, W = W T ̸= 0n×n)
+− tr ( ¯U T W ¯U) < ℓ∗
+1(( ¯U ⊥)T W ¯U ⊥),
+(184)
+then the solutions of (9) and (10) coincide. Moreover, if
+(∃W ∈ Rn×n|Avec(W) = 0m×1, W = W T ̸= 0n×n)
+− tr ( ¯U T W ¯U) ≥ ℓ∗
+1(( ¯U ⊥)T W ¯U ⊥),
+(185)
+then there is an X from the above set of the symmetric matrices with columns belonging to the span of ¯U
+such that the solutions of (9) and (10) are different.
+Proof. The proof follows literally step-by-step the proof of the corresponding theorem in [44–46] and adapts
+it to matrices or their singular/eigenvalues. For experts in the field this adaptation is highly likely to be
+viewed as trivial and certainly doesn’t need to be as detailed as we will make it to be. Nonetheless, to ensure
+a perfect clarity of all arguments we provide a step-by-step instructional derivation. For concreteness and
+without loss of generality we also assume that the eigen-decomposition of X is
+X = UΛU T =
+� ¯U
+¯U ⊥� �
+¯ΛX
+0k×(n−k)
+0(n−k)×k
+¯Λ⊥
+X
+� � ¯U
+¯U ⊥�T .
+(186)
+(i) =⇒ (the if part): Following step-by-step the proof of Theorem 2 in [46], we start by assuming that
+ˆX is the solution of (10). Then we want to show that if (184) holds then ˆX = X. As usual, we instead of that,
+assume opposite, i.e. we assume that (184) holds but ˆX ̸= X. Then since y = Avec( ˆ
+X) and y = Avec(X)
+must hold simultaneously there must exist W such that ˆX = X + W with W ̸= 0, Avec(W) = 0. Moreover,
+since ˆX is the solution of (10) one must also have
+ℓ∗
+1(X + W) = ℓ∗
+1( ˆX)
+≤
+ℓ∗
+1(X)
+⇐⇒
+ℓ∗
+1(
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+≤
+ℓ∗
+1(X)
+=⇒
+ℓ∗
+1( ¯U T (X + W) ¯U) + ℓ∗
+1(( ¯U ⊥)T (X + W) ¯U ⊥)
+≤
+ℓ∗
+1(X).
+(187)
+The last implication follows after one trivially notes
+ℓ∗
+1(
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+=
+max
+Λ∗=ΛT
+∗ ∈L∗
+tr (Λ∗
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+≥
+max
+Λ∗=ΛT
+∗ ∈L0
+∗
+tr (Λ∗
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+38
+
+=
+ℓ∗
+1( ¯U T (X + W) ¯U) + ℓ∗
+1(( ¯U ⊥)T (X + W) ¯U ⊥),
+(188)
+where
+L0
+∗
+≜
+�
+Λ∗ ∈ Rn×n|Λ∗ = ΛT
+∗ , Λ∗ΛT
+∗ ≤ I, Λ∗ =
+�
+Λ∗,1
+0k×(n−k)
+0(n−k)×k
+Λ∗,2
+��
+⊆
+�
+Λ∗ ∈ Rn×n|Λ∗ = ΛT
+∗ , Λ∗ΛT
+∗ ≤ I
+�
+≜ L∗.
+(189)
+The key observation – “Removing the absolute values”:
+Now, the key observation made in [46] comes into play. Namely, one notes that the absolute values can
+be removed in the nonzero part and that the ℓ∗
+1(·) can be “replaced” by tr (·). Such a simple observation
+is the most fundamental reason for all the success of the RDT when used for the exact performance
+characterization of the structured objects’ recovery. From (187) we then have
+ℓ∗
+1( ¯U T (X + W) ¯U) + ℓ∗
+1(( ¯U ⊥)T (X + W) ¯U ⊥)
+≤
+ℓ∗
+1(X)
+=⇒
+tr ( ¯U T (X + W) ¯U) + ℓ∗
+1(( ¯U ⊥)T (W) ¯U ⊥)
+≤
+ℓ∗
+1(X)
+⇐⇒
+tr ( ¯U T W ¯U) + ℓ∗
+1(( ¯U ⊥)T W ¯U ⊥)
+≤
+0.
+(190)
+We have arrived at a contradiction as the last inequality in (190) is exactly the opposite of (184). This
+implies that our initial assumption ˆX ̸= X cannot hold and we therefore must have ˆX = X. This is precisely
+the claim of the first part of the theorem.
+(ii) ⇐= (the only if part): We now assume that (185) holds, i.e.
+(∃W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n)
+− tr (( ¯U)T W ¯U) ≥ ℓ∗
+1(( ¯U ⊥)T W ¯U ⊥)
+(191)
+and would like to show that for such a W there is a symmetric rank-k matrix X with the columns belonging
+to the span of ¯U such that y = Avec(X), and the following holds
+ℓ∗
+1(X + W) < ℓ∗
+1(X).
+(192)
+Existence of such an X would ensure that it both, satisfies all the constraints in (10) and is not the
+solution of (10). Following the strategy of [44] one can reverse all the above steps from (191) to (187) with
+strict inequalities and arrive at the first inequality in (187) which is exactly (192). There are two implications
+that cause problems in such a reversal process, the one in (191) and the one in (187). If these implications
+were equivalences everything would be fine. We address these two implications separately.
+1) the implication in (190) – particular X to “overwhelm” W: Assume X = ¯UΛx ¯U T with Λx >
+0 being a diagonal matrix with arbitrarily large elements on the main diagonal (here it is sufficient even to
+choose diagonal of Λx so that its smallest element is larger than the maximum eigenvalue of ¯U T W ¯U). Now
+one of course sees the main idea behind the “removing the absolute values” concept from [44,46]. Namely,
+for such an X one has that ℓ∗
+1( ¯U T X + W) ¯U) = tr(ℓ∗
+1( ¯U T X + W) ¯U)) since for symmetric matrices the ℓ∗
+1(·)
+(as the sum of the argument’s absolute eigenvalues) and tr (·) (as the sum of the argument’s eigenvalues) are
+equal. That basically means that when going backwards the second inequality in (190) not only follows from
+the first one but also implies it as well. In other words, for X = ¯UΛx ¯U T (with Λx > 0 and arbitrarily large)
+tr ( ¯U T W ¯U) + ℓ∗
+1(( ¯U ⊥)T W ¯U ⊥)
+≤
+0
+⇐⇒
+tr ( ¯U T (X + W) ¯U ) + ℓ∗
+1(( ¯U ⊥)T (W) ¯U ⊥)
+≤
+ℓ∗
+1(X)
+⇐⇒
+ℓ∗
+1( ¯U T (X + W) ¯U) + ℓ∗
+1(( ¯U ⊥)T (X + W) ¯U ⊥)
+≤
+ℓ∗
+1(X),
+(193)
+which basically mans that there is an X that can “overwhelm” W (in the span of ¯U) and ensures that the
+“removing the absolute values” is not only a sufficient but also a necessary concept for creating the
+relaxation equivalence condition.
+2) the implication in (187): One would now need to somehow show that the third inequality in (187)
+not only follows from the second one but also implies it as well. This boils down to showing that inequality in
+(188) can be replaced with an equality or, alternatively, that L0 and L are provisionally equivalent. Neither
+39
+
+of these statements is generically true. However, since we have a set of X at our disposal there might be an
+X for which they actually hold. We continue to assume X = ¯UΛx ¯U T with Λx > 0 being a diagonal matrix
+with arbitrarily large entries on the main diagonal. Then the last equality in (188) gives
+ℓ∗
+1( ¯U T (X + W) ¯U) + ℓ∗
+1(( ¯U ⊥)T (X + W) ¯U ⊥)
+≤
+ℓ∗
+1(X)
+⇐⇒
+maxΛ∗=ΛT
+∗ ∈L0∗ tr (Λ∗
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+≤
+ℓ∗
+1(X).
+(194)
+Also, one has
+maxΛ∗=ΛT
+∗ ∈L0
+∗ tr (Λ∗
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+≤
+ℓ∗
+1(X)
+⇐⇒
+maxΛ∗,i=ΛT
+∗,i,Λ∗,iΛT
+∗,i≤I,i∈{1,2} tr (Λ∗,1 ¯U T X ¯U + Λ∗,2( ¯U ⊥)T W ¯U ⊥)
+≤
+ℓ∗
+1(X)
+⇐⇒
+maxΛ∗,i=ΛT
+∗,i,Λ∗,iΛT
+∗,i≤I,i∈{1,2} tr (Λ∗,1Λx + Λ∗,2( ¯U ⊥)T W ¯U ⊥)
+≤
+tr (Λx).
+(195)
+Now, if at least one of the elements on the main diagonal of Λ∗,1, diag(Λ∗,1), is smaller than 1, then the
+corresponding element on the diagonal of Λx can be made arbitrarily large compared to the other elements
+of Λx and one would have
+maxΛ∗,i=ΛT
+∗,i,Λ∗,iΛT
+∗,i≤I,i∈{1,2} tr (Λ∗,1Λx + Λ∗,2( ¯U ⊥)T W ¯U ⊥)
+<
+tr (Λx)
+⇐⇒
+maxΛ∗=ΛT
+∗ ∈L0∗ tr (Λ∗
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+<
+ℓ∗
+1(X)
+⇐⇒
+maxΛ∗=ΛT
+∗ ∈L∗ tr (Λ∗
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+<
+ℓ∗
+1(X),
+(196)
+where the last equivalence holds since the difference of the terms on the left-hand side in the last two
+inequalities is bounded independently of X. Also, the last inequality in (196) together with the first equality
+in (188) and the first inequality in (187) produces (192). Therefore the only scenario that is left as potentially
+not producing (192) is when all the elements on the main diagonal are larger than or equal to 1. However,
+the two lemmas preceding the theorem show that in such a scenario L0 = L and one consequently has an
+equality instead of the inequality in (188) which then, together with (187), implies (192). This completes
+the proof of the second (“the only if”) part of the theorem and therefore of the entire theorem.
+40
+
diff --git a/ENAyT4oBgHgl3EQf4vqx/content/tmp_files/load_file.txt b/ENAyT4oBgHgl3EQf4vqx/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..67a61df6ad883e42c17bb686febd5e7cf36d2e56
--- /dev/null
+++ b/ENAyT4oBgHgl3EQf4vqx/content/tmp_files/load_file.txt
@@ -0,0 +1,1349 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf,len=1348
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='00793v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='ML]  2 Jan 2023  Causal Inference (C-inf) — closed form worst case  typical phase transitions  Agostino Capponi ∗ Mihailo Stojnic †  Department of Industrial Engineering and Operations Research  Columbia University, New York, NY 10027, USA  Abstract  In this paper we establish a mathematically rigorous connection between Causal inference (C-inf)  and the low-rank recovery (LRR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Using Random Duality Theory (RDT) concepts developed in [46,48,50]  and novel mathematical strategies related to free probability theory, we obtain the exact explicit typical  (and achievable) worst case phase transitions (PT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' These PT precisely separate scenarios where causal  inference via LRR is possible from those where it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We supplement our mathematical analysis with  numerical experiments that confirm the theoretical predictions of PT phenomena, and further show that  the two closely match for fairly small sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We obtain simple closed form representations for the  resulting PTs, which highlight direct relations between the low rankness of the target C-inf matrix and the  time of the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Hence, our results can be used to determine the range of C-inf’s typical applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Index Terms: Causal inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Random duality theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Matrix completion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Spar-  sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  1  Introduction  The Causal inference (C-inf) discipline deals with design of methods for estimating causal effects in  panel data settings, where a subset of the units are exposed to a treatment during some time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The  goal is estimating counterfactual outcomes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=', that outcome for those units were the treatment not been  applied for that period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Casual inference plays a key role in decision making, and is essential in business decisions, network design,  medical sciences, and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' It allows answering questions of the form: What would happen to a data  center’s latency if a new congestion control algorithm were not used?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' What would have been the systolic  blood pressure of a patient if the new drug were not given to her?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  This problem of estimating the counterfactual appears in many disciplines, including economics, finance,  health, and social sciences (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' [2, 16, 18, 19, 37, 38, 58]), and computer science (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' [30–33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The  increasing availability of big data through digital services and smart sensors calls makes it possible to design  efficient algorithmic techniques to address these fundamental questions in counterfactual estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  The econometrics and social sciences communities have proposed three main approaches to causal in-  ference: 1) the unconfoundedness (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' [19, 37]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' 2) the synthetic control (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' [1, 2, 16]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' and 3)  the matrix completion (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' [3, 4, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Matrix completion methods build upon the foundation works  of [9, 11, 34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Perhaps unexpectedly, all three methods heavily rely on mathematical, statistical, and ulti-  mately algorithmic concepts with very deep roots in information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Our work is positioned within the  third line of work that mathematically resembles the matrix completion (MC) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Our main contribution is to develop a rigorous connection between Causal inference (C-inf) from  observational data and the low-rank recovery (LRR) problem in compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Methodologically,  we integrate Random Duality Theory (RDT) concepts developed in [46, 48, 50] with novel mathematical  ∗e-mail: ac3827@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='edu  †e-mail: flatoyer@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='com  1  strategies related to free probability theory, and manage to obtain the exact explicit typical (and achievable)  worst case phase transitions (PT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' These PT provide a precise separation between regions of the parameter  space where the counterfactual can be perfectly estimated via LRR from those regions where this is not  possible for large sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Our numerical study complements our theoretical predictions by showing  that theory and numerical simulation closely match even if the sample size is fairly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  2  Mathematics of causal inference  To put our contribution into a proper context, we begin by presenting the explicit causal inference (C-inf)  ↔ matrix completion (MC) connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  To introduce the most basic mathematical description of the main C-inf concepts we adopt the standard  matrix completion (MC) terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' As is well known the matrix completion problems belong to a class  of the so-called low-rank recovery problems (LRR) which themselves belong to a broader class of the so-  called structured recovery (SR) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In its most general way the description of the structured recovery  problems starts by assuming that one has access to the observation vector y ∈ Rm given by the following  yi = fi(Ai,:xsol),  (1)  where A ∈ Rm×n is a known system matrix (with rows Ai,:, 1 ≤ i ≤ m), xsol ∈ Rn is the unknown vector,  and fi(·) : R −→ R are known real functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In the treatments that will be of our interest here the functions  fi(·) will be assumed to be identically linear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' it will be assumed that fi(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then the structured  recovery assumes utilizing the xsol ’s a priori known structure to eventually recover it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Most often, that  boils down practically to finding efficient algorithmic designs that take as inputs the observation vector y  and the system matrix A and successfully output xsol or its a sufficiently close approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover,  under efficient algorithms one typically views those that run preferably provably in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' What  typically differentiates between various forms of the SR problems are the structures that xsol possesses as  well as the structure of the system matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Along the same lines, what typically differentiates the level  of success (or usefulness) that some of these forms might achieve is the ultimate theoretical and practical  capabilities of the algorithms employed for their solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  The structured recovery problems have been the subject of an extensive research for a better half of the  last century and many beautiful results appeared regarding their various theoretical and practical aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  However, it is the emergence of compressed sensing (CS) about 20 years ago that almost singlehandedly  made a key transformational change in how these problems and the surrounding research are perceived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' As a  result of such a perceptional change, the interest in the structured recovery, its importance, popularity, and  ultimate usefulness skyrocketed to the heights unseen ever before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' It is, of course, hard to pinpoint exactly  what could be the secret behind the compressed sensing meteoric rise to success, but the overall conceptual  simplicity probably contributed to some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We refer for more on the CS invention, simplicity, and  further developments to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' [7,8,13,14,44,45,47–50,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' At the same time, since (mathematically speaking)  the CS is not precisely the main topic of our interest here, we, before proceeding further, just briefly mention  that it basically assumes the above mentioned structured recovery setup with the sparsity of xsol being the  underlying structuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1  Low rank recovery (LRR)  While the mathematical problems typically seen in compressed sensing will not exactly match the key  mathematical problems of this paper, the above mentioned low-rank recovery (LRR) ones will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To introduce  the LRR problems, we first note that almost everything mentioned above for the generic structured recovery  remains in place in the LRR scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In fact, there will be only one key difference compared to the standard  CS setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The xsol (which is a sparse vector in the CS context) will within the LRR considerations be viewed  as a vectorized unknown matrix Xsol and consequently the imposed a priori known structure will be the  low-rankness of Xsol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This, of course, is nothing but the sparsity of the vector of the singular values of  Xsol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In other words, in compressed sensing the unknown vector xsol itself is sparse, whereas in the LRR  the vector of the singular values of the unknown matrix Xsol is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  2  In more mathematical terms, the LRR can then be described in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' One first starts with  the above mentioned vectorizing of matrix Xsol  xsol = vec(Xsol),  (2)  with vec(·) stacking its matrix argument columns one after another (starting from the very first one) into  a column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Assuming Xsol ∈ Rn×n, trivial dimensional adjustments then give A ∈ Rm×n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  The  low-rankness (rank(Xsol) = k ≤ n) and the corresponding sparsity are imposed via the singular value  decomposition (SVD)  X = UΣV T ,  (3)  where  σ(X) ≜ diag(Σ)  and  U T U = In×n  and  V T V = In×n,  (4)  with In×n being the identity matrix of size n × n and diag(·) being the operator that extracts the diagonal  from its matrix argument and puts it into a column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' When clear from the context, we will abbreviate  and write just σ instead of σ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover, σi(X) will stand for the i-th component of σ(X) (with σi  often being its shorter version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' As is of course well known, the elements of σ ∈ Rn are precisely the above  mentioned singular values of X and the number of nonzero such elements is precisely the rank of Xsol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' It  is probably obvious, but we state it to ensure the overall clarity, that in typical SVD treatments in the  mathematical literature one will often find in (4) Ik×k as a replacement for In×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In other words, one will  often find that the underlying identity matrix is of size k × k instead of size n × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The reason for our choice  is to ensure a complete parallelism between the LRR and the compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Basically, this choice  makes the underlying vector of the singular values visibly sparse (by this definition it will automatically have  n − k zeros) and as such more in parallel with the corresponding sparse vector structure one typically finds  in the compressed sensing setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To further maintain the parallelism with compressed sensing, we also find  it useful to introduce ℓ∗  p(X) as the so-called ℓ∗  p quasi-norm of X  ℓ∗  p(X) ≜ ℓp(σ(X)), p ∈ R+,  (5)  where, ℓp(·) is the standard vector ℓp (quasi-) norm, and to note that the following useful matrix-vector  limiting ℓp(·) connections also hold  ℓ∗  0(Xsol) ≜ ℓ0(σ(Xsol)) = ∥σ(Xsol)∥0 = lim  p−→0 ∥σ(Xsol)∥p = lim  p−→0 ℓp(σ(Xsol)) = lim  p−→0 ℓ∗  p(Xsol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (6)  One can then restate (1) adapted to fit the LRR context as  yi = Ai,:xsol = Ai,:vec(Xsol)  where  ℓ∗  0(Xsol) = ℓ0(σ(Xsol)) = ∥σ(Xsol)∥0 = k, k ∈ N,  (7)  and Ai,: being the i-th row of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Finally one can summarize the LRR mechanism as:  Generic LRR  Observations – forming y from Xsol  Given A and Xsol create y as  y = Avec(Xsol), ℓ∗  0(Xsol) = k, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (8)  Observations – forming y from Xsol  Given y and A from (8) can one efficiently  recover Xsol back?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  It is not that difficult to see that ℓ∗  0(Xsol) basically serves as a counting function that counts the number  of the nonzero singular values of Xsol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Its analogy with the function ℓ0(xsol), that appears in the compressed  sensing setup and counts the number of the nonzero elements of the unknown sparse vector, is rather obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Moreover, both such an analogy and the underlying sparsity that enables it suggest an algorithmic way that  3  one can try to employ to ultimately recover Xsol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' By the above definition, the LRR is the inverse problem  for (8) and can be posed as the so-called ℓ∗  0-minimization optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Since such a minimization  is a highly non-convex problem it is not known to be solvable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To design polynomially  solvable heuristics one then, following into the compressed sensing footsteps, introduces the tightest convex  norm relaxation concept and replaces the ℓ∗  0- with the ℓ∗  1-minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  ℓ∗  0-minimization (LRR/MC/C-inf – VMT)  min  X  ℓ∗  0(X)  subject to  y = Avec(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (9)  ℓ∗  1-minimization (LRR/MC/C-inf – VMT)  min  X  ℓ∗  1(X)  subject to  y = Avec(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (10)  More on the history, usefulness, and applicability of the above introduced generic low rank recovery (LRR)  via the tightest convex norm relaxation can be found in the introductory papers [35,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We also point out  that we might on occasion refer to the above LRR description as the “vectorized matrix terminology” (VMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Below, we wll also find it useful to introduce the very same LRR via the corresponding, so-called, “masking  matrix terminology” (MMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2  Matrix completion (MC) – a special case of LRR  The above is of course a generic LRR description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The matrix completion is a particular type of the LRR  or, in technical terms, a special case of the above described LRR mathematical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In the matrix  completion scenario one deals with a very particular type of system matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Instead of (7) one then has  yi = Avec(Xsol)  where  ℓ∗  0(Xsol) = ℓ0(σ(Xsol)) = ∥σ(Xsol)∥0 = k, k ∈ N  and  ∀p ∈ R+, ∥Ai,:∥p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (11)  This practically means that each row of system matrix A has exactly one element equal to one and all other  elements equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In a way, one can think of A as being a cardinality m subset of rows of In2×n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  The reason for the appearance of such a matrix A is of course the origin of the matrix completion itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Basically, the matrix A essentially emulates a “mask” that one puts on matrix X which allows reading out  only m of its n2 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' More on the origin of the matrix completion, its importance, and different related  algorithmic considerations can be found in the introductory papers [9,41] as well as in many further studies  that followed later on (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' [10,21–27,36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  To ensure the completeness of the overall presentation and to be in alignment with the standard rep-  resentation of the matrix completion problem usually seen throughout the literature we reformulate (11)  through the above mentioned masking matrices terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let M ∈ Rn×n be a masking matrix such that  Mi,j =  �  1,  (i, j)-th element of Xsol is observed  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (12)  One can then describe creating the linear observations in the matrix completion as the following  Y = M ◦ Xsol,  (13)  where ◦ stands for the component-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Clearly, ones in matrix M allow reading out cor-  responding elements of Xsol while zeros block (mask) them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To fit in the above linear description in (11)  one would then create A by removing all the zero rows from diag−1(vec(M))In2×n2 (diag−1(·) creates the  diagonal matrix with the elements on the main diagonal equal to its vector argument and, in a way, is the  inverse of the earlier introduced diag(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Consequently, y would be obtained as y = AXsol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='3  Causal inference (C-inf) – a special case of MC  Finally, the causal inference, which will be the main topic of our interest, is yet another special case of the  above low rank recovery framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In fact, to be a bit more precise, the matrix completion is a special case  4  of the above LRR and the causal inference is a special case of the matrix completion itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The connection  between the matrix completion (MC) and the causal inference (C-inf) was established in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover, a  very nice additional connections to the unconfoundedness and the synthetic control were established in [4]  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Here, we will also work within the context of the matrix completion-causal inference connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  To that end we start by making this connection mathematically more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Namely, if one thinks of  the matrix completion as a way of “masking” X and reading out the unmasked elements, then the causal  inference does exactly the same thing while additionally imposing a particular structure on the mask itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Let, as above, M ∈ Rn×n be a masking matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then for a fixed l ≤ n one, in a causal inference context,  has  M ≜ M (l)  and  Mi,j = M (l)  i,j =  �  1,  if min(i, j) ≤ l  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (14)  The above is the so-called block causal inference setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Figures 1 and 2 showcase the key difference  between the generic matrix completion and the causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In the generic matrix completion the mask  matrix M can have zeros and ones located within the matrix in a basically arbitrary way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' On the other  hand, in the causal inference setup the structure of M is somewhat particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In general, in a row of the  matrix M the first zero can appear not later than the first zeros appeared in any of the previous rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' After  a zero all other remaining elements in the same row must also be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' For the block case of our interest  here it is as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  0  1  1  1  0  1  1  0  1  1  1  0  0  1  0  0  1  0  0  1  M =  Matrix M – general matrix completion  0 and 1 randomly scattered  Figure 1: Matrix M – general matrix completion (MC) setup  The rationale for the use of the block causal inference is the most easily understood if one views things in  the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Namely, if the columns of the masking matrix M represent time axis then the observations  related to ceratin rows of the matrix will not be available after a fixed point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In the block scenario  this point is fixed across the affected rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' However, it does not necessarily need to be fixed (for more in this  direction we refer to [2] (in particular, the California tobacco example), [57] (in particular, the latent factor  modeling in the context of the simultaneous/staggered treatment adoption), and to [5,6,39] (in particular,  the health care applications) as excellent references for understanding the need of various C-inf scenarios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  As this is the introductory paper, where we present the overall methodology, we selected the block causal  inference scenario as probably the most representative and well-known one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  In some of our companion  papers we will show how the methodology that we are introducing here can be utilized to handle other C-inf  scenarios as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Under this casual inference assumption one then has the following for the collection of the observation Y  5  M =  1  Matrix M – block causal inference (C-inf)  1  0  1  0 and 1 grouped in blocks  l × l block of all 1s  l × (n − l) block of all 1s  (n − l) × (n − l) block of all 0s  (n − l) × l block of all 1s  Figure 2: Matrix M ≜ M (l) – block causal inference (C-inf) setup  in the matrix completion  Y = M ◦ X = M (l) ◦ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (15)  We will more often than not avoid specifying superscript to make writing easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' From the context it will  be clear if it should be there and, if so, what its value should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To fit in the linear description of (11)  one would create A in the following way: 1) start by choosing the first ln rows of In2×n2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' and 2) then for  any next set of n rows of In2×n2 continue by choosing its subset of first l rows while skipping the remaining  n − l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Similarly, y would be obtained by stacking the columns of Y and skipping the elements Yi,j where  min(i, j) > l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' As mentioned above, we will find it useful later on to have (9) and (10) rewritten in the  “masking matrix terminology” (MMT) as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  ℓ∗  0-minimization (C-inf – MMT)  min  X  ℓ∗  0(X)  subject to  Y = M ◦ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (16)  ℓ∗  1-minimization (C-inf – MMT)  min  X  ℓ∗  1(X)  subject to  Y = M ◦ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (17)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='4  C-inf ↔ MC connection via counterfactuals  To understand where such a structure may come from, it is useful to connect it to the context of treat-  ment/units/times following the terminology in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In such context, one assumes that the matrix X contains  observations about a certain set of, say, n units (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' individuals, subpopulations, and geographic regions)  over a period of say, n, time instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Assuming that the rows of X correspond to the units and the columns  to the time instances, one wants to estimate the effects that a certain treatment may have on the treated  units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' A subset of the units (say those that correspond to the rows i > l) is then at time l exposed to an  irreversible treatment (once the treatment starts its effects can not be reversed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Examples of treatments  include health therapies, socio-economic policies, and taxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To be able to appropriately assess the resulting  treatment effects, in addition to having the values of X after the treatment, one would need to have the  access to the so-called counterfactuals – the values of the treated units – had the treatment not been  applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Switching back to the matrix completion terminology, one would need to estimate (a presumably  low rank) X while not having access to its portion covered by the block-mask M = M (l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In other words,  one would need to solve (16) with M = M (l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  6  Of course, as the change in the structure of A or M does not prevent the utilization of the above ℓ∗  0 −→ ℓ∗  1  relaxation concept, one typically employs it as a provably polynomial heuristic strategy to solve the matrix  completion/causal inference problems approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' While it is somewhat intuitive that, as the closest  convex norm relaxation, the above ℓ∗  1-minimization might produce a matrix similar to the unknown Xsol it  is perhaps quite surprising that in certain scenarios it actually perfectly recovers the exact Xsol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Potential  existence of such an ℓ∗  0 − ℓ∗  1 equivalence is rather remarkable phenomenon and determining when or how  often it happens is pretty much the key task in the mathematical analysis of the convex norm relaxation  based LRR algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Along the same lines, answering this very same question will be precisely the main  mathematical contribution of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  A lot of work will need to be performed, however, before we get to the point where we can say a few  more concrete words about the ultimate mathematical contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We will utilize to a large degree some  of the Random Duality Theory (RDT) concepts presented and discussed in details in a long line of  work [42–46,48,50,51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' On top of that, quite a few additional mathematical concepts will be needed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  While we will try to explain all the needed mathematical tools in sufficient detail, a solid level of familiarity  with the RDT might be helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Before moving to a more thorough discussion particularly related to the mathematical analysis of the  causal inference we first briefly digress to address a seemingly paradoxical situation regarding the level of  simplicity/difficulty of the above LRR on the one side and the MC/C-inf, as its special cases, on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  Special cases are not necessarily simpler  The above introduction of the matrix completion and the causal inference concepts through the generic LRR  mechanism might portrait them as subproblems of a more general class of recovery problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover, one  then may be tempted to believe that as such they can be both solved and analyzed in the very same way as  the generic LRR problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' That would basically mean that all the results that one could conceivably create  for the LRR would automatically translate to hold in a similar form in the MC and the C-inf scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The  part related to the solving of these problems is indeed true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The same algorithm (say ℓ∗  1-minimization) that  can be used as a heuristic for the generic LRR can be used for the MC and the C-inf as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' On the other  hand, the part related to the analysis could not be further from the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Not only would not the results  obtained for the LRR directly translate to the MC and the C-inf but they actually often might need to be  proven in a completely different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  The key to fully understanding this paradox is in distinguishing the additional structuring of the unknown  vector Xsol from the additional structuring of the system matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' When the unknown vectors/matrices  are additionally structured it is quite likely that the corresponding performance analyses of the underlying  algorithms are translatable (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  the companion paper [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  On the other hand when the system  matrices A are additionally structured then not only that the corresponding analyses might be difficult to  translate but they also might actually need to be completely replaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Along the same lines, since the MC  and the C-inf are the special cases of the LRR with regard to the additional structuring of A, it is not a  priori clear that the ability to analytically handle the corresponding generic LRR in any way guarantees the  existence of such an ability when it comes to analytically handling the MC and the C-inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We will see some  aspects of this reasoning already in one of the sections that will follow later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  3  Causal inference – ℓ∗  0 − ℓ∗  1 relaxation equivalence  As mentioned earlier, solving the generic LRR (and consequently the C-inf as its a special case) might be  difficult due to a highly non-convex objective function in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Various heuristics can be employed depending  on the practical scenarios that one can face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In the mathematically most challenging so-called linear regime,  the above mentioned ℓ∗  1-minimization relaxation heuristic is typically viewed as the best known provably  polynomial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  We adopt the same view in what follows and take it as a current benchmark for the  algorithmic handling of the C-inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  As mentioned above, a rather remarkable feature of this heuristic is  that sometimes it can actually solve the underlying problems exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' When that happens we say that the  following ℓ∗  0 − ℓ∗  1-equivalence phenomenon occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  7  ℓ∗  0 − ℓ∗  1-equivalence (C-inf):  Let Xsol be the solution of (16) or (9) and let ˆX be a solution of (17) or (10) and set  RMSE ≜ ∥vec( ˆX) − vec(Xsol)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  If and only if ( ˆX = Xsol and RMSE = 0)  then  (ℓ∗  0 − minimization ⇐⇒ ℓ∗  1 − minimization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (18)  The above basically means that when the ℓ∗  0 − ℓ∗  1-equivalence happens the optimization problems in (16)  and (17) are equivalent and as such replaceable by each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' That would, of course, be an ideal scenario  where it would be basically possible to replace the non-convex optimization problem with the convex one  without losing anything in terms of the accuracy of the obtained solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Since the mere existence of such a  scenario is already a remarkable phenomenon we will in this paper be interested in uncovering the underlying  intricacies that enable for it ro happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover, as it will turn out that its occurrence is not an anomaly  but rather a consequence of a generic property, we will then raise the bar accordingly and attempt to provide  not only the proof of the existence but also a complete analytical characterization of this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This  will include a full characterization as to how often and in what scenarios it might happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To do so we will  combine the Random Duality Theory (RDT) tools from [42–46,48,50,51] and several advanced sophisticated  probabilistic concepts that we will introduce along the way in the sections that follow below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  In the rest of this section we will focus on some algebraic ℓ∗  0 − ℓ∗  1-equivalence preliminaries conceptually  borrowed from the RDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We start things off with a generic LRR ℓ∗  0 − ℓ∗  1-equivalence result (the result is  basically an adaptation of the general CS equivalence condition result from [44–46] to the corresponding one  for the ℓ1 norm of the singular/eigenvalues (similar adaptation can also be found in [29])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' (ℓ∗  0 − ℓ∗  1-equivalence condition (LRR) – general X) Consider a ¯U ∈ Rn×k such that  ¯U T ¯U = Ik×k and a ¯V ∈ Rn×k such that ¯V T ¯V = Ik×k and a rank− k matrix Xsol = X ∈ Rn×n with all of its  columns belonging to the span of ¯U and all of its rows belonging to the span of ¯V T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Also, let the orthogonal  spans ¯U ⊥ ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) be such that U ≜  � ¯U  ¯U ⊥�  and V ≜  � ¯V  ¯V ⊥�  and  U T U ≜  � ¯U  ¯U ⊥�T � ¯U  ¯U ⊥�  = In×n  and  V T V ≜  � ¯V  ¯V ⊥�T � ¯V  ¯V ⊥�  = In×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (19)  For a given matrix A ∈ Rm×n2 (m ≤ n2) assume that y = Avec(X) = Avec(Xsol) ∈ Rm and let ˆX be the  solution of (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' If  (∀W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n)  − tr ( ¯U T W ¯V ) < ℓ∗  1(( ¯U ⊥)T W ¯V ⊥),  (20)  then  ℓ∗  0 ⇐⇒ ℓ∗  1  and  RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0,  (21)  and the solutions of (16) (or (9)) and (17) (or (10)) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover, if  (∃W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n)  − tr ( ¯U T W ¯V ) ≥ ℓ∗  1(( ¯U ⊥)T W ¯V ⊥),  (22)  then there is an X from the above set of matrices with columns belonging to the span of ¯U and rows belonging  to the span of ¯V such that the solutions of (16) (or (9)) and (17) (or (10)) are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The proof is a trivial adaptation of the proof for symmetric matrices given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  The condition in the theorem relates matrix W to the null-space of matrix A and as such is VMT based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  In the MC and C-inf cases that are of our interest here, it is more convenient to deal with its an MMT  analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Recalling on the proof of Theorem 1 from Appendix A and the origin and role of matrix W within  that proof, one has that stating that W belongs to the null-space of A is basically equivalent to stating that  M ◦ W = 0n×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In other words, one has the equivalence between the following two sets  (W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n) ⇐⇒ (W ∈ Rn×n|M ◦ W = 0n×n, W ̸= 0n×n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (23)  8  Continuing further in the spirit of the RDT the following corollary of the above theorem can be established  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' (ℓ∗  0−ℓ∗  1-equivalence condition via masking matrix (MC/C-inf) – general X) Assume  the setup of Theorem 1 with Xsol being the unique solution of (16) (or (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let the masking matrix M ∈  Rn×n have m ones and (n2 − m) zeros and let A be generated via M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' let A be the matrix obtained after  removing all the zero rows from diag−1(vec(M))In2×n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' If and only if  min  W,W T W=1,M◦W=0n×n  tr ( ¯U T W ¯V ) + ℓ∗  1(( ¯U ⊥)T W ¯V ⊥) ≥ 0,  (24)  then  ℓ∗  0 ⇐⇒ ℓ∗  1  and  RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0,  (25)  and the solutions of (16) (or (9)) and (17) (or (10)) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Follows immediately as a combination of (20), (22), and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Remark: Carefully comparing the conditions in (20) and (24) one can observe that a strict inequality is  loosened up a bit at the expense of the uniqueness assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' With a little bit of extra effort one may  avoid this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' However, to make writings below substantially easier we will work with a non-strict inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  To analyze the optimization problem in (24) we follow into the footsteps of [45] where similar optimization  problems were handled on multiple occasions through a very generic Lagrangian mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The first step  of such a procedure is writing down explicitly the optimization from (24)  fpr(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' U, V ) ≜ min  W  tr ( ¯U T W ¯V ) + ℓ∗  1(( ¯U ⊥)T W ¯V ⊥)  subject to  M ◦ W = 0n×n  W T W = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (26)  One can then write the corresponding Lagrangian and the Lagrange dual function to obtain  L(W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' γ)  ≜  tr ( ¯U T W ¯V ) + ℓ∗  1(( ¯U ⊥)T W ¯V ⊥) + Θ(M ◦ W) + γ  �  tr (W T W) − 1  �  =  max  Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='ΛT Λ≤I  �  tr ( ¯U T W ¯V ) + tr (Λ(( ¯U ⊥)T W ¯V ⊥)) + Θ(M ◦ W) + γ  �  tr (W T W) − 1  ��  =  max  Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='ΛT Λ≤I  �  tr  �  ( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)W  �  + γtr (W T W) − γ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (27)  and  g(Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' γ)  ≜  min  W L(W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' γ)  =  min  W  max  Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='ΛT Λ≤I  �  tr  �  ( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)W  �  + γtr (W T W) − γ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (28)  Utilizing the Lagrangian duality one then further has  g(Θ, γ)  =  min  W  max  Λ,ΛT Λ≤I  �  tr  �  ( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)W  �  + γtr (W T W) − γ  �  ≥  max  Λ,ΛT Λ≤I min  W  �  tr  �  ( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)W  �  + γtr (W T W) − γ  �  =  max  Λ,ΛT Λ≤I  �  − 1  4γ tr  �  ( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)T �  − γ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (29)  Moreover,  fpr(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' U, V )  ≥  max  Θ,γ g(Θ, γ)  =  max  Λ,ΛT Λ≤I,Θ,γ  �  − 1  4γ tr  �  ( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)T �  − γ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  9  (30)  We proceed by further optimizing over γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  fpr(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' U, V )  ≥  max  Λ,ΛT Λ≤I,Θ,γ  �  − 1  4γ tr  �  ( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)T �  − γ  �  =  max  Λ,ΛT Λ≤I,Θ −  �  tr  �  ( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)T �  =  −  min  Λ,ΛT Λ≤I,Θ  �  tr  �  ( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)( ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T + Θ ◦ M)T �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (31)  We now particularize the above to the block causal inference scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In the block C-inf scenario the  matrix M is as defined in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' For the concreteness and to facilitate writings later on we will introduce  matrix I(l) and also keep in mind the following characterization of matrix M  M matrix in causal inference (C-inf):  M ≜ M (l) ≜ 1n×11T  n×1 − I(l)(I(l))T 1n×11T  n×1I(l)(I(l))T  and  I(l) ≜  �  0l×(n−l)  I(n−l)×(n−l)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (32)  In the above definition/representation of M, 1/0 stand for vectors and matrices of all ones/zeros with the  dimensions specified in their subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  To avoid overwhelming the notation, we may on occasion skip  specifying the underlying dimensions of these vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' However, they should be easy to infer from the overall  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Also, l is, of course, adjusted so that M has m nonzero elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' m elements equal to one  which basically amounts to having the following identity to hold  m  =  n2 − (n − l)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (33)  Using the above C-inf M in (31) the only terms that will be left in  � ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T �  after the  optimization are those that correspond to the zero elements of M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='.e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' the only ones where the presence of  (and consequently the optimization over) Θ can not have an effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This basically implies  fpr(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' U, V ) ≥ −  min  Λ,ΛT Λ≤I  �  tr  ��  (I(l))T � ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T �  I(l)� �  (I(l))T � ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T �  I(l)�T �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (34)  For the overall success of the whole machinery one would need that Λ in the above optimization can be chosen  so that the overall optimum is nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This is then sufficient to establish the following alternative to  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' (ℓ∗  0 − ℓ∗  1-equivalence condition via masking matrix (C-inf) – general X) Assume the  setup of Theorem 1 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let I(l) be as in (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then  C-inf perfectly succeeds: ℓ∗  0 ⇐⇒ ℓ∗  1  and  RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0  If and only if  ∃Λ|ΛTΛ ≤ I  and  (I(l))T � ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T �  I(l) = 0  (35)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The “if” part follows from Corollary 1, (34), and the above discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The “only if” part follows  after noting that all the above inequalities in (29)-(34) are written for generic instructional purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Due to  the underlying convexity and the strong duality they all actually can be replaced with equalities as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  10  For the time being we will assume k ≤ l (later on this assumption will be rigorously justified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' From (35)  one then easily has  Λ = ((I(l))T ¯V ⊥)−1(I(l))T ¯V ¯U T I(l)(( ¯U ⊥)T I(l))−1  =⇒  (I(l))T � ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T �  I(l) = 0,  (36)  where (·)−1 stands for the pseudo-inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To make writing a bit easier we can set  Λopt  ≜  ΛV ΛT  U  ΛV  ≜  ((I(l))T ¯V ⊥)−1(I(l))T ¯V  ΛU  ≜  ((I(l))T ¯U ⊥)−1(I(l))T ¯U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (37)  Let λmax(·) be the maximum eigenvalue of its symmetric matrix argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' After combining (35)-(37) we  conclude that if  λmax(ΛT  optΛopt) ≤ 1,  (38)  then (35) will be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' After basic algebraic transformations (38) can also be rewritten as  λmax(ΛT  optΛopt) = λmax(ΛoptΛT  opt) = λmax(ΛV ΛT  UΛUΛT  V ) = λmax(ΛT  V ΛV ΛT  UΛU) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (39)  From (39) it is rather clear that the spectrum of ΛT  V ΛV ΛT  UΛU as well as the spectra of ΛT  V ΛV and ΛT  UΛU  play an important role in the ℓ∗  0 − ℓ∗  1-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We first observe a worst case bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Namely, since  λmax(ΛT  optΛopt) = λmax(ΛT  V ΛV ΛT  UΛU) ≤ λmax(ΛT  V ΛV )λmax(ΛT  UΛU)),  (40)  one has that if the individual spectra of ΛT  V ΛV and ΛT  UΛU do not exceed one then the ℓ∗  0 − ℓ∗  1-equivalence  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Given the obvious importance of these spectra we will below look at them in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Clearly,  due to symmetry we need to focus on only one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To that end we start by observing  ((I(l))T ¯V ⊥)−1  =  ( ¯V ⊥)T I(l) �  (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1  ,  (41)  From (41) one quickly finds  �  ((I(l))T ¯V ⊥)−1�T  ((I(l))T ¯V ⊥)−1  =  �  (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (42)  We can then write  (I(l))T ¯V ¯V T I(l)  =  (I(l))T �  I − ¯V ⊥( ¯V ⊥)T �  I(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (43)  From (37) we have  Q1 ≜ ΛT  V ΛV  =  ��  ((I(l))T ¯V ⊥)−1(I(l))T ¯V  �T �  ((I(l))T ¯V ⊥)−1(I(l))T ¯V  ��  =  ¯V T I(l) �  ((I(l))T ¯V ⊥)−1�T  ((I(l))T ¯V ⊥)−1 �  (I(l))T ¯V  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (44)  Now, we will find it more convenient to work with a slightly change version of matrix Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Namely, after  combining (37), (42), and (43) we obtain  Q  ≜  ��  ((I(l))T ¯V ⊥)−1�T  ((I(l))T ¯V ⊥)−1 �  (I(l))T ¯V ¯V T I(l)��  =  ��  (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1 �  (I(l))T �  I − ¯V ⊥( ¯V ⊥)T �  I(l)��  =  �  (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1  − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (45)  11  Clearly, all the nonzero eigenvalues of Q1 and Q are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' When k ≤ n−l then Q has all the eigenvalues  of Q1 plus n − l − k extra zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' On the other hand, when k ≥ n − l then Q1 has all the eigenvalues of Q  plus k − (n − l) extra zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Since adding or removing zeros from the spectra will not change any of their  features of our interests here, instead of working directly with Q1, we can work with Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In particular, we  have  λmax(Q1) = λmax(ΛT  V ΛV )  =  λmax  ��  (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1�  − 1 = λmax(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (46)  We are now in position to establish a spectral alternative to Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' (ℓ∗  0 − ℓ∗  1-equivalence condition via mask-modified bases spectra (C-inf) – general  X) Assume the setup of Theorem 1 and Corollaries 1 and 2 with k ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let λV and λU be defined as in  (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then  C-inf perfectly succeeds: ℓ∗  0 ⇐⇒ ℓ∗  1  and  RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0  If and only if  λmax(ΛT  V ΛV ΛT  UΛU) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (47)  Moreover, if  �  λmax  ��  (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1�  − 1  � �  λmax  ��  (I(l))T ¯U ⊥( ¯U ⊥)T I(l)�−1�  − 1  �  ≤ 1,  (48)  then again ℓ∗  0 ⇐⇒ ℓ∗  1 and RMSE = ∥vec( ˆX − vec(Xsol)∥2 = 0 and the C-inf perfectly succeeds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The first part follows from Corollaries 1 and 2, (36), (37), (39), the above discussion and some  additional considerations while the second part follows by additional taking into account (40) and (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We  below present all the details split into three parts: the first two relate to the equivalence condition (equation  (47)) and third one to (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  1) =⇒ – The “if part” of condition (47): Choosing Λ = Λopt  Λopt  ≜  ΛV ΛT  U = −((I(l))T ¯V ⊥)−1(I(l))T ¯V ¯U T I(l)(( ¯U ⊥)T I(l))−1,  (49)  (where (·)−1 stands for the pseudo-inverse) one ensures  (I(l))T � ¯V ¯U T + ¯V ⊥Λ( ¯U ⊥)T �  I(l) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (50)  Let λmax(·) be the maximum eigenvalue of its symmetric matrix argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' A combination of (35)-(50)  ensures that if  λmax(ΛT  optΛopt) ≤ 1,  (51)  then Λopt satisfies (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' (51) is implied by (47) since  λmax(ΛT  optΛopt) = λmax(ΛUΛT  V ΛV ΛT  U) = λmax(ΛT  V ΛV ΛT  UΛU) ≤ 1,  which suffices to complete the proof of the “if part”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  2) ⇐= – The “only if part” of condition (47): Consider SVDs  B ≜ (I(l))T V ⊥ = UBΣBV T  B ,  C ≜ (I(l))T U ⊥ = UCΣCV T  C  (52)  with unitary UB, VB, UC, VC and diagonal (with no zeros on the main diagonal) ΣB, ΣC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Any Λ can be  parameterized as  Λ = VBHT + V ⊥  B DT ,  H ≜ VCE + V ⊥  C F  (53)  12  for some E, F, D and unitary V ⊥  B and V ⊥  C such that V T  B V ⊥  B = V T  C V ⊥  C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Also, one can set Λ∗ and write  the SVD of E  Λ∗ ≜ VBET V T  C ,  E = UEΣEV T  E ,  (54)  where UE, VE are unitary and ΣE is diagonal with entries on the main diagonal being nonzero and in  ascending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let ue be the last column of UE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' the eigenvector of EET that corresponds to its  largest eigenvalue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Since ∥VCue∥2 = 1,  λmax(ΛT Λ)  ≥  uT  e V T  C ΛT ΛVCue  =  uT  e V T  C HHT VCue + uT  e V T  C DDT VCue  ≥  uT  e V T  C (VCE + V ⊥  C F)(VCE + V ⊥  C F)T VCue  =  uT  e EET ue = λmax(EET )  =  λmax(VCEET V T  C ) = λmax(ΛT  ∗ Λ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (55)  If Λ satisfies the condition of (35) then a combination of (35) and (52)-(54) gives  (I(l))T ¯V ¯UI(l) + BΛ∗CT = 0,  (56)  and a combination of (37), (52), and (56) gives  ΛV ΛT  U = −B−1(I(l))T ¯V ¯UI(l)(CT )−1 = Λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (57)  Finally, for Λ that fits (35), from (55) and (57) one has  1  ≥  λmax(ΛT Λ) > λmax(ΛT  ∗ Λ∗) = λmax(Λ∗ΛT  ∗ )  =  λmax(ΛV ΛT  UΛUΛT  V ) = λmax(ΛT  V ΛV ΛT  UΛU),  (58)  which completes the proof of the “only if part”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  3) Suffciency of the condition (48): Since  λmax(ΛT  V ΛV ΛT  UΛU) ≤ λmax(ΛT  V ΛV )λmax(ΛT  UΛU),  (59)  one has that if the individual spectra of ΛT  V ΛV and ΛT  UΛU do not exceed one then the ℓ∗  0 − ℓ∗  1-equivalence  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Due to symmetry we focus only on one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' First we observe  ((I(l))T ¯V ⊥)−1 = ( ¯V ⊥)T I(l) �  (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1  ,  and quickly find  �  ((I(l))T ¯V ⊥)−1�T  ((I(l))T ¯V ⊥)−1 =  �  (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (60)  We also note  (I(l))T ¯V ¯V T I(l)  =  (I(l))T �  I − ¯V ⊥( ¯V ⊥)T �  I(l),  (61)  and  Q1  ≜  ΛT  V ΛV  =  ��  ((I(l))T ¯V ⊥)−1(I(l))T ¯V  �T �  ((I(l))T ¯V ⊥)−1(I(l))T ¯V  ��  =  ¯V T I(l) �  ((I(l))T ¯V ⊥)−1�T  ((I(l))T ¯V ⊥)−1 �  (I(l))T ¯V  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (62)  13  A combination of (60), (61), and (62) produces  Q  ≜  ��  ((I(l))T ¯V ⊥)−1�T  ((I(l))T ¯V ⊥)−1 �  (I(l))T ¯V ¯V T I(l)��  =  ��  (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1 �  (I(l))T �  I − ¯V ⊥( ¯V ⊥)T �  I(l)��  =  �  (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1  − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (63)  Since all the nonzero eigenvalues of Q1 and Q are identical  λmax(ΛT  V ΛV ) = λmax(Q1) = λmax(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (64)  Repeating the above with V replaced by U, Q1 by Q⊥  1 , and Q by Q⊥ one arrives at the following analogue  of (64)  λmax(ΛT  UΛU) = λmax(Q⊥  1 ) = λmax(Q⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (65)  A combination of (59) and (62) - (65) completes the proof of the condition (48)’s sufficiency for the ℓ∗  0 − ℓ∗  1-  equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  All the three above corollaries provide useful characterization of the ℓ∗  0 − ℓ∗  1-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Depending on  what kind of scenario one faces and what kind of numerical/computational/statistical resources might be  available each of them could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In the following sections we will focus on a particular type of analysis  that will primarily relate to the spectral characterizations given in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  4  Typical worst case analysis of the ℓ∗  0 − ℓ∗  1-equivalence  In this section we provide an analysis that sheds a bit more light on when the conditions from Corollary 3 are  indeed met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We will work in a typical statistical scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We will first assume that V and U are statistical  objects and under such an assumption we will try to see if there are regimes where the ℓ∗  0 − ℓ∗  1-equivalence  generically holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' There are of course many valid candidates for the statistics of V and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We will assume  the most generic typical uniformly random scenario from the spectral theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' That means that both ¯V and ¯U  will be Haar distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The experts in sparse recovery will quickly recognize that this is in a way analogous  to assuming that the locations of the nonzero components of a k-sparse vector in the standard compressed  sensing are uniformly randomly chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Apart from the RDT considerations from [42,43,45,46,48,50] and  the high-dimensional geometry considerations from [13,15] we are unaware of any other techniques that can  avoid assumptions of this type and still achieve the ultimate exact phase transition (PT) characterizations  for the conditions of the type similar to the one appearing in Theorem 1 and Corollaries 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Conducting the analysis assuming the statistical nature of V and U we will uncover a very interesting  and rather remarkable connection, among three a priori not necessarily related fields: 1) the compressed  sensing (CS), 2) the causal inference (C-inf), and 3) the free probability theory (FPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' As the connection  between the former two exists even outside a statistical context we were able to partially incorporate it in  our earlier discussion presented in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' On the other hand, in the sections that follow, we  will deepen our understanding of such a connection while relating it to the free probability theory and a  collection of very generic concepts from the modern spectral theory of random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1  Free probability theory (FPT) – preliminaries  Since this is the introductory paper where we are establishing the connection between the causal inference  and the free probability theory (FPT) we will find it useful to first, in this subsection, recall on some  FPT basics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' As the FPT theory is mathematically very deep and involved we will restrict ourselves to the  introduction of the basic FPT definitions, the explanations of the key technical results, and finally to a brief  description related to the practical utilization of these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' After that, in the sections that follow, we will  see how some of the introduced FPT concepts can be incorporated to strengthen our understanding of the  causal inference itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  14  The FPT is, of course, a very generic and abstract concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Below we focus on some of its key implications  related to the spectral theory of random matrices and start by sketching a bit of main motivation behind the  FPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' That effectively means that we start with the simplest possible matrices which are of course scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1  Basics of FPT – random scalar variables  It is well known that if one has two independent random variables A and B with respective pdfs fA(·)  and fB(·), then the standard way of determining the distribution of their sum or product goes through the  characteristic functions and the corresponding inverse Fourier transform considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To be a bit more  concrete, one first recognizes that the individual characteristic functions for both variables are given as  FA(jw)  ≜  EejwA =  �  ejwafA(a)da ≜ F(fA(a))  FB(jw)  ≜  EejwB =  �  ejwbfB(b)db ≜ F(fB(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (66)  Assuming that  C = A + B,  (67)  we analogously to (66) also have  FC(jw) = EejwC =  �  ejwcfC(c)dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (68)  Moreover,  FC(jw) = EejwC = Eejw(A+B) = EejwAEejwB = FA(jw)FB(jw) =  �  ejwafA(a)da  �  ejwbfB(b)db,  (69)  and finally  fC(c) = F−1(FC(jw)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (70)  It is then easy to see that (69) and (70) are sufficient to determine the pdf of C = A + B starting from the  individual pdfs of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The key that leads to the success of the above mechanism is the introduction of  the Fourier transform and the so-called characteristic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' It turns out that in the transform’s domain  the sum of random variables in a way corresponds to their product and, as a consequence, one can successfully  separate them and then rely on their individual pdfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' While this methodology is fairly simple and has been  known for almost two centuries, the existence of a similar one for matrices was not discovered until only a  couple of decades ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover, the path to its discovery turned out to be more thornier and unpredictable  than one could have ever imagined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  The work od Dan Voiculescu on group theories (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' [54–56])  uncovered it in an almost by-product type of way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Of course, due to an enormous practical importance it  immediately drew a substantial interest and in the years that followed immediately after its discovery a few  nice results appeared that helped make it presentable in a relatively simple and easily understandable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  We below follow into the same footsteps, leave all the abstractions out, and focus on presenting how the  main FPT mechanism actually works (more details can be found in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' [17,28,40,53–56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2  Basics of FPT – random matrix variables  As was the case above for scalars, we here also start with two random variables, A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This time though,  these two variables are symmetric matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' A = AT ∈ Rn×n and B = BT ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We will also assume  large n regime and that the eigenspaces of these matrices are Haar distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover, we will assume  that their individual respective spectral laws are fA(·) and fB(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Similarly to what we showed above in  the scalar case, we will here also rely on introducing distributional transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' However, differently from  the scalar case, here we will introduce not one but three different transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We start with the so-called  15  Stieltjes transform (or as we will often call it G-transform) of a pdf f(·)  G(z)  ≜  �  If  f(x)  z − xdx,  z ∈ C \\ If,  (71)  where If is the domain of f(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' One then also has the inverse relation (somewhat analogous to the above  relation between the inverse Fourier and the underlying pdf of the sum of random variables)  f(x) = lim  ǫ→0+  G(x − iǫ) − G(x + iǫ)  2iπ  or  f(x) = − lim  ǫ→0+  imag(G(x + iǫ))  π  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (72)  For the above to hold it makes things easier to implicitly assume that f(x) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We will, however,  utilize it even in discrete (or semi-discrete) scenarios since the obvious asymptotic translation to continuity  would make it fully rigorous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' A bit later though, when we see some concrete examples where things of this  nature may appear, we will say a few more words and explain more thoroughly what exactly can be discrete  and how one can deal with such a discreteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In the meantime we proceed with general principles not  necessarily worrying about all the underlying technicalities that may appear in scenarios deviating from the  typically seen ones and potentially requiring additional separate addressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To that end we continue by  considering the R(·)- and S(·)-transforms that satisfy the following  R(G(z)) +  1  G(z) = z,  (73)  and  S(z) =  1  R(zS(z))  and  R(z) =  1  S(zR(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (74)  Let fA(·) and fB(·) be the spectral distributions of A and B and let RA(z)/SA(z) and RB(z)/SB(z) be their  associated R(·)-/S(·)-transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' One then has the following  Key Voiculescu’s FPT concepts [54, 55]:  C  =  A + B  =⇒  RC(z)  =  RA(z) + RB(z)  C  =  AB  =⇒  SC(z)  =  SA(z)SB(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (75)  Now it is relatively easy to see that (71)-(75) are sufficient to determine the spectral distribution of the sum  or the product of two independent matrices with given spectral densities and the Haar distributed bases of  eigenspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The above is of course generic principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' It can be applied pretty much always as long as one  has access to the statistics of the underlying matrices A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In the following section we will raise the bar  a bit higher and show that in the case of the causal inference one can use all of the above in such a manner  that eventually all the quantities of interest are explicitly determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover, although the methodology  may, on occasion, seem a bit involved the final results will turn out to be presentable in fairly neat and  elegant closed forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2  Uncovering the C-inf ←→ FPT connection  We are now in position to finally move to one of the key aspects of this paper, namely the uncovering of a  rather unexpected connection between the C-inf and the FPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The first part of the connection is in a way  implicit and includes what we presented in in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Namely, utilizing the key compressed  sensing concepts and the Random duality theory (RDT) we connected the success of the causal inference to  behavior of ceratin algebraic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In particular, we have established the so-called ℓ∗  0 − ℓ∗  1-equivalence  as the key concept in determining the ultimate level of success of C-inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The second part of the connection  builds on the first and proceeds by analyzing the ℓ∗  0 − ℓ∗  1-equivalence via the FPT machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In the sections  that follow we provide such a very detailed and self-contained analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1  The spectral approach to the analysis of the ℓ∗  0 − ℓ∗  1-equivalence  As mentioned earlier, we below rely on the spectral characterization of the ℓ∗  0 − ℓ∗  1-equivalence provided in  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Clearly, determining the spectrum of λT  V λV λT  UλU would be sufficient to determine when the  ℓ∗  0 − ℓ∗  1-equivalence occurs (in fact, determining the edges of the spectrum is sufficient as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' While we  will in the sections that follow below indeed determine the spectrum of λT  V λV λT  UλU, here we note that in  the worst case that may not be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Namely, in the worst case one has  λmax(λT  V λV λT  UλU) ≤ λmax(λT  V λV )λmax(λT  UλU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (76)  In the large n limit due to the concentrations and identically Haar distributed V and U one also has  λmax(λT  V λV λT  UλU) ≤ λmax(λT  V λV )λmax(λT  UλU) −→  �  λmax(λT  V λV )  �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (77)  Moreover, in the worst case, U = V , the equality is actually achieved since  λmax(λT  V λV λT  V λV ) = λmax((λT  V λV )2) =  �  λmax(λT  V λV )  �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (78)  This basically means that in the worst case it is sufficient to consider only the spectrum of Q with  Q ≜ λT  V λV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (79)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2  The FPT analysis of the spectrum of Q  We start by recalling from (44) and (45)  Q1  ≜  ΛT  V ΛV  Q  ≜  �  (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1  − I  Sp(Q1)  ⇐⇒\\0  Sp(Q),  (80)  where Sp(·) stands for the spectrum of the matrix argument and ⇐⇒\\0 means the equivalence of the parts  of the spectra outside the zero eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' It is rather obvious that it will then be sufficient to handle the  spectrum of  D  ≜  (I(l))T ¯V ⊥( ¯V ⊥)T I(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (81)  Consider Haar distributed ¯U ⊥  D ∈ Rn×(n−l) with ( ¯U ⊥  D)T ¯U ⊥  D = I(n−l)×(n−l) and let  UD  =  � ¯UD  ¯U ⊥  D  �  with  U T  DUD = In×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (82)  Also, we assume that ¯U ⊥  D (and UD) are independent of ¯V ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' After setting  ¯D  ≜  (I(l))T U T  D ¯V ⊥( ¯V ⊥)T UDI(l),  (83)  we have that the spectra of D and ¯D are statistically identical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Sp(D) ≜ Sp((I(l))T ¯V ⊥( ¯V ⊥)T I(l)) ⇐⇒P Sp((I(l))T U T  D ¯V ⊥( ¯V ⊥)T UDI(l)) ≜ Sp( ¯D),  (84)  where ⇐⇒P stands for the statistical/probabilistic equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Two facts enable the above statistical iden-  tity: 1) the spectrum of the projector ¯V ⊥( ¯V ⊥)T does not change under pre- and post-unitary multiplications  on both sides;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' and 2) the Haar structure of ¯V ⊥ remains preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Modulo zero eigenvalues, we then further  have  Sp((I(l))T U T  D ¯V ⊥( ¯V ⊥)T UDI(l)) ⇐⇒P\\0 Sp( ¯V ⊥( ¯V ⊥)T UDI(l)(I(l))T U T  D) ⇐⇒ Sp( ¯V ⊥( ¯V ⊥)T ¯U ⊥  D( ¯U ⊥  D)T ), (85)  17  where, similarly as above, ⇐⇒P\\0 stands for the statistical/probabilistic equivalence in the part of the  spectrum outside the zero eignevalues (introduced due to the non-square underlying matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Clearly, the  key object of our interest below will be  ˜D  ≜  ¯V ⊥( ¯V ⊥)T ¯U ⊥  D( ¯U ⊥  D)T ,  (86)  where both ¯V ⊥ and ¯U ⊥  D are Haar distributed and independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' After setting  V  ≜  ¯V ⊥( ¯V ⊥)T  U  ≜  ¯U ⊥  D( ¯U ⊥  D)T ,  (87)  we easily have from (86)  ˜D  ≜  VU,  (88)  and below first focus on handling the spectrum and the corresponding relevant transforms of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Since we  will be working in the mathematically most challenging large n linear regime, we find it useful to introduce  the following large dimensional scalings  β ≜ lim  n→∞  k  n  and  η ≜ lim  n→∞  l  n  and  α ≜ lim  n→∞  m  n2 = lim  n→∞  n2 − (n − l)2  n2  = 1 − (1 − η)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (89)  We start with a trivial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let fV(·) be the spectral distribution of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then  fV(x) = (1 − β)δ(1 − x) + βδ(x),  (90)  where δ(·) stands for the standard delta function with nonzero value only when its argument takes value  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Using the definition of the G-transform from (71) we can find  GV(z) =  � fV(x)  z − x dx =  � (1 − β)δ(1 − x) + βδ(x)  z − x  dx = 1 − β  z − 1 + β  z = z − β  z2 − z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (91)  Also, from (85) we have  RV(y) = z − 1  y  with  y = GV(z)  and  z = G−1  V (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (92)  From (91) and (92) we further find  GV(z) = y  ⇐⇒  z − β  z2 − z = y  ⇐⇒  z2y − z(y + 1) + β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (93)  Solving for z gives  z = y + 1 ±  �  (y + 1)2 − 4βy  2y  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (94)  Combining (92) and (94) we obtain for the R-transform  RV(y) = z − 1  y = y − 1 ±  �  (y + 1)2 − 4βy  2y  ,  (95)  where we for the completeness adopt the strategy to keep both ± signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To determine the S-transform we  start by combining (74) and (95)  SV(z) =  1  RV(zSV(z)) =  1  zSV(z)−1±√  (zSV(z)+1)2−4βzSV(z)  2zSV(z)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (96)  18  After a bit of algebraic transformations we have  zSV(z) − 1 − 2z  =  ∓  �  (zSV(z) + 1)2 − 4βzSV(z)  ⇐⇒  (zSV(z) − 1 − 2z)2  =  (zSV(z) + 1)2 − 4βzSV(z)  ⇐⇒  (zSV(z))2 − 2(2z + 1)zSV(z) + (2z + 1)2  =  (zSV(z))2 + 2zSV + 1 − 4βzSV(z)  ⇐⇒  −2(2z + 1)zSV(z) + 4z2 + 4z  =  2zSV(z) − 4βzSV(z)  ⇐⇒  4z2 + 4z  =  (4z2 + 4z)SV(z) − 4βzSV(z)  ⇐⇒  z + 1  =  SV(z)(z + 1 − β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (97)  From (97) we finally have  SV(z) =  z + 1  z + 1 − β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (98)  As this is a very generic result it is useful to have it formalized in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let ¯V ⊥ ∈ Rn×(n−k) be Haar distributed unitary basis of an (n − k)-dimensional subspace of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Let V be as in (87), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  V ≜ ¯V ⊥( ¯V ⊥)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (99)  In the large n linear regime, with β ≜ limn→∞ k  n, the S-transform of the spectral density of V, fV(·), is  SV(z) =  z + 1  z + 1 − β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (100)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Follows from the above discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Since V and U are structurally identical (with the only difference being one of their dimensions) we easily  have  SU(z) =  z + 1  z + 1 − η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (101)  A combination of (75), (88), (98), and (101) gives  S ˜  D(z) =  (z + 1)2  (z + 1 − β)(z + 1 − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (102)  From (74) we also have  R ˜  D(z) =  1  S ˜  D(zR ˜  D(z)) =  1  (zR ˜  D(z)+1)2  (zR ˜  D(z)+1−β)(zR ˜  D(z)+1−η)  = (zR ˜  D(z) + 1 − β)(zR ˜  D(z) + 1 − η)  (zR ˜  D(z) + 1)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (103)  Moreover, (72) gives  R ˜  D(G ˜  D(z)) +  1  G ˜  D(z) = z,  (104)  and  G ˜  D(z)R ˜  D(G ˜  D(z)) = zG ˜  D(z) − 1,  (105)  From (103) one further finds  R ˜  D(G ˜  D(z)) = (G ˜  D(z)R ˜  D(G ˜  D(z)) + 1 − β)(G ˜  D(z)R ˜  D(G ˜  D(z)) + 1 − η)  (G ˜  D(z)R ˜  D(G ˜  D(z)) + 1)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (106)  19  After plugging (105) in (106) we have  R ˜  D(G ˜  D(z)) = (zG ˜  D(z) − 1 + 1 − β)(zG ˜  D(z) − 1 + 1 − η)  (zG ˜  D(z) − 1 + 1)2  = (zG ˜  D(z) − β)(zG ˜  D(z) − η)  (zG ˜  D(z))2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (107)  A combination of (104) and (107) further gives  z −  1  G ˜  D(z) = (zG ˜  D(z) − β)(zG ˜  D(z) − η)  (zG ˜  D(z))2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (108)  From (108) we quickly find  z3(G ˜  D(z))2 − z2G ˜  D(z) = z2(G ˜  D(z))2 − (β + η)zG ˜  D(z) + βη,  (109)  and  (G ˜  D(z))2(z3 − z2) − G ˜  D(z)(z2 − z(β + η)) − βη = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (110)  Solving for G ˜  D(z) finally gives  G±  ˜  D(z) = z2 − z(β + η) ±  �  (z2 − z(β + η))2 + 4βη(z3 − z2)  2(z3 − z2)  ,  (111)  or  G±  ˜  D(z) = z − (β + η) ±  �  (z − (β + η))2 + 4βη(z − 1)  2(z2 − z)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (112)  The above is sufficient to establish the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let ¯V ⊥ ∈ Rn×(n−k) and ¯U ⊥  D ∈ Rn×(n−k) be Haar distributed unitary bases of (n−k)-dimensional  subspaces of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let V and U be as in (87) and ˜D as in (88), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  V  ≜  ¯V ⊥( ¯V ⊥)T  U  ≜  ¯U ⊥  D( ¯U ⊥  D)T  ˜D  ≜  VU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (113)  In the large n linear regime, with β ≜ limn→∞ k  n, the G-transform of the spectral density of ˜D, f ˜  D(·), is  G±  ˜  D(z) = z − (β + η) ±  �  (z − (β + η))2 + 4βη(z − 1)  2(z2 − z)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (114)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Follows from the above discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The “+/−” signs are taken for negative/positive imaginary part  under the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  One then relies on (72) to determine f ˜  D(x) as  f ˜  D(x) = − lim  ǫ→0+  imag(G ˜  D(x + iǫ))  π  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (115)  The above is a generic procedure and we in Figure 3 show the results that one can get for two concrete values  β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' One should note that it is not clear a priori which of the two ± signs should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  As Figure 3 indicates one most definitely has to be fairly careful and account for both signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' From Figure 3  one further observes that there are four critical points in the spectrum itself: the locations of the two delta  functions, zero and one, and two edges of the spectrum’s bulk, xl and xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The values of these points are  shown in the plots on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In general one can actually determine their closed forms as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  20  Moreover, it turns out that one can determine the closed form of the entire spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The section  that follows analyzes the spectrum of ˜D in more details and eventually provides the closed form expressions  for all the relevant spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  1  f ˜D(x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  f ˜D(x) obtained using G+  ˜D(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6  simulated  theory  xl  xc  xu  f ˜D(x) = −limǫ→0+ imag(G+  ˜D(x+iǫ))  π  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  1  f ˜D(x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  f ˜D(x) obtained using G ˜D(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6  simulated  theory  xu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='9519  xl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1681  xl = β + η − 2βη = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='56  G ˜D = G−  ˜D  G ˜D = G+  ˜D  f ˜D(x) = −limǫ→0+ imag(G ˜D(x+iǫ))  π  Figure 3: Both G+  ˜  D(z) and G−  ˜  D(z) need to be taken into account  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1  The spectrum of ˜D – closed form expressions  As one of our main concerns in this paper is the utilization of the final results that we will get in this  section and not necessarily the presentation of the tiny details needed to get them, we will sketch all the key  arguments and leave out all the unnecessary minute details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' However, we do emphasize that the sketch will  contain all the key pointers so that with a little bit of effort one, if in a need, can fill in all the missing pieces  of the overall mosaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  As mentioned above, looking at the denominator of (112) and keeping in mind the f ←→ G connection  from (115) one observes that the pdf of interest, f ˜  D(x), potentially has two delta functions, one at zero and  the other one at one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover, the bulk of the spectrum will be in the range where the real part under  the root is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' It also goes almost without saying that the entire spectrum will be located between  zero and one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Finally, the breaking point, xc, where one needs to switch from G+  ˜  D(z) to G−  ˜  D(z) in (115) is  determined as the value where the imaginary part under the root changes its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Equipped with these  observations one can then proceed to actually concretely determine some of the relevant quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Based on what we have just observed above, we first express f ˜  D(x) as the sum of its three key constitutive  parts (two delta functions and the bulk)  f ˜  D(x) = f0δ(x − 0) + f (b)  ˜  D (x) + f1δ(x − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (116)  From (116) one has that f ˜  D(x) will be fully specified if one can determine the delta multipliers f0 and f1,  and the bulk pdf f (b)  ˜  D (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  1) Finding f0: To determine f0 we start by observing from (115) for x = 0  f ˜  D(0) = − lim  ǫ→0+  imag(G ˜  D(iǫ))  π  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (117)  Utilizing (112) we further have  f ˜  D(0)  =  − 1  π lim  ǫ→0+ imag  �  iǫ − (β + η) ±  �  (iǫ − (β + η))2 + 4βη(iǫ − 1)  2((iǫ)2 − iǫ)  �  21  =  − 1  π lim  ǫ→0+ imag  �  −(β + η) ±  �  −ǫ2 + (β + η)2 − 4βη − 2iǫ(β + η − 2βη)  2(−ǫ2 − iǫ)  �  =  − 1  π lim  ǫ→0+ imag  �  −(β + η) ±  �  (β − η)2  2(−iǫ)  �  =  − 1  π lim  ǫ→0+ imag  �−(β + η) − |β − η|  −2iǫ  �  =  (β + η + |β − η|)  �  − 1  π lim  ǫ→0+ imag  � 1  iǫ  ��  =  max(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' η)δ(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (118)  where the fourth equality (the choice of the “−” sign in ±) follows since 0 ≤ xc (the spectrum belongs to  the interval [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' 1] and xc must be in the spectrum) and the last equality follows since by convention  δ(0) =  �  − 1  π lim  ǫ→0+ imag  � 1  iǫ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (119)  To see the rationale behind (119) we briefly digress and start with  g(x) = δ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (120)  Then from (71)  G(z) =  �  x  δ(x)dx  z − x = 1  z ,  (121)  and from (72)  δ(x) = − lim  ǫ→0+  imag(G(x + iǫ))  π  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (122)  For x = 0 then  δ(0) = − lim  ǫ→0+  imag(G(iǫ))  π  = − lim  ǫ→0+ imag  � 1  πiǫ  �  = − 1  π lim  ǫ→0+ imag  � 1  iǫ  �  ,  (123)  which is identical to (119).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The above description of the delta function may not necessarily be the most  adequate one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' However, for what we need here it is conceptually sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Namely, we are here interested  in determining the proportionality constants that multiply the delta functions rather than the functions’  expressions themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' One way to make everything more adequate would be to translate everything into  the continuous domain by choosing a continuous function as an asymptotic replacement for δ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  For  example, one can use the Gaussian continual approximation  δ(x) = lim  σ→0+  e− x2  2σ2  √  2πσ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (124)  Then all the above holds for small σ = ǫ√π/2 and  δ(x) → lim  σ→0+  e− x2  2σ2  √  2πσ2 → lim  σ→0+  e− x2  πǫ2  πǫ  and  δ(0) → lim  σ→0+  1  πǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (125)  The difference though would be that when computing and maneuvering with all the above transforms one  would need to account for the resulting/induced ǫ-differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' These are of course practically and concep-  tually negligible and all the results that we presented would continue to hold in the limit of small σ or ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  However, the writing would be substantially more tedious and a tone of additional minute details would  need to be added to express all the ǫ-type of modifications and to show that their contributions are indeed  marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' These things are conceptually highly trivial but require a tedious detail-oriented work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Since, on  22  the other hand, they contribute exactly nothing to the essence of the arguments and final results we chose to  operate in a semi-discrete domain with the delta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' As a consequence one has the expressions given  in (119) and (123).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We believe that a little bit of conventional inadequacy is better than to overwhelm the  presentation with a tone of details which would avoid it but at the same time make the overall content less  accessible and potentially even less understandable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  2) Finding f1: To determine f1 we follow the above methodology and start by observing from (115)  for x = 1  f ˜  D(1) = − lim  ǫ→0+  imag(G ˜  D(1 + iǫ))  π  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='(126)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='Further utilization of (112) gives  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='f ˜  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='D(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='π lim  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='ǫ→0+ imag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1 + iǫ − (β + η) ±  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='(1 + iǫ − (β + η))2 + 4βηiǫ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2((1 + iǫ)2 − 1 − iǫ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='π lim  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='ǫ→0+ imag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1 − (β + η) ±  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='−ǫ2 + (1 − (β + η))2 − 2iǫ(−1 + β + η − 2βη)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2(−ǫ2 + iǫ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='π lim  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='ǫ→0+ imag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1 − (β + η) ±  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='(1 − (β − η))2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2(iǫ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='π lim  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='ǫ→0+ imag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='�1 − (β + η) + |1 − (β + η)|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2iǫ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='(1 − (β + η) + |1 − (β + η)|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='π lim  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='ǫ→0+ imag  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='� 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='iǫ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='max(1 − (β + η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' 0)δ(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (127)  where the fourth equality (the choice of the “+” sign in ±) follows since now xc ≤ 1 and the last equality  follows by the above discussed δ(0) convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  3) Finding f (b)  ˜  D (x): To determine f (b)  ˜  D (x) for x /∈ {0, 1} we again start with (115) and wrte the following  for a general x from the bulk of the spectrum  f (b)  ˜  D (x) = − lim  ǫ→0+  imag(G ˜  D(x + iǫ))  π  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (128)  Relying once again on (112) we,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' for x /∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' 1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' have  f (b)  ˜  D (x)  =  − 1  π lim  ǫ→0+ imag  �  x + iǫ − (β + η) ±  �  (x + iǫ − (β + η))2 + 4βη(x + iǫ − 1  2((x + iǫ)2 − x − iǫ)  �  =  − 1  π lim  ǫ→0+ imag  �  x − (β + η) ±  �  −ǫ2 + (x − (β + η))2 + 4βη(x − 1) − 2iǫ(−x + β + η − 2βη)  2(x2 − x − ǫ2 + iǫ(2x − 1))  �  =  − 1  π lim  ǫ→0+ imag  �  x − (β + η) ±  �  (x − (β + η))2 + 4βη(x − 1) − 2iǫ(−x + β + η − 2βη)  2(x2 − x)  �  =  − 1  π lim  ǫ→0+ imag  �  ±  �  (x − (β + η))2 + 4βη(x − 1) − 2iǫ(−x + β + η − 2βη)  2(x2 − x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (129)  Now, since one is interested in the imaginary part of interest is the region of x where the real part under the  root is negative (outside that region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e in the region of x where the real part under the root is nonnegative  f (b)  ˜  D (x) is zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To determine the region of interest we start by setting  T ˜  D ≜ {x ∈ R|(x − (β + η))2 + 4βη(x − 1) ≤ 0}  and  xc ≜ β + η − 2βη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (130)  23  To explicitly characterize T ˜  D we look at the following  (x − (β + η))2 + 4βη(x − 1)  =  0  ⇐⇒  x2 − 2x(β + η − 2βη) + (β + η)2 − 4βη  =  0  ⇐⇒  x2 − 2x(β + η − 2βη) + (β − η)2  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (131)  Solving for x one finds  x = 2(β + η − 2βη) ±  �  (2(β + η − 2βη))2 − 4(β − η)2  2  = β +η −2βη ±  �  (β + η − 2βη)2 − (β − η)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' (132)  Setting  xl  ≜  β + η − 2βη −  �  (β + η − 2βη)2 − (β − η)2  xu  ≜  β + η − 2βη +  �  (β + η − 2βη)2 − (β − η)2,  (133)  one has  T ˜  D = {x ∈ R|x ∈ [xl, xu]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (134)  Moreover, from (133), one also has  0 ≤ xl ≤ xu ≤ 1,  (135)  with  xl = 0  if  β = η  and  xu = 1  if  β = η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (136)  The first two inequalities in (133) are trivial, whereas the third one follows after noting  β + η − 2βη ≤ max(β, 1 − β) ≤ 1,  (137)  and observing the following sequence  β + η − 2βη +  �  (β + η − 2βη)2 − (β − η)2  ≤  1  ⇐⇒  (β + η − 2βη)2 − (β − η)2  ≤  (1 − (β + η − 2βη))2  ⇐⇒  −(β − η)2  ≤  1 − 2(β + η − 2βη)  ⇐⇒  −(β − η)2 + 2(β + η − 2βη) − 1  ≤  0  ⇐⇒  −(β + η)2 + 2(β + η) − 1  ≤  0  ⇐⇒  −(1 − (β + η))2  ≤  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (138)  Returning to (129) we further have for x ∈ T ˜  D = [xl, xu]  f (b)  ˜  D (x)  =  − 1  π lim  ǫ→0+ imag  �  ±  �  (x − (β + η))2 + 4βη(x − 1) − 2iǫ(−x + β + η − 2βη)  2(x2 − x)  �  =  − 1  π lim  ǫ→0+ imag  �  ±  �  (x − (β + η))2 + 4βη(x − 1) − 2iǫ(xc − x)  2(x2 − x)  �  =  − 1  π lim  ǫ→0+ imag  �  i  �  −(x − (β + η))2 − 4βη(x − 1)  2(x2 − x)  �  ,  (139)  where the “+” plus sign is chosen if xc ≤ x ≤ xu and the “−” sign is chosen if xl ≤ x ≤ xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Finally, from  (139) one easily finds  f (b)  ˜  D (x)  =  �  −(x − (β + η))2 − 4βη(x − 1)  2π(x − x2)  if  xl ≤ x ≤ xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (140)  24  The above is then sufficient to completely characterize the spectral distribution f ˜  D(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We summarize  the results in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Assume large n linear regime with  β ≜ lim  n→∞  k  n  and  η ≜ lim  n→∞  l  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (141)  Let ¯V ⊥ ∈ Rn×(n−k) be a Haar distributed basis of an n − k-dimensional subspace of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Analogously,  let ¯U ⊥  D ∈ Rn×(n−k) be a Haar distributed basis of an n − l-dimensional subspace of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Moreover, let  ¯V ⊥ ∈ Rn×(n−k) and ¯U ⊥  D ∈ Rn×(n−k) be independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Also, let V, U, and ˜D be as defined in  (87) and (88), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' let  V  ≜  ¯V ⊥( ¯V ⊥)T  U  ≜  ¯U ⊥  D( ¯U ⊥  D)T  ˜D  ≜  VU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (142)  Set xl and xu as in (133), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  xl  ≜  β + η − 2βη −  �  (β + η − 2βη)2 − (β − η)2  xu  ≜  β + η − 2βη +  �  (β + η − 2βη)2 − (β − η)2,  (143)  Then the limiting spectral distribution of ˜D, f ˜  D(x), is  f ˜  D(x) = f0δ(x) + f (b)  ˜  D (x) + f1f0δ(x − 1) = max(β, η)δ(x) + f (b)  ˜  D (x) + max(1 − (β + η), 0)δ(x − 1),  (144)  with  f (b)  ˜  D (x)  =  �√  −(x−(β+η))2−4βη(x−1)  2π(x−x2)  ,  if xl ≤ x ≤ xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (145)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Follows through a combination of (116), (118), (127), (140), and the above discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  In Figure 4 we show the spectral function obtained based on the above lemma for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  We observe a very strong agreement between the simulated results and the above theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Simulation results were obtained using moderately large n = 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  In Figure 5 we show the spectral function obtained based on the above lemma for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Due to a remarkable property of the underlying functions the spectrum is identical as in Figure 4 apart  from the fact that the multiplier of the delta function at zero is increased from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='9 at the expense of  removing the delta function at one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We also again observe a very strong agreement between the simulated  results and the theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' As in Figure 4, Simulation results were again obtained for n = 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2  The spectrum of ¯D/D – closed form expressions  We recall on (82) and (83) to set  ¯D  ≜  (I(l))T U T  D ¯V ⊥( ¯V ⊥)T UDI(l) = ( ¯U (⊥)  D )T ¯V ⊥( ¯V ⊥)T ¯U (⊥)  D ,  (146)  Comparing (86) and (146) we observe that their spectra are modulo scalings basically identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To be a bit  more precise, ¯D has all eigenvalues that ˜D has with ηn zeros less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' That basically means that one needs to  adjust the multiplier of δ(x) in f ˜  D(x) and to scale everything by (1 − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The following lemma summarizes  the final results of such a procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Assume the setup of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let ¯D be as in (146), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  ¯D  ≜  ( ¯U (⊥)  D )T ¯V ⊥( ¯V ⊥)T ¯U (⊥)  D ,  (147)  25  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  1  f ˜D(x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  f ˜  D(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  simulated  theory  xu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='98 (with “+” sign)  xl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5 (with “−” sign)  Bulk  xl/u = β + η − 2βη ± �(β + η − 2βη)2 − (β − η)2  f ˜  D(0) = max(β, η)δ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8δ(0)  f ˜  D(1) = max(1 − (β + η), 0)δ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1δ(0)  f (b)  ˜  D (x) =  √  −(x−(β+η))2−4βη(x−1)  2π(x−x2)  Figure 4: f ˜  D(x) – spectral function of ˜D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  and let xl and xu be as in (143).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then the limiting spectral distribution of ¯D, f ¯  D(x), is  f ¯  D(x) = (max(β, η) − η)  1 − η  δ(x) + f (b)  ¯  D (x) + max(1 − (β + η), 0)  1 − η  δ(x − 1),  (148)  with  f (b)  ¯  D (x)  =  �√  −(x−(β+η))2−4βη(x−1)  2π(x−x2)(1−η)  ,  if xl ≤ x ≤ xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (149)  Moreover, let D be as in (81), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  D  ≜  (I(l))T ¯V ⊥( ¯V ⊥)T I(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (150)  The limiting spectral distribution of D, fD(x), is  fD(x) = f ¯  D(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (151)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The part that relates to the spectral distribution of ¯D follows by removing l = ηn zeros from the  spectrum of ˜D and appropriately scaling the residual pdf by (1 − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The part that relates to the spectral  distribution of D follows from (84) which itself is a consequence of the spectral invariance under unitary  multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  In Figure 6 we show the spectral function obtained based on the above lemma for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  We observe a very strong agreement between the simulated results and the above theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Simulation results were obtained using moderately large n = 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='3  The spectrum of Q – closed form expressions  We start by recalling on (81)  Q = D−1 − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (152)  26  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  1  f ˜D(x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  f ˜  D(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='9  simulated  theory  xu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='98 (with “+” sign)  xl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5 (with “−” sign)  Bulk  xl/u = β + η − 2βη ± �(β + η − 2βη)2 − (β − η)2  f ˜  D(0) = max(β, η)δ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='9δ(0)  f (b)  ˜  D (x) =  √  −(x−(β+η))2−4βη(x−1)  2π(x−x2)  Figure 5: f ˜  D(x) – spectral function of ˜D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='9  Almost all of the above holds without explicitly assuming k ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' From this point on such an assumption is  needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Since k ≤ l means β ≤ η one has that D has no zeros in its spectrum and can be inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Since  the portion of the spectrum at one remains the same after the inversion, one then basically has that the  spectrum of Q is the same as the spectrum of D modulo the inversion of the bulk of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' After a change of  variables y = 1  x one has for the spectral distribution of the inverted bulk  f (b)  ¯  D−1(y)  =  \uf8f1  \uf8f2  \uf8f3  −  �  −( 1  y −(β+η))2−4βη( 1  y −1)  2π( 1  y −( 1  y )2)(1−η)y2  ,  if  1  xu ≤ y ≤  1  xl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  0,  otherwise,  (153)  which after elementary algebraic transformations becomes  f (b)  ¯  D−1(x)  =  �√  −(1−x(β+η))2−4βηx(1−x)  2π(1−x)(1−η)x  ,  if  1  xu ≤ x ≤  1  xl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  0,  otherwise,  (154)  Noting that β ≤ η implies max(β, η) − η = 0, one can combine (154 together with (148) to obtain  f ¯  D−1(x) = f (b)  ¯  D−1(x) + max(1 − (β + η), 0)  1 − η  δ(x − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (155)  Finally adjusting for a subtracted identity matrix corresponds to subtracting one from any point in the  spectrum (or basically to shifting the entire spectral distribution to the left by one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In other words  fQ(x) = f ¯  D−1−I(x) = f (b)  ¯  D−1−I(x) + max(1 − (β + η), 0)  1 − η  δ(x),  (156)  with  f (b)  Q (x) ≜ f (b)  ¯  D−1−I(x)  =  �√  −(1−(x+1)(β+η))2−4βηx(x+1)  2πx(x+1)(1−η)  ,  if  1  xu − 1 ≤ x ≤  1  xl − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  0,  otherwise,  (157)  The following lemma summarizes the key results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  27  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  1  fD(x)  2  1  0  1  2  3  4  5  6  fD(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  simulated  theory  xu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='98 (with “+” sign)  xl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5 (with “−” sign)  Bulk  xl/u = β + η − 2βη ± �(β + η − 2βη)2 − (β − η)2  f (b)  D (x) =  √  −(x−(β+η))2−4βη(x−1)  2π(x−x2)(1−η)  fD(1) = max(1−(β+η),0)  1−η  δ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5δ(0)  Figure 6: fD(x) – spectral function of D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Assume the setup of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let Q be as in (81), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Q ≜ D−1 − I =  �  (I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1  − I,  (158)  and let xl and xu be as in (143).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then the limiting spectral distribution of Q, fQ(x), is  fQ(x) = f (b)  Q (x) + max(1 − (β + η), 0)  1 − η  δ(x),  (159)  with  f (b)  Q (x)  =  �√  −(1−(x+1)(β+η))2−4βηx(x+1)  2πx(x+1)(1−η)  ,  if  1  xu − 1 ≤ x ≤  1  xl − 1,  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (160)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Follows from the above discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  In Figure 7 we show the spectral function, fQ(x), obtained based on the above lemma for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1 and  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' One observes a very strong agreement between what the theory predicts and what the simulations  produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' As in all earlier experiments n = 4000 was used here again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='3  ℓ∗  0 − ℓ∗  1-equivalence via the spectral limit  From Corollary 3, (47), (48), and (77) one has in the worst case  ℓ∗  0 − ℓ∗  1 − equivalence  ⇐⇒  λmax(Q) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (161)  From Lemma 5 we have  λmax(Q) = 1  xl  − 1,  (162)  28  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  1  fQ(x)  2  1  0  1  2  3  4  5  6  fQ(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  simulated  theory  xl/u = β + η − 2βη ± �(β + η − 2βη)2 − (β − η)2  Bulk  fQ(0) = max(1−(β+η),0)  1−η  δ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5δ(0)  x∗ =  1  xu − 1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='02  x∗ =  1  xl − 1 = 1  f (b)  Q (x) =  √  −(1−(x+1)(β+η))2−4βηx(x+1)  2πx(x+1))(1−η)  Figure 7: fQ(x) – spectral function of Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  with xl as in (143).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover, from (161) and (162), one arrives at the following necessary and sufficient  condition to achieve the ℓ∗  0 − ℓ∗  1-equivalence  ℓ∗  0 − ℓ∗  1 − equivalence  ⇐⇒  1  xl  − 1 ≤ 1,  (163)  or  ℓ∗  0 − ℓ∗  1 − equivalence  ⇐⇒  1  2 ≤ xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (164)  Recalling on (143)  xl  ≜  β + η − 2βη −  �  (β + η − 2βη)2 − (β − η)2,  (165)  and combining further with (164) we have  1  2  ≤  xl  ⇐⇒  1  2  =  β + η − 2βη −  �  (β + η − 2βη)2 − (β − η)2  ⇐⇒  (β + η − 2βη)2 − (β − η)2  ≤  �  β + η − 2βη − 1  2  �2  ⇐⇒  (β + η − 2βη)2 − (β − η)2  ≤  (β + η − 2βη)2 − (β + η − 2βη) + 1  4  ⇐⇒  −(β − η)2  ≤  − (β + η − 2βη) + 1  4  ⇐⇒  0  ≤  β2 + η2 − (β + η) + 1  4  ⇐⇒  η − η2  ≤  � 1  2 − β  �2  ⇐⇒  β  ≤  1  2 −  �  η − η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (166)  From (164) and (166) we finally have  ℓ∗  0 − ℓ∗  1 − equivalence  ⇐⇒  β ≤ 1  2 −  �  η − η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (167)  We are now in position to formalize the key causal inference results that establish the so-called phase-  transition phenomenon as well as its a precise worst case location in a typical statistical scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' (ℓ∗  1 – phase transition – C-inf (typical worst case)) Consider a rank-k matrix Xsol =  29  X ∈ Rn×n with the Haar distributed ( not necessarily independent) bases of its orthogonal row and column  spans ¯U ⊥ ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) (XT  sol ¯U ⊥ = Xsol ¯V ⊥ = 0n×(n−k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let M ≜ M (l) ∈ Rn×n be as  defined in (15) or (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Assume a large n linear regime with β ≜ limn→∞ k  n and η ≜ limn→∞ l  n and let βwc  and η satisfy the following  C-inf ℓ∗  1 worst case phase transition (PT) characterization  ξ(wc)  η  (β) ≜ β − 1  2 +  �  η − η2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (168)  If and only if β ≤ βwc  lim  n→∞ P(ℓ∗  0 ⇐⇒ ℓ∗  1) =  lim  n→∞ P(RMSE = 0) = 1,  (169)  and the solutions of (16) and (17) coincide with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The “if” part follows through a combination of Theorem 1, Corollary 3, Lemma 5, and the above  discussion from (161) to (167).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The “only if” part additionally assumes ¯U ⊥ = ¯V ⊥ and then relying on (77)  and (78) ensures that the results are in the worst case achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  The results obtained based on the above theorem are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' As can be seen from the figure,  the phase transition curve splits the entire (β, η) region into two subregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The first of the subregions is  below (or to the right of) the curve and in that region the ℓ∗  0 − ℓ∗  1-equivalence phenomenon occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This  means that one can recover Xsol masked by M as in (16) via the ℓ∗  1 heuristic from (17) with the residual  mean square error (RMSE) equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In other words, for the system parameters (β, η) that belong to  the subregion below the curve one has a perfect recovery with Xsol and ˆX (the respective solutions of (16)  and (17)) being equal to each other and consequently with RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' On the  other hand, in the subregion above the curve, the ℓ∗  1 heuristic fails and one can even find an Xsol for which  RMSE → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='95  1  β  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
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+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  (η, β) region of success/failure — C-inf ℓ∗1 PT  RMSE −→ ∞, ℓ∗1  fails  RMSE = 0, ℓ∗1 succeeds  ℓ∗1’s PT: ξ(wc)  η  (β) = β − 1  2 + �η − η2 = 0  Figure 8: Causal inference (C-inf) – typical worst case ℓ∗  1 phase transition  We make a few remarks that relate to some general features of matrix completion, and their differences  with respect with standard compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Even in the so-called random-masking scenario (where  elements of M take values 0/1 with probability one half), one may have troubles recovering low-rank matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  30  For example, in such a scenario, it is statistically unlikely that one can guarantee universal recovery even  of rank-1 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To see this, one can choose rank-1 Xsol with one in the upper left corner and all other  elements equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then such a matrix will be recoverable from M ◦ Xsol only if the element in the  upper left corner of M is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Since that happens with probability 1/2, one can not have a reliable recovery  with probability going to one as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' It is the discreteness in the process of acquiring observations Y that  enables scenarios like this, and makes the matrix completion (MC) substantially different from its compressed  sensing vector analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In that light, it is somewhat surprising that any form of the phase-transition can  be established in the linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The key behind the success of the above machinery is the focus on the  typical worst case and the causal inference internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In the vector compressed sensing setup, the  above described scenario cannot happen and one consequently does not need to resort to the typicality and  instead can formulate more generic phase transition concepts (more on the non-typical compressed sensing  approaches can be found in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' [13,45,48] and on the corresponding typical ones in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' [7,14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' (ℓ∗  1 – phase transition – C-inf (typical worst case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' standard (α, β) representation))  Assume the setup of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let m be the total number of ones in matrix M and let α ≜ limn→∞ m  n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Let β and αw satisfy the  C-inf ℓ∗  1 worst case PT (standard (α, β) representation)  ξ(wc,s)  β  (α) ≜ β − 1  2 +  �√  1 − α − 1 + α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (170)  If and only if α ≥ αw  lim  n→∞ P(ℓ∗  0 ⇐⇒ ℓ∗  1) =  lim  n→∞ P(RMSE = 0) = 1,  (171)  and the solutions of (16) and (17) coincide with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Follows immediately from Theorem 2 after observing that m = n2 − (n − l)2 and consequently  α = 1 − (1 − η)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Figure 9 shows the results obtained based on the above corollary in the standard (α, β) region format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  As earlier, in the subregion to the right of (or below) the curve RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In  the subregion to the left (or above) the curve, the ℓ∗  1 heuristic generally fails and RMSE → ∞ can even be  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='3  Numerical results  We conducted a set of numerical experiments to complement the above theoretical findings and see how well  the entire theoretical machinery characterizes the utilization of the ℓ∗  1-minimization heuristic in the causal  inference problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Figure 10 shows the performance obtained through the numerical experiments as well  as the corresponding theoretical worst case predictions discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We clearly observe the existence  of the phase transition and a solid agreement between the theoretical predictions and the results obtained  through the numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  We should also add that we conducted the numerical experiments for fairly small matrix sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' On the  other hand, the theoretical predictions assume large n (basically an n → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In particular, we chose n = 80  and η in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Such a choice shows that even though the theory is predicated on the large n  assumption, its conclusions may be applicable for smaller values of n as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Viewed a bit alternatively, the  large n regime, needed for the theory to properly operate, practically may start ro kick in already for not  necessarily super large values of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This also means that the presented theory may actually be of a practical  use as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We do, however, mention that for larger values of n an even better fit between the theoretical  and simulated results might be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Finally, we should emphasize that in the numerical experiments here (as well as in a significant portion of  the paper) we considered the so-called typical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Also, it should be mentioned that we followed into  the footsteps of our theoretical analyses from the previous sections and presented the simulations results for  the square matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' With a little bit of extra effort, all of our theoretical considerations can be repeated  31  α  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='95  1  β/α  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
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+page_content='5  (α, β) region of success/failure — C-inf ℓ∗1 PT  RMSE = 0, ℓ∗1 succeeds  ℓ∗1’s PT: ξ(wc,s)  β  (α) = β − 1  2 +  �√1 − α − 1 + α = 0  RMSE −→ ∞, ℓ∗1  fails  Figure 9: Causal inference (C-inf) – typical worst case ℓ∗  1 phase transition ((α, β) region)  in the non-square scenarios as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Since the writings would be a bit more involved we found it useful  to preserve the clarity of the presentation at the expense of rather trivial generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Finally, in all  numerical experiments that we present we chose the unknown matrices with the singular values equal to  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' While a thorough discussion regarding this choice goes well beyond the scope of this paper, we just  briefly recall that these types of structures typically serve as the worst case examples in establishing the  reversal ℓ0 − ℓ1-equivalence conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In other words, they are usually the examples that make many of  our key results/theorems hold as equivalences rather than just as implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We also conducted numerical  experiments where singular values were randomly chosen with results being identical to the ones shown in  Figure 10 or better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  5  Conclusion  In this paper, we have explored the Causal inference (C-inf) ↔ low-rank recovery (LRR) connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  We have analyzed how the best known convex type of heuristic, called ℓ∗  1-minimization, fairs when used for  solving the C-inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We have shown both theoretically and numerically that in a typical statistical context,  causal inference exhibits the so-called phase transition (PT) phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This ensures that for certain  range of system parameters ℓ∗  1 succeeds in recovering the unobserved potential outcomes, and outside such  a range it fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover, we have obtained the exact explicit functional characterization for the location  of the worst case phase transition (PT) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' As a byproduct of our analysis, we have obtained (somewhat  surprisingly) that the underlying functional characterization admits a fairly simple form, which elegantly  pins down the relation between the low rankness of the target C-inf matrix and the time when the treatment  is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We also emphasize that, while establishing the mathematical methodology to provide the C-  inf PT characterizations, we uncovered a rather interesting connection between C-inf via LRR and the  free probability theory (FPT) from modern spectral theory of random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' After establishing the  connection between C-inf and the compressed sensing via the Random duality theory (RDT), we have  proceeded to recognize the role that FPT plays in the overall mosaic that ultimately enables handling the  C-inf problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  On a path to achieving complete handling of the causal inference we have created quite a few mathematical  results that are of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' To ensure a completeness of the overall treatment, we for all of  them also ran the corresponding numerical experiments and again observed a rather overwhelming agreement  between the theoretical predictions and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  32  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='9  1  β  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='5  (η, β) region of success/failure — ℓ∗1’s PT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' simulated/theory  ℓ∗  1’s PT – simulated  ℓ∗  1’s PT – theory  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
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+page_content='9  1  Failure  Success  Figure 10: C-inf ℓ∗  1’s phase transition (PT)  While there exists different approaches that can be explored to attack the problems considered here  (with some of them being even conceptually simpler), our choice is partly motivated by the ability to handle  more complicated problems in future extensions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Being this the introductory paper, we stopped  short of showcasing how the introduced theory fairs when applied to many such specific, more complex  instances (the simplest among them would be the noisy and approximately low-rank corresponding ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  In our companion papers, we will establish results along these directions, which all rely on the mathematical  framework presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Abadie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Using synthetic controls: Feasibility, data requirements, and methodological aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Journal  of Economic Literature, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Abadie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Diamond, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Hainmueller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Synthetic control methods for comparative case stud-  ies: Estimating the effect of california’s tobacco control program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Journal of the American Statistical  Association, 105:493–505, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Agarwal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Dahel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Shah, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Shen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Causal matrix completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' available online at arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Athey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Bayati, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Doudchenko, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Imbens, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Khosravi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Matrix completion methods for  causal panel data models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Journal of the American Statistical Association, 116(536):1716–1730, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Athey and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Imbens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Design-based analysis in differencein- differences settings with staggered  adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Technical Report, National Bureau of Economic Research, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Athey and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The impact of information technology on emergency health care outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The  RAND Journal of Economics, 33:399–432, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Bayati and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Montanari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The dynamics of message passing on dense graphs, with applications to  compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' available online at http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='org/abs/1001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='3448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  33  [8] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Candes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Romberg, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Robust uncertainty principles: exact signal reconstruction from  highly incomplete frequency information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' on Information Theory, 52:489–509, December  2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [9] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Candes and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Recht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Exact matrix completion via convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Foundations of Compu-  tational Mathematics, (9):717, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Candes and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The power of convex relaxation: Near-optimalmatrix completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' IEEE  Transactions on Information Theory, 56:2053–2080, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Cand`es and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Matrix completion with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Proceedings of the IEEE, 98:925–936, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Capponi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stojnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Causal inference (C-inf) — asymmetric scenario of typical phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  available online at arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Donoho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' High-dimensional centrally symmetric polytopes with neighborlines proportional to dimen-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Geometry, 35(4):617–652, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Donoho, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Maleki, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Montanari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Message-passing algorithms for compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  National Academy of Sciences, 106(45):18914–18919, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Donoho and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Tanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Sparse nonnegative solutions of underdetermined linear equations by linear  programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' National Academy of Sciences, 102(27):9446–9451, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [16] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Doudchenko and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Imbens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Balancing, regression, difference-in-differences and synthetic control  methods: A synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Technical Report, National Bureau of Economic Research, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [17] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Haagerup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' On Voiculescu’s R- and S-transforms for free non-commuting random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In: Free  Probability Theory, Fields Institute Communications, 12:127–148, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Hernan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Robins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Causal Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Boca Raton, FL: CRC Press, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [19] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Imbens and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Rubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Causal Inference in Statistics, Social, and Biomedical Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' New  York: Cambridge University Press, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [20] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Kallus, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Mao, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Udell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Causal inference with noisy and missing covariates via matrix  factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Proceedings of the 32rd International Conference on Neural Information Processing, pages  6921—-6932, December 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Keshavan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Montanari, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Matrix completion from a few entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' IEEE Transactions  on Information Theory, 56:2980–2998, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Keshavan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Montanari, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Matrix completion from noisy entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Journal of Machine  Learning Research, 11:2057–2078, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [23] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Klopp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Noisy low-rank matrix completion with general sampling distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Bernoulli, 20:282–303,  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [24] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Koltchinskii, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Lounici, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Tsybakov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Nuclear-norm penalization and optimal rates for noisy  low-rank matrix completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The Annals of Statistics, 39:2302–2329, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Mazumder, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Hastie, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Tibshirani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Spectral regularization algorithms for learning large  incomplete matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Journal of Machine Learning Research, 11:2287–2322, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Negahban and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Wainwright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Estimation of (near) low-rank matriceswith noise and high-  dimensional scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The Annals of Statistics, 39:1069–1097, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Negahban and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Wainwright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Restricted strong convexity and weighted matrix completion:  Optimal bounds with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Journal of Machine Learning Research, 13:1685–1697, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Nica and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Speicher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  London Mathematical Society Lecture Note Series, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' 335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Cambridge  University Press, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  34  [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Oymak and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Hassibi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' New null space results and recovery thresholds for matrix rank minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Nov 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' available online at http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='org/abs/1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='6326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Pearl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Causal inference in statistics: An overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Statistics Surveys, 3:96–146, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Pearl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Causality: Models, Reasoning, and Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' 2nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Cambridge University Press, New York,  2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Pearl and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Bareinboim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' A note on ‘generalizability of study results’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Journal of Epidemiology,  30:186–188, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Pearl and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Mackenzie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The Book of Why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Basic Books, New York, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [34] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Recht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' A simpler approach to matrix completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Journal of Machine Learning Research, 12:3413–  3430, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [35] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Recht, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Fazel, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Parrilo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Guaranteed minimum-rank solution of linear matrix equations  via nuclear norm minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' available online at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='dsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='ece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='rice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='edu/cs/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Rohde and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Tsybakov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Estimation of high-dimensional low-rank matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  The Annals of  Statistics, 39:887–930, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [37] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Rosenbaum and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Rubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Thecentral role of the propensity score in observational studies for  causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Biometrika, 70, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [38] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Rubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Matched Sampling for Causal Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Cambridge: Cambridge University Press, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [39] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Shaikh and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Toulis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Randomization tests in observational studies with staggered adoption of  treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  University of Chicago, Becker Friedman Institute for Economics Working Paper, (144),  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [40] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Speicher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Free probability and random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' ICM, III:477–501, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [41] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Srebro, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Alon, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Jaakkola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Generalization error bounds for collaborative prediction with  low-rank matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' in Advances in Neural Information Processing Systems, 17:1321–1328, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Saul, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='Weiss, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Bottou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stojnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' A framework for perfromance characterization of LASSO algortihms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
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+page_content='  [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stojnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
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+page_content='  [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stojnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Upper-bounding ℓ1-optimization weak thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
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+page_content='  [45] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stojnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Various thresholds for ℓ1-optimization in compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
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+page_content='org/abs/0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='3666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [46] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stojnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' A simple performance analysis of ℓ1-optimization in compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' ICASSP, Inter-  national Conference on Acoustics, Signal and Speech Processing, pages 3021–3024, 15-19 April 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Taipei, Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [47] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stojnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Block-length dependent thresholds for ℓ2/ℓ1-optimization in block-sparse compressed sens-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' ICASSP, IEEE International Conference on Acoustics, Signal and Speech Processing, pages 3918–  3921, 14-19 March 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Dallas, TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stojnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' ℓ1 optimization and its various thresholds in compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' ICASSP, IEEE Inter-  national Conference on Acoustics, Signal and Speech Processing, pages 3910–3913, 14-19 March 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Dallas, TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stojnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' ℓ2/ℓ1-optimization in block-sparse compressed sensing and its strong thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' IEEE  Journal of Selected Topics in Signal Processing, 4(2):350–357, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  35  [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stojnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Recovery thresholds for ℓ1 optimization in binary compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' ISIT, IEEE Inter-  national Symposium on Information Theory, pages 1593 – 1597, 13-18 June 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Austin, TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [51] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stojnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Towards improving ℓ1 optimization in compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' ICASSP, IEEE International  Conference on Acoustics, Signal and Speech Processing, pages 3938–3941, 14-19 March 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Dallas,  TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [52] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Stojnic, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Parvaresh, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Hassibi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' On the reconstruction of block-sparse signals with an optimal  number of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' on Signal Processing, 57(8):3075–3085, August 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [53] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Tulino and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Verd´u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Random Matrix Theory and Wireless Communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Foundations and  Trends® in Communications and Information Theory: Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Now Publishers, Hanover, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [54] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Voiculescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Addition of certain non-commuting random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=', 66(3):323–346,  1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [55] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Voiculescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Multiplication of certain noncommuting random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Operator Theory, 18:2223–  2235, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [56] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Voiculescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Limit laws for random matrices and free products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=', 104(1):201–220, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [57] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Xiong and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Pelger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Large dimensional latent factor modeling with missing observations and appli-  cations to causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Electronic copy available at: https://ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='com/abstract=3465357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  [58] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Generalized synthetic control method: Causal inference with interactive fixed effects models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Political Analysis, 25:57–76, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  A  Proof of Theorem 1  As mentioned earlier, the proof of Theorem 1 is conceptually identical to the corresponding proof when  matrix X is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' A detailed proof for the symmetric matrices is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Before being able to  present the proof we need a couple of technical lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let C = CT ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Also let all eigenvalues of C belong to the interval [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Finally, let the  first k entries on the main diagonal, Ci,i, 1 ≤ i ≤ k, be larger than or equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then the upper k × k left  block of C, C1:k,1:k, is an identity matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  C1:k,1:k  =  Ik×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (172)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Let λmax(C) be the maximum eigenvalue of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then  λmax(C)  ≜  max  ∥c∥2=1 cT Cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (173)  Since by assumptions 1 ≤ Ci,i, 1 ≤ i ≤ k and λmax(C) ≤ 1 we also have for any 1 ≤ i ≤ k  1 ≤ Ci,i ≤ max  ∥c∥2=1 cT Cc ≜ λmax(C) ≤ 1,  (174)  which implies C(i, i) = 1, 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The proof that all other elements of C1:k,1:k are equal to zero proceeds  inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  1) Induction move from l = 1 to l = 2: First we look at the upper block of size 2 × 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' at C1:2,1:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  We then have  1 ≥ max  ∥c∥2=1 cT Cc ≥  max  ∥c1:2∥2=1 cT  1:2C1:2,1:2c1:2  ≥  max  ∥c1:2∥2=1 (∥c1:2∥2 + 2|c1c2C1,2|)  ≥  max  ∥c1:2∥2=1 (1 + 2|c1c2C1,2|) ≥ 1,  (175)  36  which implies C1,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  2) Induction move from l to l + 1: Now we look at the upper block of size (l + 1) × (l + 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' at  C1:l+1,1:l+1 while assuming that C1:l,1:l = Il×l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We then have  1  ≥  max  ∥c∥2=1 cT Cc  ≥  max  ∥c1:l+1∥2=1 cT  1:l+1C1:l+1,1:l+1c1:l+1  ≥  max  ∥c1:l+1∥2=1  �  ∥c1:l+1∥2 + 2|cT  1:lC1:l,l+1cl+1|  �  ≥  max  ∥c1:l+1∥2=1  �  1 + 2|cT  1:lC1:l,l+1cl+1|  �  ≥  1,  (176)  which implies C1:l,l+1 = 0l×1 and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Assume the setup of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then the upper k × k left block of C, C1:k,1:k, is an identity  matrix and the upper k × (n − k) right block of C, C1:k,n−k+1:n is a zero matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  C1:k,1:k  =  Ik×k  C1:k,n−k+1:n  =  0k×(n−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (177)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The first part follows by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We now focus on the second part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Consider the following partition  of matrix C  C  =  �  C1:k,1:k  C1:k,n−k+1:n  Cn−k+1:n,1:k  Cn−k+1:n,n−k+1:n  �  =  �  Ik×k  C1:k,n−k+1:n  Cn−k+1:n,1:k  Cn−k+1:n,n−k+1:n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (178)  Then assuming that the largest nonzero singular value of C1:k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='n−k+1:n is equal to b > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' we have  1  ≥  max  ∥c∥2=1 cT Cc  ≥  max  ∥c1:k∥2=a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='cn−k+1:n  �  cT  1:kC1:k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1:kc1:k + 2|cT  1:kC1:k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='n−k+1:ncn−k+1:n| + cT  n−k+1:nCn−k+1:n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='n−k+1:ncn−k+1:n  �  ≥  max  ∥c1:k∥2=a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='cn−k+1:n  �  a2 + 2|cT  1:kC1:k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='n−k+1:ncn−k+1:n| + cT  n−k+1:nCn−k+1:n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='n−k+1:ncn−k+1:n  �  ≥  max  ∥c1:k∥2=a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='cn−k+1:n  �  a2 + 2|cT  1:kC1:k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='n−k+1:ncn−k+1:n| − cT  n−k+1:ncn−k+1:n  �  ≥  max  a∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1]  �  a2 + 2ba  �  1 − a2 − (1 − a2)  �  =  max  a∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1]  �  2a2 − 1 + 2ba  �  1 − a2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (179)  where the fourth inequality follows since the minimum eigenvalue of Cn−k+1:n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='n−k+1:n is larger than or equal  to the minimum eigenvalue of C which is by the lemma’s assumption larger than or equal to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Now, we  further have  c ≜ 2a  �  1 − a2  and  2a2 − 1 + 2ba  �  1 − a2 =  �  1 − c2 + bc,  (180)  and  d(  √  1 − c2 + bc)  dc  =  −c  √  1 − c2 + b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (181)  37  From (181) we then easily obtain  c =  b  √  1 + b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (182)  A combination of (179), (180), and (182) gives  1 ≥ max  ∥c∥2=1 cT Cc ≥ max  a∈[0,1]  �  2a2 − 1 + 2ba  �  1 − a2  �  =  √  1 + b2,  (183)  which implies b = 0 and automatically C1:k,n−k+1:n = 0k×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Now we can consider the above mentioned theorem that adapts the general ℓ1 equivalence condition  result from [44–46] to the corresponding one for the ℓ1 norm of the singular/eigenvalues (similar adaptation  can also be found in [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' (ℓ∗  0 − ℓ∗  1-equivalence condition (LRR) – symmetric X) Consider a ¯U ∈ Rn×k such that  ¯U T ¯U = Ik×k and a rank − k a priori known to be symmetric matrix Xsol = X ∈ Rn×n with all of its  columns belonging to the span of ¯U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' For concreteness, and without loss of generality, assume that X has only  positive nonzero eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' For a given matrix A ∈ Rm×n2 (m ≤ n2) assume that y = Avec(X) ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' If  (∀W ∈ Rn×n|Avec(W) = 0m×1, W = W T ̸= 0n×n)  − tr ( ¯U T W ¯U) < ℓ∗  1(( ¯U ⊥)T W ¯U ⊥),  (184)  then the solutions of (9) and (10) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover, if  (∃W ∈ Rn×n|Avec(W) = 0m×1, W = W T ̸= 0n×n)  − tr ( ¯U T W ¯U) ≥ ℓ∗  1(( ¯U ⊥)T W ¯U ⊥),  (185)  then there is an X from the above set of the symmetric matrices with columns belonging to the span of ¯U  such that the solutions of (9) and (10) are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' The proof follows literally step-by-step the proof of the corresponding theorem in [44–46] and adapts  it to matrices or their singular/eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' For experts in the field this adaptation is highly likely to be  viewed as trivial and certainly doesn’t need to be as detailed as we will make it to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Nonetheless, to ensure  a perfect clarity of all arguments we provide a step-by-step instructional derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' For concreteness and  without loss of generality we also assume that the eigen-decomposition of X is  X = UΛU T =  � ¯U  ¯U ⊥� �  ¯ΛX  0k×(n−k)  0(n−k)×k  ¯Λ⊥  X  � � ¯U  ¯U ⊥�T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (186)  (i) =⇒ (the if part): Following step-by-step the proof of Theorem 2 in [46], we start by assuming that  ˆX is the solution of (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then we want to show that if (184) holds then ˆX = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' As usual, we instead of that,  assume opposite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' we assume that (184) holds but ˆX ̸= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then since y = Avec( ˆ  X) and y = Avec(X)  must hold simultaneously there must exist W such that ˆX = X + W with W ̸= 0, Avec(W) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Moreover,  since ˆX is the solution of (10) one must also have  ℓ∗  1(X + W) = ℓ∗  1( ˆX)  ≤  ℓ∗  1(X)  ⇐⇒  ℓ∗  1(  � ¯U  ¯U ⊥�T (X + W)  � ¯U  ¯U ⊥�  )  ≤  ℓ∗  1(X)  =⇒  ℓ∗  1( ¯U T (X + W) ¯U) + ℓ∗  1(( ¯U ⊥)T (X + W) ¯U ⊥)  ≤  ℓ∗  1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (187)  The last implication follows after one trivially notes  ℓ∗  1(  � ¯U  ¯U ⊥�T (X + W)  � ¯U  ¯U ⊥�  )  =  max  Λ∗=ΛT  ∗ ∈L∗  tr (Λ∗  � ¯U  ¯U ⊥�T (X + W)  � ¯U  ¯U ⊥�  )  ≥  max  Λ∗=ΛT  ∗ ∈L0  ∗  tr (Λ∗  � ¯U  ¯U ⊥�T (X + W)  � ¯U  ¯U ⊥�  )  38  =  ℓ∗  1( ¯U T (X + W) ¯U) + ℓ∗  1(( ¯U ⊥)T (X + W) ¯U ⊥),  (188)  where  L0  ∗  ≜  �  Λ∗ ∈ Rn×n|Λ∗ = ΛT  ∗ , Λ∗ΛT  ∗ ≤ I, Λ∗ =  �  Λ∗,1  0k×(n−k)  0(n−k)×k  Λ∗,2  ��  ⊆  �  Λ∗ ∈ Rn×n|Λ∗ = ΛT  ∗ , Λ∗ΛT  ∗ ≤ I  �  ≜ L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (189)  The key observation – “Removing the absolute values”:  Now, the key observation made in [46] comes into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Namely, one notes that the absolute values can  be removed in the nonzero part and that the ℓ∗  1(·) can be “replaced” by tr (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Such a simple observation  is the most fundamental reason for all the success of the RDT when used for the exact performance  characterization of the structured objects’ recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' From (187) we then have  ℓ∗  1( ¯U T (X + W) ¯U) + ℓ∗  1(( ¯U ⊥)T (X + W) ¯U ⊥)  ≤  ℓ∗  1(X)  =⇒  tr ( ¯U T (X + W) ¯U) + ℓ∗  1(( ¯U ⊥)T (W) ¯U ⊥)  ≤  ℓ∗  1(X)  ⇐⇒  tr ( ¯U T W ¯U) + ℓ∗  1(( ¯U ⊥)T W ¯U ⊥)  ≤  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (190)  We have arrived at a contradiction as the last inequality in (190) is exactly the opposite of (184).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This  implies that our initial assumption ˆX ̸= X cannot hold and we therefore must have ˆX = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This is precisely  the claim of the first part of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (ii) ⇐= (the only if part): We now assume that (185) holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (∃W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n)  − tr (( ¯U)T W ¯U) ≥ ℓ∗  1(( ¯U ⊥)T W ¯U ⊥)  (191)  and would like to show that for such a W there is a symmetric rank-k matrix X with the columns belonging  to the span of ¯U such that y = Avec(X), and the following holds  ℓ∗  1(X + W) < ℓ∗  1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (192)  Existence of such an X would ensure that it both, satisfies all the constraints in (10) and is not the  solution of (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Following the strategy of [44] one can reverse all the above steps from (191) to (187) with  strict inequalities and arrive at the first inequality in (187) which is exactly (192).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' There are two implications  that cause problems in such a reversal process, the one in (191) and the one in (187).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' If these implications  were equivalences everything would be fine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We address these two implications separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  1) the implication in (190) – particular X to “overwhelm” W: Assume X = ¯UΛx ¯U T with Λx >  0 being a diagonal matrix with arbitrarily large elements on the main diagonal (here it is sufficient even to  choose diagonal of Λx so that its smallest element is larger than the maximum eigenvalue of ¯U T W ¯U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Now  one of course sees the main idea behind the “removing the absolute values” concept from [44,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Namely,  for such an X one has that ℓ∗  1( ¯U T X + W) ¯U) = tr(ℓ∗  1( ¯U T X + W) ¯U)) since for symmetric matrices the ℓ∗  1(·)  (as the sum of the argument’s absolute eigenvalues) and tr (·) (as the sum of the argument’s eigenvalues) are  equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' That basically means that when going backwards the second inequality in (190) not only follows from  the first one but also implies it as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' In other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' for X = ¯UΛx ¯U T (with Λx > 0 and arbitrarily large)  tr ( ¯U T W ¯U) + ℓ∗  1(( ¯U ⊥)T W ¯U ⊥)  ≤  0  ⇐⇒  tr ( ¯U T (X + W) ¯U ) + ℓ∗  1(( ¯U ⊥)T (W) ¯U ⊥)  ≤  ℓ∗  1(X)  ⇐⇒  ℓ∗  1( ¯U T (X + W) ¯U) + ℓ∗  1(( ¯U ⊥)T (X + W) ¯U ⊥)  ≤  ℓ∗  1(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (193)  which basically mans that there is an X that can “overwhelm” W (in the span of ¯U) and ensures that the  “removing the absolute values” is not only a sufficient but also a necessary concept for creating the  relaxation equivalence condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  2) the implication in (187): One would now need to somehow show that the third inequality in (187)  not only follows from the second one but also implies it as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This boils down to showing that inequality in  (188) can be replaced with an equality or, alternatively, that L0 and L are provisionally equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Neither  39  of these statements is generically true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' However, since we have a set of X at our disposal there might be an  X for which they actually hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' We continue to assume X = ¯UΛx ¯U T with Λx > 0 being a diagonal matrix  with arbitrarily large entries on the main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Then the last equality in (188) gives  ℓ∗  1( ¯U T (X + W) ¯U) + ℓ∗  1(( ¯U ⊥)T (X + W) ¯U ⊥)  ≤  ℓ∗  1(X)  ⇐⇒  maxΛ∗=ΛT  ∗ ∈L0∗ tr (Λ∗  � ¯U  ¯U ⊥�T (X + W)  � ¯U  ¯U ⊥�  )  ≤  ℓ∗  1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (194)  Also, one has  maxΛ∗=ΛT  ∗ ∈L0  ∗ tr (Λ∗  � ¯U  ¯U ⊥�T (X + W)  � ¯U  ¯U ⊥�  )  ≤  ℓ∗  1(X)  ⇐⇒  maxΛ∗,i=ΛT  ∗,i,Λ∗,iΛT  ∗,i≤I,i∈{1,2} tr (Λ∗,1 ¯U T X ¯U + Λ∗,2( ¯U ⊥)T W ¯U ⊥)  ≤  ℓ∗  1(X)  ⇐⇒  maxΛ∗,i=ΛT  ∗,i,Λ∗,iΛT  ∗,i≤I,i∈{1,2} tr (Λ∗,1Λx + Λ∗,2( ¯U ⊥)T W ¯U ⊥)  ≤  tr (Λx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (195)  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' if at least one of the elements on the main diagonal of Λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' diag(Λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' is smaller than 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' then the  corresponding element on the diagonal of Λx can be made arbitrarily large compared to the other elements  of Λx and one would have  maxΛ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='i=ΛT  ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='Λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='iΛT  ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='i≤I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='i∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2} tr (Λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='1Λx + Λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='2( ¯U ⊥)T W ¯U ⊥)  <  tr (Λx)  ⇐⇒  maxΛ∗=ΛT  ∗ ∈L0∗ tr (Λ∗  � ¯U  ¯U ⊥�T (X + W)  � ¯U  ¯U ⊥�  )  <  ℓ∗  1(X)  ⇐⇒  maxΛ∗=ΛT  ∗ ∈L∗ tr (Λ∗  � ¯U  ¯U ⊥�T (X + W)  � ¯U  ¯U ⊥�  )  <  ℓ∗  1(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  (196)  where the last equivalence holds since the difference of the terms on the left-hand side in the last two  inequalities is bounded independently of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Also, the last inequality in (196) together with the first equality  in (188) and the first inequality in (187) produces (192).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' Therefore the only scenario that is left as potentially  not producing (192) is when all the elements on the main diagonal are larger than or equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' However,  the two lemmas preceding the theorem show that in such a scenario L0 = L and one consequently has an  equality instead of the inequality in (188) which then, together with (187), implies (192).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content=' This completes  the proof of the second (“the only if”) part of the theorem and therefore of the entire theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
+page_content='  40' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQf4vqx/content/2301.00793v1.pdf'}
diff --git a/EdFRT4oBgHgl3EQfyTjG/content/tmp_files/2301.13645v1.pdf.txt b/EdFRT4oBgHgl3EQfyTjG/content/tmp_files/2301.13645v1.pdf.txt
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+arXiv:2301.13645v1  [math.AP]  31 Jan 2023
+Existence, uniqueness and
+L2
+t(H2
+x) ∩ L∞
+t (H1
+x) ∩ H1
+t (L2
+x) regularity of the
+gradient flow of the Ambrosio-Tortorelli functional
+Tommaso Cortopassi ∗
+Abstract
+We consider the gradient flow of the Ambrosio-Tortorelli functional at
+fixed ǫ > 0, proving existence and uniqueness of a solution in dimension 2.
+The strategy of the proof essentially follows the one in the first part of [9],
+but as suggested in a footnote by the authors of [9] it employs a different
+and simpler technique, which is used for a different equation in [4] and in the
+end it allows to prove better estimates than the ones obtained in the original
+article. In particular we prove that if U ⊂ R2 is a bounded Lipshitz domain,
+the initial data (u0, z0) ∈ [H1(U)]2 and 0 ≤ z0 ≤ 1, then for every T > 0
+there exists a unique gradient flow (u(t), z(t)) of the Ambrosio-Tortorelli
+functional such that
+(u, z) ∈ [L2(0, T; H2(U)) ∩ L∞(0, T; H1(U)) ∩ H1(0, T; L2(U))]2.
+The basic difference from [9], as already said, is a better regularity result
+and a simpler proof: while in [9] they used a localization argument based
+on an idea by Struwe (see [19]), here crucial estimates on the fourth powers
+of the L4
+t (L4
+x) norms of the gradients will be obtained employing a suitable
+version of the Meyers theorem due to Gallouet and Monier (see [15], [11]).
+1
+Introduction
+The Mumford-Shah functional, introduced in [16], is defined as
+E(u, Γ) = 1
+2
+�
+U\Γ |∇u|2 + (u − g)2dx + H1(Γ),
+(1.1)
+where u ∈ H1(U \ Γ), U ⊂ R2 open and bounded, Γ ⊂ U is closed and H1
+is the one dimensional Hausdorff measure. This functional has been extensively
+studied for its applications in image segmentation and fracture mechanics (a de-
+tailed survey can be found in [5]). The definition of the functional depends on the
+model it is used for, in (1.1) we gave the original definition of [16] which is suited
+∗Scuola Normale Superiore, 56126 Pisa, Italy. E-mail: tommaso.cortopassi@sns.it
+1
+
+for image segmentation models, the interested reader can see the seminal paper
+[10] about the fracture mechanic case. The idea is rather simple: given a gray
+image g (i.e. a scalar function), we want to find (˜u, ˜Γ) such that
+(˜u, ˜Γ) = arg min
+u∈H1(U\Γ)
+Γ⊂U closed
+E(u, Γ).
+(1.2)
+In this case, the function u will approximate in a “smooth way” the image g
+while Γ will be the contour set. From a theoretical point of view, in [6] the authors
+proved the existence of a solution for (1.2) by restricting to functions u ∈ SBV (U)
+(see [1]) and Γ = Su, i.e. the set of discontinuity jump points of u.
+From a numerical point perspective, since the functional involves the measure
+of the singular jump set of a (unknown) function u, its direct numerical implemen-
+tation is often not possible or not feasible. A standard approach is to minimize a
+more regular functional proposed by Ambrosio and Tortorelli in [2] defined as
+1
+2
+�
+U[(ηǫ + z2)|∇u|2 + (u − g)2] +
+�
+U
+�(1 − z)2
+4ǫ
++ ǫ|∇z|2
+�
+(1.3)
+which approximates the Mumford-Shah functional in Γ-convergence as ǫ → 0
+in an even more general setting, i.e. not being restricted to the 2 dimensional case,
+but considering U ⊂ Rn open and the (n − 1)-dimensional Hausdorff measure for
+Γ. The Ambrosio-Tortorelli functional is the way with which one usually finds
+(approximate) minima points for (1.1), in particular one can use a gradient flow
+approach for (1.3). As in [9], we’ll consider the gradient flow of (1.3), which is
+given by
+
+
+
+
+
+
+
+
+
+
+
+
+
+∂tu = div((ηǫ + z2)∇u) − (u − g) in (0, T) × U
+∂tz = 2ǫ∆z − z|∇u|2 + 1−z
+2ǫ in (0, T) × U
+∂
+∂nu =
+∂
+∂nz = 0 in (0, T) × ∂U
+u(0, ·) = u0 and z(0, ·) = z0 in {0} × U
+(1.4)
+with ηǫ, ǫ > 0 fixed. As already mentioned, our goal is to study the existence,
+uniqueness and regularity of solutions of (1.4), and the novelty lies in an approach
+(already suggested in a footnote of [9] and used in [4] for a different equation)
+which simplifies the proof of [9] while also gaining better regularity results. In
+particular in [9] they only manage to prove the L2
+t(H2
+x) regularity for a short time
+T1, since the crucial estimate in their argument is a local energy estimate, inspired
+by a technique due to Struwe [19], which only holds for sufficiently small times.
+2
+
+2
+Main result
+Let U ⊂ R2 be a Lipshitz bounded domain and let (u0, z0) ∈ [H1(U)]2 with
+0 ≤ z0 ≤ 1, g ∈ L2(U). As stated in the introduction, we want to prove existence,
+uniqueness and regularity of solutions of the gradient flow of (1.3), given by
+
+
+
+
+
+
+
+
+
+
+
+
+
+∂tu = div((ηǫ + z2)∇u) − (u − g) in (0, T) × U
+∂tz = 2ǫ∆z − z|∇u|2 + 1−z
+2ǫ ) in (0, T) × U
+∂
+∂nu =
+∂
+∂nz = 0 in (0, T) × ∂U
+u(0, ·) = u0 and z(0, ·) = z0 in {0} × U.
+(2.1)
+However we will mostly work with the slightly modified system
+
+
+
+
+
+
+
+
+
+
+
+
+
+∂tu = div((ηǫ + φ(z)2)∇u) − (u − g) in (0, T) × U
+∂tz = 2ǫ∆z − φ′(z)φ(z)|∇u|2 + 1−z
+2ǫ in (0, T) × U
+∂
+∂nu =
+∂
+∂nz = 0 in (0, T) × ∂U
+u(0, ·) = u0 and z(0, ·) = z0 in {0} × U
+(2.2)
+where φ is a cutoff. The reason to introduce such a cutoff, as we will show later,
+is to bound the L∞ norm of ηǫ+φ(zN)2 when considering Galerkin approximations
+zN. To be precise, a good working definition of φ might be
+φ(s) =
+
+
+
+
+
+
+
+−1 if s ≤ −1
+s if − 1 < s < 2
+2 if s ≥ 2
+but you could actually do the cuts at levels −δ1 and 1 + δ2 for every δ1, δ2 > 0
+and nothing would change. A modified Ambrosio-Tortorelli functional (of which
+(2.2) is the gradient flow of) will be denoted as ATǫ and defined as
+ATǫ(u(t), z(t)) = 1
+2
+�
+U[(ηǫ+φ(z(t))2)|∇u(t)|2+(u(t)−g)2]+
+�
+U
+�(1 − z(t))2
+4ǫ
++ ǫ|∇z(t)|2
+�
+.
+(2.3)
+First of all, let’s define the notion of strong solution, following [9].
+Definition 2.1. Strong solution
+A couple (u, z) is a strong solution of (2.1) if it satisfies the system in a dis-
+tributional sense, i.e. for every ψ ∈ H1(U) it holds
+�
+U
+u(t)ψdx =
+�
+U
+u0ψdx −
+� t
+0
+�
+U
+(ηǫ + z(s)2)⟨∇u(s), ∇ψ⟩dxds −
+� t
+0
+�
+U
+(u(s) − g)ψdxds
+�
+U
+z(t)ψdx =
+�
+U
+z0ψdx − 2ǫ
+� t
+0
+�
+U
+⟨∇z(s), ∇ψ⟩dxds −
+� t
+0
+�
+U
+z(s)ψ|∇u(s)|2 + 1 − z(s)
+2ǫ
+ψdxds
+for almost every t ∈ (0, T), and moreover we require
+(u, z) ∈ [L2(0, T; H2(U)) ∩ L∞(0, T; H1(U)) ∩ H1(0, T; L2(U))]2.
+We start with a proposition which shows how a solution of (2.2) is also solution
+of (2.1) if 0 ≤ z0 ≤ 1.
+3
+
+Proposition 2.1. Let 0 ≤ z0 ≤ 1. If a strong solution (u, z) for (2.2) exists, it
+must be 0 ≤ z(t, x) ≤ 1 for every t ∈ (0, T) and for a.e. x ∈ U. So (u, z) will also
+be a solution of (2.1).
+Proof. If we test the equation for z in (2.2) with z itself it gives
+1
+2
+d
+dt||z(t)||2
+L2(U) = −2ǫ||∇z(t)||2
+L2(U)−
+�
+U φ′(z(t))φ(z(t))z(t)|∇u(t)|2ds+
+�
+U
+1 − z(t)
+2ǫ
+z(t)ds
+(2.4)
+where the existence of the time derivative is ensured by Lions-Magenes lemma
+(see [14]). Consider f0(z) = max{−z, 0} and f1(z) = max{z − 1, 0}, so f0 and f1
+are defined as
+f0(s) =
+
+
+
+−s if s ≤ 0
+0 if s > 0
+and f1(s) =
+
+
+
+0 if s ≤ 1
+s − 1 if s > 1
+.
+It’s easy to check that:
+1
+2
+d
+dt||f0(z(t))||2
+L2(U) =
+�
+U f0(z(t))f ′
+0(z(t))∂tz(t)dx =
+�
+U χ{z(t)<0} z(t) ∂tz(t)dx,
+and noting that χ{z(t)<0}z(t) = −f0(z(t)) is a Sobolev function (see Lemma 7.6
+in [12]) we are allowed to use it as a test function in (2.2), getting:
+1
+2
+d
+dt||f0(z(t))||2
+L2(U) = ⟨∂tz(t), −f0(z(t))⟩L2(U) =
+= χ{z(t)<0}
+�
+−2ǫ||∇z(t)||2
+L2(U) −
+�
+U φ′(z(t))φ(z(t))z(t)|∇u(t)|2 +
+�
+U
+1 − z(t)
+2ǫ
+z(t)
+�
+≤ 0.
+Since f0(z(0)) = 0 because z0 ≥ 0, we have f0(z(t)) ≡ 0 for every t. In the
+same way we can prove z ≤ 1 by considering f1(z(t)).
+The strategy will be to make use of Galerkin approximates. Consider an or-
+thogonal basis of H1(U) composed of eigenfunctions of −∆ on U with homoge-
+neous Neumann boundary conditions normalized with respect to the L2(U) norm,
+and denote it as {ei}i∈N. So
+
+
+
+−∆ei = λiei in U
+∂nei = 0 in ∂U
+and ||ei||L2(U) = 1 for every i.
+We want to find Galerkin approximates (uN, zN) such that they solve (in dis-
+tributional sense) in
+VN = Span({e1, . . . , eN})
+(2.5)
+the system
+4
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+∂tuN = πN[div((ηǫ + φ(zN)2)∇uN)] − (uN − gN) in (0, T) × U
+∂tzN = 2ǫ∆zN − πN[φ′(zN)φ(zN)|∇uN|2] + 1−zN
+2ǫ
+in (0, T) × U
+∂
+∂nuN =
+∂
+∂nzN = 0 in (0, T) × ∂U
+uN(0, ·) = πN[u0] and zN(0, ·) = πN[z0] in {0} × U
+(2.6)
+with πN the orthogonal projection on VN. Notice that by the orthogonality
+properties of {ei}N
+i=1 this is actually a 2N system in u(1)(t), . . . , u(N)(t), z(1)(t), . . . , z(N)(t),
+with
+uN(t, x) =
+N
+�
+i=1
+u(i)(t)ei(x) and zN(t, x) =
+N
+�
+i=1
+z(i)(t)ei(x)
+which by Cauchy-Lipshitz admits a unique local solution. Indeed if we test
+with ei it holds that:
+u(i)(t) = (u0)(i) −
+� t
+0
+��
+U(ηǫ + φ(zN(s))2)⟨∇uN(s), ∇ei⟩ + (uN(s) − gN)eidx
+�
+ds
+z(i)(t) = (z0)(i) −
+� t
+0
+��
+U 2ǫ⟨∇zN(s), ∇ei⟩ − φ′(zN(s))φ(zN(s))|∇uN(s)|2ei + 1 − zN(s)
+2ǫ
+eidx
+�
+ds
+for every 1 ≤ i ≤ N. However there are strong non linearities at play, so a
+priori for every N we only have a local solution in [0, tN) without being able to
+extend it immediately to [0, T]. In order to gain existence in [0, T] of Galerkin
+approximates we’ll use the following a priori estimates, which hold in a slightly
+more general situation with less regular initial data:
+Proposition 2.2. Existence of weak approximate solutions in [0,T]
+Given (u0, z0) ∈ [L2(U)]2 and a solution (uN, zN) of (2.6) in VN, it holds
+sup
+0≤t≤T[||uN(t)||2
+L2(U)] +
+� T
+0 ||∇uN(s)||2
+L2(U)ds ≤ C
+and
+sup
+0≤t≤T[||zN(t)||2
+L2(U)] +
+� T
+0 ||∇zN(s)||2
+L2(U)ds ≤ C
+with C a positive constant independent of N.
+Proof. Test the equation for uN in (2.6) with uN itself, to get
+d
+dt||uN||2
+L2(U) = −
+�
+U(ηǫ + φ(zN)2)|∇uN|2dx −
+�
+U uN(uN − gN)dx.
+(2.7)
+So
+d
+dt||uN||2
+L2(U) ≤ ||gN||L2(U)||uN||L2(U) ≤ ||g||L2(U)(1 + ||uN||2
+L2(U))
+5
+
+and you get uniform boundedness of ||uN||L2(U) by Gronwall’s lemma. Going
+back to (2.7) you easily conclude by integrating in time. The same holds if we test
+the equation in zN with zN itself, obtaining
+sup
+0≤t≤T[||zN(t)||2
+L2(U)] + 2ǫ
+� T
+0
+≤ T
+8ǫ.
+By orthogonality of the ei s this can be rewritten as
+||uN(t)||2
+L2(U) =
+N
+�
+i=1
+[u(i)(t)]2 and ||zN(t)||2
+L2(U) =
+N
+�
+i=1
+[z(i)(t)]2
+so we cannot have a blow-up in finite time and we have thus proved existence
+up to time T for every T > 0.
+Now that we have existence of solutions of (2.6) in [0, T] for every N, let’s prove
+stronger inequalities exploiting the variational characterization of the problem.
+Proposition 2.3. A priori energy estimates
+Assume (u0, z0) ∈ [H1(U)]2, then
+sup
+t∈[0,T]
+[ATǫ(uN(t), zN(t))] +
+� T
+0 ||∂tuN(s)||2
+L2(U) + ||∂tzN(s)||2
+L2(U)ds =
+= sup
+t∈[0,T]
+[ATǫ(uN(t), zN(t))] +
+� T
+0 ||πN[∇ATǫ(uN, zN)]||2
+L2(U)ds ≤
+≲ ATǫ(u0, z0) + ||g||2
+L2(U).
+In particular we have (∂tuN, ∂tzN) ∈ [L2(0, T; L2(U))]2 and (uN, zN) ∈ [L∞(0, T; H1(U))]2,
+both uniformly bounded independently from N.
+Proof. Derive ATǫ(uN, zN) in time and get:
+d
+dtATǫ(uN, zN) = −||πN∇ATǫ||2
+L2(U) = −||∂tuN||2
+L2(U) − ||∂tzN||2
+L2(U).
+Integrating this equality:
+ATǫ(uN, zN) +
+� T
+0 ||∂tuN(s)||2
+L2(U) + ||∂tzN(s)||2
+L2(U)ds = ATǫ(πNu0, πNz0) ≤ C.
+(2.8)
+Notice that a priori we have no control in N for the quantity
+�
+U(ηǫ + πN[z0]2)|∇πN[u0]|2dx.
+But since we truncated with |φ| ≤ 2 we have
+�
+U(ηǫ + 4)|∇πN[u0]|2dx ≤
+�
+U(ηǫ + 4)|∇u0|2dx.
+6
+
+At this point we still can’t prove the weak convergence of the non linear parts of
+the equation, in particular div((ηǫ+φ(zN)2)∇uN) and φ′(zN)φ(zN)|∇u|2. Stronger
+estimates are needed. In the next Proposition we’ll prove uniform L2(0, T; H2(U))
+boundedness of (uN, zN).
+Proposition 2.4. Uniform L2(0, T; H2(U)) estimates
+Let U ⊂ R2 be a bounded Lipshitz domain and let (u0, z0) ∈ [H1(U)]2. Consider
+solutions (uN, zN) of (2.6). It holds that
+sup
+t∈[0,T]
+[||uN(t)||2
+H1(U) + ||zN(t)||2
+H1(U)] +
+� T
+0 ||∆uN||2
+L2(U) + ||∆zN||2
+L2(U) ≤ C
+for some C > 0 independent of N.
+Proof. First of all, we want to prove an estimate like
+sup
+0≤t≤T[ATǫ(uN(t), zN(t))] +
+� T
+0 ||∆uN(s)||2
+L2(U) + ||∆zN(s)||2
+L2(U)ds ≲
+≲ C +
+� T
+0 ||∇uN(s)||4
+L4(U) + ||∇zN(s)||4
+L4(U)ds.
+(2.9)
+The idea is to expand the energy equality (2.8) obtained in Proposition 2.3
+with ||πN∇ATǫ(uN, zN)||2
+L2(U). For the sake of readability we omit writing time
+dependence, abbreviate φ(z) as φ and omit the subscripts too:
+|∇ATǫ(u, z)|2 = ∂uATǫ(u, z)2 + ∂zATǫ(u, z)2 =
+= [(η + φ2)∆u + 2φφ′⟨∇z, ∇u⟩ − (u − g)]2 +
+�
+2ǫ∆z − φφ′|∇u|2 + 1 − z
+2ǫ
+�2
+=
+= (η + φ2)2(∆u)2
+�
+��
+�
+1
++ 4φ2(φ′)2⟨∇z, ∇u⟩2
+�
+��
+�
+2
++(u − g)2 + 4φφ′(η + φ2)⟨∇z, ∇u⟩∆u
+�
+��
+�
+3
+−
+−4φφ′(u − g)⟨∇z, ∇u⟩
+�
+��
+�
+4
+−2(u − g)(η + φ2)∆u
+�
+��
+�
+5
++ 4ǫ2(∆z)2
+�
+��
+�
+6
++(φ′)2φ2|∇u|4 + (1 − z)2
+4ǫ2
+−
+−4ǫφφ′|∇u|2∆z
+�
+��
+�
+7
+−φφ′(1 − z)|∇u|2
+ǫ
+�
+��
+�
+8
++ 2(1 − z)∆z
+�
+��
+�
+9
+,
+where we highlighted all the terms we will manipulate. The strategy is very
+simple: use estimates like
+ab ≥ − 1
+2δa2 − δ
+2b2
+(2.10)
+to get
+7
+
+C ≥ sup
+0≤t≤T[ATǫ(u(t), z(t))] +
+� T
+0 ||πN∇ATǫ(uN(s), zN(s))||2
+L2(U)ds ≳
+sup
+0≤t≤T[ATǫ(u(t), z(t))] +
+� T
+0 ||∆uN(s)||2
+L2(U) + ||∆zN(s)||2
+L2(U)ds−
+−
+� T
+0 ||∇uN(s)||4
+L4(U) + ||∇zN(s)||4
+L4(U)ds − 1,
+and from this recover the desired inequality (2.9). The idea is to use (2.10)
+with different suitable δ on all highlighted terms except
+1
+and
+6
+which will
+absorb squared laplacians. You can easily see that each term can be estimated
+with a sum like in (2.10) of two of the following quantities:
+• A term −(∆u)2 and/or −(∆z)2;
+• A term −|∇u|4 and/or −|∇z|4;
+• A term which by Proposition 2.3 we know to be uniformly bounded.
+The only tedious part (which we skip) is to choose δ wisely each time so that
+in the end you remain with
+c1(∆u)2 + c2(∆z)2 − C1|∇u|4 − C2|∇z|4 + h
+with c1, c2 > 0 and h a sum of functions in L∞(0, T; L2(U)). In fact we can
+reduce to estimating the L2(0, T; H2(U)) norm of uN. Testing the equation in zN
+with −∆zN in (2.6) we get
+1
+2
+d
+dt||∇zN||2
+L2(U) = −2ǫ||∆zN||2
+L2(U) +
+�
+U φ′(zN)φ(zN)|∇uN|2∆zNdx −
+�
+U
+1 − zN
+2ǫ
+∆zNdx
+and from this, using again ab ≤ (δ/2)a2 + (1/2δ)b2, it’s clear we can estimate
+the L2(0, T; H2(U)) norm of zN with the L4(0, T; L4(U)) norm of ∇uN. Moreover
+by Gagliardo-Niremberg inequality:
+||∇zN||4
+L4(U) ≤ C(1 + ||∇zN||2
+L2(U)||∇2zN||2
+L2(U)) ≤ C(1 + ||∇2zN||2
+L2(U))
+thanks to the L∞(0, T; H1(U)) estimates on zN. Then:
+� T
+0 ||uN(t)||2
+H2(U)dt ≲ 1 + sup
+t∈[0,T]
+[ATǫ(uN(t), zN(t))]+
++
+� T
+0 ||∆uN(t)||2
+L2(U) + ||∆zN(t)||2
+L2(U)dt ≲ 1 +
+� T
+0 ||∇uN(t)||4
+L4(U) + ||∇zN(t)||4
+L4(U)dt ≲
+≲ 1 +
+� T
+0 ||∇uN(t)||4
+L4(U) + ||∇2zN(t)||2
+L2(U)dt ≲ 1 +
+� T
+0 ||∇uN(t)||4
+L4(U)dt.
+(2.11)
+8
+
+The goal will be to obtain an estimate like
+� T
+0 ||∇uN(t)||4
+L4(U) ≲
+�� T
+0 ||uN(t)||2
+H2(U)
+�q/2
+(2.12)
+for some q < 2 so that we can get uniform bounds in (2.11) and conclude.
+Notice that in (2.12) the estimate is non homogeneous, i.e. we are estimating
+a fourth power with something of homogeneity q < 2. The reason why this is
+possible is that the constants we are omitting in (2.12) actually depend on uN in
+a way such that the homogeneity is preserved, as we will see later.
+Considering the time fixed (we will thus omit writing the dependence on t
+for the moment) we focus on the first equation of (2.6) and we consider uN the
+solution of:
+
+
+
+−div((ηǫ + φ(zN)2)∇uN) = f
+∂nuN = 0
+where f = −∂tuN − (uN − gN).
+Now we use a procedure used in [4] and suggested as a possible alternative
+proof in a footnote in [9], that is using Meyers theorem (see [15]) to get H2 esti-
+mates. Meyers theorem was originally proved for homogeneous Dirichlet boundary
+conditions on ∂U, but in [11] it has been generalised (among others) to the case
+of homogeneous Neumann boundary conditions. The proof in [11] for the Neu-
+mann case consists in a series of strategies (i.e. partition of unity, extension of the
+functions, etc.) in order to go back to the case of the original Meyers theorem. In
+particular we want to use Theorem 2 in [11], so we consider
+G : L2
+m(U) �→ H1
+m(U)
+such that G(f) = ϕ with
+
+
+
+−∆ϕ = f in U
+∂nϕ = 0 in ∂U
+(2.13)
+and
+f ∈ L2
+m(U) =
+�
+g ∈ L2(U)
+����
+�
+U g = 0
+�
+;
+ϕ ∈ H1
+m(U) = H1(U) ∩ L2
+m(U).
+Notice that problem (2.13) admits a unique solution in H1
+m(U) if and only if
+�
+U f = 0 (see [8]), which is our case. In particular it holds
+⟨∇G(f), ∇φ⟩L2 = ⟨f, φ⟩L2 for all φ ∈ H1(U).
+(2.14)
+By Theorem 2 in [11] (up to multiplicative constants we are neglecting):
+||∇uN||Lp(U) ≤ ||∇G(f)||Lp(U) for some p ∈ (2, +∞).
+(2.15)
+9
+
+Remark 1. Actually, the precise statement of Theorem 2 in [11] would give the
+estimate:
+||uN||W 1,p(U) ≲ ||f||W 1,q(U)′,
+with 1
+p + 1
+q = 1, 2 < p < +∞ and W 1,q(U)′ denoting the dual. But then we
+readily have:
+||f||W 1,q(U)′ =
+sup
+||φ||W 1,q(U)=1
+⟨f, φ⟩L2 =
+sup
+||φ||W 1,q(U)=1
+⟨∇G(f), ∇φ⟩L2 ≤
+≤
+sup
+||φ||W 1,q(U)=1
+||∇G(f)||Lp(U)||∇φ||Lq(U) ≤ ||∇G(f)||Lp(U)
+and so we have the estimate (2.15). The reason why we consider ∇G(f) instead
+of dealing with f is because we’ll make use of Gagliardo-Niremberg inequality and
+estimates from elliptic regularity theory on ∇G(f).
+Using Gagliardo-Niremberg inequality we get
+||∇uN||Lp(U) ≤ C(1 + ||∇G(f)||2/p
+L2(U)||∇2G(f)||
+p−2
+p
+L2(U)).
+(2.16)
+Now notice that up to modification by an additive constant we can consider
+without loss of generality −∇G(f) = (ηǫ + φ(zN)2)∇uN. Indeed:
+−∆G(f) = div(−∇G(f)) = f = div((ηǫ + φ(zN)2)∇uN),
+so −∇G(f) = (ηǫ + φ(zN)2)∇uN ∈ L∞(0, T; L2(U)) thanks to Proposition 2.3.
+Integrate in time the inequality (2.16) raised to the power 2p/(p − 2) to get:
+� T
+0 ||∇uN(t)||
+2p
+p−2
+Lp(U)dt ≲ 1 +
+� T
+0 ||∇2G(f)(t)||2
+L2(U)dt ≲ 1 +
+� T
+0 ||f(t)||2
+L2(U)dt ≤ C,
+(2.17)
+where we used standard elliptic regularity theory to pass from the L2 norm of
+∇2G(f) to the L2 norm of f. We can assume without loss of generality that 2 < p <
+4, otherwise if we had p > 4 we could conclude directly by the above estimates,
+indeed 2p/(p − 2) < 4 and by Hölder, (2.17) and the uniform L∞(0, T; L2(U))
+bounds on ∇uN:
+� T
+0 ||∇uN(t)||4
+L4(U)dt =
+� T
+0
+��
+U |∇uN(t)|
+2p
+p−2|∇uN(t)|
+2p−8
+p−2 dx
+�
+dt ≤
+≤
+� T
+0 ||∇uN(t)||2p/(p−2)
+Lp(U)
+��
+U |∇uN(t)|2dx
+� p−4
+p−2 dt ≤
+≤ C
+� T
+0 ||∇uN(t)||2p/(p−2)
+Lp(U)
+dt ≤ C.
+(2.18)
+Of course we also assume p ̸= 4, or the thesis would follow trivially. So assume
+2 < p < 4. Applying again Gagliardo-Nirenberg, Hölder and (2.17):
+10
+
+� T
+0 ||∇uN(t)||4
+L4(U)dt ≤
+� T
+0 ||∇uN(t)||p
+Lp(U)||uN(t)||4−p
+H2(U)dt ≤
+≤
+�� T
+0 ||∇uN||
+2p
+p−2
+Lp(U)
+� p−2
+p
+�� T
+0 ||uN||2
+H2(U)
+� 4−p
+2
+≲
+�� T
+0 ||uN||2
+H2(U)
+� 4−p
+2
+,
+(2.19)
+and we can conclude since (4 − p)/2 < 2.
+Remark 2. Notice the assumption n = 2 is needed in order to have the necessary
+Gagliardo-Niremberg estimates in the previous Proposition. Also, notice how the
+homogeneity of degree 4 is preserved both in (2.18) and in (2.19), where to conclude
+we uniformly bound some quantities depending on uN, namely (
+�
+U |∇uN(t)|2dx)
+p−4
+p−2
+and
+�� T
+0 ||∇uN||
+2p
+p−2
+Lp(U)
+� p−2
+p
+.
+The estimates obtained in the previous Proposition actually yield uniform esti-
+mates of uN and zN in L2(0, T; H2(U)) thanks to the classical fact that ||u||L2(U) +
+||∆u||L2(U) is an equivalent norm for H2(U). To recapitulate, we have (up to a
+subsequence we will not rename):
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(uN, zN) weakly- ∗ converging in L∞(0, T; H1(U))
+(uN, zN) weakly converging in L2(0, T; H2(U))
+(∂tuN, ∂tzN) weakly converging in L2(0, T; L2(U))
+(uN, zN) converging in C(0, T; L2(U))
+(uN, zN) converging in the strong topology in L2(0, T; H1(U))
+(2.20)
+where the compact embeddings in C(0, T; L2(U)) and L2(0, T; H1(U)) are ob-
+tained by applying the Aubin-Lions lemma (see [3], [13], [18]). We are now ready
+to prove the main result.
+Theorem 2.1. Existence and uniqueness of strong solutions
+Let U ⊂ R2 be a bounded Lipshitz domain and let (u0, z0) ∈ [H1(U)]2 with
+0 ≤ z0 ≤ 1. Then there exists a unique strong solution (u, z) of (2.1).
+Proof. Let (u, z) be the weak limit of (uN, zN) in L2(0, T; H2(U)), let’s see how
+the pair is a solution of (2.2). This will be sufficient to prove the thesis thanks to
+Proposition 2.1. Let ψ ∈ VM = Span{e1, . . . , eM} be a test function for (2.6) with
+N > M, so it holds:
+�
+U uN(t)ψ
+�
+��
+�
+1
+=
+�
+U πN[u0]ψ −
+� t
+0
+�
+U(ηǫ + φ(zN)2)∇uN∇ψ
+�
+��
+�
+2
+−
+� t
+0
+�
+U(uN − gN)ψ
+�
+U zN(t)ψ
+�
+��
+�
+3
+=
+�
+U πN[z0]ψ − 2ǫ
+� t
+0
+�
+U ∇zN∇ψ −
+� t
+0
+�
+U φ′(zN)φ(zN)|∇uN|2ψ
+�
+��
+�
+4
++
+� t
+0
+�
+U
+1 − zN
+2ǫ
+ψ,
+(2.21)
+11
+
+and we want to show we can pass to the limit in every highlighted term, since
+for the others it’s trivial by weak convergence.
+• As for 1 and 3 , we can pass to the limit thanks to the compactness in
+C(0, T; L2(U)).
+• For 2 , we have (by dominated convergence) strong convergence in L2(0, T; L2(U))
+of (ηǫ + φ(zN)2), and weak convergence of ∇uN. So their product weakly
+converges and we can pass to the limit.
+• We already saw in Proposition 2.4 how ∇uN is uniformly bounded in L4(0, T; L4(U)),
+which is the same as saying |∇uN|2 is uniformly bounded in L2(0, T; L2(U)).
+Then, up to taking another subsequence, |∇uN|2 ⇀ |∇u|2 in L2(0, T; L2(U)).
+Since φ′(zN)φ(zN) → φ′(z)φ(z) in the strong L2(0, T; L2(U)) topology by
+dominated convergence, their product weakly converges and we can pass to
+the limit in 4 .
+It only remains to prove that (u, z) satisfy the homogeneous Neumann bound-
+ary conditions of (2.1).
+To do that we first have to make sense of ∂nu for any u ∈ H2(U). We define
+∂n : H2(U) → H1/2(∂U)
+as
+∂nu(ψ) =
+�
+U ∆uΨdx +
+�
+U ∇u∇Ψdx,
+where ψ ∈ H1/2(∂U) and Ψ ∈ H1(U) is an extension of ψ to the whole U. In
+particular Ψ will be chosen according to the trace extension operator, i.e. Ψ = Eψ,
+where E is defined as:
+Theorem 2.2. Trace extension operator, see [17]
+Given a bounded, Lipshitz domain Ω ⊂ Rn and 1 < p < +∞, there exists a
+linear and bounded trace extension operator
+E : W 1− 1
+p ,p(∂Ω) → W 1,p(Ω)
+such that Tr(Eu) = u for every u ∈ W 1− 1
+p(∂Ω).
+The operator ∂n just defined is continuous, indeed using ||Eψ||H1(U) ≤ C||ψ||H1/2(U):
+||∂nu||H1/2(∂U) =
+sup
+ψ∈H1/2(∂U)
+�
+∂nu(ψ)
+||ψ||H1/2(∂U)
+�
+≤
+≤ C
+sup
+ψ∈H1/2(∂U)
+�
+1
+||Eψ||H1(U)
+�
+U ∆uEψdx +
+�
+U ∇u∇Eψdx
+�
+≤
+≤ C(||∆u||L2(U) + ||∇u||L2(U)).
+Moreover it is known that W 1−1/p,p(∂U) compactly embeds into Lp(∂U) (see
+[7]), so
+12
+
+∂n : H2(U) �→ L2(∂U) is weak-strong continuous,
+meaning it sends weakly converging sequences in strong converging ones. We
+have to prove that ∂nu(t) = ∂nz(t) = 0 for almost every t. Since the argument is
+the same we’ll only show that the boundary conditions hold for u, moreover for
+simplicity we assume that u(t) ∈ H2(U) for every t. By weak-strong continuity of
+∂n we have that for every t ∈ [0, T], modulo a subsequence (which depends on t):
+uN(t)
+H2(U)
+−−−⇀ u(t) =⇒ ∂nuN(t)
+L2(∂U)
+−−−−→ ∂nu(t) as N → +∞,
+but since ∂nuN(t) ≡ 0 for every N we have the thesis.
+References
+[1] Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Functions of bounded vari-
+ation and free discontinuity problems. Courier Corporation, 2000.
+[2] Luigi Ambrosio and Vincenzo Maria Tortorelli. Approximation of functional
+depending on jumps by elliptic functional via gamma-convergence. Commu-
+nications on Pure and Applied Mathematics, 43(8):999–1036, 1990.
+[3] Jean-Pierre Aubin.
+Analyse mathematique-un theoreme de compacite.
+Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences,
+256(24):5042, 1963.
+[4] John W Barrett, Xiaobing Feng, and Andreas Prohl. Convergence of a fully
+discrete finite element method for a degenerate parabolic system modelling
+nematic liquid crystals with variable degree of orientation. ESAIM: Mathe-
+matical Modelling and Numerical Analysis, 40(1):175–199, 2006.
+[5] Guy David. Singular sets of minimizers for the Mumford-Shah functional,
+volume 233. Springer Science & Business Media, 2006.
+[6] E De Giorgi, M Carriero, and A Leaci. Existence theorem for a minimum
+problem with free discontinuity set. Ennio De Giorgi, page 654, 1989.
+[7] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci. Hitchhiker’s
+guide to the fractional sobolev spaces. Bulletin des sciences mathématiques,
+136(5):521–573, 2012.
+[8] Lawrence C Evans. Partial differential equations, volume 19. American Math-
+ematical Soc., 2010.
+[9] Xiaobing Feng and Andreas Prohl. Analysis of gradient flow of a regular-
+ized mumford-shah functional for image segmentation and image inpaint-
+ing. ESAIM: Mathematical Modelling and Numerical Analysis, 38(2):291–320,
+2004.
+13
+
+[10] Gilles A Francfort and J-J Marigo.
+Revisiting brittle fracture as an en-
+ergy minimization problem. Journal of the Mechanics and Physics of Solids,
+46(8):1319–1342, 1998.
+[11] Thierry Gallouet and Alexis Monier. On the regularity of solutions to elliptic
+equations. Rend. Mat. Appl.(7), 19(4):471–488, 1999.
+[12] David Gilbarg, Neil S Trudinger, David Gilbarg, and NS Trudinger. Elliptic
+partial differential equations of second order, volume 224. Springer, 1977.
+[13] Jacques-Louis Lions.
+Quelques méthodes de résolution de problemes aux
+limites non linéaires. 1969.
+[14] Jacques Louis Lions and Enrico Magenes. Non-homogeneous boundary value
+problems and applications: Vol. 1, volume 181. Springer Science & Business
+Media, 2012.
+[15] Norman G Meyers. An Lp-estimate for the gradient of solutions of second
+order elliptic divergence equations. Annali della Scuola Normale Superiore di
+Pisa-Classe di Scienze, 17(3):189–206, 1963.
+[16] David Bryant Mumford and Jayant Shah. Optimal approximations by piece-
+wise smooth functions and associated variational problems. Communications
+on pure and applied mathematics, 1989.
+[17] Jindrich Necas. Les méthodes directes en théorie des équations elliptiques.
+1967.
+[18] Jacques Simon. Compact sets in the space Lp(0, T; B). Annali di Matematica
+pura ed applicata, 146(1):65–96, 1986.
+[19] Michael Struwe. Geometric evolution problems. Nonlinear partial differential
+equations in differential geometry, 2:257–339, 1996.
+14
+
diff --git a/EdFRT4oBgHgl3EQfyTjG/content/tmp_files/load_file.txt b/EdFRT4oBgHgl3EQfyTjG/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,352 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf,len=351
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='13645v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='AP]  31 Jan 2023  Existence, uniqueness and  L2  t(H2  x) ∩ L∞  t (H1  x) ∩ H1  t (L2  x) regularity of the  gradient flow of the Ambrosio-Tortorelli functional  Tommaso Cortopassi ∗  Abstract  We consider the gradient flow of the Ambrosio-Tortorelli functional at  fixed ǫ > 0, proving existence and uniqueness of a solution in dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  The strategy of the proof essentially follows the one in the first part of [9],  but as suggested in a footnote by the authors of [9] it employs a different  and simpler technique, which is used for a different equation in [4] and in the  end it allows to prove better estimates than the ones obtained in the original  article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In particular we prove that if U ⊂ R2 is a bounded Lipshitz domain,  the initial data (u0, z0) ∈ [H1(U)]2 and 0 ≤ z0 ≤ 1, then for every T > 0  there exists a unique gradient flow (u(t), z(t)) of the Ambrosio-Tortorelli  functional such that  (u, z) ∈ [L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U)) ∩ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  The basic difference from [9], as already said, is a better regularity result  and a simpler proof: while in [9] they used a localization argument based  on an idea by Struwe (see [19]), here crucial estimates on the fourth powers  of the L4  t (L4  x) norms of the gradients will be obtained employing a suitable  version of the Meyers theorem due to Gallouet and Monier (see [15], [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  1  Introduction  The Mumford-Shah functional, introduced in [16], is defined as  E(u, Γ) = 1  2  �  U\\Γ |∇u|2 + (u − g)2dx + H1(Γ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1)  where u ∈ H1(U \\ Γ), U ⊂ R2 open and bounded, Γ ⊂ U is closed and H1  is the one dimensional Hausdorff measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' This functional has been extensively  studied for its applications in image segmentation and fracture mechanics (a de-  tailed survey can be found in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The definition of the functional depends on the  model it is used for, in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1) we gave the original definition of [16] which is suited  ∗Scuola Normale Superiore, 56126 Pisa, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' E-mail: tommaso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='cortopassi@sns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='it  1  for image segmentation models, the interested reader can see the seminal paper  [10] about the fracture mechanic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The idea is rather simple: given a gray  image g (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' a scalar function), we want to find (˜u, ˜Γ) such that  (˜u, ˜Γ) = arg min  u∈H1(U\\Γ)  Γ⊂U closed  E(u, Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2)  In this case, the function u will approximate in a “smooth way” the image g  while Γ will be the contour set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' From a theoretical point of view, in [6] the authors  proved the existence of a solution for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2) by restricting to functions u ∈ SBV (U)  (see [1]) and Γ = Su, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' the set of discontinuity jump points of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  From a numerical point perspective, since the functional involves the measure  of the singular jump set of a (unknown) function u, its direct numerical implemen-  tation is often not possible or not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' A standard approach is to minimize a  more regular functional proposed by Ambrosio and Tortorelli in [2] defined as  1  2  �  U[(ηǫ + z2)|∇u|2 + (u − g)2] +  �  U  �(1 − z)2  4ǫ  + ǫ|∇z|2  �  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3)  which approximates the Mumford-Shah functional in Γ-convergence as ǫ → 0  in an even more general setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' not being restricted to the 2 dimensional case,  but considering U ⊂ Rn open and the (n − 1)-dimensional Hausdorff measure for  Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The Ambrosio-Tortorelli functional is the way with which one usually finds  (approximate) minima points for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1), in particular one can use a gradient flow  approach for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' As in [9], we’ll consider the gradient flow of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3), which is  given by  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∂tu = div((ηǫ + z2)∇u) − (u − g) in (0, T) × U  ∂tz = 2ǫ∆z − z|∇u|2 + 1−z  2ǫ in (0, T) × U  ∂  ∂nu =  ∂  ∂nz = 0 in (0, T) × ∂U  u(0, ·) = u0 and z(0, ·) = z0 in {0} × U  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='4)  with ηǫ, ǫ > 0 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' As already mentioned, our goal is to study the existence,  uniqueness and regularity of solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='4), and the novelty lies in an approach  (already suggested in a footnote of [9] and used in [4] for a different equation)  which simplifies the proof of [9] while also gaining better regularity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In  particular in [9] they only manage to prove the L2  t(H2  x) regularity for a short time  T1, since the crucial estimate in their argument is a local energy estimate, inspired  by a technique due to Struwe [19], which only holds for sufficiently small times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  2  2  Main result  Let U ⊂ R2 be a Lipshitz bounded domain and let (u0, z0) ∈ [H1(U)]2 with  0 ≤ z0 ≤ 1, g ∈ L2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' As stated in the introduction, we want to prove existence,  uniqueness and regularity of solutions of the gradient flow of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3), given by  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∂tu = div((ηǫ + z2)∇u) − (u − g) in (0, T) × U  ∂tz = 2ǫ∆z − z|∇u|2 + 1−z  2ǫ ) in (0, T) × U  ∂  ∂nu =  ∂  ∂nz = 0 in (0, T) × ∂U  u(0, ·) = u0 and z(0, ·) = z0 in {0} × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1)  However we will mostly work with the slightly modified system  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∂tu = div((ηǫ + φ(z)2)∇u) − (u − g) in (0, T) × U  ∂tz = 2ǫ∆z − φ′(z)φ(z)|∇u|2 + 1−z  2ǫ in (0, T) × U  ∂  ∂nu =  ∂  ∂nz = 0 in (0, T) × ∂U  u(0, ·) = u0 and z(0, ·) = z0 in {0} × U  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2)  where φ is a cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The reason to introduce such a cutoff, as we will show later,  is to bound the L∞ norm of ηǫ+φ(zN)2 when considering Galerkin approximations  zN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' To be precise, a good working definition of φ might be  φ(s) =  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  −1 if s ≤ −1  s if − 1 < s < 2  2 if s ≥ 2  but you could actually do the cuts at levels −δ1 and 1 + δ2 for every δ1, δ2 > 0  and nothing would change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' A modified Ambrosio-Tortorelli functional (of which  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2) is the gradient flow of) will be denoted as ATǫ and defined as  ATǫ(u(t), z(t)) = 1  2  �  U[(ηǫ+φ(z(t))2)|∇u(t)|2+(u(t)−g)2]+  �  U  �(1 − z(t))2  4ǫ  + ǫ|∇z(t)|2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3)  First of all, let’s define the notion of strong solution, following [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Strong solution  A couple (u, z) is a strong solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1) if it satisfies the system in a dis-  tributional sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' for every ψ ∈ H1(U) it holds  �  U  u(t)ψdx =  �  U  u0ψdx −  � t  0  �  U  (ηǫ + z(s)2)⟨∇u(s), ∇ψ⟩dxds −  � t  0  �  U  (u(s) − g)ψdxds  �  U  z(t)ψdx =  �  U  z0ψdx − 2ǫ  � t  0  �  U  ⟨∇z(s), ∇ψ⟩dxds −  � t  0  �  U  z(s)ψ|∇u(s)|2 + 1 − z(s)  2ǫ  ψdxds  for almost every t ∈ (0, T), and moreover we require  (u, z) ∈ [L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U)) ∩ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  We start with a proposition which shows how a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2) is also solution  of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1) if 0 ≤ z0 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  3  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Let 0 ≤ z0 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' If a strong solution (u, z) for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2) exists, it  must be 0 ≤ z(t, x) ≤ 1 for every t ∈ (0, T) and for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' So (u, z) will also  be a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' If we test the equation for z in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2) with z itself it gives  1  2  d  dt||z(t)||2  L2(U) = −2ǫ||∇z(t)||2  L2(U)−  �  U φ′(z(t))φ(z(t))z(t)|∇u(t)|2ds+  �  U  1 − z(t)  2ǫ  z(t)ds  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='4)  where the existence of the time derivative is ensured by Lions-Magenes lemma  (see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Consider f0(z) = max{−z, 0} and f1(z) = max{z − 1, 0}, so f0 and f1  are defined as  f0(s) =  \uf8f1  \uf8f2  \uf8f3  −s if s ≤ 0  0 if s > 0  and f1(s) =  \uf8f1  \uf8f2  \uf8f3  0 if s ≤ 1  s − 1 if s > 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  It’s easy to check that:  1  2  d  dt||f0(z(t))||2  L2(U) =  �  U f0(z(t))f ′  0(z(t))∂tz(t)dx =  �  U χ{z(t)<0} z(t) ∂tz(t)dx,  and noting that χ{z(t)<0}z(t) = −f0(z(t)) is a Sobolev function (see Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6  in [12]) we are allowed to use it as a test function in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2), getting:  1  2  d  dt||f0(z(t))||2  L2(U) = ⟨∂tz(t), −f0(z(t))⟩L2(U) =  = χ{z(t)<0}  �  −2ǫ||∇z(t)||2  L2(U) −  �  U φ′(z(t))φ(z(t))z(t)|∇u(t)|2 +  �  U  1 − z(t)  2ǫ  z(t)  �  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Since f0(z(0)) = 0 because z0 ≥ 0, we have f0(z(t)) ≡ 0 for every t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In the  same way we can prove z ≤ 1 by considering f1(z(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  The strategy will be to make use of Galerkin approximates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Consider an or-  thogonal basis of H1(U) composed of eigenfunctions of −∆ on U with homoge-  neous Neumann boundary conditions normalized with respect to the L2(U) norm,  and denote it as {ei}i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' So  \uf8f1  \uf8f2  \uf8f3  −∆ei = λiei in U  ∂nei = 0 in ∂U  and ||ei||L2(U) = 1 for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  We want to find Galerkin approximates (uN, zN) such that they solve (in dis-  tributional sense) in  VN = Span({e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' , eN})  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='5)  the system  4  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∂tuN = πN[div((ηǫ + φ(zN)2)∇uN)] − (uN − gN) in (0, T) × U  ∂tzN = 2ǫ∆zN − πN[φ′(zN)φ(zN)|∇uN|2] + 1−zN  2ǫ  in (0, T) × U  ∂  ∂nuN =  ∂  ∂nzN = 0 in (0, T) × ∂U  uN(0, ·) = πN[u0] and zN(0, ·) = πN[z0] in {0} × U  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6)  with πN the orthogonal projection on VN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Notice that by the orthogonality  properties of {ei}N  i=1 this is actually a 2N system in u(1)(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' , u(N)(t), z(1)(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' , z(N)(t),  with  uN(t, x) =  N  �  i=1  u(i)(t)ei(x) and zN(t, x) =  N  �  i=1  z(i)(t)ei(x)  which by Cauchy-Lipshitz admits a unique local solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Indeed if we test  with ei it holds that:  u(i)(t) = (u0)(i) −  � t  0  ��  U(ηǫ + φ(zN(s))2)⟨∇uN(s), ∇ei⟩ + (uN(s) − gN)eidx  �  ds  z(i)(t) = (z0)(i) −  � t  0  ��  U 2ǫ⟨∇zN(s), ∇ei⟩ − φ′(zN(s))φ(zN(s))|∇uN(s)|2ei + 1 − zN(s)  2ǫ  eidx  �  ds  for every 1 ≤ i ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' However there are strong non linearities at play, so a  priori for every N we only have a local solution in [0, tN) without being able to  extend it immediately to [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In order to gain existence in [0, T] of Galerkin  approximates we’ll use the following a priori estimates, which hold in a slightly  more general situation with less regular initial data:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Existence of weak approximate solutions in [0,T]  Given (u0, z0) ∈ [L2(U)]2 and a solution (uN, zN) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6) in VN, it holds  sup  0≤t≤T[||uN(t)||2  L2(U)] +  � T  0 ||∇uN(s)||2  L2(U)ds ≤ C  and  sup  0≤t≤T[||zN(t)||2  L2(U)] +  � T  0 ||∇zN(s)||2  L2(U)ds ≤ C  with C a positive constant independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Test the equation for uN in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6) with uN itself, to get  d  dt||uN||2  L2(U) = −  �  U(ηǫ + φ(zN)2)|∇uN|2dx −  �  U uN(uN − gN)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='7)  So  d  dt||uN||2  L2(U) ≤ ||gN||L2(U)||uN||L2(U) ≤ ||g||L2(U)(1 + ||uN||2  L2(U))  5  and you get uniform boundedness of ||uN||L2(U) by Gronwall’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Going  back to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='7) you easily conclude by integrating in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The same holds if we test  the equation in zN with zN itself, obtaining  sup  0≤t≤T[||zN(t)||2  L2(U)] + 2ǫ  � T  0  ≤ T  8ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  By orthogonality of the ei s this can be rewritten as  ||uN(t)||2  L2(U) =  N  �  i=1  [u(i)(t)]2 and ||zN(t)||2  L2(U) =  N  �  i=1  [z(i)(t)]2  so we cannot have a blow-up in finite time and we have thus proved existence  up to time T for every T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Now that we have existence of solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6) in [0, T] for every N, let’s prove  stronger inequalities exploiting the variational characterization of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' A priori energy estimates  Assume (u0, z0) ∈ [H1(U)]2, then  sup  t∈[0,T]  [ATǫ(uN(t), zN(t))] +  � T  0 ||∂tuN(s)||2  L2(U) + ||∂tzN(s)||2  L2(U)ds =  = sup  t∈[0,T]  [ATǫ(uN(t), zN(t))] +  � T  0 ||πN[∇ATǫ(uN, zN)]||2  L2(U)ds ≤  ≲ ATǫ(u0, z0) + ||g||2  L2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  In particular we have (∂tuN, ∂tzN) ∈ [L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))]2 and (uN, zN) ∈ [L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U))]2,  both uniformly bounded independently from N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Derive ATǫ(uN, zN) in time and get:  d  dtATǫ(uN, zN) = −||πN∇ATǫ||2  L2(U) = −||∂tuN||2  L2(U) − ||∂tzN||2  L2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Integrating this equality:  ATǫ(uN, zN) +  � T  0 ||∂tuN(s)||2  L2(U) + ||∂tzN(s)||2  L2(U)ds = ATǫ(πNu0, πNz0) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='8)  Notice that a priori we have no control in N for the quantity  �  U(ηǫ + πN[z0]2)|∇πN[u0]|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  But since we truncated with |φ| ≤ 2 we have  �  U(ηǫ + 4)|∇πN[u0]|2dx ≤  �  U(ηǫ + 4)|∇u0|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  6  At this point we still can’t prove the weak convergence of the non linear parts of  the equation, in particular div((ηǫ+φ(zN)2)∇uN) and φ′(zN)φ(zN)|∇u|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Stronger  estimates are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In the next Proposition we’ll prove uniform L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U))  boundedness of (uN, zN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Uniform L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)) estimates  Let U ⊂ R2 be a bounded Lipshitz domain and let (u0, z0) ∈ [H1(U)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Consider  solutions (uN, zN) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' It holds that  sup  t∈[0,T]  [||uN(t)||2  H1(U) + ||zN(t)||2  H1(U)] +  � T  0 ||∆uN||2  L2(U) + ||∆zN||2  L2(U) ≤ C  for some C > 0 independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' First of all, we want to prove an estimate like  sup  0≤t≤T[ATǫ(uN(t), zN(t))] +  � T  0 ||∆uN(s)||2  L2(U) + ||∆zN(s)||2  L2(U)ds ≲  ≲ C +  � T  0 ||∇uN(s)||4  L4(U) + ||∇zN(s)||4  L4(U)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='9)  The idea is to expand the energy equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='8) obtained in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3  with ||πN∇ATǫ(uN, zN)||2  L2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' For the sake of readability we omit writing time  dependence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' abbreviate φ(z) as φ and omit the subscripts too:  |∇ATǫ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' z)|2 = ∂uATǫ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' z)2 + ∂zATǫ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' z)2 =  = [(η + φ2)∆u + 2φφ′⟨∇z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' ∇u⟩ − (u − g)]2 +  �  2ǫ∆z − φφ′|∇u|2 + 1 − z  2ǫ  �2  =  = (η + φ2)2(∆u)2  �  ��  �  1  + 4φ2(φ′)2⟨∇z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' ∇u⟩2  �  ��  �  2  +(u − g)2 + 4φφ′(η + φ2)⟨∇z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' ∇u⟩∆u  �  ��  �  3  −  −4φφ′(u − g)⟨∇z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' ∇u⟩  �  ��  �  4  −2(u − g)(η + φ2)∆u  �  ��  �  5  + 4ǫ2(∆z)2  �  ��  �  6  +(φ′)2φ2|∇u|4 + (1 − z)2  4ǫ2  −  −4ǫφφ′|∇u|2∆z  �  ��  �  7  −φφ′(1 − z)|∇u|2  ǫ  �  ��  �  8  + 2(1 − z)∆z  �  ��  �  9  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  where we highlighted all the terms we will manipulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The strategy is very  simple: use estimates like  ab ≥ − 1  2δa2 − δ  2b2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='10)  to get  7  C ≥ sup  0≤t≤T[ATǫ(u(t), z(t))] +  � T  0 ||πN∇ATǫ(uN(s), zN(s))||2  L2(U)ds ≳  sup  0≤t≤T[ATǫ(u(t), z(t))] +  � T  0 ||∆uN(s)||2  L2(U) + ||∆zN(s)||2  L2(U)ds−  −  � T  0 ||∇uN(s)||4  L4(U) + ||∇zN(s)||4  L4(U)ds − 1,  and from this recover the desired inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The idea is to use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='10)  with different suitable δ on all highlighted terms except  1  and  6  which will  absorb squared laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' You can easily see that each term can be estimated  with a sum like in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='10) of two of the following quantities:  A term −(∆u)2 and/or −(∆z)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  A term −|∇u|4 and/or −|∇z|4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  A term which by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3 we know to be uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  The only tedious part (which we skip) is to choose δ wisely each time so that  in the end you remain with  c1(∆u)2 + c2(∆z)2 − C1|∇u|4 − C2|∇z|4 + h  with c1, c2 > 0 and h a sum of functions in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In fact we can  reduce to estimating the L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)) norm of uN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Testing the equation in zN  with −∆zN in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6) we get  1  2  d  dt||∇zN||2  L2(U) = −2ǫ||∆zN||2  L2(U) +  �  U φ′(zN)φ(zN)|∇uN|2∆zNdx −  �  U  1 − zN  2ǫ  ∆zNdx  and from this, using again ab ≤ (δ/2)a2 + (1/2δ)b2, it’s clear we can estimate  the L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)) norm of zN with the L4(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L4(U)) norm of ∇uN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Moreover  by Gagliardo-Niremberg inequality:  ||∇zN||4  L4(U) ≤ C(1 + ||∇zN||2  L2(U)||∇2zN||2  L2(U)) ≤ C(1 + ||∇2zN||2  L2(U))  thanks to the L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U)) estimates on zN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Then:  � T  0 ||uN(t)||2  H2(U)dt ≲ 1 + sup  t∈[0,T]  [ATǫ(uN(t), zN(t))]+  +  � T  0 ||∆uN(t)||2  L2(U) + ||∆zN(t)||2  L2(U)dt ≲ 1 +  � T  0 ||∇uN(t)||4  L4(U) + ||∇zN(t)||4  L4(U)dt ≲  ≲ 1 +  � T  0 ||∇uN(t)||4  L4(U) + ||∇2zN(t)||2  L2(U)dt ≲ 1 +  � T  0 ||∇uN(t)||4  L4(U)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='11)  8  The goal will be to obtain an estimate like  � T  0 ||∇uN(t)||4  L4(U) ≲  �� T  0 ||uN(t)||2  H2(U)  �q/2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='12)  for some q < 2 so that we can get uniform bounds in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='11) and conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Notice that in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='12) the estimate is non homogeneous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' we are estimating  a fourth power with something of homogeneity q < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The reason why this is  possible is that the constants we are omitting in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='12) actually depend on uN in  a way such that the homogeneity is preserved, as we will see later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Considering the time fixed (we will thus omit writing the dependence on t  for the moment) we focus on the first equation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6) and we consider uN the  solution of:  \uf8f1  \uf8f2  \uf8f3  −div((ηǫ + φ(zN)2)∇uN) = f  ∂nuN = 0  where f = −∂tuN − (uN − gN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Now we use a procedure used in [4] and suggested as a possible alternative  proof in a footnote in [9], that is using Meyers theorem (see [15]) to get H2 esti-  mates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Meyers theorem was originally proved for homogeneous Dirichlet boundary  conditions on ∂U, but in [11] it has been generalised (among others) to the case  of homogeneous Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The proof in [11] for the Neu-  mann case consists in a series of strategies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' partition of unity, extension of the  functions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=') in order to go back to the case of the original Meyers theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In  particular we want to use Theorem 2 in [11], so we consider  G : L2  m(U) �→ H1  m(U)  such that G(f) = ϕ with  \uf8f1  \uf8f2  \uf8f3  −∆ϕ = f in U  ∂nϕ = 0 in ∂U  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='13)  and  f ∈ L2  m(U) =  �  g ∈ L2(U)  ����  �  U g = 0  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  ϕ ∈ H1  m(U) = H1(U) ∩ L2  m(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Notice that problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='13) admits a unique solution in H1  m(U) if and only if  �  U f = 0 (see [8]), which is our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In particular it holds  ⟨∇G(f), ∇φ⟩L2 = ⟨f, φ⟩L2 for all φ ∈ H1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='14)  By Theorem 2 in [11] (up to multiplicative constants we are neglecting):  ||∇uN||Lp(U) ≤ ||∇G(f)||Lp(U) for some p ∈ (2, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='15)  9  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Actually, the precise statement of Theorem 2 in [11] would give the  estimate:  ||uN||W 1,p(U) ≲ ||f||W 1,q(U)′,  with 1  p + 1  q = 1, 2 < p < +∞ and W 1,q(U)′ denoting the dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' But then we  readily have:  ||f||W 1,q(U)′ =  sup  ||φ||W 1,q(U)=1  ⟨f, φ⟩L2 =  sup  ||φ||W 1,q(U)=1  ⟨∇G(f), ∇φ⟩L2 ≤  ≤  sup  ||φ||W 1,q(U)=1  ||∇G(f)||Lp(U)||∇φ||Lq(U) ≤ ||∇G(f)||Lp(U)  and so we have the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The reason why we consider ∇G(f) instead  of dealing with f is because we’ll make use of Gagliardo-Niremberg inequality and  estimates from elliptic regularity theory on ∇G(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Using Gagliardo-Niremberg inequality we get  ||∇uN||Lp(U) ≤ C(1 + ||∇G(f)||2/p  L2(U)||∇2G(f)||  p−2  p  L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='16)  Now notice that up to modification by an additive constant we can consider  without loss of generality −∇G(f) = (ηǫ + φ(zN)2)∇uN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Indeed:  −∆G(f) = div(−∇G(f)) = f = div((ηǫ + φ(zN)2)∇uN),  so −∇G(f) = (ηǫ + φ(zN)2)∇uN ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)) thanks to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Integrate in time the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='16) raised to the power 2p/(p − 2) to get:  � T  0 ||∇uN(t)||  2p  p−2  Lp(U)dt ≲ 1 +  � T  0 ||∇2G(f)(t)||2  L2(U)dt ≲ 1 +  � T  0 ||f(t)||2  L2(U)dt ≤ C,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='17)  where we used standard elliptic regularity theory to pass from the L2 norm of  ∇2G(f) to the L2 norm of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' We can assume without loss of generality that 2 < p <  4, otherwise if we had p > 4 we could conclude directly by the above estimates,  indeed 2p/(p − 2) < 4 and by Hölder, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='17) and the uniform L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))  bounds on ∇uN:  � T  0 ||∇uN(t)||4  L4(U)dt =  � T  0  ��  U |∇uN(t)|  2p  p−2|∇uN(t)|  2p−8  p−2 dx  �  dt ≤  ≤  � T  0 ||∇uN(t)||2p/(p−2)  Lp(U)  ��  U |∇uN(t)|2dx  � p−4  p−2 dt ≤  ≤ C  � T  0 ||∇uN(t)||2p/(p−2)  Lp(U)  dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='18)  Of course we also assume p ̸= 4, or the thesis would follow trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' So assume  2 < p < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Applying again Gagliardo-Nirenberg, Hölder and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='17):  10  � T  0 ||∇uN(t)||4  L4(U)dt ≤  � T  0 ||∇uN(t)||p  Lp(U)||uN(t)||4−p  H2(U)dt ≤  ≤  �� T  0 ||∇uN||  2p  p−2  Lp(U)  � p−2  p  �� T  0 ||uN||2  H2(U)  � 4−p  2  ≲  �� T  0 ||uN||2  H2(U)  � 4−p  2  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='19)  and we can conclude since (4 − p)/2 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Notice the assumption n = 2 is needed in order to have the necessary  Gagliardo-Niremberg estimates in the previous Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Also, notice how the  homogeneity of degree 4 is preserved both in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='18) and in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='19), where to conclude  we uniformly bound some quantities depending on uN, namely (  �  U |∇uN(t)|2dx)  p−4  p−2  and  �� T  0 ||∇uN||  2p  p−2  Lp(U)  � p−2  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  The estimates obtained in the previous Proposition actually yield uniform esti-  mates of uN and zN in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)) thanks to the classical fact that ||u||L2(U) +  ||∆u||L2(U) is an equivalent norm for H2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' To recapitulate, we have (up to a  subsequence we will not rename):  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  (uN, zN) weakly- ∗ converging in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U))  (uN, zN) weakly converging in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U))  (∂tuN, ∂tzN) weakly converging in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))  (uN, zN) converging in C(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))  (uN, zN) converging in the strong topology in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U))  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='20)  where the compact embeddings in C(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)) and L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U)) are ob-  tained by applying the Aubin-Lions lemma (see [3], [13], [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' We are now ready  to prove the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Existence and uniqueness of strong solutions  Let U ⊂ R2 be a bounded Lipshitz domain and let (u0, z0) ∈ [H1(U)]2 with  0 ≤ z0 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Then there exists a unique strong solution (u, z) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Let (u, z) be the weak limit of (uN, zN) in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)), let’s see how  the pair is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' This will be sufficient to prove the thesis thanks to  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Let ψ ∈ VM = Span{e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' , eM} be a test function for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6) with  N > M, so it holds:  �  U uN(t)ψ  �  ��  �  1  =  �  U πN[u0]ψ −  � t  0  �  U(ηǫ + φ(zN)2)∇uN∇ψ  �  ��  �  2  −  � t  0  �  U(uN − gN)ψ  �  U zN(t)ψ  �  ��  �  3  =  �  U πN[z0]ψ − 2ǫ  � t  0  �  U ∇zN∇ψ −  � t  0  �  U φ′(zN)φ(zN)|∇uN|2ψ  �  ��  �  4  +  � t  0  �  U  1 − zN  2ǫ  ψ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='21)  11  and we want to show we can pass to the limit in every highlighted term, since  for the others it’s trivial by weak convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  As for 1 and 3 , we can pass to the limit thanks to the compactness in  C(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  For 2 , we have (by dominated convergence) strong convergence in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))  of (ηǫ + φ(zN)2), and weak convergence of ∇uN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' So their product weakly  converges and we can pass to the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  We already saw in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='4 how ∇uN is uniformly bounded in L4(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L4(U)),  which is the same as saying |∇uN|2 is uniformly bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Then, up to taking another subsequence, |∇uN|2 ⇀ |∇u|2 in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Since φ′(zN)φ(zN) → φ′(z)φ(z) in the strong L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)) topology by  dominated convergence, their product weakly converges and we can pass to  the limit in 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  It only remains to prove that (u, z) satisfy the homogeneous Neumann bound-  ary conditions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  To do that we first have to make sense of ∂nu for any u ∈ H2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' We define  ∂n : H2(U) → H1/2(∂U)  as  ∂nu(ψ) =  �  U ∆uΨdx +  �  U ∇u∇Ψdx,  where ψ ∈ H1/2(∂U) and Ψ ∈ H1(U) is an extension of ψ to the whole U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In  particular Ψ will be chosen according to the trace extension operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Ψ = Eψ,  where E is defined as:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Trace extension operator, see [17]  Given a bounded, Lipshitz domain Ω ⊂ Rn and 1 < p < +∞, there exists a  linear and bounded trace extension operator  E : W 1− 1  p ,p(∂Ω) → W 1,p(Ω)  such that Tr(Eu) = u for every u ∈ W 1− 1  p(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  The operator ∂n just defined is continuous, indeed using ||Eψ||H1(U) ≤ C||ψ||H1/2(U):  ||∂nu||H1/2(∂U) =  sup  ψ∈H1/2(∂U)  �  ∂nu(ψ)  ||ψ||H1/2(∂U)  �  ≤  ≤ C  sup  ψ∈H1/2(∂U)  �  1  ||Eψ||H1(U)  �  U ∆uEψdx +  �  U ∇u∇Eψdx  �  ≤  ≤ C(||∆u||L2(U) + ||∇u||L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Moreover it is known that W 1−1/p,p(∂U) compactly embeds into Lp(∂U) (see  [7]), so  12  ∂n : H2(U) �→ L2(∂U) is weak-strong continuous,  meaning it sends weakly converging sequences in strong converging ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' We  have to prove that ∂nu(t) = ∂nz(t) = 0 for almost every t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Since the argument is  the same we’ll only show that the boundary conditions hold for u, moreover for  simplicity we assume that u(t) ∈ H2(U) for every t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' By weak-strong continuity of  ∂n we have that for every t ∈ [0, T], modulo a subsequence (which depends on t):  uN(t)  H2(U)  −−−⇀ u(t) =⇒ ∂nuN(t)  L2(∂U)  −−−−→ ∂nu(t) as N → +∞,  but since ∂nuN(t) ≡ 0 for every N we have the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  References  [1] Luigi Ambrosio, Nicola Fusco, and Diego Pallara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Functions of bounded vari-  ation and free discontinuity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Courier Corporation, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [2] Luigi Ambrosio and Vincenzo Maria Tortorelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Approximation of functional  depending on jumps by elliptic functional via gamma-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Commu-  nications on Pure and Applied Mathematics, 43(8):999–1036, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [3] Jean-Pierre Aubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Analyse mathematique-un theoreme de compacite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences,  256(24):5042, 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [4] John W Barrett, Xiaobing Feng, and Andreas Prohl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Convergence of a fully  discrete finite element method for a degenerate parabolic system modelling  nematic liquid crystals with variable degree of orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' ESAIM: Mathe-  matical Modelling and Numerical Analysis, 40(1):175–199, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [5] Guy David.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Singular sets of minimizers for the Mumford-Shah functional,  volume 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Springer Science & Business Media, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [6] E De Giorgi, M Carriero, and A Leaci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Existence theorem for a minimum  problem with free discontinuity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Ennio De Giorgi, page 654, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [7] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Hitchhiker’s  guide to the fractional sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Bulletin des sciences mathématiques,  136(5):521–573, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [8] Lawrence C Evans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Partial differential equations, volume 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' American Math-  ematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=', 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [9] Xiaobing Feng and Andreas Prohl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Analysis of gradient flow of a regular-  ized mumford-shah functional for image segmentation and image inpaint-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' ESAIM: Mathematical Modelling and Numerical Analysis, 38(2):291–320,  2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  13  [10] Gilles A Francfort and J-J Marigo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Revisiting brittle fracture as an en-  ergy minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Journal of the Mechanics and Physics of Solids,  46(8):1319–1342, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [11] Thierry Gallouet and Alexis Monier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' On the regularity of solutions to elliptic  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' (7), 19(4):471–488, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [12] David Gilbarg, Neil S Trudinger, David Gilbarg, and NS Trudinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Elliptic  partial differential equations of second order, volume 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Springer, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [13] Jacques-Louis Lions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Quelques méthodes de résolution de problemes aux  limites non linéaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [14] Jacques Louis Lions and Enrico Magenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Non-homogeneous boundary value  problems and applications: Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' 1, volume 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Springer Science & Business  Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [15] Norman G Meyers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' An Lp-estimate for the gradient of solutions of second  order elliptic divergence equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Annali della Scuola Normale Superiore di  Pisa-Classe di Scienze, 17(3):189–206, 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [16] David Bryant Mumford and Jayant Shah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Optimal approximations by piece-  wise smooth functions and associated variational problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Communications  on pure and applied mathematics, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [17] Jindrich Necas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Les méthodes directes en théorie des équations elliptiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [18] Jacques Simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Compact sets in the space Lp(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Annali di Matematica  pura ed applicata, 146(1):65–96, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [19] Michael Struwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Geometric evolution problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Nonlinear partial differential  equations in differential geometry, 2:257–339, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
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+Adversarial training with informed data selection
+Marcele O. K. Mendonc¸a∗, Javier Maroto⋆, Pascal Frossard⋆ and Paulo S. R. Diniz∗
+∗ SMT - Signals, Multimedia, and Telecommunications Lab.
+Universidade Federal do Rio de Janeiro, DEL/Poli & PEE/COPPE/UFRJ
+P.O. Box 68504, Rio de Janeiro, RJ, 21941-972, Brazil,
+⋆ ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Switzerland.
+emails: {marcele.kuhfuss,diniz}@smt.ufrj.br, {javier.marotomorales,pascal.frossard}@epfl.ch
+Abstract—With the increasing amount of available data and
+advances in computing capabilities, deep neural networks (DNNs)
+have been successfully employed to solve challenging tasks in var-
+ious areas, including healthcare, climate, and finance. Neverthe-
+less, state-of-the-art DNNs are susceptible to quasi-imperceptible
+perturbed versions of the original images – adversarial examples.
+These perturbations of the network input can lead to disastrous
+implications in critical areas where wrong decisions can directly
+affect human lives. Adversarial training is the most efficient
+solution to defend the network against these malicious attacks.
+However, adversarial trained networks generally come with lower
+clean accuracy and higher computational complexity. This work
+proposes a data selection (DS) strategy to be applied in the
+mini-batch training. Based on the cross-entropy loss, the most
+relevant samples in the batch are selected to update the model
+parameters in the backpropagation. The simulation results show
+that a good compromise can be obtained regarding robustness
+and standard accuracy, whereas the computational complexity of
+the backpropagation pass is reduced.
+Index Terms—data-selection, sampling strategy, adversarial
+training, robustness-accuracy tradeoff
+I. INTRODUCTION
+Over the past decade, the amount of available digital data
+has exponentially increased. Thanks to the advances in com-
+puting capabilities, deep neural networks (DNNs) have been
+successfully employed to solve challenging image and natural
+language processing tasks. However, state-of-the-art DNNs are
+known to be highly vulnerable to adversarial examples [1],
+[2]. These small but malicious perturbations of the network
+input can manipulate the trained model to produce incorrect
+predictions with high confidence, and some perturbations can
+even fool different network models [3]. Since adversarial
+attacks might lead to disastrous implications in critical areas
+like healthcare [4], climate [5] and finance [6], defending
+against them is critical.
+So far, adversarial training is the most effective approach to
+mitigate the effect of strong attacks like the Projected Gradient
+Descent (PGD) attack [7], DeepFool [8], and AutoAttack [9].
+Training the DNN with perturbed versions of the original
+samples makes it possible to improve the accuracy on unseen
+adversarial examples, also known as robustness accuracy [10].
+However, generating adversarial examples during training can
+be highly computationally intense since each sample is usually
+built with several steps in the direction of the gradient as
+This study was financed in part by the Coordenac¸˜ao de Aperfeic¸oamento
+de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001. This
+work was also supported by the Swiss Government Excellence Scholarships
+for Foreign Students.
+the model is trained. Moreover, adversarial training generally
+decreases the standard accuracy, that is, the accuracy on clean
+samples [11]. This robustness-accuracy tradeoff is reported
+to be highly data-dependent, especially regarding the data
+distribution [12] and its quality [13]. Furthermore, we only
+have access to a training dataset which is not necessarily
+representative for the problem we aim to learn. In this case,
+we could avoid using the entire training data. Since the
+dataset is reduced, we can save several computations during
+backpropagation and speed-up training. This hypothesis was
+already investigated for standard training in [14], [15]. In this
+work, we extend the work in [14], [15] and apply it to the
+adversarial training case. From each mini-batch composed
+of both clean and adversarial samples, the proposed data
+selection algorithm selects the most relevant samples based
+on the cross-entropy loss. Since only the selected samples are
+used to update the model parameters in the backpropagation,
+the training time is reduced. The selection also balances the
+necessary amount of clean and adversarial samples required
+to yield satisfactory robustness and standard accuracy.
+The paper is organized as follows. Section II presents a
+brief overview of the adversarial training method and some
+notations. In section III, we propose a data selection technique
+for adversarial training. The proposed approach is tested via
+simulation results in section IV. Finally, section V includes
+some conclusion remarks.
+II. ADVERSARIAL TRAINING
+Adversarial training continually creates and incorporates
+adversarial examples into the training process of a deep neural
+network classifier
+fθ(x) : RN → {1 · · · C},
+(1)
+with θ weights, which maps an input image x to a label y
+from a dataset
+D = {(x(1), y(1)), (x(2), y(2)), · · · , (x(M), y(M))},
+(2)
+with C possible classes. Adversarial training attempts to solve
+the min-max optimization problem
+minθ
+1
+|D|
+�
+x,y∈D
+maxη L(fθ(x + η), y)
+s.t ||η||p ≤ ϵ,
+(3)
+where L(fθ(x + η), y) is the loss function on the adversarial
+sample and η is a small perturbation constrained by ϵ.
+arXiv:2301.04472v1  [cs.LG]  7 Jan 2023
+
+Creating adversarial samples involves solving the inner
+maximization problem in equation (3), in which the loss
+function L is maximized in an effort to change the prediction,
+that is, fθ(x+η) ̸= fθ(x). The optimization constraints ensure
+that the distance between the adversarial and original example
+should be less than ϵ under a particular norm, ||η||p ≤ ϵ.
+The norms aim to quantify how imperceptible to humans an
+adversarial example is. Some examples of norms are the l0
+norm, l2 norm, and l∞. We then briefly review the most
+popular methods to create adversarial examples.
+Introduced by [2], the Fast Gradient Sign Method (FGSM)
+attack generates adversarial examples by modifying the input
+towards the direction where the loss L increases
+x′ = x + ϵsign(∇xL(θ, x, y)),
+(4)
+with sign(·) the sign function, and ∇xL(θ, x, y) the loss
+gradient with respect to x. One of the strongest l∞-bounded at-
+tacks, the PGD attack [7] tries to solve the inner maximization
+problem in equation (3) following an iterative procedure. At
+each step i, the adversarial example is updated as
+x′
+i = clipx+ϵ(xi−1 + αsign(∇xL(θ, x, y))),
+(5)
+in which function clipx+ϵ(·) clips the input at the positions
+around the predefined perturbation range. In the context of l2-
+bounded attacks, Deepfool [8] is an iterative attack optimized
+for the l2-norm based on a linear approximation of the
+classifier. Using geometry concepts, DeepFool searches within
+the region of the space that describes the output of the classifier
+(polyhedron) for the minimal perturbation that can change the
+classifiers decision. Among black-box attacks, one pixel attack
+[16] is a l0-bounded attack that employs differential evolution
+to create adversarial examples without knowing the network
+gradients and its parameters. Finally, the AutoAttack [9]
+method consists of an ensemble of four attacks: two versions
+of the PGD attack, the targeted version of the Fast Adaptive
+Boundary (FAB) attack [17] and the black-box Square Attack
+[18]. Currently, AutoAttack and PGD attack are the most
+popular methods to test adversarial robustness. Since the PGD
+attack is less computationally intense than AutoAttack, we
+consider the PGD attack in this work. However, other attacks
+can be used with the proposed data selection.
+With the inner maximization problem addressed, the outer
+minimization problem in equation (3) is then solved to find
+the model parameters that minimize the loss on the generated
+adversarial examples. The original dataset D is split into small
+batches B and stochastic gradient descent (SGD) is employed
+to update the model parameters
+θt = θt−1 + µ 1
+|B|
+�
+x,y∈B
+∇θL(fθ(x + η∗), y),
+(6)
+where the gradient is evaluated at the maximum point η∗ found
+in the inner maximization problem, thanks to the Danskin’s
+theorem [19].
+III. PROPOSED DATA SELECTION FOR ADVERSARIAL
+TRAINING
+When performing adversarial training, we are interested in
+learning a process or function f(·) that maps a data space X
+into an output space Y. However, we do not have direct access
+to samples from X in order to train the model according to the
+adversarial objective. We only have access to a subset D which
+is split into batches used to update the model parameters in
+equation (6). However, there is no guarantee that this available
+subset or its batches consist of a good representation of the
+process f(·). In this regard, we propose a sampling strategy
+to select the most relevant samples to compose the batches in
+adversarial training.
+We first consider the entire original dataset D of input-
+output pairs in equation (2). Then, at each mini-batch iteration,
+b′ clean samples are selected from the whole dataset to form
+the batch set B′. By using PGD, b′ adversarial examples are
+generated from the samples in the set B′ using equation (5).
+The resulting mini-batch B is then composed of b = 2b′ sam-
+ples. The samples in the mini-batch flow through the network,
+the gradients are computed, and we obtain the network output
+as a one-hot-encoded vector y, as shown in Figure 1. In order
+to quantify the relevance of the samples in the mini-batch, we
+define the error signal
+E(ˆy, y) =
+C
+�
+c=1
+e(ˆyc, yc),
+(7)
+which is based on the cross-entropy loss
+e(ˆyc, yc) = log
+� C
+�
+c=1
+exp(ˆyc)
+�
+− yc,
+(8)
+where C is the number of classes.
+As a rule, the closer to zero the error signal is, the less
+informative or relevant will be the contribution of the corre-
+spondent data pair to the parameter update in equation (6). We
+then propose to select a portion Pup of the samples in B based
+on the higher error values in equation (7), forming a selection
+set S. After the forward propagation is completed, only the
+samples in S are used in the backpropagation to update the
+network parameters θ, as depicted in Figure 2. Since only a
+portion Pup of the samples are used to update the parameters,
+we can save some computations and we alleviate the training
+burden.
+Mini-batch of size 2b
+forward propagation
+clean sample
+adversarial sample
+x1
+ˆy1
+y1
+E(ˆy1, y1)
+error
+Fig. 1: Forward propagation and error signal computation.
+One question remains about how to choose an adequate Pup
+for our problem. As Pup → 0, fewer samples are selected and
+we save more computations in the backpropagation. In this
+case, however, the selected samples might be insufficient lo
+
+clean sample
+adversarial sample
+E(ˆy1, y1)
+E(ˆy3, y3)
+E(ˆyk, yk)
+greatest errors
+back-propagation
+Fig. 2: Selected samples being used in the backpropagation.
+learn the problem. For standard training, the most favorable
+Pup choice mainly depends on the dataset complexity [14].
+Simpler datasets like MINIST requires Pup = 0.3, whereas
+for more complex datasets as CIFAR10, Pup = 0.5 is a
+better choice. Thus, one option is to set a fixed Pup for the
+whole training process. In this way, we can set the amount of
+saved computations from the beginning. Nevertheless, in cases
+where the dataset complexity is unknown and it is difficult to
+prescribe a Pup for all the epochs, an automatic Pup can be
+advantageous. In this way, we can obtain the Pup for each
+epoch in an adaptive manner as the training is performed. This
+can be achieved by considering the accuracy at each epoch as
+a criterion. Hence, we can estimate the number of selected
+samples Pup at each epoch t.
+P (t)
+up = (1 − λ(t−1)
+acc
+)P (t−1)
+up
+(9)
+where P (0)
+up
+= 1 and λ(t−1)
+acc
+is the last available accuracy.
+We need more samples in the mini-batch to improve learning
+when the accuracy is low, whereas fewer samples are required
+to continue the learning process when the accuracy increases.
+As it will be shown in the simulations, updating the Pup
+using equation (9) accelerates the convergence for P (0)
+up
+=
+1 because, in this case, it selects more samples in the first
+epochs. Our motivation was to provide more samples to the
+model at the beginning to improve and accelerate its learning.
+Therefore, early stopping methods [20] can be employed to
+further reduce the training time. Since we do not consider
+the early stopping approach in the simulations, we propose
+using a fixed prescribed Pup in this work. The main proposed
+algorithm is detailed in Algorithm 1.
+IV. SIMULATION RESULTS
+In this section, we assess the performance of the proposed
+data selection method in the CIFAR10 dataset using the
+Resnet18 model. The PGD attack with ϵ = 8/255, α = 0.01
+and 20 iterations is employed to build the adversarial exam-
+ples. We consider the following methods in the simulations.
+The standard method trains only with clean samples with a
+mini-batch B of size b = 256. Also using B with b = 256, the
+robust method is trained only with adversarial examples. The
+DS robust method is trained with the selection set S of size
+b = 256, which is composed of both clean and adversarial
+samples, and it is obtained using our selection strategy with
+Algorithm 1 Proposed Data Selection for adversarial training
+1: Given dataset D, mini-batch size b′, and prescribed Pup
+2: for epoch = 1 · · · T do
+3:
+for mini-batch B ⊂ D do
+4:
+Create adversarial examples {x′
+1, · · · x′
+b′} from
+clean samples {x1, · · · xb′} using current state of the
+network and obtain B′ = {x′
+1, · · · x′
+b′, x1, · · · xb′};
+5:
+Forward propagation with samples in B′;
+6:
+Compute the error signal for each sample in B′
+using equation (7);
+7:
+Select the Pup × 100% of the samples in B′ with
+greatest error values;
+8:
+Update model parameters by back propagation
+using only the data samples in S;
+Pup fixed or varying. The random robust method is trained
+with a mini-batch of size b = 256, composed of clean and
+adversarial samples selected at random. We also consider the
+selection method proposed in [13] in which the samples are
+selected based on their learning stability. In this case, we used
+50% of the samples with high quality in order to perform a
+fair comparison in terms of number of samples used.
+First, we vary the portion of selected samples Pup in
+Figure 3 to investigate the impact on the standard and ro-
+bustness accuracy at the last epoch. By using Pup = 0.5, we
+slightly outperform the approach that consider all the samples
+(Pup = 1) in terms of standard accuracy, with the benefit of
+requiring only 50% of the samples in the batch. In terms of
+robustness, the methods with 0.5 ≤ Pup < 1 perform quite
+close to the method with Pup = 1. If we reduce Pup even
+further, we do not observe a gain in performance. In such
+case, the model would require more epochs to achieve the
+same performance or it would need more samples to learn the
+problem.
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Percentage of update , Pup
+30
+40
+50
+60
+70
+80
+90
+100
+Accuracy (%)
+standard accuracy
+robustness accuracy
+Fig. 3: Evolution of the standard and robustness accuracy as
+the portion of selected samples Pup is varied.
+We then evaluate the proposed DS robust method with
+varying Pup and compare it with the fixed Pup = 0.5,
+the standard and robust methods in terms of standard and
+robust accuracy in Figures 4 and 5. We show in Figure 6
+the obtained Pup for each epoch following equation (9). By
+using both a varying Pup and Pup = 0.5, we observe an
+
+improvement in terms of standard accuracy when compared
+with the standard and robust methods. Moreover, reducing the
+number of samples in the mini-batch does not affect the robust
+accuracy, as shown in Figure 5.
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Number of epochs , t
+0
+20
+40
+60
+80
+100
+Standard accuracy (%)
+Standard
+Robust
+DS robust (P_up = 1)
+DS robust (varying P_up)
+DS robust (P_up = 0.5)
+Fig. 4: Standard accuracy as a function the number of epochs.
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Number of epochs , t
+0
+20
+40
+60
+80
+100
+Robust accuracy (%)
+Standard
+Robust
+DS robust (P_up = 1)
+DS robust (varying P_up)
+DS robust (P_up = 0.5)
+Fig. 5: Robust accuracy as a function the number of epochs.
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Number of epochs , t
+0.6
+0.7
+0.8
+0.9
+1.0
+P_up
+Fig. 6: Portion of selected samples Pup obtained for each
+epoch following equation (9).
+This improvement in robustness-accuracy tradeoff is rea-
+sonable since our method includes the most potential relevant
+clean and adversarial samples in the mini-batch. Some claim
+that such a tradeoff exists because the standard and robust
+objectives conflict [21], [22]. We can then observe in Figure 7
+that the model trained with Pup = 0.5 starts by selecting more
+adversarial samples than clean samples. However, after a few
+epochs, this behavior changes, and the number of selected
+clean samples increases. This feature potentially suggests
+that the model tries to learn the adversarial problem first.
+When it is done, the DS method attempts to improve the
+clean accuracy. Moreover, the number of selected minimum
+adversarial examples increases as the model is trained, as
+depicted in Figure 8. The minimum adversarial examples are
+generated by slowly increasing the perturbation constraint ϵ
+until the prediction changes.
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Number of epochs , t
+75
+100
+125
+150
+175
+200
+Amount of selected samples
+Clean samples
+Adv samples
+Fig. 7: Averaged amount of selected clean and adversarial
+samples at each epoch for Pup = 0.5.
+0
+20
+40
+60
+80
+100
+120
+Number of epochs , t
+60
+70
+80
+90
+100
+Averaged amount of selected 
+ minimum adversarial examples (%)
+Fig. 8: Averaged amount of selected minimum adversarial
+examples at each epoch for Pup = 0.5.
+Finally, our methods are compared with other selection
+methods in terms of standard and robust accuracy in Figures
+9 and 10, respectively. The DS approach outperforms both
+the random method and the selection method with 50% of
+high quality samples from [13], especially in terms of standard
+accuracy.
+The benefits of the proposed methods in terms of perfor-
+mance are followed by a reduction in computational complex-
+ity. Since only Pup samples in the mini-batch are backprop-
+agated through the network to update its parameters; we can
+save some computations. For example, we present the total
+training time after 200 epochs in Table I. The simulations were
+performed in a computer with two GTX-1080 GPUs. With
+Pup = 0.5, the training time is reduced when compared with
+Pup = 1 and varying Pup. However, if we stop the training
+by the 150th epoch, the training time for the varying Pup can
+be reduced to 15261.29s. Therefore, the varying Pup strategy
+can be applied if an early stopping method is also employed.
+We also outperform the method introduced in [13] in terms
+of total training time as their method needs a pre-training to
+rank the samples by the learning stability values.
+
+TABLE I: Total training time after 200 epochs.
+Method
+Time (s)
+Selection approach from [13] with 50% of samples removed
+39970.71
+Robust with Pup = 1
+20200.33
+DS Robust with Pup = 0.5
+19770.51
+DS Robust with Pup varying as in equation (9)
+20161.29
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Number of epochs , t
+0
+20
+40
+60
+80
+100
+Standard accuracy (%)
+DS robust (varying P_up)
+DS robust (P_up = 0.5)
+Random robust
+Selection approach from [13]
+Fig. 9: Comparing the proposed method with other selection
+methods in terms of standard accuracy.
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Number of epochs , t
+0
+20
+40
+60
+80
+100
+Robust accuracy (%)
+DS robust (varying P_up)
+DS robust (P_up = 0.5)
+Random robust
+Selection approach from [13]
+Fig. 10: Comparing the proposed method with other selection
+methods in terms of robust accuracy.
+V. CONCLUSION
+Adversarial training is the most popular solution to mitigate
+the effect of malicious attacks on the deep neural networks.
+Although adversarial training is able to improve the robustness
+accuracy, it usually sacrifices standard accuracy in its way.
+Motivated by this drawback and also seeking to reduce the
+computational complexity during training, we proposed a data
+selection strategy to include the data samples that bring about
+a novelty to the learning process. The simulation results with
+CIFAR10 using the Resnet18 model indicate that the method
+is beneficial to improve the robustness-accuracy tradeoff and
+reduce the computational complexity of the training. In the
+future investigation, one can employ the data selection method
+to other CNNs models and other datasets.
+REFERENCES
+[1] C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. Goodfellow,
+and R. Fergus, “Intriguing properties of neural networks,” arXiv preprint
+arXiv:1312.6199, 2013.
+[2] I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and harnessing
+adversarial examples,” arXiv preprint arXiv:1412.6572, 2014.
+[3] S.-M. Moosavi-Dezfooli, A. Fawzi, O. Fawzi, and P. Frossard, “Univer-
+sal adversarial perturbations,” in Proceedings of the IEEE conference
+on computer vision and pattern recognition, 2017, pp. 1765–1773.
+[4] A. Esteva, A. Robicquet, B. Ramsundar, V. Kuleshov, M. DePristo,
+K. Chou, C. Cui, G. Corrado, S. Thrun, and J. Dean, “A guide to deep
+learning in healthcare,”
+Nature medicine, vol. 25, no. 1, pp. 24–29,
+2019.
+[5] Y. Liu, E. Racah, J. Correa, A. Khosrowshahi, D. Lavers, K. Kunkel,
+M. Wehner, W. Collins, et al., “Application of deep convolutional neural
+networks for detecting extreme weather in climate datasets,”
+arXiv
+preprint arXiv:1605.01156, 2016.
+[6] M. Dixon, D. Klabjan, and J. H. Bang, “Classification-based financial
+markets prediction using deep neural networks,” Algorithmic Finance,
+vol. 6, no. 3-4, pp. 67–77, 2017.
+[7] A. Madry, A. Makelov, L. Schmidt, D. Tsipras, and A. Vladu, “Towards
+deep learning models resistant to adversarial attacks,” arXiv preprint
+arXiv:1706.06083, 2017.
+[8] S.-M. Moosavi-Dezfooli, A. Fawzi, and P. Frossard, “Deepfool: a simple
+and accurate method to fool deep neural networks,” in Proceedings of
+the IEEE conference on computer vision and pattern recognition, 2016,
+pp. 2574–2582.
+[9] F. Croce and M. Hein, “Reliable evaluation of adversarial robustness
+with an ensemble of diverse parameter-free attacks,” in International
+conference on machine learning. PMLR, 2020, pp. 2206–2216.
+[10] G. Ortiz-Jim´enez, A. Modas, S.-M. Moosavi-Dezfooli, and P. Frossard,
+“Optimism in the face of adversity: Understanding and improving deep
+learning through adversarial robustness,” Proceedings of the IEEE, vol.
+109, no. 5, pp. 635–659, 2021.
+[11] H. Zhang, Y. Yu, J. Jiao, E. Xing, L. El Ghaoui, and M. Jordan,
+“Theoretically principled trade-off between robustness and accuracy,” in
+International conference on machine learning. PMLR, 2019, pp. 7472–
+7482.
+[12] G. W. Ding, K. Y. C. Lui, X. Jin, L. Wang, and R. Huang, “On the
+sensitivity of adversarial robustness to input data distributions.,” in ICLR
+(Poster), 2019.
+[13] C. Dong, L. Liu, and J. Shang, “Data quality matters for adversarial
+training: An empirical study,” arXiv preprint arXiv:2102.07437, 2021.
+[14] J. O. Ferreira, M. O. K. Mendonc¸a, and P. S. R. Diniz, “Data selection
+in neural networks,” IEEE Open Journal of Signal Processing, vol. 2,
+pp. 522–534, 2021.
+[15] M. O. K. Mendonc¸a, J. O. Ferreira, and P. S. R. Diniz, “Data selective
+deep neural networks for image classification,” in 2021 29th European
+Signal Processing Conference (EUSIPCO). IEEE, 2021, pp. 1376–1380.
+[16] J. Su, D. V. Vargas, and K. Sakurai, “One pixel attack for fooling deep
+neural networks,” IEEE Transactions on Evolutionary Computation, vol.
+23, no. 5, pp. 828–841, 2019.
+[17] F. Croce and M. Hein, “Minimally distorted adversarial examples with a
+fast adaptive boundary attack,” in International Conference on Machine
+Learning. PMLR, 2020, pp. 2196–2205.
+[18] M. Andriushchenko, F. Croce, N. Flammarion, and M. Hein, “Square
+attack: a query-efficient black-box adversarial attack via random search,”
+in European Conference on Computer Vision. Springer, 2020, pp. 484–
+501.
+[19] J. M. Danskin,
+“The theory of max-min, with applications,”
+SIAM
+Journal on Applied Mathematics, vol. 14, no. 4, pp. 641–664, 1966.
+[20] L. Prechelt, “Early stopping-but when?,” in Neural Networks: Tricks of
+the trade, pp. 55–69. Springer, 1998.
+[21] A. Raghunathan, S. M. Xie, F. Yang, J. C. Duchi, and P. Liang,
+“Adversarial
+training
+can
+hurt
+generalization,”
+arXiv
+preprint
+arXiv:1906.06032, 2019.
+[22] A. Javanmard, M. Soltanolkotabi, and H. Hassani, “Precise tradeoffs in
+adversarial training for linear regression,” in Conference on Learning
+Theory. PMLR, 2020, pp. 2034–2078.
+
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+page_content='kuhfuss,diniz}@smt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
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+page_content='br, {javier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='marotomorales,pascal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='frossard}@epfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='ch  Abstract—With the increasing amount of available data and  advances in computing capabilities, deep neural networks (DNNs)  have been successfully employed to solve challenging tasks in var-  ious areas, including healthcare, climate, and finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Neverthe-  less, state-of-the-art DNNs are susceptible to quasi-imperceptible  perturbed versions of the original images – adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  These perturbations of the network input can lead to disastrous  implications in critical areas where wrong decisions can directly  affect human lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Adversarial training is the most efficient  solution to defend the network against these malicious attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  However, adversarial trained networks generally come with lower  clean accuracy and higher computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' This work  proposes a data selection (DS) strategy to be applied in the  mini-batch training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Based on the cross-entropy loss, the most  relevant samples in the batch are selected to update the model  parameters in the backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The simulation results show  that a good compromise can be obtained regarding robustness  and standard accuracy, whereas the computational complexity of  the backpropagation pass is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  Index Terms—data-selection, sampling strategy, adversarial  training, robustness-accuracy tradeoff  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' INTRODUCTION  Over the past decade, the amount of available digital data  has exponentially increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Thanks to the advances in com-  puting capabilities, deep neural networks (DNNs) have been  successfully employed to solve challenging image and natural  language processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' However, state-of-the-art DNNs are  known to be highly vulnerable to adversarial examples [1],  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' These small but malicious perturbations of the network  input can manipulate the trained model to produce incorrect  predictions with high confidence, and some perturbations can  even fool different network models [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Since adversarial  attacks might lead to disastrous implications in critical areas  like healthcare [4], climate [5] and finance [6], defending  against them is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  So far, adversarial training is the most effective approach to  mitigate the effect of strong attacks like the Projected Gradient  Descent (PGD) attack [7], DeepFool [8], and AutoAttack [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  Training the DNN with perturbed versions of the original  samples makes it possible to improve the accuracy on unseen  adversarial examples, also known as robustness accuracy [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  However, generating adversarial examples during training can  be highly computationally intense since each sample is usually  built with several steps in the direction of the gradient as  This study was financed in part by the Coordenac¸˜ao de Aperfeic¸oamento  de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' This  work was also supported by the Swiss Government Excellence Scholarships  for Foreign Students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  the model is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Moreover, adversarial training generally  decreases the standard accuracy, that is, the accuracy on clean  samples [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' This robustness-accuracy tradeoff is reported  to be highly data-dependent, especially regarding the data  distribution [12] and its quality [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Furthermore, we only  have access to a training dataset which is not necessarily  representative for the problem we aim to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In this case,  we could avoid using the entire training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Since the  dataset is reduced, we can save several computations during  backpropagation and speed-up training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' This hypothesis was  already investigated for standard training in [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In this  work, we extend the work in [14], [15] and apply it to the  adversarial training case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' From each mini-batch composed  of both clean and adversarial samples, the proposed data  selection algorithm selects the most relevant samples based  on the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Since only the selected samples are  used to update the model parameters in the backpropagation,  the training time is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The selection also balances the  necessary amount of clean and adversarial samples required  to yield satisfactory robustness and standard accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Section II presents a  brief overview of the adversarial training method and some  notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In section III, we propose a data selection technique  for adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The proposed approach is tested via  simulation results in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Finally, section V includes  some conclusion remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' ADVERSARIAL TRAINING  Adversarial training continually creates and incorporates  adversarial examples into the training process of a deep neural  network classifier  fθ(x) : RN → {1 · · · C},  (1)  with θ weights, which maps an input image x to a label y  from a dataset  D = {(x(1), y(1)), (x(2), y(2)), · · · , (x(M), y(M))},  (2)  with C possible classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Adversarial training attempts to solve  the min-max optimization problem  minθ  1  |D|  �  x,y∈D  maxη L(fθ(x + η), y)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='t ||η||p ≤ ϵ,  (3)  where L(fθ(x + η), y) is the loss function on the adversarial  sample and η is a small perturbation constrained by ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='04472v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='LG]  7 Jan 2023  Creating adversarial samples involves solving the inner  maximization problem in equation (3), in which the loss  function L is maximized in an effort to change the prediction,  that is, fθ(x+η) ̸= fθ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The optimization constraints ensure  that the distance between the adversarial and original example  should be less than ϵ under a particular norm, ||η||p ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  The norms aim to quantify how imperceptible to humans an  adversarial example is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Some examples of norms are the l0  norm, l2 norm, and l∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' We then briefly review the most  popular methods to create adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  Introduced by [2], the Fast Gradient Sign Method (FGSM)  attack generates adversarial examples by modifying the input  towards the direction where the loss L increases  x′ = x + ϵsign(∇xL(θ, x, y)),  (4)  with sign(·) the sign function, and ∇xL(θ, x, y) the loss  gradient with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' One of the strongest l∞-bounded at-  tacks, the PGD attack [7] tries to solve the inner maximization  problem in equation (3) following an iterative procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' At  each step i, the adversarial example is updated as  x′  i = clipx+ϵ(xi−1 + αsign(∇xL(θ, x, y))),  (5)  in which function clipx+ϵ(·) clips the input at the positions  around the predefined perturbation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In the context of l2-  bounded attacks, Deepfool [8] is an iterative attack optimized  for the l2-norm based on a linear approximation of the  classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Using geometry concepts, DeepFool searches within  the region of the space that describes the output of the classifier  (polyhedron) for the minimal perturbation that can change the  classifiers decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Among black-box attacks, one pixel attack  [16] is a l0-bounded attack that employs differential evolution  to create adversarial examples without knowing the network  gradients and its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Finally, the AutoAttack [9]  method consists of an ensemble of four attacks: two versions  of the PGD attack, the targeted version of the Fast Adaptive  Boundary (FAB) attack [17] and the black-box Square Attack  [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Currently, AutoAttack and PGD attack are the most  popular methods to test adversarial robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Since the PGD  attack is less computationally intense than AutoAttack, we  consider the PGD attack in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' However, other attacks  can be used with the proposed data selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  With the inner maximization problem addressed, the outer  minimization problem in equation (3) is then solved to find  the model parameters that minimize the loss on the generated  adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The original dataset D is split into small  batches B and stochastic gradient descent (SGD) is employed  to update the model parameters  θt = θt−1 + µ 1  |B|  �  x,y∈B  ∇θL(fθ(x + η∗), y),  (6)  where the gradient is evaluated at the maximum point η∗ found  in the inner maximization problem, thanks to the Danskin’s  theorem [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' PROPOSED DATA SELECTION FOR ADVERSARIAL  TRAINING  When performing adversarial training, we are interested in  learning a process or function f(·) that maps a data space X  into an output space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' However, we do not have direct access  to samples from X in order to train the model according to the  adversarial objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' We only have access to a subset D which  is split into batches used to update the model parameters in  equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' However, there is no guarantee that this available  subset or its batches consist of a good representation of the  process f(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In this regard, we propose a sampling strategy  to select the most relevant samples to compose the batches in  adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  We first consider the entire original dataset D of input-  output pairs in equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Then, at each mini-batch iteration,  b′ clean samples are selected from the whole dataset to form  the batch set B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' By using PGD, b′ adversarial examples are  generated from the samples in the set B′ using equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  The resulting mini-batch B is then composed of b = 2b′ sam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The samples in the mini-batch flow through the network,  the gradients are computed, and we obtain the network output  as a one-hot-encoded vector y, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In order  to quantify the relevance of the samples in the mini-batch, we  define the error signal  E(ˆy, y) =  C  �  c=1  e(ˆyc, yc),  (7)  which is based on the cross-entropy loss  e(ˆyc, yc) = log  � C  �  c=1  exp(ˆyc)  �  − yc,  (8)  where C is the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  As a rule, the closer to zero the error signal is, the less  informative or relevant will be the contribution of the corre-  spondent data pair to the parameter update in equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' We  then propose to select a portion Pup of the samples in B based  on the higher error values in equation (7), forming a selection  set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' After the forward propagation is completed, only the  samples in S are used in the backpropagation to update the  network parameters θ, as depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Since only a  portion Pup of the samples are used to update the parameters,  we can save some computations and we alleviate the training  burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  Mini-batch of size 2b  forward propagation  clean sample  adversarial sample  x1  ˆy1  y1  E(ˆy1, y1)  error  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 1: Forward propagation and error signal computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  One question remains about how to choose an adequate Pup  for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' As Pup → 0, fewer samples are selected and  we save more computations in the backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In this  case, however, the selected samples might be insufficient lo  clean sample  adversarial sample  E(ˆy1, y1)  E(ˆy3, y3)  E(ˆyk, yk)  greatest errors  back-propagation  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 2: Selected samples being used in the backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  learn the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' For standard training, the most favorable  Pup choice mainly depends on the dataset complexity [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  Simpler datasets like MINIST requires Pup = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='3, whereas  for more complex datasets as CIFAR10, Pup = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5 is a  better choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Thus, one option is to set a fixed Pup for the  whole training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In this way, we can set the amount of  saved computations from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Nevertheless, in cases  where the dataset complexity is unknown and it is difficult to  prescribe a Pup for all the epochs, an automatic Pup can be  advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In this way, we can obtain the Pup for each  epoch in an adaptive manner as the training is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' This  can be achieved by considering the accuracy at each epoch as  a criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Hence, we can estimate the number of selected  samples Pup at each epoch t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  P (t)  up = (1 − λ(t−1)  acc  )P (t−1)  up  (9)  where P (0)  up  = 1 and λ(t−1)  acc  is the last available accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  We need more samples in the mini-batch to improve learning  when the accuracy is low, whereas fewer samples are required  to continue the learning process when the accuracy increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  As it will be shown in the simulations, updating the Pup  using equation (9) accelerates the convergence for P (0)  up  =  1 because, in this case, it selects more samples in the first  epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Our motivation was to provide more samples to the  model at the beginning to improve and accelerate its learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  Therefore, early stopping methods [20] can be employed to  further reduce the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Since we do not consider  the early stopping approach in the simulations, we propose  using a fixed prescribed Pup in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The main proposed  algorithm is detailed in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' SIMULATION RESULTS  In this section, we assess the performance of the proposed  data selection method in the CIFAR10 dataset using the  Resnet18 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The PGD attack with ϵ = 8/255, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='01  and 20 iterations is employed to build the adversarial exam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' We consider the following methods in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  The standard method trains only with clean samples with a  mini-batch B of size b = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Also using B with b = 256, the  robust method is trained only with adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The  DS robust method is trained with the selection set S of size  b = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' which is composed of both clean and adversarial  samples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' and it is obtained using our selection strategy with  Algorithm 1 Proposed Data Selection for adversarial training  1: Given dataset D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' mini-batch size b′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' and prescribed Pup  2: for epoch = 1 · · · T do  3:  for mini-batch B ⊂ D do  4:  Create adversarial examples {x′  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' · · · x′  b′} from  clean samples {x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' · · · xb′} using current state of the  network and obtain B′ = {x′  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' · · · x′  b′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' · · · xb′};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  5:  Forward propagation with samples in B′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  6:  Compute the error signal for each sample in B′  using equation (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  7:  Select the Pup × 100% of the samples in B′ with  greatest error values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  8:  Update model parameters by back propagation  using only the data samples in S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  Pup fixed or varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The random robust method is trained  with a mini-batch of size b = 256, composed of clean and  adversarial samples selected at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' We also consider the  selection method proposed in [13] in which the samples are  selected based on their learning stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In this case, we used  50% of the samples with high quality in order to perform a  fair comparison in terms of number of samples used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  First, we vary the portion of selected samples Pup in  Figure 3 to investigate the impact on the standard and ro-  bustness accuracy at the last epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' By using Pup = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5, we  slightly outperform the approach that consider all the samples  (Pup = 1) in terms of standard accuracy, with the benefit of  requiring only 50% of the samples in the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In terms of  robustness, the methods with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5 ≤ Pup < 1 perform quite  close to the method with Pup = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' If we reduce Pup even  further, we do not observe a gain in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In such  case, the model would require more epochs to achieve the  same performance or it would need more samples to learn the  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='0  Percentage of update , Pup  30  40  50  60  70  80  90  100  Accuracy (%)  standard accuracy  robustness accuracy  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 3: Evolution of the standard and robustness accuracy as  the portion of selected samples Pup is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  We then evaluate the proposed DS robust method with  varying Pup and compare it with the fixed Pup = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5,  the standard and robust methods in terms of standard and  robust accuracy in Figures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' We show in Figure 6  the obtained Pup for each epoch following equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' By  using both a varying Pup and Pup = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5, we observe an  improvement in terms of standard accuracy when compared  with the standard and robust methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Moreover, reducing the  number of samples in the mini-batch does not affect the robust  accuracy, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  0  25  50  75  100  125  150  175  200  Number of epochs , t  0  20  40  60  80  100  Standard accuracy (%)  Standard  Robust  DS robust (P_up = 1)  DS robust (varying P_up)  DS robust (P_up = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 4: Standard accuracy as a function the number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  0  25  50  75  100  125  150  175  200  Number of epochs , t  0  20  40  60  80  100  Robust accuracy (%)  Standard  Robust  DS robust (P_up = 1)  DS robust (varying P_up)  DS robust (P_up = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 5: Robust accuracy as a function the number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  0  25  50  75  100  125  150  175  200  Number of epochs , t  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='0  P_up  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 6: Portion of selected samples Pup obtained for each  epoch following equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  This improvement in robustness-accuracy tradeoff is rea-  sonable since our method includes the most potential relevant  clean and adversarial samples in the mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Some claim  that such a tradeoff exists because the standard and robust  objectives conflict [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' We can then observe in Figure 7  that the model trained with Pup = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5 starts by selecting more  adversarial samples than clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' However, after a few  epochs, this behavior changes, and the number of selected  clean samples increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' This feature potentially suggests  that the model tries to learn the adversarial problem first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  When it is done, the DS method attempts to improve the  clean accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Moreover, the number of selected minimum  adversarial examples increases as the model is trained, as  depicted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The minimum adversarial examples are  generated by slowly increasing the perturbation constraint ϵ  until the prediction changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  0  25  50  75  100  125  150  175  200  Number of epochs , t  75  100  125  150  175  200  Amount of selected samples  Clean samples  Adv samples  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 7: Averaged amount of selected clean and adversarial  samples at each epoch for Pup = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  0  20  40  60  80  100  120  Number of epochs , t  60  70  80  90  100  Averaged amount of selected  minimum adversarial examples (%)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 8: Averaged amount of selected minimum adversarial  examples at each epoch for Pup = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  Finally, our methods are compared with other selection  methods in terms of standard and robust accuracy in Figures  9 and 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The DS approach outperforms both  the random method and the selection method with 50% of  high quality samples from [13], especially in terms of standard  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  The benefits of the proposed methods in terms of perfor-  mance are followed by a reduction in computational complex-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Since only Pup samples in the mini-batch are backprop-  agated through the network to update its parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' we can  save some computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' For example, we present the total  training time after 200 epochs in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The simulations were  performed in a computer with two GTX-1080 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' With  Pup = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5, the training time is reduced when compared with  Pup = 1 and varying Pup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' However, if we stop the training  by the 150th epoch, the training time for the varying Pup can  be reduced to 15261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='29s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Therefore, the varying Pup strategy  can be applied if an early stopping method is also employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  We also outperform the method introduced in [13] in terms  of total training time as their method needs a pre-training to  rank the samples by the learning stability values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  TABLE I: Total training time after 200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  Method  Time (s)  Selection approach from [13] with 50% of samples removed  39970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='71  Robust with Pup = 1  20200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='33  DS Robust with Pup = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5  19770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='51  DS Robust with Pup varying as in equation (9)  20161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='29  0  25  50  75  100  125  150  175  200  Number of epochs , t  0  20  40  60  80  100  Standard accuracy (%)  DS robust (varying P_up)  DS robust (P_up = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5)  Random robust  Selection approach from [13]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 9: Comparing the proposed method with other selection  methods in terms of standard accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  0  25  50  75  100  125  150  175  200  Number of epochs , t  0  20  40  60  80  100  Robust accuracy (%)  DS robust (varying P_up)  DS robust (P_up = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='5)  Random robust  Selection approach from [13]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 10: Comparing the proposed method with other selection  methods in terms of robust accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' CONCLUSION  Adversarial training is the most popular solution to mitigate  the effect of malicious attacks on the deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  Although adversarial training is able to improve the robustness  accuracy, it usually sacrifices standard accuracy in its way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  Motivated by this drawback and also seeking to reduce the  computational complexity during training, we proposed a data  selection strategy to include the data samples that bring about  a novelty to the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' The simulation results with  CIFAR10 using the Resnet18 model indicate that the method  is beneficial to improve the robustness-accuracy tradeoff and  reduce the computational complexity of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' In the  future investigation, one can employ the data selection method  to other CNNs models and other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  REFERENCES  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Szegedy, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Zaremba, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Sutskever, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Bruna, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Erhan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Goodfellow,  and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Fergus, “Intriguing properties of neural networks,” arXiv preprint  arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='6199, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [2] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Goodfellow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Shlens, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Szegedy, “Explaining and harnessing  adversarial examples,” arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='6572, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Moosavi-Dezfooli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Fawzi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Fawzi, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Frossard, “Univer-  sal adversarial perturbations,” in Proceedings of the IEEE conference  on computer vision and pattern recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 1765–1773.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Esteva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Robicquet, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Ramsundar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Kuleshov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' DePristo,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Chou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Cui, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Corrado, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Thrun, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Dean, “A guide to deep  learning in healthcare,”  Nature medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 24–29,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Liu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Racah, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Correa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Khosrowshahi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Lavers, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Kunkel,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Wehner, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Collins, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=', “Application of deep convolutional neural  networks for detecting extreme weather in climate datasets,”  arXiv  preprint arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='01156, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Dixon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Klabjan, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Bang, “Classification-based financial  markets prediction using deep neural networks,” Algorithmic Finance,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 3-4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 67–77, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Madry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Makelov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Schmidt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Tsipras, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Vladu, “Towards  deep learning models resistant to adversarial attacks,” arXiv preprint  arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='06083, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Moosavi-Dezfooli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Fawzi, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Frossard, “Deepfool: a simple  and accurate method to fool deep neural networks,” in Proceedings of  the IEEE conference on computer vision and pattern recognition, 2016,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 2574–2582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Croce and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Hein, “Reliable evaluation of adversarial robustness  with an ensemble of diverse parameter-free attacks,” in International  conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' PMLR, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 2206–2216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [10] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Ortiz-Jim´enez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Modas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Moosavi-Dezfooli, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Frossard,  “Optimism in the face of adversity: Understanding and improving deep  learning through adversarial robustness,” Proceedings of the IEEE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  109, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 635–659, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Jiao, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Xing, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' El Ghaoui, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Jordan,  “Theoretically principled trade-off between robustness and accuracy,” in  International conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' PMLR, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 7472–  7482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Ding, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Lui, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Jin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Wang, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Huang, “On the  sensitivity of adversarial robustness to input data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=',” in ICLR  (Poster), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Dong, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Liu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Shang, “Data quality matters for adversarial  training: An empirical study,” arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='07437, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Ferreira, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Mendonc¸a, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Diniz, “Data selection  in neural networks,” IEEE Open Journal of Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 2,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 522–534, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Mendonc¸a, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Ferreira, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Diniz, “Data selective  deep neural networks for image classification,” in 2021 29th European  Signal Processing Conference (EUSIPCO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 1376–1380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Su, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Vargas, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Sakurai, “One pixel attack for fooling deep  neural networks,” IEEE Transactions on Evolutionary Computation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 828–841, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [17] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Croce and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Hein, “Minimally distorted adversarial examples with a  fast adaptive boundary attack,” in International Conference on Machine  Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' PMLR, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 2196–2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Andriushchenko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Croce, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Flammarion, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Hein, “Square  attack: a query-efficient black-box adversarial attack via random search,”  in European Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Springer, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 484–  501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Danskin,  “The theory of max-min, with applications,”  SIAM  Journal on Applied Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 641–664, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [20] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Prechelt, “Early stopping-but when?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=',” in Neural Networks: Tricks of  the trade, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 55–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Springer, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Raghunathan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Xie, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Duchi, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Liang,  “Adversarial  training  can  hurt  generalization,”  arXiv  preprint  arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='06032, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content='  [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Javanmard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Soltanolkotabi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' Hassani, “Precise tradeoffs in  adversarial training for linear regression,” in Conference on Learning  Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' PMLR, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
+page_content=' 2034–2078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E3T4oBgHgl3EQfWwp_/content/2301.04472v1.pdf'}
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+arXiv:2301.01527v1  [math.OC]  4 Jan 2023
+2D AND 3D CONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS
+PERTURBED BY A SUBDIFFERENTIAL AND APPLICATIONS TO CONTROL
+PROBLEMS
+SAGAR GAUTAM1, KUSH KINRA2 AND MANIL T. MOHAN3*
+Abstract. The following convective Brinkman-Forchheimer (CBF) equations (or damped
+Navier-Stokes (NS) equations) with potential
+∂y
+∂t − µ∆y + (y · ∇)y + αy + β|y|r−1y + ∇p + Ψ(y) ∋ g, ∇ · y = 0,
+in a d-dimensional torus is considered in this work, where d ∈ {2, 3}, µ, α, β > 0 and
+r ∈ [1, ∞). For d = 2 with r ∈ [1, ∞) and d = 3 with r ∈ [3, ∞) (2βµ ≥ 1 for d = r = 3), we
+establish the existence of a unique global strong solution for the above multivalued problem
+with the help of the abstract theory of m-accretive operators. Moreover, we demonstrate
+that the same results hold local in time for the case d = 3 with r ∈ [1, 3]. We explored
+the m-accretivity of the nonlinear as well as multivalued operators, Yosida approximations
+and their properties, and several higher order energy estimates in the proofs. For r ∈ [1, 3],
+we quantize the NS nonlinearity (y · ∇)y to establish the existence and uniqueness results,
+while for r ∈ [3, ∞) (2βµ ≥ 1 for r = 3), we handle the NS nonlinearity by the nonlinear
+damping term |y|r−1y. Finally, we discuss the applications of the above developed theory
+in feedback control problems like flow invariance, time optimal control and stabilization.
+1. Introduction
+1.1. The model. Let Td =
+�
+R/LZ
+�d be a d-dimensional torus (d = 2, 3). The convective
+Brinkman-Forchheimer (CBF) equations describe the motion of incompressible fluid flows
+in a saturated porous medium ([33]). With control applications in mind, we consider the
+following CBF equations with potential (perturbed by a subdifferential, see Hypothesis 1.1
+below):
+
+
+
+
+
+
+
+
+
+∂y
+∂t − µ∆y + (y · ∇)y + αy + β|y|r−1y + ∇p + Ψ(y) ∋ g,
+in Td × (0, ∞),
+∇ · y = 0,
+in Td × (0, ∞),
+y(0) = y0 in Td,
+(1.1)
+1,2,3Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar High-
+way, Roorkee, Uttarakhand 247667, INDIA.
+e-mail: Manil T. Mohan:
+maniltmohan@ma.iitr.ac.in, maniltmohan@gmail.com.
+e-mail: Kush Kinra:
+kkinra@ma.iitr.ac.in.
+e-mail: Sagar Gautam:
+sagar_g@ma.iitr.ac.in.
+*Corresponding author.
+Key words:
+Convective Brinkman-Forchheimer equations, monotone operators, strong solution, stabi-
+lization, feedback control, time optimal control.
+Mathematics Subject Classification (2020): Primary 49J20, 49N35, 93D15; Secondary 35Q35, 76D03.
+1
+
+2
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+where y(x, t) : Td × (0, ∞) → Rd represents the velocity field at time t and position x,
+p(x, t) : Td ×(0, ∞) → R denotes the pressure field, g(x, t) : Td ×(0, ∞) → Rd is an external
+forcing and Ψ(·) ⊂ L2(Td) ×L2(Td) is a multivalued map. Moreover, y(·, ·), p(·, ·) and g(·, ·)
+satisfies the following periodic conditions:
+y(x + Lei, ·) = y(x, ·), p(x + Lei, ·) = p(x, ·) and g(x + Lei, ·) = g(x, ·),
+(1.2)
+for every x ∈ Rd and i = 1, . . . , d, where {e1, . . . , ed} is the canonical basis of Rd. The constant
+µ > 0 denotes the Brinkman coefficient (effective viscosity), the positive constants α and
+β represent the Darcy (permeability of porous medium) and Forchheimer (proportional to
+the porosity of the material) coefficients, respectively. The absorption exponent r ∈ [1, ∞)
+and r = 3 is known as the critical exponent. The critical homogeneous CBF equations ((1.1)
+without potential, r = 3 and g = 0) have the same scaling as Navier-Stokes (NS) equations
+only when α = 0 ([27]). We refer the case r < 3 as subcritical and r > 3 as supercritical
+(or fast growing nonlinearities). The model is accurate when the flow velocity is too large
+for Darcy’s law to be valid, and apart from that the porosity is not too small ([33]). If one
+considers (1.1) without potential and if α = β = 0, then we obtain the classical NS equations,
+and if α, β > 0, then it can be considered as damped NS equations. A discussion on NS
+equations with potential can be accessed from [30].
+1.2. Literature survey. In the literature, CBF equations are also known as tamed Navier-
+Stokes equations or Navier-Stokes equations modified with an absorption term, cf. [3, 42] etc.,
+and references therein. The damping αy+β|y|r−1y arises from the resistance to the motion of
+the flow, which describes several physical phenomena such as drag or friction effects, porous
+media flow, some dissipative mechanisms, cf. [17, 27, 33, 49] etc., and references therein.
+The continuous data assimilation problem for CBF model is described in [33]. The global
+solvability results for CBF model (for fast growing nonlinearities in 3D) can be accessed
+from [3, 29, 33, 27, 34], etc. Similar to 3D Navier-Stokes equations, the existence of a unique
+global (in time) weak solution of 3D CBF equations with r ∈ [1, 3) (for any β, µ > 0) and
+r = 3 (for 2βµ < 1) is also an open problem.
+The theory of monotone operators is an important tool in the study of nonlinear operator
+equations, we refer the readers to [5, 6, 19, 28], etc., for more details. When the operator has
+some kind of monotonicity properties, then one can pass the limit in the Galerkin and Faedo-
+Galerkin approximations of the original equation, with a-priori estimates that are in general
+weaker than those necessary in the compactness methods ([21]). In particular, monotone
+operators are suitable tools for studying variational inequalities ([6]).
+Local and global
+solvability of NS equations with potential (or perturbed by a subdifferential) is established
+in [30].
+Control of ordinary/partial differential equations associated with fluid flow motions have
+numerous applications in science, engineering and technology. Behavior and control of turbu-
+lent flows are some of the most difficult problems in fluid mechanics. By control of turbulent
+flows, we meant to determine an optimal action which minimizes the turbulence inside the
+flow, (cf.
+[1, 23, 26, 44]).
+One of the interesting control problem is the flow invariance
+preserving feedback controllers for fluid flows. The authors in [13] developed a procedure to
+design feedback controllers that ensure the resultant dynamics of turbulence preserve some
+prescribed physical constraints such as enstrophy, helicity, etc. Flow invariance of controlled
+
+CBF EQUATIONS WITH POTENTIAL
+3
+flux sets with respect to Navier-Stokes equations is discussed in [12]. The existence prob-
+lem of the variational inequality for Stokes and Navier-Stokes equations with constraints of
+obstacle type is considered in [24, 25], respectively.
+An another interesting feedback control problem is the time optimal control problem,
+where one finds a control of bang-bang type to reach a fixed state from an arbitrary state
+in minimal time (cf. [6, 48]). As far as the time optimal control of fluid flow models are
+concerned, the time optimal control problem for 2D NS equations, Boussinesq equations, 3D
+Navier-Stokes-Voigt equations, 2D CBF equations with r ∈ [1, 3], and 3D NS-α model is
+considered in [7, 32, 2, 37, 43], respectively, and references therein.
+Stabilization of NS equations is dealt to stabilize the equilibrium solution of NS equations
+by using finite dimensional feedback controllers having support either in interior or on the
+boundary of the domain (cf. [8, 9, 10], etc.). The internal stabilizability of NS equations
+(with slip and non-slip Dirichlet boundary conditions) is developed in [11, 14]. The author
+in [31] discussed the feedback stabilization of NS equations preserving the invariance of a
+given convex set. Recently the authors in [15] established feedback stabilization of 2D NS
+equations by using the Taylor approximation of the value function.
+1.3. Main results. The main objective of this work is to establish the solvability results of
+the inclusion problem (1.1) and discuss their applications in the context of control problems.
+Since α is not playing a major role in this work, so we fix α = 0 in the rest of the paper. Let
+us state the main results of this work for the problem (1.1) in an abstract framework (see
+(1.4) below). We will prove these results in the subsequent sections. Let us denote f = Pg
+and Φ(·) = PΨ(·), where P is the Helmholtz-Hodge (or Leray) projection. The functional
+setting has been provided in Section 2. The following assumption is imposed on Φ(·) to
+achieve our goals.
+Hypothesis 1.1. Let Φ be a maximal monotone operator on H × H satisfying the following
+hypothesis [30]:
+(H.1) Φ = ∂ϕ, where ϕ : H → R := R ∪ {+∞} is a lower semicontinuous proper convex
+function.
+(H.2) 0 ∈ D(Φ).
+(H.3) There exists two constants γ ≥ 0, ς ∈ (0, 1
+µ) such that
+(Ay, Φλ(y)) ≥ −γ(1 + ∥y∥2
+H)
+−
+�
+ς∥Φλ(y)∥2
+H,
+for d = 2 with r ∈ [1, ∞) and d = 3 with r ∈ [1, 5),
+0,
+for d = 3 with r ∈ [5, ∞),
+for all λ > 0 and y ∈ D(A), where Φλ =
+1
+λ(I − (I + λΦ)−1) : H → H is the Yosida
+approximation of Φ. We observe from here that
+Φλ(y) ∈ Φ((I + λΦ)−1)(y)),
+for every y ∈ H and λ > 0.
+Remark 1.2.
+(1) The results of this work hold true if one replaces (1+∥y∥2
+H) by (1+∥y∥2
+V)
+in (H.3).
+(2) Using condition (H.1), the system (1.4) can be considered as CBF equations perturbed
+by a subdifferential.
+
+4
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+Theorem 1.3. Let T > 0 and assume that Φ ⊂ H × H satisfies Hypothesis 1.1. Let y0 ∈
+D(A) ∩ D(Φ) and f ∈ W1,1(0, T; H). For d = 2 with r ∈ [1, ∞) and d = 3 with r ∈ [3, ∞)
+(2βµ ≥ 1 for r = 3), there exists a unique strong solution
+y ∈ W1,∞(0, T; H) ∩ L∞(0, T; D(A)) ∩ C([0, T]; V),
+(1.3)
+such that in H
+
+
+
+dy(t)
+dt
++ µAy(t) + B(y(t)) + βC(y(t)) + Φ(y(t)) ∋ f(t),
+a.e. t ∈ (0, T),
+y(0) = y0.
+(1.4)
+Furthermore, y is right differentiable, d+y
+dt
+is right continuous, and
+d+y(t)
+dt
++ (µAy(t) + B(y(t)) + βC(y(t)) + Φ(y(t)) − f(t))0 = 0,
+for all t ∈ [0, T]. (1.5)
+For d = 3 with r ∈ [1, 3] and 2βµ < 1 for r = 3 the solution y exists on some interval [0, T0),
+where
+T0 = T0
+�
+∥y0∥V, ∥f∥L2(0,T;H)
+�
+≤ T.
+Theorem 1.4. Let T > 0 and assume that Φ ⊂ H × H satisfies Hypothesis 1.1. Let y0 ∈
+D(A) ∩ D(Φ) and f ∈ W1,2(0, T; H) (y0 ∈ V ∩ D(Φ) and f ∈ L2(0, T; H) for d = 2, 3 and
+r ∈ [1, 3]). For d = 2 with r ∈ [1, ∞) and d = 3 with r ∈ [3, ∞), there exists a unique strong
+solution
+y ∈ C([0, T]; V) ∩ L2(0, T; D(A)) ∩ Lr+1(0, T; �L3(r+1)) ∩ W1,2(0, T; V),
+with dy
+dt , B(y), C(y) ∈ L2(0, T; H) such that in H
+
+
+
+dy(t)
+dt
++ µAy(t) + B(y(t)) + βC(y(t)) + Φ(y(t)) ∋ f(t),
+a.e. t ∈ (0, T),
+y(0) = y0.
+(1.6)
+For d = 3 with r ∈ [1, 3), there exists a time T0 = T0
+�
+∥y0∥V, ∥f∥L2(0,T;H)
+�
+≤ T such that
+the solution y exists on some interval [0, T0).
+1.4. Difficulties, approaches and novelties. The main concern for considering the CBF equa-
+tions (1.1) in a d-dimensional torus is as follows. In the torus Td, the Helmholtz-Hodge
+projection P and −∆ is commute ([41, Theorem 2.22]). So, the equality ([27, Lemma 2.1])
+�
+Td(−∆y(x)) · |y(x)|r−1y(x)dx
+=
+�
+Td |∇y(x)|2|y(x)|r−1dx + 4
+� r − 1
+(r + 1)2
+� �
+Td |∇|y(x)|
+r+1
+2 |2dx,
+(1.7)
+is quite useful in obtaining regularity results. It is also noticed in the literature that the
+above equality may not be useful in domains other than the whole domain or a d-dimensional
+torus (see [29, 34], etc. for a detailed discussion). Recently, the authors in [45] addressed
+this regularity problem for Dirichlet’s boundary conditions and the well-posedness of CBF
+equations with potential in bounded domains will be a future work.
+The main difficulty with nonlinear terms arises when we multiply them by a generalized
+function to get m-accretivity. In the literature for NS equations with potential (cf. [30, 31])
+or for feedback control problems (cf. [13, 7]), V-quantization of the nonlinear term (y · ∇)y
+is used to obtain the m-accretivity of the operators. Whereas, for the supercritical CBF
+
+CBF EQUATIONS WITH POTENTIAL
+5
+equations (1.1) (that is, for r > 3), one can handle the NS nonlinearity (y · ∇)y by the
+Forchheimer nonlinearity |y|r−1y (see steps (3.13), (3.44), etc.
+below).
+Along with this
+fact, the monotonicity of the nonlinear term |y|r−1y helps to obtain the m-accretivity of
+the operators without using a quantization technique (see Proposition 3.1 below). The same
+results hold true for r = 3 with 2βµ ≥ 1 also without quantization, but for r ∈ [1, 3] (2βµ < 1
+for r = 3), we need an �L4-quantization technique (Appendix A).
+The condition (Ay, Φλ(y)) ≥ −γ(1 + ∥y∥2
+H) is considered in Hypothesis 1.1 (H.3) for
+the case d = 3 with r ∈ [5, ∞). This is required to handle the term |(C(yλ), Φλ(yλ))| in
+Proposition 3.3, while taking the inner product with Φλ(yλ) for the Yoisida approximated
+stationary problem. The Sobolev embedding of V ⊂ �Lp for any p ∈ [1, ∞) helps us to resolve
+this problem in 2D, whereas in 3D, the embedding is true only for p ∈ [2, 6]. Moreover,
+for the supercritical case, by choosing F1(·) = µ(1 − δ1)A + β(1 − δ2)C(·) and F2(·) =
+µδ1A + B(·) + βδ2C(·) + κI, for some δ1, δ2 ∈ (0, 1) and κ ≥ ̺ =
+r−3
+2µ(r−1)
+�
+2
+βµ(r−1)
+�
+2
+r−3, we used
+the well-known perturbation theorem for nonlinear m-accretive operators ([5, Theorem 3.5,
+Chapter II]) to show that the operator F1 + F2 = µA + B(·) + βC(·) + κI with the domain
+D(A) is m-accretive in H.
+For NS equations with potential, the authors in [30] proved a result similar to Theorem
+1.4 by assuming that y0 ∈ V ∩ D(Φ) and f ∈ L2(0, T; H). Under the same assumptions,
+we are able to prove Theorem 1.4 for the case d = 2, 3 and r ∈ [1, 3] only (Appendix
+A). For d = 2, 3 and r ∈ (3, ∞), we need y0 ∈ D(A) ∩ D(Φ) and f ∈ W1,2(0, T; H) to
+control the term
+� T
+0 ∥Φλ(y(t))∥2
+Hdt (Step IV, Proposition 4.1). In order to do this, we first
+obtain the regularity estimates for
+��� d+yλ(·)
+dt
+���
+H and
+� T
+0
+��� dyλ(t)
+dt
+���
+2
+Vdt (Step II, Proposition 4.1)
+by taking the difference of Yosida approximated CBF equations (see (3.74) below) at t + h
+and t for h > 0 and t ∈ [0, T] and then using the monotonicity of Φλ(·). Due to the lack
+of Gateaux derivative of Φλ(y(·)), one cannot differentiate the equation (3.74) and get the
+required estimates by taking inner product with dyλ(·)
+dt
+in the resulting equation. This kind
+of difficulty is not appearing in the case of NS equations.
+Flow invraince preserving feedback controllers for 2D as well 3D NS equations with normal
+cone as potential were considered in [13]. The results obtained in the work [13] were global
+for d = 2 and local for d = 3. But the presence of the damping term |y|r−1y helps us
+to obtain global results in 3D as well for supercritical CBF equations. The author in [37]
+discussed the time optimal control problem for 2D CBF equations with r ∈ [1, 3] by using a
+V-quantization and m-accretivity of the nonlinear operators. Hypothesis 1.1 and the results
+in Theorems 1.3-1.4 help us to study the time optimal control problem of CBF equations for
+d = 2, 3 with r > 3 also. Moreover, the author in [31] examined the feedback stabilization
+of 2D and 3D NS equations preserving the invariance of a given convex set by deducing
+the existence of weak solutions (uniqueness only in 2D) for the NS system perturbed by a
+subdifferential. Whereas, for the CBF equations (1.4), one can address similar problems for
+d = 2, 3 with r > 3 by establishing uniqueness results also.
+1.5. Outline of the paper. The rest of the paper is organized as follows: The next section is
+devoted for the functional settings, definition and properties of linear, bilinear and nonlinear
+operators.
+Proof of Theorem 1.3 for the case r ∈ [3, ∞) (2βµ ≥ 1 for r = 3) is provided in Section 3. In
+order to do this, we apply the abstract theory of m-accretive operators available in [5, 6], etc.,
+
+6
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+in Propositions 3.1 and 3.3. We prove the m-accretivity of the operator F(·) = µA + B(·) +
+βC(·) + κI for some κ > 0 in Proposition 3.1 by showing the monotonicity, demicontinuity
+and coercivity of operator F(·). Furthermore, in Proposition 3.3, we establish the maximal
+monotonicity of the multivalued operator A(·), where A(·) := µA + B(·) + βC(·) + Φ(·) + κI,
+by showing the range condition R(I + A) = H. The primary tool in establishing the range
+condition is the well-posedness of a Yosida approximated problem (see (3.33) below for yλ).
+Then we establish the necessary stationary energy estimates and obtain uniform bounds of
+the sequence {yλ}λ>0 (see Step II). By passing to limit and applying the abstract theory
+for maximal monotone operators (cf. [5, 6], etc.), we finally deduce the m-accretivity of the
+operator A(·) (see Step III). A result similar to Theorem 1.3 for the Yosida approximated
+problem (3.74) is established in Proposition 3.4.
+We first derive necessary higher order energy estimates to prove Theorem 1.4 in Section 4
+(see Proposition 4.1). Then we prove some convergence results using the Banach-Alaoaglu
+theorem and Aubin-Lions compactness lemma (see Proposition 4.2). Lastly, we conclude
+the proof of Theorem 1.4 by using Proposition 3.4, and the uniqueness result is provided in
+Proposition 4.3.
+In Section 5, we discuss three applications of Theorems 1.3 and 1.4, namely, flow invariance
+feedback controllers, time optimal control problem and feedback stabilization. A brief sketch
+of proofs of Theorems 1.3 and 1.4 is provided for the case r ∈ [1, 3] in Appendix A by using a
+quantization of the NS nonlinearity (y·∇)y and the abstract theory of m-accretive operators.
+2. Functional settings and Preliminaries
+In this section, we provide the necessary functional setting needed to obtain the results of
+this work. We consider the problem (1.1) on a d-dimensional torus Td =
+�
+R/LZ
+�d (d = 2, 3),
+with periodic boundary conditions and zero-mean value constraint for the functions, that is,
+�
+Td y(x)dx = 0.
+2.1. Function spaces. Let
+˙C∞
+p (Td; Rd) denote the space of all infinitely differentiable func-
+tions (Rd-valued) such that
+�
+Td y(x)dx = 0 and satisfy periodic boundary conditions (1.2).
+The Sobolev space ˙Hk
+p(Td) := ˙Hk
+p(Td; Rd) is the completion of ˙C∞
+p (Td; Rd) with respect to
+the Hs norm
+∥y∥ ˙Hsp :=
+
+ �
+0≤|α|≤s
+∥Dαy∥2
+L2(Td)
+
+
+1/2
+.
+The Sobolev space of periodic functions with zero mean ˙Hk
+p(Td) is the same as [39, Proposition
+5.39]
+�
+y : y =
+�
+k∈Zd
+yke2πik·x/L, y0 = 0, ¯yk = y−k, ∥y∥ ˙Hs
+f :=
+�
+k∈Zd
+|k|2s|yk|2 < ∞
+�
+.
+From [39, Proposition 5.38], we infer that the norms ∥ · ∥ ˙Hsp and ∥ · ∥ ˙Hs
+f are equivalent. Let
+us define
+V := {y ∈ ˙C∞
+p (Td; Rd) : ∇ · y = 0}.
+The spaces H and �Lp are the closure of V in the Lebesgue spaces L2(Td; Rd) and Lp(Td; Rd)
+for p ∈ (2, ∞), respectively. The space V is the closure of V in the Sobolev space H1(Td; Rd).
+
+CBF EQUATIONS WITH POTENTIAL
+7
+The zero mean condition provides the well-known Poincar´e inequality,
+λ1∥y∥2
+H ≤ ∥y∥2
+V,
+(2.1)
+where λ1 = 4π2
+L2 ([39, Lemma 5.40]). Then, we characterize the spaces H, �Lp and V with the
+norms
+∥y∥2
+H :=
+�
+Td |y(x)|2dx,
+∥y∥p
+�Lp =
+�
+Td |y(x)|pdx and ∥y∥2
+V :=
+�
+Td |∇y(x)|2dx,
+respectively. Let (·, ·) denote the inner product in the Hilbert space H and ⟨·, ·⟩ represent the
+induced duality between the spaces V and its dual V′ as well as �Lp and its dual �Lp′, where
+1
+p + 1
+p′ = 1. Note that H can be identified with its own dual H′. The sum space V′ + �Lp′ is
+well defined (see [20, Subsection 2.1]). Furthermore, we have
+(V′ + �Lp′)′ = V ∩ �Lp and (V ∩ �Lp)′ = V′ + �Lp′,
+where ∥y∥V∩�Lp = max{∥y∥V, ∥y∥�Lp}, which is equivalent to the norms ∥y∥V + ∥y∥�Lp and
+�
+∥y∥2
+V + ∥y∥2
+�Lp, and
+∥y∥V′+�Lp′ = inf{∥y1∥V′ + ∥y2∥�Lp′ : y = y1 + y2, y1 ∈ V′ and y2 ∈ �Lp′}
+= sup
+�|⟨y1 + y2, f⟩|
+∥f∥V∩�Lp
+: 0 ̸= f ∈ V ∩ �Lp
+�
+.
+Note that V ∩ �Lp and V′ + �Lp′ are Banach spaces.
+Moreover, we have the continuous
+embedding V ∩ �Lp ֒→ V ֒→ H ֒→ V′ ֒→ V′ + �Lp′.
+2.2. Linear operator. Let Pp : Lp(Td) → �Lp, p ∈ [1, ∞) be the Helmholtz-Hodge (or Leray)
+projection (cf. [4, 22], etc.). Note that Pp is a bounded linear operator and for p = 2, P := P2
+is an orthogonal projection ([41, Section 2.1]). We define the Stokes operator
+Ay := −P∆y, y ∈ D(A) := V ∩ ˙H2
+p(Td).
+Note that D(A) can also be written as D(A) =
+�
+y ∈ ˙H2
+p(Td) : ∇·y = 0
+�
+. It should be noted
+that P and ∆ commutes in a torus ([41, Lemma 2.9]).
+Remark 2.1. Note that for d ≤ 4, by Sobolev’s inequality, one has D(A) ⊂ H2 ⊂ Lp, for all
+p ∈ [1, ∞).
+2.3. Bilinear operator. Let us define the trilinear form b(·, ·, ·) : V × V × V → R by
+b(y, z, w) =
+�
+Td(y(x) · ∇)z(x) · w(x)dx =
+d
+�
+i,j=1
+�
+Td yi(x)∂zj(x)
+∂xi
+wj(x)dx.
+If y, z are such that the linear map b(y, z, ·) is continuous on V, the corresponding element
+of V′ is denoted by B(y, z). We also denote B(y) = B(y, y) = P[(y · ∇)y]. An integration
+by parts yields
+�b(y, z, w) = −b(y, w, z),
+for all y, z, w ∈ z,
+b(y, z, z) = 0,
+for all y, z ∈ z.
+(2.2)
+Remark 2.2. We need the following estimates on the trilinear form b(·, ·, ·) in the sequel (see
+[46, Chapter 2, Section 2.3]):
+
+8
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+(i) For d = 2,
+|b(y, z, w)| ≤ C ×
+�
+∥y∥1/2
+H ∥y∥1/2
+V ∥z∥V∥w∥1/2
+H ∥w∥1/2
+V ,
+for all y, z, w ∈ V,
+∥y∥1/2
+H ∥y∥1/2
+V ∥z∥1/2
+V ∥Az∥1/2
+H ∥w∥H,
+for all y ∈ V, z ∈ D(A), w ∈ H.
+(2.3)
+(ii) For d = 3,
+|b(y, z, w)| ≤ C ×
+�
+∥y∥1/4
+H ∥y∥3/4
+V ∥z∥V∥w∥1/4
+H ∥w∥3/4
+V ,
+for all y, z, w ∈ V,
+∥y∥V∥z∥1/2
+V ∥Az∥1/2
+H ∥w∥H,
+for all y ∈ V, z ∈ D(A), w ∈ H.
+(2.4)
+2.4. Nonlinear operator. Let us now consider the operator C(y) := P(|y|r−1y). It is imme-
+diate that ⟨C(y), y⟩ = ∥y∥r+1
+�Lr+1 and the map C(·) : �Lr+1 → �L
+r+1
+r
+is Gateaux differentiable
+with Gateaux derivative
+C′(y)z =
+
+
+
+
+
+
+
+
+
+P(z),
+for r = 1,
+�
+P(|y|r−1z) + (r − 1)P
+�
+y
+|y|3−r (y · z)
+�
+,
+if y ̸= 0,
+0,
+if y = 0,
+for 1 < r < 3,
+P(|y|r−1z) + (r − 1)P(y|y|r−3(y · z)),
+for r ≥ 3,
+(2.5)
+for all y, z ∈ �Lr+1.
+From [36, Subsection 2.4], we have
+⟨C(y) − C(z), y − z⟩ ≥ 1
+2∥|y|
+r−1
+2 (y − z)∥2
+H + 1
+2∥|z|
+r−1
+2 (y − z)∥2
+H
+≥
+1
+2r−1∥y − z∥r+1
+�Lr+1 ≥ 0,
+(2.6)
+for r ≥ 1.
+Remark 2.3. In periodic domain (cf. [36, Subsection 3.5]), we have
+∥y∥r+1
+�L3(r+1) ≤ C
+�
+Td |∇y(x)|2|y(x)|r−1dx,
+(2.7)
+for d = 3 and r ≥ 1. Also from [40, Lemma 2.2], we obtain
+∥y∥r+1
+�Lp(r+1) = ∥|y|
+r+1
+2 ∥2
+�L2p(Td) ≤ C
+�
+Td |∇|y|
+r+1
+2 |2dxC
+�
+Td |∇y(x)|2|y(x)|r−1dx,
+(2.8)
+for d = 2 and for all p ∈ [2, ∞).
+3. Proof of Theorem 1.3
+We prove Theorem 1.3 in this section for the case r ∈ [3, ∞) (2βµ ≥ 1 for r = 3) by using
+the abstract theory available in the works [5, 6], etc.
+Proposition 3.1. For d = 2, 3 with r > 3, define the operator G(·) : D(G) → H by
+G(·) = µA + B(·) + βC(·),
+
+CBF EQUATIONS WITH POTENTIAL
+9
+where D(G) = {y ∈ V ∩ �Lr+1 : Ay ∈ H}. Then G + κI is m-accretive in H × H for some
+κ ≥ ̺, where
+̺ =
+r − 3
+2µ(r − 1)
+�
+2
+βµ(r − 1)
+�
+2
+r−3
+.
+(3.9)
+Proof. We shall first show that G + κI is a monotone operator for κ ≥ ̺ > 0. Then we
+will show that G + κI is coercive and demicontinuous, which imply the m-accretivity of the
+operator G + κI. The proof is divided into following steps:
+Step I: G + κI is monotone for some κ > 0. We estimate ⟨Ay − Az, y − z⟩ by using an
+integration by parts as
+⟨Ay − Az, y − z⟩ = ∥y − z∥2
+V.
+(3.10)
+Note that ⟨B(y, y−z), y−z⟩ = 0 which implies along with H¨older’s and Young’s inequalities
+that
+|⟨B(y) − B(z), y − z⟩| = |⟨B(y − z, y − z), z⟩| ≤ µ
+2 ∥y − z∥2
+V + 1
+2µ∥z(y − z)∥2
+H.
+(3.11)
+We take the term ∥z(y − z)∥2
+H from (3.11) and use H¨older’s and Young’s inequalities to
+estimate it as (see [27] also)
+�
+Td |z(x)|2|y(x) − z(x)|2dx =
+�
+Td |z(x)|2|y(x) − z(x)|
+4
+r−1|y(x) − z(x)|
+2(r−3)
+r−1 dx
+≤
+��
+Td |z(x)|r−1|y(x) − z(x)|2dx
+�
+2
+r−1��
+Td |y(x) − z(x)|2dx
+� r−3
+r−1
+≤ βµ∥|z|
+r−1
+2 (y − z)∥2
+H + r − 3
+r − 1
+�
+2
+βµ(r − 1)
+�
+2
+r−3
+∥y − z∥2
+H,
+(3.12)
+for r > 3. Using (3.12) in (3.11), we find
+|⟨B(y) − B(z), y − z⟩|
+≤ µ
+2 ∥y − z∥2
+V + β
+2 ∥|z|
+r−1
+2 (y − z)∥2
+H +
+r − 3
+2µ(r − 1)
+�
+2
+βµ(r − 1)
+�
+2
+r−3
+∥y − z∥2
+H.
+(3.13)
+From (2.6), we easily have
+β⟨C(y) − C(z), y − z⟩ ≥ β
+2 ∥|z|
+r−1
+2 (y − z)∥2
+H.
+(3.14)
+Combining (3.10) and (3.13)-(3.14), we conclude that
+⟨(G + κI)(y) − (G + κI)(z), y − z⟩ ≥ µ
+2 ∥y − z∥2
+V + (κ − ̺)∥y − z∥2
+H ≥ µ
+2 ∥y − z∥2
+V, (3.15)
+for κ ≥ ̺ and r > 3. Thus G + κI is monotone.
+Step II: G + κI is demicontinuous. Let us take a sequence yn → y in V ∩ �Lr+1, so that
+∥yn − y∥V + ∥yn − y∥�Lr+1 → 0 as n → ∞. For any z ∈ V ∩ �Lr+1, we consider
+⟨(F + κI)(yn) − (F + κI)(y), z⟩
+= µ⟨A(yn) − A(y), z⟩ + ⟨B(yn) − B(y), z⟩ − β⟨C(yn) − C(y), z⟩ + κ⟨yn − y, z⟩.
+(3.16)
+
+10
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+Note that
+|µ⟨A(yn − y), z⟩ + κ⟨yn − y, z⟩| ≤ µ∥yn − y∥V∥z∥V + κ∥yn − y∥H∥z∥H → 0 as n → ∞,
+since yn → y strongly in V ∩ �Lr+1. We estimate the term |⟨B(yn) − B(y), z⟩| by using the
+H¨older’s and interpolation inequalities
+|⟨B(yn) − B(y), z⟩| = |⟨B(yn, yn − y), z⟩ + ⟨B(yn − y, y), z⟩|
+≤ |⟨B(yn, z), yn − y⟩| + |⟨B(yn − y, z), y⟩|
+≤
+�
+∥yn∥�L
+2(r+1)
+r−1 + ∥y∥�L
+2(r+1)
+r−1
+�
+∥yn − y∥�Lr+1∥z∥V
+≤
+�
+∥yn∥
+r−3
+r−1
+H
+∥yn∥
+2
+r−1
+�Lr+1 + ∥y∥
+r−3
+r−1
+H
+∥y∥
+2
+r−1
+�Lr+1
+�
+∥yn − y∥�Lr+1∥z∥V
+→ 0,
+as n → ∞,
+since yn → y strongly in V ∩ �Lr+1 and yn, y ∈ V ∩ �Lr+1. We estimate the term |⟨C(yn) −
+C(y), z⟩| using the Taylor’s formula ([18, Theorem 7.9.1]) as
+|⟨C(yn) − C(y), z⟩| ≤ sup
+0<θ<1
+r∥(yn − y)|θyn + (1 − θ)y|r−1∥�L
+r+1
+r ∥z∥�Lr+1
+≤ r∥yn − y∥�Lr+1
+�
+∥yn∥�Lr+1 + ∥y∥�Lr+1
+�r−1∥z∥�Lr+1 → 0 as n → ∞,
+since yn → y strongly in V ∩ �Lr+1 and yn, y ∈ V ∩ �Lr+1. From the above convergences, it is
+immediate that ⟨(G+κI)(yn)−(G+κI)(y), z⟩ → 0, for all z ∈ V∩ �Lr+1. Hence the operator
+G + κI : V ∩ �Lr+1 → V′ + �L
+r+1
+r
+is demicontinuous and hence it is hemicontinuous.
+Step III: G + κI is coercive. We consider
+⟨(G + κI)(y), y⟩
+∥y∥V∩�Lr+1
+=
+µ∥y∥2
+V + β∥y∥r+1
+�Lr+1 + κ∥y∥2
+H
+�
+∥y∥2
+V + ∥y∥2
+�Lr+1
+≥
+min{µ, β}
+�
+∥y∥2
+V + ∥y∥2
+�Lr+1
+�
+− 1
+�
+∥y∥2
+V + ∥y∥2
+�Lr+1
+,
+where we have used the fact that x2 ≤ xr+1 + 1, for x ≥ 0 and r ≥ 1. Thus, we have
+lim
+∥y∥V∩�Lr+1→∞
+⟨(G + κI)(y), y⟩
+∥y∥V∩�Lr+1
+= ∞,
+and it shows that the operator G + κI is coercive.
+Step IV: F(·) := G(·) + κI is m-accretive in H × H. Let us define an operator
+F(y) = µAy + B(y) + βC(y) + κy,
+where D(A) = {y ∈ V ∩ �Lr+1 : µAy + B(y) + βC(y) ∈ H}. Note that the space V ∩ �Lr+1 is
+reflexive. Since G+κI is monotone, hemicontinuous and coercive from V∩ �Lr+1 to V′ + �L
+r+1
+r ,
+then by an application of [16, Example 2.3.7], we obtain that G+ κI is maximal monotone in
+H with domain D(F) ⊇ D(A). In fact, we shall prove that F is m-accretive for κ sufficiently
+large with D(F) = D(A).
+Let us consider the operators for some δ1, δ2 ∈ (0, 1) as
+F1(·) = µ(1 − δ1)A + β(1 − δ2)C(·),
+(3.17)
+F2(·) = µδ1A + B(·) + βδ2C(·) + κI,
+(3.18)
+
+CBF EQUATIONS WITH POTENTIAL
+11
+where D(F1) = {y ∈ V ∩ �Lr+1 : F1(·) ∈ H} and D(F2) = {y ∈ V ∩ �Lr+1 : F2(·) ∈ H}. Taking
+the inner product with y in (3.17), we obtain
+µ(1 − δ1)∥y∥2
+V + β(1 − δ2)∥y∥r+1
+�Lr+1 ≤ (F1(y), y) ≤ ∥F1(y)∥H∥y∥H,
+so that
+∥y∥2
+V ≤
+1
+µ(1 − δ1)∥F1(y)∥H∥y∥H.
+(3.19)
+Taking the inner product with Ay in (3.17) and using (1.7), we get
+µ(1 − δ1)∥Ay∥2
+H + β(1 − δ2)
+�
+∥|y|
+r−1
+2 ∇y∥2
+H + 4(r − 1)
+(r + 1)2 ∥|∇|y|
+r+1
+2 |∥2
+H
+�
+= (F1(y), Ay).
+Therefore, we have
+∥Ay∥H ≤
+1
+µ(1 − δ1)∥F1(y)∥H which implies D(F1) ⊆ D(A).
+(3.20)
+Moreover, using Sobolev’s inequality, we infer
+∥F1(y)∥H ≤ µ(1 − δ1)∥Ay∥H + Cβ(1 − δ2)∥y∥r
+�L2r
+≤ µ(1 − δ1)∥Ay∥H + Cβ(1 − δ2)∥Ay∥r
+H,
+which gives D(F1) ⊇ D(A) and therefore D(A) = D(F1). Taking the inner product with C(y)
+in (3.17), we find
+µ(1 − δ1)
+�
+∥|y|
+r−1
+2 ∇y∥2
+H + 4(r − 1)
+(r + 1)2 ∥|∇|y|
+r+1
+2 |∥2
+H
+�
++ β(1 − δ2)∥C(y)∥2
+H = (F1(y), C(y)),
+so that
+∥C(y)∥H ≤
+1
+β(1 − δ2)∥F1(y)∥H.
+(3.21)
+For r > 3, we estimate ∥B(y)∥H using H¨older’s inequality as follows:
+∥B(y)∥2
+H ≤
+�
+Td |y(x)|2|∇y(x)|2dx =
+�
+Td |y(x)|2|∇y(x)|
+4
+r−1|∇y(x)|
+2(r−3)
+r−1 dx
+≤ ∥|y|
+r−1
+2 ∇y∥
+4
+r−1
+H
+∥y∥
+2(r−3)
+r−1
+V
+.
+(3.22)
+Note that (C(y), Ay) =
+�
+Td(−∆y(x)) · |y(x)|r−1y(x)dx. Using the estimate (3.19) and the
+equality (1.7) in (3.22), we find
+∥B(y)∥2
+H ≤ [(C(y), Ay)]
+2
+r−1
+�
+1
+µ(1 − δ1)∥F1(y)∥H∥y∥H
+� r−3
+r−1
+.
+Therefore, we estimate ∥B(y)∥H as
+∥B(y)∥H ≤ ∥C(y)∥
+1
+r−1
+H
+∥Ay∥
+1
+r−1
+H
+�
+1
+µ(1 − δ1)∥F1(y)∥H∥y∥H
+�
+r−3
+2(r−1)
+.
+(3.23)
+Using the estimates (3.20)-(3.21) in (3.23), then using Young’s inequality, we get
+∥B(y)∥H ≤
+�
+∥F1(y)∥2
+H
+βµ(1 − δ1)(1 − δ2)
+�
+1
+r−1�∥F1(y)∥H∥y∥H
+µ(1 − δ1)
+�
+r−3
+2(r−1)
+
+12
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+=
+1
+�
+µ(1 − δ1)
+�
+1
+β(1 − δ2)
+�
+1
+r−1
+∥F1(y)∥
+r+1
+2(r−1)
+H
+∥y∥
+r−3
+2(r−1)
+H
+≤
+δ1
+1 − δ1
+∥F1(y)∥H + Cδ1,δ2,µ,β∥y∥H,
+(3.24)
+where Cδ1,δ2,µ,β =
+r−3
+2(r−1)
+�
+1−δ1
+µ
+r−1
+2
+β(1−δ2)
+�
+2
+r−3�
+r+1
+2δ1(r−1)
+� r+1
+r−3. Now using the estimates (3.20)-(3.21)
+and (3.24) in (3.18), we deduce
+∥F2(y)∥H ≤ µδ1∥Ay∥H + βδ2∥C(y)∥H + ∥B(y)∥H + κ∥y∥H
+≤
+� 2δ1
+1 − δ1
++
+δ2
+1 − δ2
+�
+∥F1(y)∥H + (Cδ1,δ2,µ,β + κ)∥y∥H.
+Let us choose δ1 and δ2 in such a way that ρ =
+2δ1
+1−δ1 +
+δ2
+1−δ2 < 1, for example, one can choose
+δ1 = 1
+9, δ2 = 1
+5, so that ρ = 1
+2. Then by the well-known perturbation theorem for nonlinear
+m-accretive operators ([5, Chapter II, Theorem 3.5]), we conclude that the operator F1 + F2
+with the domain D(A) is m-accretive in H. Since F1 + F2 = G + κI , the operator G + κI is
+m-accretive in H.
+□
+Remark 3.2. 1. For d = 2 with r ∈ [1, ∞) and d = 3 with r ∈ [1, 5], Sobolev’s embedding
+yields V ⊂ �Lr+1, so that V ∩ �Lr+1 = V.
+2.
+For d = r = 3 and 2βµ ≥ 1, one can obtain global monotonicity of the operator
+G(·) : V → V′ in the following way:
+We estimate |⟨B(y − z, y − z), z⟩| using H¨older’s and Young’s inequalities as
+|⟨B(y − z, y − z), z⟩| ≤ ∥z(y − z)∥H∥y − z∥V ≤ µ∥y − z∥2
+V + 1
+4µ∥z(y − z)∥2
+H.
+(3.25)
+Combining (3.10), (3.14) and (3.25), we obtain
+⟨G(y) − G(z), y − z⟩ ≥ 1
+2
+�
+β − 1
+2µ
+�
+∥z(y − z)∥2
+H ≥ 0,
+(3.26)
+provided 2βµ ≥ 1.
+Moreover, other properties like demicontinuity and coercivity can be
+proved in similar way as r > 3 case (see the proof of Proposition 3.1).
+Proposition 3.3. Let Φ ⊂ H × H be a maximal monotone operator satisfying Hypothesis 1.1.
+Define the multivalued operator A : D(A) → H by
+A(·) = µA + B(·) + βC(·) + Φ(·) + κI,
+with the domain D(A) = {y ∈ H : ∅ ̸= A(y) ⊂ H}. Then D(A) = D(A) ∩ D(Φ) and A is a
+maximal monotone operator in H × H, where κ is as in Proposition 3.1.
+Furthermore, the following estimates holds
+∥Aw∥2
+H ≤ C(1 + ∥w∥2
+H + ∥µAw + B(w) + βC(w) + Φλ(w)∥2
+H)ϑ,
+(3.27)
+for every w ∈ D(A), λ > 0 and
+∥Aw∥2
+H ≤ C(1 + ∥w∥2
+H + ∥µAw + B(w) + βC(w) + ξ∥2
+H)ϑ,
+(3.28)
+
+CBF EQUATIONS WITH POTENTIAL
+13
+for every w ∈ D(A) ∩ D(Φ) and ξ ∈ Φ(w), where
+ϑ =
+
+
+
+
+
+
+
+
+
+r,
+when d = 2 with r ∈ (3, ∞),
+r+3
+5−r,
+when d = 3 with r ∈ (3, 5),
+3,
+when d = r = 3 with 2βµ ≥ 1,
+1,
+when d = 3 with r ∈ [5, ∞).
+(3.29)
+Proof. It has been shown in Proposition 3.1 that the operator F(·) = µA + B(·) + βC(·) + κI
+is maximal monotone with domain D(F) = D(A) in H × H. Note that A = F + Φ implies
+D(A) ∩ D(Φ) ⊆ D(A) and since A is the sum of two monotone operators, it is monotone.
+In order to prove A is maximal monotone, we need to show that
+R(I + A) = H.
+(3.30)
+Step I: Well-posedness of the Yosida approximated problem. Let f ∈ H be arbitrary
+but fixed. We approximate the inclusion problem
+y + µAy + B(y) + βC(y) + Φ(y) + κy ∋ f,
+(3.31)
+by the equation
+yλ + µAyλ + B(yλ) + βC(yλ) + Φλ(yλ) + κyλ = f,
+(3.32)
+where Φλ is the Yosida approximation of Φ. By the properties of Yosida approximation, Φλ
+is demicontinuous and monotone (see [5, Chapter 2, Proposition 1.3]). Therefore the sum
+F(·) + Φλ(·) is maximal monotone (see [6, Chapter 2, Corollary 1.1]). This guarantees the
+existence of a solution yλ ∈ D(A) for (3.32). Let �κ = κ + 1. Then (3.32) can be written as
+µAyλ + B(yλ) + βC(yλ) + Φλ(yλ) + �κyλ = f.
+(3.33)
+We shall now prove the uniqueness. Let yλ and zλ be two solutions of the equation (3.33)
+with the same data f and let wλ = yλ − zλ. Then we have
+µAwλ + B(yλ) − B(zλ) + β(C(yλ) − C(zλ)) + Φλ(yλ) − Φλ(zλ) + �κwλ = 0.
+(3.34)
+Taking the inner product with wλ in (3.34), we get
+µ∥wλ∥2
+V + (Φλ(yλ) − Φλ(zλ), wλ) + �κ∥wλ∥2
+H
+= −(B(yλ) − B(zλ), wλ) − β(C(yλ) − C(zλ), wλ).
+(3.35)
+By similar calculations as in (3.13) and (2.6), we obtain
+−(B(yλ) − B(zλ) − β(C(yλ) − C(zλ)), wλ) ≤ µ
+2 ∥wλ∥2
+V + η∥wλ∥2
+H,
+(3.36)
+where ̺ =
+r−3
+2µ(r−1)
+�
+2
+βµ(r−1)
+�
+2
+r−3. By [6, Chapter 2, Proposition 1.3, part (i)], we know that Φλ
+is monotone, so that (Φλ(yλ) − Φλ(zλ), wλ) ≥ 0 for any λ > 0. Therefore, we conclude from
+(3.35) that
+µ
+2∥wλ∥2
+V + (�κ − ̺)∥wλ∥2
+H ≤ 0.
+Since ̺ < �κ, we get wλ = 0 and thus yλ = zλ.
+Step II: Uniform bounds for yλ. Let us take the inner product with yλ in (3.33) to get
+µ∥yλ∥2
+V + β(C(yλ), yλ) + (Φλ(yλ), yλ) + �κ∥yλ∥2
+H = (f, yλ),
+(3.37)
+
+14
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+since (B(yλ), yλ) = 0. As the operator Φλ is monotone with 0 ∈ D(Φλ) = H, we infer
+(Φλ(yλ), yλ) ≥ (Φλ(0), yλ).
+(3.38)
+By applying Young’s inequality and by of [6, Chapter 2, Proposition 1.3, part (ii)], we have
+− (Φλ(0), yλ) ≤ ∥Φλ(0)∥H∥yλ∥H ≤ 1
+�κ∥Φ(0)∥2
+H + �κ
+4∥yλ∥2
+H.
+(3.39)
+Then equation (3.37) yields
+µ∥yλ∥2
+V + �κ
+2∥yλ∥2
+H + β∥yλ∥r+1
+�Lr+1 ≤ 1
+�κ∥f∥2
+H + 1
+�κ∥Φ(0)∥2
+H,
+which gives
+∥yλ∥2
+H + ∥yλ∥2
+V + ∥yλ∥r+1
+�Lr+1 ≤ C(1 + ∥f∥2
+H),
+for all λ > 0,
+(3.40)
+where the constant C = C(µ, β, �κ, ∥Φ(0)∥H) does not depend on λ.
+Taking the inner product of (3.33) with Ayλ, we get
+µ∥Ayλ∥2
+H + (B(yλ), Ayλ) + β(C(yλ), Ayλ) + (Φλ(yλ), Ayλ) + �κ∥yλ∥2
+V = (f, Ayλ).
+(3.41)
+By [47, Chapter VI, pp. 404], we have (B(yλ), Ayλ) = 0 for d = 2. For d = 3, we consider
+the cases r > 3 and r = 3 with 2βµ ≥ 1 separately.
+Case I: r > 3. From Cauchy-Schwarz and Young’s inequalities, we obtain
+(f, Ayλ) ≤ ∥f∥H∥Ay∥H ≤ µ
+4∥Ay∥2
+H + 1
+µ∥f∥2
+H.
+(3.42)
+We estimate |(B(yλ), Ayλ)| using H¨older’s, and Young’s inequalities as
+|(B(yλ), Ayλ)| ≤ ∥|yλ||∇yλ|∥H∥Ayλ∥H ≤ µ
+2 ∥Ayλ∥2
+H + 1
+2µ∥|yλ||∇yλ|∥2
+H.
+(3.43)
+We estimate the final term from (3.43) using H¨older’s and Young’s inequalities as
+�
+Td |yλ(x)|2|∇yλ(x)|2dx =
+�
+Td |yλ(x)|2|∇yλ(x)|
+4
+r−1|∇yλ(x)|
+2(r−3)
+r−1 dx
+≤
+��
+Td |yλ(x)|r−1|∇yλ(x)|2dx
+�
+2
+r−1��
+Td |∇yλ(x)|2dx
+� r−3
+r−1
+≤ βµ
+��
+Td |yλ(x)|r−1|∇yλ(x)|2dx
+�
++ 2µ̺
+��
+Td |∇yλ(x)|2dx
+�
+,
+(3.44)
+where ̺ =
+r−3
+2µ(r−1)
+�
+2
+βµ(r−1)
+�
+2
+r−3. From (1.7), we can write
+(C(yλ), Ayλ) = ∥|∇yλ||yλ|
+r−1
+2 ∥2
+H + 4
+� r − 1
+(r + 1)2
+�
+∥|∇|yλ|
+r+1
+2 |∥2
+H.
+(3.45)
+Using the condition (H.3) of Hypothesis 1.1, estimates (3.40), (3.42)-(3.43) and (3.45) in
+(3.41), it yields for all λ > 0
+µ
+4 ∥Ayλ∥2
+H + β
+2 ∥|∇yλ||yλ|
+r−1
+2 ∥2
+H + 4β
+� r − 1
+(r + 1)2
+�
+∥|∇|yλ|
+r+1
+2 |∥2
+H
+
+CBF EQUATIONS WITH POTENTIAL
+15
+≤ C(1 + ∥f∥2
+H) +
+�
+ς∥Φλ(yλ)∥2
+H,
+for d = 2 with r ∈ (3, ∞) and d = 3 with r ∈ (3, 5),
+0,
+for d = 3 with r ∈ [5, ∞).
+(3.46)
+This completes the proof of energy estimates for d = 3 with r ∈ [5, ∞).
+Case II: r = 3 with 2βµ ≥ 1. From (1.7) and Cauchy-Schwarz and Young’s inequalities, we
+calculate
+|(B(yλ), Ayλ)| ≤ ∥|yλ||∇yλ|∥H∥Ayλ∥H ≤ µ
+2 ∥Ayλ∥2
+H + 1
+2µ∥|yλ||∇yλ|∥2
+H,
+(3.47)
+(C(yλ), Ayλ) = ∥|yλ||∇yλ|∥2
+H + 1
+2∥|∇|yλ|2|∥2
+H,
+(3.48)
+|(f, Ayλ)| ≤ ∥f∥H∥Ayλ∥H ≤ µ
+8 ∥Ayλ∥2
+H + 1
+2µ∥f∥2
+H.
+(3.49)
+Using (3.40) and (3.47)-(3.49) in (3.41), we get
+3µ
+8 ∥Ayλ∥2
+H +
+�
+β − 1
+2µ
+�
+∥|yλ||∇yλ|∥2
+H + β
+2 ∥|∇|yλ|2|∥2
+H
+≤ C(1 + ∥f∥2
+H) + ς∥Φλ(yλ)∥2
+H.
+(3.50)
+Estimate for ∥Φλ(yλ)∥2
+H. Taking the inner product with Φλ(yλ) in (3.33), we have
+∥Φλ(yλ)∥2
+H = (f, Φλ(yλ)) − (B(yλ), Φλ(yλ)) − β(C(yλ), Φλ(yλ)) − µ(Ayλ, Φλ(yλ))
+− �κ(yλ, Φλ(yλ)).
+(3.51)
+Similar to (3.39), we have
+(yλ, Φλ(yλ)) ≥ −1
+2(∥Φ(0)∥2
+H + ∥yλ∥2
+H).
+(3.52)
+We calculate |(B(yλ), Φλ(yλ)| using (2.3), Agmon’s and Young’s inequalities as
+|(B(yλ), Φλ(yλ)| = |b(yλ, yλ, Φλ(yλ))|
+≤ C
+�
+∥yλ∥
+1
+2
+H∥yλ∥V∥Ayλ∥
+1
+2
+H∥Φλ(yλ)∥H,
+for d = 2,
+∥yλ∥
+3
+2
+V∥Ayλ∥
+1
+2
+H∥Φλ(yλ)∥H,
+for d = 3,
+≤ 1 − µς
+8
+∥Φλ(yλ)∥2
+H + µ(1 − µς)
+8ς
+∥Ayλ∥2
+H + C(1 + ∥f∥2
+H)3.
+(3.53)
+For d = 3 with r ∈ (3, 5), by using the Cauchy-Schwarz, interpolation and Young’s
+inequalities, and (3.40) and (3.46), we obtain
+|(C(yλ), Φλ(yλ))| ≤ ∥C(yλ∥H∥Φλ(yλ)∥H ≤ ∥yλ∥r
+�L2r∥Φλ(yλ)∥H
+≤ ∥yλ∥
+r+3
+4
+�Lr+1∥yλ∥
+3(r−1)
+4
+�L3(r+1)∥Φλ(yλ)∥H
+≤ C∥yλ∥
+r+3
+4
+�Lr+1∥|yλ|
+r−1
+2 ∇yλ∥
+3(r−1)
+2(r+1)
+H
+∥Φλ(yλ)∥H
+≤ C(1 + ∥f∥2
+H)
+r+3
+4(r+1)
+�2ς
+β ∥Φλ(yλ)∥2
+H + 2C
+β (1 + ∥f∥2
+H)
+� 3(r−1)
+4(r+1)
+∥Φλ(yλ)∥H
+
+16
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+≤ C
+�
+∥Φλ(yλ)∥
+5r−1
+2(r+1)
+H
+(1 + ∥f∥2
+H)
+r+3
+4(r+1) + ∥Φλ(yλ)∥H(1 + ∥f∥2
+H)
+r
+r+1
+�
+≤ 1 − µς
+8β
+∥Φλ(yλ)∥2
+H + C(1 + ∥f∥2
+H)
+r+3
+5−r + C(1 + ∥f∥2
+H)
+2r
+r+1
+≤ 1 − µς
+8β
+∥Φλ(yλ)∥2
+H + C(1 + ∥f∥2
+H)
+r+3
+5−r ,
+(3.54)
+where we have used the fact that r+3
+5−r >
+2r
+r+1. Using the Sobolev embedding (for d = 2), we
+deduce
+|(C(yλ), Φλ(yλ))| ≤ ∥C(yλ∥H∥Φλ(yλ)∥H ≤ ∥yλ∥r
+�L2r∥Φλ(yλ)∥H
+≤ ∥yλ∥r
+V∥Φλ(yλ)∥H ≤ C(1 + ∥f∥2
+H)
+r
+2∥Φλ(yλ)∥H
+≤ 1 − µς
+8β
+∥Φλ(yλ)∥2
+H + C(1 + ∥f∥2
+H)r,
+(3.55)
+for d = 2 with r ∈ (3, ∞). Also, by the Cauchy-Schwarz and Young’s inequalities, we get
+|(f, Φλ(yλ))| ≤
+1
+1 − µς ∥f∥2
+H + 1 − µς
+4
+∥Φλ(yλ)∥2
+H.
+(3.56)
+Using the estimates (3.40) and (3.52)-(3.56) in (3.51), we arrive at
+ς∥Φλ(yλ)∥2
+H ≤ µ
+4∥Ay∥2
+H +
+
+
+
+
+
+C(1 + ∥f∥2
+H)r,
+for d = 2 with r ∈ (3, ∞),
+C(1 + ∥f∥2
+H)
+r+3
+5−r ,
+for d = 3 with r ∈ (3, 5),
+C(1 + ∥f∥2
+H)3,
+for d = r = 3.
+(3.57)
+Uniform boundedness of sequqences. It implies from (3.46) and (3.57) that
+µ
+4 ∥Ayλ∥2
+H + β
+2 ∥|∇yλ||yλ|
+r−1
+2 ∥2
+H + 4β
+� r − 1
+(r + 1)2
+�
+∥|∇|yλ|
+r+1
+2 |∥2
+H
+≤
+
+
+
+
+
+C(1 + ∥f∥2
+H)r,
+for d = 2 with r ∈ (3, ∞),
+C(1 + ∥f∥2
+H)
+r+3
+5−r ,
+for d = 3 with r ∈ (3, 5),
+C(1 + ∥f∥2
+H),
+for d = 3 with r ∈ (5, ∞).
+(3.58)
+For d = r = 3 with 2βµ ≥ 1, using (3.57) in (3.50), we obtain
+µ
+8∥Ayλ∥2
+H +
+�
+β − 1
+2µ
+�
+∥|yλ||∇yλ|∥2
+H + β
+2 ∥|∇|yλ|2|∥2
+H ≤ C(1 + ∥f∥H)3.
+(3.59)
+Thus under Hypothesis 1.1 (condition (H.3)), we have
+∥Ayλ∥H ≤ C, ∥|∇yλ||yλ|
+r−1
+2 ∥H ≤ C and ∥|∇|yλ|
+r+1
+2 |∥H ≤ C,
+(3.60)
+for d = 2, 3 with r ∈ (3, ∞) and d = r = 3 with 2βµ ≥ 1 for all yλ ∈ D(A). Using
+interpolation inequality and estimates (2.7) and (2.8), we have
+∥C(yλ)∥H ≤ ∥yλ∥r
+�L2r ≤ ∥yλ∥
+r+3
+4
+�Lr+1∥yλ∥
+3(r−1)
+4
+�L3(r+1) ≤ C,
+for all yλ ∈ D(A).
+Also, by using H¨older’s and Agmon’s inequalities, we obtain
+∥B(yλ)∥H ≤ ∥(yλ · ∇)yλ∥H ≤ ∥yλ∥V∥yλ∥
+1− d
+4
+H
+∥A(yλ)∥
+d
+4
+H ≤ C,
+for all yλ ∈ D(A).
+
+CBF EQUATIONS WITH POTENTIAL
+17
+Now, the equation (3.32) can be rewritten as
+yλ + F(yλ) + Φλ(yλ) = f,
+(3.61)
+where F(·) = µA + B(·) + βC(·) + �κI. Hence from (3.60), we conclude that
+∥F(yλ)∥H ≤ C and ∥Φλ(yλ)∥H ≤ C,
+for all yλ ∈ D(A).
+(3.62)
+Step III: Convergence of yλ and proof of (3.30). The estimates (3.40), (3.60) and (3.62),
+and the Banach-Alaoglu theorem guarantee the existence of a weakly convergent subsequence
+{yλj} of {yλ} such that as j → ∞
+� yλj ⇀ y,
+in V,
+Ayλj ⇀ Ay,
+in H,
+�
+Φλ(yλj) ⇀ f 1,
+in H,
+F(yλj) ⇀ f 2,
+in H.
+(3.63)
+Since the embedding D(A) ֒→ V is compact, we get the following strong convergence also:
+yλj → y in V.
+(3.64)
+Passing weak limit in (3.61), we get
+y + f 1 + f 2 = f in H.
+(3.65)
+In order to prove (3.30), we need to show that f 2 = F(y) and f 1 ∈ Φ(y). For this, we
+rewrite equation (3.61) for λ and �λ, subtract and then take the inner product with yλ − y�λ
+to find
+(Φλ(yλ) − Φ�λ(y�λ), yλ − y�λ) + ((F + I)(yλ) − (F + I)(y�λ), yλ − y�λ) = 0.
+(3.66)
+By the monotonicity of F + I (cf. Proposition 3.1), we conclude that
+(Φλ(yλ) − Φ�λ(y�λ), yλ − y�λ) ≤ 0,
+(3.67)
+for all λ, �λ > 0. By [6, Proposition 1.3, part (iv), pp. 49] (see (3.63)-(3.64) and (3.67)), we
+conclude that (y, f 1) ∈ Φ and
+lim
+λ,�λ→0
+(Φλ(yλ) − Φ�λ(y�λ), yλ − y�λ) = 0.
+This also implies from (3.66) that
+lim
+λ,�λ→0
+((F + I)(yλ) − (F + I)(y�λ), yλ − y�λ) = 0.
+(3.68)
+Since yλ → y, F(yλ) ⇀ f 2 in H (see (3.63)-(3.64)), F + I is maximal monotone (cf.
+Proposition 3.1) and (3.68) holds, then by [6, Lemma 1.3, pp. 49], we deduce that (y, y +
+f 2) ∈ F + I, and this implies that F(y) = f 2. Hence it follows that f ∈ y + F(y) + Φ(y),
+as claimed in (3.30).
+It also follows that y ∈ D(F) ∩ D(Φ) = D(A) ∩ D(Φ) and hence
+D(A) = D(A) ∩ D(Φ).
+Step IV: Proof of (3.27) and (3.28). From (3.58), it implies that
+∥Ayλ∥2
+H ≤ C(1 + ∥f∥2
+H)ϑ,
+(3.69)
+where ϑ is given as in (3.29). For a fixed λ > 0 and w ∈ D(A), let
+gλ = µAw + B(w) + βC(w) + Φλ(w) + �κw.
+(3.70)
+Then
+∥gλ∥2
+H ≤ 2�κ2∥w∥2
+H + 2∥µAw + B(w) + βC(w) + Φλ(w)∥2
+H.
+(3.71)
+
+18
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+Analogous to (3.69) (for the solution yλ of (3.32) with f ∈ H), it yields from (3.71) that the
+solution w of (3.70) with gλ ∈ H satiesfies (3.27). Now, for w ∈ D(A)∩D(Φ) and ξ ∈ Φ(w),
+let
+g = µAw + B(w) + βC(w) + ξ + �κw.
+Since g ∈ H, we obtain a sequence {wλ}λ>0 ⊂ H such that wλ is a solution of
+µAwλ + B(wλ) + βC(wλ) + Φλ(wλ) + �κwλ = g,
+for all λ > 0.
+Then as similar to Step III, we get wλ → w in V and Awλ ⇀ Aw in H. Now we calculate
+the estimate ∥Awλ∥2
+H as we calculate above and then passing the limit as λ → 0, we obtain
+that
+∥Aw∥2
+H ≤ C(1 + ∥w∥2
+H + ∥µAw + B(w) + βC(w) + ξ∥2
+H)ϑ,
+where ϑ is defined as in (3.29) and this completes the proof of (3.28).
+□
+3.1. Proof of Theorem 1.3. From Proposition 3.3 and [6, Theorems 1.4-1.6, pp. 214-216],
+the problem (1.4) has unique solution y ∈ W1,∞(0, T; H) satisfying the equation (1.5). Let
+ξ(t) ∈ Φ(y(t)) be such that
+dy(t)
+dt
++ µAy(t) + B(y(t)) + βC(y(t)) + ξ(t) = f(t),
+(3.72)
+for a.e. t ∈ [0, T]. Since f ∈ W1,1(0, T; H), so f is absolutely continuous and subsequently
+f ∈ L∞(0, T; H) and therefore f − dy
+dt ∈ L∞(0, T; H). Then from (3.72), we have
+µAy(t) + B(y(t)) + βC(y(t)) + ξ(t) ∈ L∞(0, T; H),
+and thus from (3.28), we conclude that Ay ∈ L∞(0, T; H). Moreover, we have the Gelfand
+triplet D(A) ⊂ V ⊂ H, and
+y ∈ L∞(0, T; D(A)) and dy(t)
+dt
+∈ L∞(0, T; H),
+which imply that y ∈ C([0, T]; V).
+A similar result holds for the system (1.4), when one replaces Φ with the Yosida approxi-
+mation Φλ.
+Proposition 3.4. Let Φ ⊂ H × H satisfy Hypothesis 1.1. Let f ∈ W1,1(0, T; H) and y0 ∈
+D(A) ∩ D(Φ). Then there exists a unique strong solution
+yλ ∈ W1,∞(0, T; H) ∩ L∞(0, T; D(A)) ∩ C([0, T]; V)
+(3.73)
+to the problem
+
+
+
+dyλ(t)
+dt
++ µAyλ(t) + B(yλ(t)) + βC(yλ(t)) + Φλ(yλ(t)) = f(t),
+a.e. t ∈ (0, T),
+yλ(0) = y0.
+(3.74)
+Furthermore, yλ is right differentiable, d+yλ
+dt
+is right continuous, and
+d+yλ(t)
+dt
++ µAyλ(t) + B(yλ(t)) + βC(yλ(t)) + Φλ(yλ(t)) = f(t),
+for all t ∈ [0, T). (3.75)
+
+CBF EQUATIONS WITH POTENTIAL
+19
+Proof. From Proposition 3.1, we know that the operator y �→ F(y) = µAy +B(y)+βC(y)+
+κy is maximal monotone (for κ sufficiently large) in H × H. Since Φλ(·) is single-valued,
+monotone and demicontinuous in H × H (cf. [6, Proposition 1.3]), then by [6, Chapter 2,
+Corollary 1.1] (see [5] also), then the sum F(·) + Φλ(·) = µA + B(·) + βC(·) + κI + Φλ(·) is
+maximal monotone in H×H. Since f ∈ W1,1(0, T; H) and y0 ∈ D(A)∩D(Φ), an application
+of [6, Chapter 4, Theorem 1.8] yields the existence of a unique solution yλ ∈ W1,∞(0, T; H) to
+the problem (3.74). Arguing similarly as in the proof of Theorem 1.3 and using the estimate
+(3.27), one can conclude the proof of (3.73)-(3.75).
+□
+4. Proof of Theorem 1.4
+The aim of this section is to prove Theorem 1.4 using the solvability results obtained
+in Theorem 1.3. We first provide some uniform energy estimates for the solutions of the
+problem (3.74).
+4.1. Energy estimates for the solution of the problem (3.74). From Theorem 1.3, we infer
+that the problem (1.4) has a unique strong solution with the regularity given in (1.3). Our
+aim in this subsection is to obtain some energy estimates for the solution of the problem
+(1.6). In order to do this, we first obtain suitable energy estimates for the solution yλ(·) for
+the approximate problem (3.74), which also has a unique strong solution with the regularity
+given in (1.3).
+Proposition 4.1. Let yλ(·) be the unique strong solution of the problem (3.74) obtained in
+Proposition 3.4. Then for f ∈ W1,2(0, T; H) and y0 ∈ D(A) ∩ D(Φ), the solution yλ(·)
+satisfies the following energy estimates:
+sup
+t∈[0,T]
+∥yλ(t)∥2
+H + µ
+� T
+0
+∥yλ(t)∥2
+Vdt + β
+� T
+0
+∥yλ(t)∥r+1
+�Lr+1dt
+≤ C
+�
+∥y0∥H, ∥f∥L2(0,T;H), ∥Φ(0)∥H
+�
+,
+(4.1)
+where C is independent of λ. Furthermore, we have
+sup
+t∈[0,T]
+∥yλ(t)∥2
+V + µ
+� T
+0
+∥Ayλ(t)∥2
+Hdt + β
+� T
+0
+∥|∇yλ(t)||yλ(t)|
+r−1
+2 ∥Hdt
+≤ C
+�
+µ, β, T, ∥Ay0∥H, ϕ(y0), ∥Φ(y0)∥H, ∥Φ(0)∥H, ∥f(0)∥H, ∥f∥W1,2(0,T;H)
+�
+,
+(4.2)
+where C is independent of λ.
+Proof. We prove (4.1) and (4.2) in the following steps:
+Step I: Proof of (4.1). Taking the inner product with yλ(·) in (3.74), we get for a.e. t ∈ [0, T],
+1
+2
+d
+dt∥yλ(t)∥2
+H + µ∥yλ(t)∥2
+V + β∥yλ(t)∥r+1
+�Lr+1 + (Φλ(yλ(t)), yλ(t)) = (f(t), yλ(t)),
+(4.3)
+since (B(yλ), yλ) = 0. By the monotonicity of Φλ(·), [6, Chapter 2, Proposition 1.3, part(ii)]
+and using the Cauchy-Schwarz and Young’s inequalities, we have
+(Φλ(yλ(·)), yλ(·)) ≥ (Φλ(0), yλ(·)) ≥ −∥Φ(0)∥2
+H − 1
+4∥yλ(·)∥2
+H.
+(4.4)
+
+20
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+Using the estimate (4.4) and (2.1), and Cauchy-Schwarz and Young’s inequalities in (4.3),
+we deduce
+∥yλ(t)∥2
+H + µ
+� t
+0
+∥yλ(s)∥2
+Vds + 2β
+� t
+0
+∥yλ(s)∥r+1
+�Lr+1ds
+≤ ∥y0∥2
+H +
+1
+µλ1
+� t
+0
+∥f(s)∥2
+Hds + t∥Φ(0)∥2
+H,
+(4.5)
+for all t ∈ [0, T].
+Step II: Regularity estimates. In order to obtain the energy estimate (4.2), we first need fur-
+ther regularity estimates on the solution. This is due to the (H.3) assumption in Hypothesis
+1.1. We observe that yλ(·) satisfies for any h > 0
+dyλ(t + h)
+dt
++ µAyλ(t + h) + B(yλ(t + h)) + βC(yλ(t + h)) + Φλ(yλ(t + h)) = f(t + h),
+for a.e.
+t ∈ (0, T). Then subtracting above equation from (3.74), and taking the inner
+product with yλ(· + h) − yλ(·) and then using (2.6), we get for a.e. t ∈ (0, T)
+1
+2
+d
+dt∥yλ(t + h) − yλ(t)∥2
+H + µ∥yλ(t + h) − yλ(t)∥2
+V + β
+2 ∥|y(t)|
+r−1
+2 (y(t + h) − y(t))∥2
+H
+≤ (f(t + h) − f(t), yλ(t + h) − yλ(t)) − (B(yλ(t + h)) − B(yλ(t)), yλ(t + h) − yλ(t)),
+(4.6)
+where we have used the monotonicity property of the Yosida approximation Φλ(·) also, that
+is, (Φλ(yλ(· + h)) − Φλ(yλ(·)), yλ(· + h) − yλ(·)) ≥ 0. We consider the follwing cases:
+For r > 3. From (3.13) and the Cauchy-Schwarz inequality, (4.6) yields
+1
+2
+d
+dt∥yλ(t + h) − yλ(t)∥2
+H + µ
+2∥yλ(t + h) − yλ(t)∥2
+V
+≤ 1
+2∥f(t + h) − f(t)∥2
+H + 1
+2∥yλ(t + h) − yλ(t)∥2
+H + ̺∥yλ(t + h) − yλ(t)∥2
+H,
+(4.7)
+or we can write
+d
+dt∥yλ(t + h) − yλ(t)∥2
+H ≤ ∥f(t + h) − f(t)∥2
+H + (2̺ + 1)∥yλ(t + h) − yλ(t)∥2
+H,
+for a.e. t ∈ (0, T). By Gronwall’s inequality, we have
+∥yλ(t + h) − yλ(t)∥2
+H ≤ e(2̺+1)t
+�
+∥yλ(h) − yλ(0)∥2
+H +
+� t
+0
+∥f(s + h) − f(s)∥2
+Hds
+�
+,
+for all t ∈ [0, T]. On dividing by h2 and then taking limit as h → 0, we obtain for all t ∈ [0, T]
+����
+d+yλ(t)
+dt
+����
+2
+H
+≤ e(2̺+1)T
+�����
+d+yλ(0)
+dt
+����
+2
+H
++
+� T
+0
+����
+df
+dt (t)
+����
+2
+H
+dt
+�
+≤ Ce(2̺+1)T �
+µ∥Ayλ(0)∥2
+H + ∥B(yλ(0))∥2
+H + β∥C(yλ(0))∥2
+H + ∥Φλ(yλ(0))∥2
+H
++∥f(0)∥2
+H + ∥f∥W1,2(0,T;H)
+�
+≤ C
+�
+µ, β, T, ∥Ay0∥H, ∥Φ(y0)∥H, ∥f(0)∥H, ∥f∥W1,2(0,T;H)
+�
+,
+(4.8)
+
+CBF EQUATIONS WITH POTENTIAL
+21
+where we have used (3.75) and the fact that f ∈ W1,2(0, T; H) implies f ∈ C([0, T]; H). Now
+on integrating (4.7), we get
+∥yλ(t + h) − yλ(t)∥2
+H + µ
+� t
+0
+∥yλ(s + h) − yλ(s)∥2
+Vds
+≤ ∥yλ(h) − yλ(0)∥2
+H + (2̺ + 1)
+� t
+0
+∥yλ(s + h) − yλ(s)∥2
+Hds +
+� t
+0
+∥f(s + h) − f(s)∥2
+Hds,
+or we can write
+µ
+� t
+0
+∥yλ(s + h) − yλ(s)∥2
+Vds
+≤ ∥yλ(h) − yλ(0)∥2
+H + (2̺ + 1)
+� t
+0
+∥yλ(s + h) − yλ(s)∥2
+Hds +
+� t
+0
+∥f(s + h) − f(s)∥2
+Hds.
+On dividing both sides by h2 and then passing limit as h → 0, we get
+� T
+0
+����
+dyλ(s)
+ds
+����
+2
+V
+ds ≤ C
+�
+µ, β, T, ∥Ay0∥H, ∥Φ(y0)∥H, ∥f(0)∥H, ∥f∥W1,2(0,T;H)
+�
+.
+(4.9)
+For r = 3 with 2βµ ≥ 1. Once again by using Cauchy-Schwarz and Young’s inequalities,
+we get
+|(B(yλ(t + h)) − B(yλ(t)), yλ(t + h) − yλ(t))|
+≤ ∥yλ(t)(yλ(t + h) − yλ(t))∥H∥yλ(t + h) − yλ(t)∥V
+≤ 1
+2β∥yλ(t + h) − yλ(t)∥2
+V + β
+2 ∥yλ(t)(yλ(t + h) − yλ(t))∥2
+H.
+Thus we get from (4.6)
+d
+dt∥yλ(t + h) − yλ(t)∥2
+H + 2
+�
+µ − 1
+2β
+�
+∥yλ(t + h) − yλ(t)∥2
+V
+≤ ∥f(t + h) − f(t)∥2
+H + ∥yλ(t + h) − yλ(t)∥2
+H.
+Using the similar calculations as we have done in the case r > 3, we get the similar estimates
+as in (4.9).
+Taking the inner product with
+dyλ
+dt
+in (3.74) and then using the Cauchy-Schwarz and
+Young’s inequalities, we get for a.e. t ∈ (0, T)
+����
+dyλ(t)
+dt
+����
+2
+H
++ µ
+2
+d
+dt∥yλ(t)∥2
+V +
+β
+r + 1
+d
+dt∥yλ(t)∥r+1
+�Lr+1 +
+�dyλ(t)
+dt
+, Φλ(yλ(t))
+�
+=
+�
+f(t), dyλ(t)
+dt
+�
++
+�
+B(yλ(t)), dyλ(t)
+dt
+�
+.
+(4.10)
+We calculate
+�
+B(yλ), dyλ
+dt
+�
+by using the interpolation, H¨older’s and Young’s inequalities as
+�
+B(yλ), dyλ
+dt
+�
+= −b
+�
+yλ, dyλ
+dt , yλ
+�
+≤ ∥yλ∥2
+�L4
+����
+dyλ
+dt
+����
+V
+≤ 1
+2
+����
+dyλ
+dt
+����
+2
+V
++ 1
+2∥yλ∥
+2(r+1)
+r−1
+�Lr+1 ∥yλ∥
+2(r−3)
+r−1
+H
+.
+
+22
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+Therefore from (4.10), it is immediate that
+µ
+2 ∥yλ(t)∥2
+V +
+β
+r + 1∥yλ(t)∥r+1
+�Lr+1 + 1
+2
+� t
+0
+����
+dyλ(s)
+ds
+����
+2
+H
+ds +
+� t
+0
+�dyλ(s)
+ds
+, Φλ(yλ(s))
+�
+ds
+≤ µ
+2 ∥y0∥2
+V +
+β
+r + 1∥y0∥r+1
+�Lr+1 + 1
+2
+� t
+0
+∥f(s)∥2
+Hds + 1
+2
+� t
+0
+����
+dyλ(s)
+ds
+����
+2
+V
+ds
++ 1
+2t
+r−3
+r−1 sup
+s∈[0,t]
+∥yλ(s)∥
+2(r−3)
+r−1
+H
+�� t
+0
+∥yλ(s)∥r+1
+�Lr+1ds
+�
+2
+r−1
+,
+(4.11)
+for all t ∈ [0, T]. From Hypothesis 1.1, we know that Φ = ∂ϕ, where ϕ : H → ¯R is a lower
+semicontinuous proper convex function. Then by an application of [6, Chapter 2, Theorem
+2.2] yields that the Yosida approximation Φλ is the Gateaux derivative of ϕλ, for all λ > 0,
+that is, Φλ = ∇ϕλ, where ϕλ is the regularization of ϕ [6, pp. 64], given by
+ϕλ(y) = inf
+�∥y − z∥2
+H
+2λ
++ ϕ(z) : z ∈ H
+�
+,
+for all y ∈ H.
+(4.12)
+Moreover by a standard calculation, we have
+d
+ds[ϕλ(yλ(·))] =
+�dyλ(·)
+ds
+, (∇ϕλ)(yλ(·))
+�
+,
+(4.13)
+and
+� t
+0
+�dyλ(s)
+ds
+, Φλ(yλ(s))
+�
+ds = ϕλ(yλ(t)) − ϕλ(y0),
+(4.14)
+for all t ∈ [0, T]. From [6, Chapter 2, Proposition 1.3], we infer that Jλ := (I + λΦ)−1 is
+bounded on bounded subsets of H. Furthermore, from [6, Chapter 2, Theorem 2.2], we also
+have
+ϕ(Jλ(y)) ≤ ϕλ(y) ≤ ϕ(y),
+for all λ > 0, y ∈ H.
+(4.15)
+From [6, Chapter 2, Proposition 2.1], we know that any proper lower semicontinuous convex
+function is bounded from below by an affine function. Therefore, there exists w ∈ H and
+q ∈ R such that
+ϕ(y) ≥ (y, w) + q,
+for all y ∈ H.
+(4.16)
+From (4.15), (4.16) and application of the Cauchy-Schwarz inequality yield
+−ϕλ(yλ) ≤ −ϕ(Jλ(yλ)) ≤ −(Jλ(yλ), w) − q ≤ ∥Jλ(yλ)∥H∥w∥H + |q| ≤ C,
+(4.17)
+where C is independent of λ. Thus using ϕλ(y0) ≤ ϕ(y0) in (4.14), we deduce
+−
+� t
+0
+�dyλ(s)
+ds
+, Φλ(yλ(s))
+�
+ds ≤ C,
+(4.18)
+where the constant C depends on ϕ(y0). Thus from (4.1), (4.9) and (4.18), we get for all
+t ∈ [0, T]
+µ
+2 ∥yλ(t)∥2
+V +
+β
+r + 1∥yλ(t)∥r+1
+�Lr+1 + 1
+2
+� t
+0
+����
+dyλ(s)
+ds
+����
+2
+H
+ds ≤ C,
+(4.19)
+where C = C
+�
+µ, β, T, ∥Ay0∥H, ϕ(y0), ∥Φ(y0)∥H, ∥Φ(0)∥H, ∥f(0)∥H, ∥f∥W1,2(0,T;H)
+�
+.
+
+CBF EQUATIONS WITH POTENTIAL
+23
+Step III: Proof of (4.2). We take the inner product with Ayλ(·) in (3.74) to obtain
+1
+2
+d
+dt∥yλ(t)∥2
+V + µ∥Ayλ(t)∥2
+H + β(C(yλ(t)), Ayλ(t))
+= (f(t), Ayλ(t)) − (B(yλ(t)), Ayλ(t)) − (Φλ(yλ(t)), Ayλ(t)),
+(4.20)
+for a.e. t ∈ [0, T]. This yields
+∥yλ(t)∥2
+V + 2µ
+� t
+0
+∥Ayλ(s)∥2
+Hds + 2β
+� t
+0
+(C(yλ(s)), Ayλ(s))ds
+= ∥y0∥2
+V + 2
+� t
+0
+(f(s), Ayλ(s))ds − 2
+� t
+0
+(B(yλ(s)), Ayλ(s))ds
+− 2
+� t
+0
+(Φλ(yλ(s)), Ayλ(s))ds,
+(4.21)
+for all t ∈ [0, T]. We consider the cases r > 3 and for d = r = 3 separately.
+Case I: For r > 3. From the equality (1.7), we infer
+(C(yλ), Ayλ) = ∥|∇yλ||yλ|
+r−1
+2 ∥2
+H + 4
+� r − 1
+(r + 1)2
+�
+∥|∇|yλ|
+r+1
+2 |∥2
+H.
+(4.22)
+From [47, Lemma 3.1, pp. 404], for d = 2, we have (B(yλ), Ayλ) = 0. For d = 3 with
+r ∈ (3, ∞), we estimate |(B(yλ), Ayλ)| as in (3.43)-(3.44) as
+|(B(yλ), Ayλ)| ≤ µ
+2∥Ayλ∥2
+H + β
+2 ∥|∇yλ||yλ|
+r−1
+2 ∥2
+H + ̺∥yλ∥2
+V,
+(4.23)
+where ̺ =
+r−3
+2µ(r−1)
+�
+2
+βµ(r−1)
+�
+2
+r−3. From condition (H.3) of Hypothesis 1.1, (4.5) and (4.22)-
+(4.23), we obtain from (4.21) that
+∥yλ(t)∥2
+V + µ
+2
+� t
+0
+∥Ayλ(s)∥2
+Hds + 3β
+2
+� t
+0
+∥|∇yλ(s)||yλ(s)|
+r−1
+2 ∥2
+Hds
+≤ C +
+�
+ς
+� t
+0 ∥Φλ(yλ(s))∥2
+Hds,
+for d = 2 with r ∈ (3, ∞) and d = 3 with r ∈ (3, 5),
+0,
+for d = 3 with r ∈ [5, ∞).
+(4.24)
+where C = C(µ, β, T, ∥y0∥H, ∥f∥L2(0,T;H), ∥Φ(0)∥H). This completes the proof of (4.2) for
+d = 3 with r ∈ [5, ∞).
+Case II: For d = r = 3 with 2βµ ≥ 1. We calculate
+(C(yλ), Ayλ) = ∥|∇yλ||yλ|∥2
+H + 1
+2∥|∇|yλ|2|∥2
+H,
+|(B(yλ), Ayλ)| ≤ µ
+2 ∥Ayλ∥2
+H + 1
+2µ∥|yλ||∇yλ|∥2
+H,
+|(f, Ayλ)| ≤ µ
+8 ∥Ayλ∥2
+H + 1
+2µ∥f∥2
+H.
+Using above estimates and Hypothesis 1.1 in (4.21), we obtain
+∥yλ(t)∥2
+V + 3µ
+4
+� t
+0
+∥Ayλ(s)∥2
+Hds + 2
+�
+β − 1
+2µ
+� � t
+0
+∥|∇yλ(s)||yλ(s)|∥2
+Hds
+
+24
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+≤ C + ς
+� t
+0
+∥Φλ(yλ(s))∥2
+Hds,
+(4.25)
+where constant C is same as given in (4.24).
+Step IV: An estimate for
+� t
+0 ∥Φλ(yλ(s))∥2
+Hds. Let us now find a bound for
+� t
+0 ∥Φλ(yλ(s))∥2
+Hds.
+For this, taking the inner product of (3.74) with Φλ(yλ(·)), we get for a.e. t ∈ (0, T)
+�dyλ(t)
+ds
+, Φλ(yλ(t))
+�
++ µ(Φλ(yλ(t)), Ayλ(t)) + (B(yλ(t)), Φλ(yλ(t))
++ β(C(yλ(t)), Φλ(yλ(t))) + ∥Φλ(yλ(t))∥2
+H = (f(t), Φλ(yλ(t))).
+(4.26)
+Integrating (4.26) and using the condition (H3) of Hypothesis 1.1 and (4.18), we obtain
+(1 − µς)
+� t
+0
+∥Φλ(yλ(s))∥2
+Hds ≤ C +
+� t
+0
+(f(s), Φλ(yλ(s)))ds −
+� t
+0
+(B(yλ(s)), Φλ(yλ(s))ds
+− β
+� t
+0
+(C(yλ(s)), Φλ(yλ(s)))ds,
+(4.27)
+for all t ∈ [0, T]. A calculation similar to (3.53) yields
+����
+� t
+0
+(B(yλ(s)), Φλ(yλ(s))ds
+���� ≤ C + 1 − µς
+8
+� t
+0
+∥Φλ(yλ(s))∥2
+Hds
++ µ(1 − µς)
+8ς
+� t
+0
+∥Ayλ(s)∥2
+Hds,
+(4.28)
+where we have used (4.19) also. Using the estimates (4.19) and (4.24), we find
+����
+� t
+0
+(C(yλ(s)), Φλ(yλ(s)))ds
+���� ≤ C
+� t
+0
+∥yλ(s)∥r
+�L2r∥Φλ(yλ(s))∥Hds
+≤ C sup
+s∈[0,t]
+∥yλ(s)∥
+r+3
+4
+�Lr+1
+� t
+0
+∥yλ(s)∥
+3(r−1)
+4
+�L3(r+1)∥Φλ(yλ(s))∥Hds
+≤ Ct
+5−r
+4(r+1)
+�� t
+0
+∥Φλ(yλ(s))∥2
+Hds
+� 1
+2�� t
+0
+∥yλ(s)∥r+1
+�L3(r+1)ds
+� 3(r−1)
+4(r+1)
+≤ C
+�
+C + ς
+� t
+0
+∥Φλ(yλ(s))∥2
+Hds
+� 3(r−1)
+4(r+1)�� t
+0
+∥Φλ(yλ(s))∥2
+Hds
+� 1
+2
+≤ 1 − µς
+8β
+� t
+0
+∥Φλ(yλ(s))∥2
+Hds + C,
+(4.29)
+for d = 3 and r ∈ (3, 5). We further calculate by using Sobolev’s embedding V ⊂ �L2r and
+(4.19), for d = 2 with r ∈ (3, ∞) as
+����
+� t
+0
+(C(yλ(s)), Φλ(yλ(s)))ds
+���� ≤ C
+� t
+0
+∥yλ(s)∥r
+�L2r∥Φλ(yλ(s))∥Hds
+≤ C
+� t
+0
+∥yλ(s)∥2r
+V ds + 1 − µς
+8β
+� t
+0
+∥Φλ(yλ(s))∥2
+Hds
+≤ C + 1 − µς
+8β
+� t
+0
+∥Φλ(yλ(s))∥2
+Hds.
+(4.30)
+
+CBF EQUATIONS WITH POTENTIAL
+25
+Using the Cauchy-Schwarz and Young’s inequailities, we further have
+����
+� t
+0
+(f(s), Φλ(yλ(s)))ds
+���� ≤ C
+� t
+0
+∥f(s)∥2
+Hds + 1 − µς
+4
+� t
+0
+∥Φλ(yλ(s))∥2
+Hds.
+(4.31)
+By using (4.28)-(4.31), we conclude from (4.27) that
+(1 − µς)
+� t
+0
+∥Φλ(yλ(s))∥2
+Hds ≤ C + C
+� t
+0
+∥f(s)∥2
+Hds + 1 − µς
+2
+� t
+0
+∥Φλ(yλ(s))∥2
+Hds
++ µ(1 − µς)
+8ς
+� t
+0
+∥Ayλ(s)∥2
+Hds,
+or we can write
+ς
+� t
+0
+∥Φλ(yλ(s))∥2
+Hds ≤ C + C
+� t
+0
+∥f(s)∥2
+Hds + µ
+4
+� t
+0
+∥Ayλ(s)∥2
+Hds.
+(4.32)
+Thus from (4.24), for d = 2, 3 with r > 3, we get
+∥yλ(t)∥2
+V + µ
+4
+� t
+0
+∥Ayλ(s)∥2
+Hds + 3β
+2
+� t
+0
+∥|∇yλ(s)||yλ(s)|
+r−1
+2 ∥2
+Hds ≤ C.
+(4.33)
+Also from (4.25), for d = r = 3 with 2βµ ≥ 1, we find
+∥yλ(t)∥2
+V + µ
+2
+� t
+0
+∥Ayλ(s)∥2
+Hds + 2
+�
+β − 1
+2µ
+� � t
+0
+∥|∇yλ(s)||yλ(s)|∥2
+Hds ≤ C.
+(4.34)
+Combining the above estimates with (4.32), we deduce
+� t
+0
+∥Φλ(yλ(s))∥2
+Hds ≤ C,
+(4.35)
+which completes the proof.
+□
+4.2. Passing to the limit as λ → 0. Let us now pass λ → 0 and obtain the energy estimates
+for the solution of the problem (1.6).
+Proposition 4.2. The limit of the sequence (yλ)λ>0 satisfies the problem (1.6) for a.e. t ∈
+(0, T) in H.
+Proof. We prove (yλ) satisfies the problem (1.6) for a.e. t ∈ (0, T) in H in the following
+steps:
+Step I: For d = 2 with r ∈ (3, ∞) and d = 3 with r ∈ (3, 5). From the Proposition 3.4, we
+have
+yλ ∈ W1,∞(0, T; H) ∩ L∞(0, T; D(A)) ∩ C([0, T]; V).
+(4.36)
+From (4.33)-(4.35), we have uniform bounds for the sequences
+(Ayλ)λ>0 and (Φλ(yλ))λ>0 in L2(0, T; H).
+Then from (2.3)-(2.4), we have the sequence (B(yλ))λ is bounded in L2(0, T; H), since
+� T
+0
+∥B(yλ(t))∥2
+Hdt ≤ C
+
+
+
+
+
+
+
+T 1/2 sup
+t∈[0,T]
+(∥yλ(t)∥H∥yλ(t)∥2
+V)
+�� T
+0 ∥Ayλ(t)∥2
+Hdt
+�1/2
+,
+for d = 2,
+T 1/2 sup
+t∈[0,T]
+∥yλ(t)∥3
+V
+�� T
+0 ∥Ayλ(t)∥2
+Hdt
+�1/2
+,
+for d = 3,
+
+26
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+≤ C.
+(4.37)
+Moreover, from (4.19), we have
+�
+dyλ
+dt
+�
+λ>0 is uniformly bounded in L2(0, T; H). Using (3.74)
+and the energy estimates (4.19), (4.33)-(4.35) and (4.37), we find
+� T
+0
+∥C(yλ(t))∥2
+Hdt ≤ C
+� T
+0
+�����
+dyλ(t)
+dt
+����
+2
+H
++ ∥Ayλ(t)∥2
+H + ∥B(yλ(t))∥2
+H + ∥Φλ(yλ(t))∥2
+H
++ ∥f(t)∥2
+H
+�
+dt
+≤ C.
+Therefore the sequence (C(yλ))λ>0 is bounded in L2(0, T; H). Thus by making the use of the
+Banach-Alaoglu theorem, we infer
+
+
+
+
+
+
+
+
+
+yλ
+∗⇀ y
+in L∞(0, T; V ∩ �Lr+1),
+yλ ⇀ y
+in Lr+1(0, T; �L3(r+1)),
+dyλ
+dt ⇀ dy
+dt
+in L2(0, T; V),
+
+
+
+
+
+
+
+
+
+
+
+Ayλ ⇀ Ay
+in L2(0, T; H),
+B(yλ) ⇀ ζ
+in L2(0, T; H),
+C(yλ) ⇀ ϑ
+in L2(0, T; H),
+Φλ(yλ) ⇀ φ
+in L2(0, T; H).
+(4.38)
+Since V ֒→ H ֒→ V′, the embedding of V ֒→ H is compact, and the fact that y ∈ L∞(0, T; V),
+dy
+dt ∈ L2(0, T; H) ֒→ L2(0, T; V′) imply
+yλ → y in C([0, T]; H),
+(4.39)
+by an application of the Aubin-Lions compactness lemma. Since D(A) ֒→ V ֒→ H, (yλ)λ>0
+is bounded in L2(0, T; D(A)) and
+�
+dyλ
+dt
+�
+λ>0 is bounded in L2(0, T; H), and the embedding
+D(A) ֒→ V is compact, it implies once again from Aubin-Lions compactness lemma that
+yλ → y in L2(0, T; V).
+(4.40)
+From [6, Chapter 2, Proposition 1.4, part(i)], we know that (I + λΦ)−1 is nonexpansive, that
+is, Lipschitz with Lipschitz constant 1 and from [6, Chapter 2, Proposition 1.3, part (iii)],
+we have
+� T
+0
+∥(I + λΦ)−1yλ(t) − y(t)∥2
+Hdt
+≤ 2
+� T
+0
+∥(I + λΦ)−1(yλ(t)) − (I + λΦ)−1y(t)∥2
+Hdt + 2
+� T
+0
+∥(I + λΦ)−1y(t) − y(t)∥2
+Hdt
+≤ 2
+� T
+0
+∥yλ(t) − y(t)∥2
+Hdt + 2
+� T
+0
+∥(I + λΦ)−1y(t) − y(t)∥2
+Hdt
+→ 0 as λ → 0,
+so that (I + λΦ)−1(yλ) → y in L2(0, T; H) and (I + λΦ)−1(yλ) → (I + λΦ)−1y, for a.e.
+t ∈ (0, T) in H (along a subsequence, which is still denoted by the same). From [6, Chapter
+2, Proposition 1.1, part (i)] and [38, Proposition 1.7], we know that the maximal monotone
+operator Φ is weak-strong and strong-weak closed in H × H, that is, if Φλ(yλ) ∈ Φ(I +
+λΦ)−1(yλ), (I + λΦ)−1(yλ) → y in L2(0, T; H) and Φλ(yλ) ⇀
+φ in
+L2(0, T; H), then
+φ ∈ Φ(y) for a.e. t ∈ (0, T) in H.
+
+CBF EQUATIONS WITH POTENTIAL
+27
+Convergence of Bilinear operator B(·). We have from (2.3) and (2.4)
+∥(B(yλ) − B(y)∥H ≤ C ×
+
+
+
+
+
+
+
+
+
+
+
+
+
+∥yλ − y∥
+1
+2
+H∥yλ − y∥
+1
+2
+V∥yλ∥
+1
+2
+V∥Ayλ∥
+1
+2
+H
++∥y∥
+1
+2
+H∥y∥
+1
+2
+V∥yλ − y∥
+1
+2
+V∥Ayλ − Ay∥
+1
+2
+H,
+for d = 2,
+∥yλ − y∥H∥yλ∥
+1
+2
+V∥Ayλ∥
+1
+2
+H
++∥y∥H∥yλ − y∥
+1
+2
+V∥Ayλ − Ay∥
+1
+2
+H,
+for d = 3.
+For d = 2, we calculate
+� T
+0
+∥B(yλ(t)) − B(y(t))∥2
+Hdt
+≤ C
+� T
+0
+∥yλ(t) − y(t)∥H∥yλ(t) − y(t)∥V∥yλ(t)∥V∥Ayλ(t)∥Hdt
++ C
+� T
+0
+∥y(t)∥H∥y(t)∥V∥yλ(t) − y(t)∥V∥Ayλ(t) − Ay(t)∥Hdt
+≤ C∥yλ − y∥L∞(0,T;H)∥yλ∥L∞(0,T;V)∥yλ − y∥L2(0,T;V)∥Ayλ∥L2(0,T;H)
++ ∥y∥L∞(0,T;H)∥y∥L∞(0,T;V)∥yλ − y∥L2(0,T;V)∥Ayλ − Ay∥L2(0,T;H)
+→ 0, as λ → 0,
+(4.41)
+where we have used the H¨older’s inequality, strong convergences (4.39)-(4.40) and (4.2).
+For d = 3, we estimate
+� T
+0
+∥B(yλ(t)) − B(y(t))∥2
+Hdt ≤ C
+� T
+0
+∥yλ(t) − y(t)∥2
+V∥yλ(t)∥V∥Ayλ(t)∥Hdt
++ C
+� T
+0
+∥y(t)∥2
+V∥yλ(t) − y(t)∥V∥Ayλ(t) − Ay(t)∥Hdt
+≤ ∥yλ − y∥2
+L∞(0,T;V)∥yλ∥L2(0,T;V)∥Ayλ∥L2(0,T;H)
++ ∥y∥2
+L∞(0,T;V)∥yλ − y∥L2(0,T;V)∥Ayλ − Ay∥L2(0,T;H)
+→ 0 as λ → 0,
+(4.42)
+where we have used the H¨older’s inequality, strong convergences (4.39)-(4.40) and (4.2).
+Convergence of nonlinear operator C(·). By using Taylor’s formula [18, Theorem 7.9.1] and
+H¨older’s inequality, we have
+� T
+0
+∥C(yλ(t)) − C(y(t))∥
+r+1
+r
+H
+dt
+≤
+� T
+0
+�� 1
+0
+∥C′(θyλ(t) + (1 − θ)y(t))(yλ(t) − y(t))∥
+r+1
+r
+H
+dθ
+�
+dt
+≤ r
+� T
+0
+�
+∥yλ(t)∥r−1
+�L2r + ∥y(t)∥r−1
+�L2r
+� r+1
+r ∥yλ(t) − y(t)∥
+r+1
+r
+�L2r dt
+≤ C
+�� T
+0
+∥yλ(t)∥
+(r−1)(r+1)
+r
+�L2r
+∥yλ(t) − y(t)∥
+r+1
+r
+�L2r dt +
+� T
+0
+∥y(t)∥
+(r−1)(r+1)
+r
+�L2r
+∥yλ(t) − y(t)∥
+r+1
+r
+�L2r dt
+�
+.
+(4.43)
+
+28
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+Using the interpolation in 2 ≤ 2r ≤ 3(r + 1) and H¨older’s inequalities, we obtain
+� T
+0
+∥yλ(t)∥
+(r−1)(r+1)
+r
+�L2r
+∥yλ(t) − y(t)∥
+r+1
+r
+�L2r dt
+≤
+� T
+0
+∥yλ(t)∥
+(r+3)(r−1)(r+1)
+r2(3r+1)
+H
+∥yλ(t)∥
+3(r−1)2(r+1)2
+r2(3r+1)
+�L3(r+1)
+∥yλ(t) − y(t)∥
+(r+3)(r+1)
+r2(3r+1)
+H
+× ∥y(t)λ − y(t)∥
+3(r−1)(r+1)2
+r2(3r+1)
+�L3(r+1)
+dt
+≤
+�
+∥yλ∥
+(r+3)(r−1)(r+1)
+r2(3r+1)
+L∞(0,T;H)
+��
+sup
+t∈[0,T]
+∥yλ(t) − y(t)∥
+(r+3)(r+1)
+r2(3r+1)
+H
+�
+×
+� T
+0
+∥yλ(t)∥
+3(r−1)2(r+1)2
+r2(3r+1)
+�L3(r+1)
+∥yλ(t) − y(t)∥
+3(r−1)(r+1)2
+r2(3r+1)
+�L3(+1)
+dt
+≤
+�
+∥yλ∥
+(r+3)(r−1)(r+1)
+r2(3r+1)
+L∞(0,T;H)
+��
+sup
+t∈[0,T]
+∥yλ(t) − y(t)∥
+(r+3)(r+1)
+r2(3r+1)
+H
+�
+× T
+r(3r+1)
+r+3 ∥yλ∥Lr+1(0,T;�L3(r+1))∥yλ − y∥Lr+1(0,T;�L3(r+1))
+→ 0 as λ → 0,
+where we have used the H¨older’s inequality, strong convergence (4.39) and energy estimates
+(4.1)-(4.2), and for H¨older’s inequality, we use the exponents
+3(r−1)2(r+1)
+r2(3r+1)
++ 3(r−1)(r+1)
+r2(3r+1)
++
+r+3
+r(3r+1) = 1. Similarly,
+� T
+0
+∥yλ(t)∥
+(r−1)(r+1)
+r
+�L2r
+∥yλ(t) − y(t)∥
+r+1
+r
+�L2r dt → 0, as λ → 0.
+Hence from (4.43), we conclude that
+C(yλ) → C(y) strongly in L
+r+1
+r (0, T; H).
+(4.44)
+Letting λ → 0 in (3.74), we obtain that y satisfies the problem (1.6).
+□
+4.3. Uniqueness of solution to the problem (1.6). Let us now prove that the solution ob-
+tained by passing to the limit with λ → 0 is unique.
+Proposition 4.3. The solution for the problem (1.6) is unique.
+Proof. Let y1(·) and y2(·) be two solutions of (1.6) satisfying (4.1)-(4.2). Then we have
+for a.e. t ∈ (0, T),
+1
+2
+d
+dt∥y1(t) − y2(t)∥2
+H + µ∥y1(t) − y2(t)∥2
+V + (B(y1(t)) − B(y2(t)), y1(t) − y2(t))
++ β(C(y1(t)) − C(y2(t)), y1(t) − y2(t)) + (ξ1(t) − ξ2(t), y1(t) − y2(t)) = 0,
+where ξj(·) ∈ Φ(yj(·)), for j = 1, 2. Integrating the above equality and using the monotonicity
+of Φ, we can write
+∥y1(t) − y2(t)∥2
+H + 2µ
+� t
+0
+∥y1(s) − y2(s)∥2
+Vds
+
+CBF EQUATIONS WITH POTENTIAL
+29
+≤ ∥y1(0) − y2(0)∥2
+H − 2
+� t
+0
+(B(y1(s)) − B(y2(s)), y1(s) − y2(s))ds
+− 2β
+� t
+0
+(C(y1(s)) − C(y2(s)), y1(s) − y2(s))ds.
+(4.45)
+Using calculations similar to (3.12)-(3.13), we find
+|(B(y1) − B(y2), y1 − y2)| ≤ µ
+2 ∥y1 − y2∥2
+V + β
+2 ∥|y2|
+r−1
+2 (y1 − y2)∥2
+H + ̺∥y1 − y2∥2
+H. (4.46)
+Also calculation similar to (2.6) gives
+β(C(y1) − C(y2), y1 − y2) ≥ β
+2 ∥|y2|
+r−1
+2 (y1 − y2)∥2
+H.
+(4.47)
+Using (4.46)-(4.47) in (4.45), we get
+∥y1(t) − y2(t)∥2
+H + µ
+� t
+0
+∥y1(s) − y2(s)∥2
+Vds
+≤ ∥y1(0) − y2(0)∥2
+H + ̺
+� t
+0
+∥y1(s) − y2(s)∥2
+Hds,
+(4.48)
+for all t ∈ (0, T). Applying Gronwall’s inequality in (4.48), we obtain for all t ∈ (0, T)
+∥y1(t) − y2(t)∥2
+H ≤ ∥y1(0) − y2(0)∥2
+He̺T ,
+which proves the uniqueness.
+□
+4.4. Proof of Theorem 1.4. From Proposition 3.3, we know that the operator A(·) is m-
+accretive in H × H for sufficiently large κ ≥ ̺. So by using the abstract theory, we obtain
+the regularity (1.3) given in Theorem 1.3. Moreover, from Proposition 4.2, the solution
+y ∈ L2(0, T; D(A)) ∩ Lr+1(0, T; �L3(r+1)) ∩ W1,2(0, T; V)
+satisfies (1.6) in H for a.e. in t ∈ (0, T). The uniqueness of the problem (1.6) follows from
+Proposition 4.3 and this completes the proof.
+5. Applications
+We discuss some applications of the results obtained in Theorems 1.3 and 1.4. These
+include flow invariance preserving feedback controllers, a time optimal control problem and
+stabilizing feedback controllers for 2D and 3D CBF equations, etc.
+5.1. Flow invariance preserving feedback controllers ([13]). Let us consider the following
+controlled CBF equations:
+
+
+
+dy(t)
+dt
++ µAy(t) + B(y(t)) + βC(y(t)) = f(t) + U(t), t ∈ (0, T],
+y(0) = y0,
+(5.1)
+where U(·) is distributed control acting on the system, f ∈ W1,1(0, T; H) and y0 ∈ D(A).
+Consider a closed and convex set K ⊂ H such that 0 ∈ K and
+(I + λA)−1K ⊂ K, for all λ > 0.
+(5.2)
+
+30
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+Our aim is to search for a feedback control U = Ψ(y) such that y(t) ∈ K, for all t ∈ [0, T], if
+y0 ∈ K. That is, we have to find a feedback controller for which the set K is invariant with
+respect to CBF flow. We establish this by solving the following CBF inclusion problem:
+dy(t)
+dt
++ µAy(t) + B(y(t)) + βC(y(t)) − f(t) + NK(y(t)) ∋ 0, t ∈ (0, T],
+(5.3)
+where NK(y) = {w ∈ H : (w, y − z) ≥ 0, for all z ∈ K} is the well-known Clark’s normal
+cone to K at y. We consider the indicator function IK : H → R ([6]) given by
+IK(x) =
+�
+0,
+if x ∈ K,
++∞,
+if x /∈ K,
+whose subdifferential is given by
+∂IK(x) =
+
+
+
+
+
+∅,
+if x /∈ K,
+{0},
+if x ∈ int(K),
+NK(x) = {y ∈ H : (y, x − z) ≥ 0, for all z ∈ K},
+if x ∈ ∂K,
+where int(K) and ∂K denote the interior and boundary of K, respectively. Then regulariza-
+tion of IK is given by [6, Chapter 2, Theorem 2.2]
+(IK)λ(x) = 1
+2λ∥x − PK(x)∥2
+H,
+and its Gateaux derivative
+(∂IK)λ(x) = 1
+λ(x − PK(x)),
+where PK : L2(Td) → K is the projection operator of x onto K which is equal to the resolvent
+(I + λ∂IK)−1. It implies that the above derivative is equal to the Yosida approximation of
+∂IK, that is
+(∂IK)λ(x) = 1
+λ(x − (I + λ∂IK)−1(x)), for all x ∈ H.
+We observe that the multi valued operator Φ := NK is a maximal monotone operator with
+0 ∈ D(Φ) = D(∂IK) = K. Also, the operator A is single-valued maximal monotone, and thus
+from [5, Chapter IV, Proposition 1.1, part(iv)], we have
+(Ay, (∂IK)λ(y)) ≥ 0,
+for all y ∈ D(A), λ > 0.
+Thus the multi valued operator NK = ∂IK satisfies all the assumptions (H1)-(H3) of Hypoth-
+esis 1.1 and therefore we can apply the main result Theorem 1.3 to the inclusion problem
+(5.3) to determine a feedback controller U ∈ L∞(0, T; H) with U(t) ∈ −NK(y(t)) for a.e.
+t ∈ [0, T] which is given by
+U(t) = − f(t) + µAy(t) + B(y(t)) + βC(y(t))
+− (−f(t) + µAy(t) + B(y(t)) + βC(y(t)) + NK(y(t)))0,
+for all t ∈ [0, T),
+(5.4)
+where NK(y) is the H-valued normal cone to K at y.
+Flow invariance for the estrophy of the system. We consider the constraint set
+K = {y ∈ V : ∥∇ × y∥H = ∥∇y∥H = ∥A
+1
+2y∥H ≤ ̟}.
+
+CBF EQUATIONS WITH POTENTIAL
+31
+Let f be any arbitrary element of K such that y + λAy = f. Taking the inner product with
+Ay and using the Cauchy-Schwarz and Young’s inequalities, we obtain
+∥y∥2
+V + λ∥Ay∥2
+H ≤ 1
+2∥f∥2
+V + 1
+2∥y∥2
+V ⇒ ∥y∥V ≤ ∥f∥V,
+for all λ > 0. Thus from the definition of K we have y ∈ K and this imply (I+λA)−1K ⊂ K.
+We will find a feedback control so that enstrophy of the system kept inside this constraint
+set K. The normal cone corresponding to the convex set K is
+NK(y) =
+
+
+
+0,
+if ∥∇y∥H < ̟,
+�
+λ>0
+λAy,
+if ∥∇y∥H = ̟.
+The feedback control is given by
+U(·) ∈ −NK(y(·)).
+For ∥∇y∥H < ̟, that is, when the flow remain inside the constraint set K, we have
+U(t) = 0. For ∥∇y∥H = ̟,
+U(t) = −λ0Ay(t), for a.e. t ∈ [0, T],
+(5.5)
+for some λ0 > 0. Then from (5.4), we have
+U(t) = − f(t) + µAy(t) + B(y(t)) + βC(y(t)) + d+y(t)
+dt
+.
+Taking the inner product with Ay and using (5.5), we obtain
+λ0 =
+−1
+∥Ay∥2
+H
+�
+(f, Ay) − µ∥Ay∥2
+H − b(y, y, Ay) − (C(y), Ay)
+�
+,
+where we have used the fact that
+�d+y(t)
+dt
+, Ay(t)
+�
+= d+
+dt ∥∇y(t)∥2
+H = 0.
+Therefore the feedback control becomes
+U(t) = −Ay(t)
+∥Ay(t)∥2
+H
+�
+(f(t), Ay(t)) − µ∥Ay(t)∥2
+H − b(y(t), y(t), Ay(t)) − (C(y(t)), Ay(t))
+�
+,
+for all t ∈ [0, T).
+Thus, for y0 ∈ D(A) ∩ K and f ∈ W1,1(0, T; H), and the feedback
+control given above, the closed loop problem (5.1) has a unique solution y ∈ W1,∞(0, T; H)∩
+L∞(0, T; D(A)) which satisfies
+y(t) ∈ K,
+for all t ∈ [0, T].
+We refer the interested readers to [13] for some other important flow invariance problems
+like localized dissipation, pointwise velocity constraints, pointwise vorticity contraint, helicity
+invariance, etc.
+
+32
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+5.2. A time optimal control problem ([7, 37]). Let us discuss the following time optimal
+control for CBF equations
+
+
+
+dy(t)
+dt
++ µAy(t) + B(y(t)) + βC(y(t)) = U(t), for a.e. t > 0, in H,
+y(0) = y0,
+(5.6)
+Let κ > 0 and we define the class of controls
+Uκ = {U(·) ∈ L∞(R+; H) : ∥U(t)∥H ≤ κ, a.e. t > 0}.
+Let y0 and y1 be arbitrary but fixed. A control U(·) ∈ Uκ is said to be admissible if it
+steers from the (initial state) y0 to the (target) y1 in a finite time T along the trajectory
+y(t; y0, U(·)) of (5.6) which starts from y0. We assume that the class of all such controls
+(admissible class) is nonempty. Let T(y0, y1) be the infimum of all such times and it is called
+minimal time, that is,
+T(y0, y1) := inf
+T∈R+{T : y(T; y0, U(·)) = y1, U(·) ∈ Uκ}.
+A control U∗(·) such that y(T(y0, y1); y0, U∗(·)) = y1 is called time optimal control and the
+time T(y0, y1) is said to be optimal time. The pair (y∗, U∗) is called the time optimal pair,
+where y∗ = y(t; y0, U∗).
+We define a multivalued operator sgn : H → H by
+sgny =
+�
+y
+∥y∥H,
+if y ̸= 0,
+{z ∈ H : ∥z∥H ≤ 1},
+if y = 0,
+which is the subdifferential of ∥y∥H and hence it is maximal monotone in H×H ([6, Theorem
+2.1, Chapter 2]). From [6, Proposition 2.4, Chapter 4], the Yosida approximation of B :=
+κ sgn(·) is given by
+Bλ(y) = 1
+λ
+�
+y − (I + λB)−1y
+�
+=
+�
+κy
+∥y∥H,
+if ∥y∥H ≥ λ,
+κ
+λy,
+if ∥y∥H < λ.
+From the above definition, we conclude that
+(Ay, Bλ(y − y1)) ≥ 0,
+for all y ∈ D(A), λ > 0,
+and therefore all the assumptions of Hypothesis 1.1 are satisfied. Thus we can apply the
+existence and uniqueness result (see Theorem 1.3) of strong solutions for the system
+
+
+
+dy(t)
+dt
++ µAy(t) + B(y(t)) + βC(y(t)) + κ(sgn(y(t) − y1)) ∋ 0, for a.e. t > 0,
+y(0) = y0.
+(5.7)
+Then the feedback law U(t) ∈ −κ(sgn(y(t) − y1)), for t > 0, ensures the existence of an
+admissible control U(·) ∈ Uκ for the system (5.6), under the assumption that
+∥µAy1 + B(y1) + βC(y1)∥H < κ,
+(5.8)
+and
+∥y0 − y1∥H ≤ κ − ∥µAy1 + B(y1) + βC(y1)∥H
+̺
+,
+(5.9)
+
+CBF EQUATIONS WITH POTENTIAL
+33
+for y0, y1 ∈ D(A), where ̺ is given as in (3.1). In order to prove this, we show that the
+system (5.7) has finite extinction property in H, that is, y(T) = y1 for some T > 0 (see [6,
+section 5.3, Chapter 5]). Let us set z(·) = y(·) − y1. Then z(·) satisfies
+
+
+
+
+
+
+
+dz(t)
+dt
++ µAz(t) + B(z(t) + y1) − B(y1) + β(C(z(t) + y1) − C(y1))
++κ sgn(z(t)) ∋ −(µAy1 + B(y1) + βC(y1)), for a.e. t > 0,
+z(0) = y0 − y1.
+We assume that there exists no T such that z(T) = 0. Taking the inner product with
+sgn(z(·)) (using a smooth approximation of sgn(z(·)) [6], one can justify), we get
+1
+2
+d
+dt∥z(t)∥2
+H + µ∥z(t)∥2
+V + κ∥z(t)∥H + (C(z(t) + y1) − C(y1), z(t))
+= (B(y1) − B(z(t) + y1), z(t)) − (µAy1 + B(y1) + βC(y1), z(t)).
+From (3.13), (2.6) and using a calculation similar to (3.36), we obtain
+1
+2
+d
+dt∥z(t)∥2
+H + µ
+2 ∥z(t)∥2
+V + κ∥z(t)∥H ≤ ̺∥z(t)∥2
+H + ∥µAy1 + B(y1) + βC(y1)∥H∥z(t)∥H,
+and we can rewrite
+d
+dt∥z(t)∥H + η ≤ ̺∥z(t)∥H,
+where η = κ − ∥µAy1 + B(y1) + βC(y1)∥H > 0 and ̺ is given in (3.1). By using variation of
+constant formula, we get
+e−̺t∥z(t)∥H ≤
+�
+∥z(0)∥H − η
+̺
+�
++ η
+̺e−̺t.
+This shows as t → ∞ we are getting contradiction to the assumption (5.9). This implies
+that z = z(t) has finite extinction property in time T > 0 and this proves the existence of
+an admissible control U(·) ∈ Uκ.
+5.3. Stabilizing feedback controllers. Let us consider the following controlled CBF equa-
+tions:
+
+
+
+dy(t)
+dt
++ µAy(t) + B(y(t)) + βC(y(t)) = f e + U(t), for a.e. t > 0,
+y(0) = y0.
+(5.10)
+Let ye ∈ D(A) be the steady-state (equilibrium) solution of (5.10), that is, ye satisfies
+µAye + B(ye) + βC(ye) = f e in Td,
+(5.11)
+whose solvability results are available in [35, Theorem 4.1]. Let K ⊂ H be a closed and
+convex set with 0 ∈ K such that (5.2) is satisfied.
+We set z(·) = y(·) − ye, then (5.10) becomes
+
+
+
+dz(t)
+dt
++ µAz(t) + B(z(t) + ye) − B(ye) + βC(z(t) + ye) − C(ye) = U(t), for a.e. t > 0,
+z(0) = y0 − ye.
+(5.12)
+
+34
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+Let �B(z(·)) := B(z(·)+ye)−B(ye) and �C(z(·)) := C(z(·)+ye)−C(ye). Then (5.12) becomes
+
+
+
+dz(t)
+dt
++ µAz(t) + �B(z(t)) + �C(z(t)) = U(t), for a.e. t > 0,
+z(0) = y0 − ye.
+(5.13)
+Using Step IV in the proof of Proposition 3.1, it is clear that �B(·) and �C(·) map from D(A)
+to H. Therefore the operator
+µA + �B(·) + β�C(·) + θI + ∂IK(·),
+is m-accretive in H × H for θ > 0 is sufficiently large. Let
+Φ(w) := θw + ∂IK(w),
+for all w ∈ H. Since D(∂IK) = K and K ⊂ H, then from [6, Chapter 2, Theorem 2.3], we
+can write for all w ∈ H
+Φ(w) = ∂
+�θ
+2∥w∥2
+H + IK(w)
+�
+.
+Clearly 0 ∈ D(Φ) = K. Also, from [5, Chapter IV, Proposition 1.1, part (iv)], we have
+(Φ(w), Aw) ≥ 0.
+Thus the Hypothesis 1.1 is satisfied. Therefore, we can apply the existence and uniqueness
+result (see Theorem 1.3) for the inclusion problem
+
+
+
+dz(t)
+dt
++ µAz(t) + �B(z(t)) + �C(z(t)) + θz(t) + ∂IK(z(t)) ∋ 0, for a.e. t > 0,
+z(0) = y0 − ye.
+(5.14)
+We intend to find a feedback controller U(·) given by U(t) ∈ −θz(t) − ∂IK(z(t) which
+statbilizes the equilibrium solution ye exponentially under the invariance condition that
+y0 −ye ∈ K, then y(t) −ye ∈ K for all t ≥ 0. The stability part will be discussed in a future
+work.
+Appendix A. The case of d = 2, 3 and r ∈ [1, 3]
+The case of d = 2, 3 and r ∈ [1, 3] is considered in this section. We quantize the Navier-
+Stokes nonlinearity B(·) and prove monotonicity property. The authors in [13] took a V-ball
+for quantization, while we are taking an �L4-ball. Define the quantized nonlinearity as
+BN(y) =
+
+
+
+B(y),
+if ∥y∥�L4 ≤ N,
+�
+N
+∥y∥�L4
+�4
+B(y),
+if ∥y∥�L4 > N,
+(A.1)
+where N ∈ N∗ := N ∪ {0}.
+Lemma A.1. The operator BN(·) : V → V′ satisfies
+|⟨BN(y) − BN(z), y − z⟩| ≤ µ
+2∥y − z∥2
+V + CN∥y − z∥2
+H,
+for all y, z ∈ V,
+(A.2)
+where µ > 0 is the same as in (1.1).
+
+CBF EQUATIONS WITH POTENTIAL
+35
+Proof. Without loss of generality, one may assume that ∥y∥�L4 ≤ ∥z∥�L4. Therefore, we need
+to consider the following three cases:
+Case I: ∥y∥�L4, ∥z∥�L4 ≤ N. Using (2.2), H¨older’s, Ladyzhenskaya’s and Young’s inequalities,
+we have
+|⟨BN(y) − BN(z), y − z⟩|
+= |⟨B(y) − B(z), y − z⟩| = |⟨B(y − z), z⟩|
+≤ C∥y − z∥�L4∥y − z∥V∥z∥�L4 ≤ CN∥y − z∥�L4∥y − z∥V
+≤ CN∥y − z∥
+1− d
+4
+H
+∥y − z∥
+1+ d
+4
+V
+≤ µ
+2∥y − z∥2
+V + CN∥y − z∥2
+H.
+(A.3)
+Case II: ∥y∥�L4, ∥z∥�L4 > N. Let us first consider
+⟨BN(y) − BN(z), y − z⟩
+=
+��
+N
+∥y∥�L4
+�4
+B(y) −
+�
+N
+∥z∥�L4
+�4
+B(z), y − z
+�
+=
+��
+N
+∥y∥�L4
+�4
+−
+�
+N
+∥z∥�L4
+�4�
+⟨B(y), y − z⟩ +
+�
+N
+∥z∥�L4
+�4
+⟨B(y) − B(z), y − z⟩.
+(A.4)
+Now by using (2.2) and H¨older’s inequality, we calculate
+|⟨B(y), y − z⟩| = |⟨B(y, y − z), z⟩| ≤ ∥y∥�L4∥y − z∥V∥z∥�L4,
+(A.5)
+By Taylor’s formula, (A.5), H¨older’s, Ladyzhenskaya’s and Young’s inequalities, we obtain
+����
+�
+N
+∥y∥�L4
+�4
+−
+�
+N
+∥z∥�L4
+�4����|⟨B(y), y − z⟩|
+≤ 4
+��
+N
+∥y∥�L4
+�
++
+�
+N
+∥z∥�L4
+��3����
+N
+∥y∥�L4
+−
+N
+∥z∥�L4
+����|⟨B(y, y − z), z⟩|
+≤ CN∥y − z∥�L4∥y − z∥V
+≤ µ
+4 ∥y − z∥2
+V + CN∥y − z∥2
+H.
+A calculation similar to (A.3) yields
+�����
+�
+N
+∥z∥�L4
+�4
+⟨B(y) − B(z), y − z⟩
+����� ≤ µ
+4 ∥y − z∥2
+V + CN∥y − z∥2
+H.
+(A.6)
+Combining the above estimates imply (A.2).
+Case III: ∥y∥�L4 ≤ N and ∥z∥�L4 > N. One can rewrite
+⟨BN(y) − BN(z), y − z⟩
+=
+�
+B(y) −
+�
+N
+∥z∥�L4
+�4
+B(z), y − z
+�
+=
+�
+1 −
+�
+N
+∥z∥�L4
+�4�
+⟨B(y), y − z⟩ +
+�
+N
+∥z∥�L4
+�4
+⟨B(y) − B(z), y − z⟩.
+
+36
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+As 1 −
+�
+N
+∥z∥�L4
+�4
+=
+∥z∥4
+�L4−N4
+∥z∥4
+�L4
+≤
+∥z∥4
+�L4−∥y∥4
+�L4
+∥z∥4
+�L4
+, one can use the estimates in the previous cases to
+conclude (A.2).
+□
+Proposition A.2. For d = 2, 3 and 1 ≤ r ≤ 3, define the operator ΥN : D(ΥN) → H by
+ΥN(·) := µA + BN(·) + βC(·),
+with D(ΥN) = D(A). Moreover, there exists ηN > 0 such that ΥN + ηNI is m-accretive in
+H × H.
+Proof. Using Lemma A.1, proof follows in a similar way as the proof of Proposition 3.1 with
+some minor modifications.
+□
+One can prove the following result similar to Proposition 3.3.
+Proposition A.3. Let N ∈ N∗ be fixed. Let Φ ⊂ H × H be a maximal monotone operator
+satisfying Hypothesis 1.1. Define the multi-valued operator AN : D(AN) → H by
+AN(·) = µA + BN(·) + βC(·) + Φ(·) + ηNI
+with domain D(AN) = {y ∈ H : ∅ ̸= AN(y) ⊂ H}. Then D(AN) = D(A) ∩ D(Φ) and AN is
+a maximal monotone operator in H × H, where ηN is as in Proposition A.2.
+Moreover, there exists a constant C such that
+∥Aw∥2
+H ≤ C(1 + ∥w∥2
+H + ∥µAw + BN(w) + βC(w) + Φλ(w)∥2
+H)3,
+(A.7)
+for every w ∈ D(A) and for every λ > 0. Furthermore, we have
+∥Aw∥2
+H ≤ C(1 + ∥w∥2
+H + ∥µAw + BN(w) + βC(w) + ξ∥2
+H)3,
+(A.8)
+for every w ∈ D(A) ∩ D(Φ) and for every ξ ∈ Φ(w).
+Let us now consider the following approximate equation:
+
+
+
+dyN(t)
+dt
++ µAyN(t) + BN(yN(t)) + βC(yN(t)) + Φ(yN(t)) ∋ f(t),
+a.e. t ∈ (0, T),
+yN(0) = y0.
+Using Proposition A.3, one can establish the following results in a similar way as in the
+proof of Proposition 3.4.
+Proposition A.4. Let Φ ⊂ H × H satisfy Hypothesis 1.1. Let f ∈ W1,1(0, T; H) and y0 ∈
+D(A) ∩ D(Φ). Then there exists a unique strong solution
+yN ∈ W1,∞(0, T; H) ∩ L∞(0, T; D(A)) ∩ C([0, T]; V)
+to the problem.
+Furthermore, yN is right differentiable, d+yN
+dt
+is right continuous, and
+d+yN(t)
+dt
++ (µAyN(t) + BN(yN(t)) + βC(yN(t)) + Φ(yN(t)) − f(t))0 = 0,
+for all t ∈ [0, T].
+Proposition A.5. Let Φ ⊂ H × H satisfy Hypothesis 1.1. Let f ∈ W1,1(0, T; H) and y0 ∈
+D(A) ∩ D(Φ). Then there exists a unique strong solution
+yλ
+N ∈ W1,∞(0, T; H) ∩ L∞(0, T; D(A)) ∩ C([0, T]; V)
+
+CBF EQUATIONS WITH POTENTIAL
+37
+to the problem
+
+
+
+dyλ
+N(t)
+dt
++ µAyλ
+N(t) + BN(yλ
+N(t)) + βC(yλ
+N(t)) + Φλ(yλ
+N(t)) = f(t),
+a.e. t ∈ (0, T),
+yλ
+N(0) = y0.
+Furthermore, yλ
+N is right differentiable, d+yλ
+N
+dt
+is right continuous, and
+d+yλ
+N(t)
+dt
++ µAyλ
+N(t) + BN(yλ
+N(t)) + βC(yλ
+N(t)) + Φλ(yλ
+N(t)) = f(t),
+for all t ∈ [0, T).
+Proofs of Theorems 1.3 and 1.4. For the case d = 2, 3 and r ∈ [1, 3], calculations similar to
+the energy estimates (4.1) yields
+∥yλ
+N(t)∥2
+H + µ
+� t
+0
+∥yλ
+N(s)∥2
+Vds + 2β
+� t
+0
+∥yλ
+N(s)∥r+1
+�Lr+1ds
+≤ ∥y0∥2
+H +
+1
+µλ1
+� t
+0
+∥f(s)∥2
+Hds + t∥Φ(0)∥2
+H,
+(A.9)
+for all t ∈ [0, T]. We take the inner product with Ayλ
+N(·) in (3.74) to obtain
+1
+2
+d
+dt∥yλ
+N(t)∥2
+V + µ∥Ayλ
+N(t)∥2
+H + (BN(yλ
+N(t)) + βC(yλ
+N(t)) + Φλ(yλ
+N(t)), Ayλ
+N(t))
+= (f(t), Ayλ
+N(t)),
+for a.e. t ∈ [0, T]. From (1.7), we infer that
+(C(yλ
+N), Ayλ
+N) = ∥|∇yλ
+N||yλ
+N|
+r−1
+2 ∥2
+H + 4
+� r − 1
+(r + 1)2
+�
+∥|∇|yλ
+N|
+r+1
+2 |∥2
+H.
+Using [47, Lemma 3.1, pp. 404] and (2.4), we find
+|(BN(yλ
+N), Ayλ
+N)| ≤
+�
+0,
+for d = 2,
+µ
+4∥Ayλ
+N∥2
+H + C∥yλ
+N∥6
+V,
+for d = 3.
+Using Hypothesis 1.1 (H.3), and the above estimates, we deduce
+∥yλ
+N(t)∥2
+V + µ
+� t
+0
+∥Ayλ
+N(s)∥2
+H + 2β
+� t
+0
+∥|∇yλ
+N(s)||yλ
+N(s)|
+r−1
+2 ∥2
+Hds
+≤ ∥y0∥2
+V + 2γ
+� t
+0
+(1 + ∥yλ
+N(s)∥2
+H) + 2ς
+� t
+0
+∥Φλ(yλ
+N(s))∥2
+Hds + 2
+µ
+� t
+0
+∥f(s)∥2
+Hds
++
+�
+0,
+for d = 2,
+� t
+0 ∥yλ
+N(s)∥6
+Vds,
+for d = 3,
+for all t ∈ [0, T]. From (4.5), we obtain
+∥yλ
+N(t)∥2
+V + µ
+� t
+0
+∥Ayλ
+N(s)∥2
+H + 2β
+� t
+0
+∥|∇yλ
+N(s)||yλ
+N(s)|
+r−1
+2 ∥2
+Hds
+≤ C + ς
+� t
+0
+∥Φλ(yλ
+N(s))∥2
+Hds +
+�
+0,
+for d = 2,
+� t
+0 ∥yλ
+N(s)∥6
+Vds,
+for d = 3,
+(A.10)
+
+38
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+where C = C
+�
+∥y0∥V, ∥f∥L2(0,T;H), ∥Φ(0)∥H
+�
+. Calculation similar to (4.29) and (4.32) yield
+����
+� t
+0
+(C(yλ
+N(s)), Φλ(yλ
+N(s)))ds
+����
+≤ C
+� t
+0
+∥yλ
+N(s)∥r
+�L2r∥Φλ(yλ
+N(s))∥Hds
+≤ 1 − µς
+8β
+� t
+0
+∥Φλ(yλ
+N(s))∥2
+Hds + C sup
+s∈[0,t]
+∥yλ
+N(s)∥r(2−d)+d
+H
+� t
+0
+∥yλ
+N(s)∥(r−1)d
+V
+ds,
+and
+ς
+� t
+0
+∥Φλ(yλ
+N(s))∥2
+Hds ≤ C + C sup
+s∈[0,t]
+∥yλ
+N(s)∥r(2−d)+d
+H
+� t
+0
+∥yλ
+N(s)∥(r−1)d
+V
+ds
++ C
+� t
+0
+∥f(s)∥2
+Hds + µ
+4
+� t
+0
+∥Ayλ
+N(s)∥2
+Hds.
+Therefore, from (A.10) one can conclude
+∥yλ
+N(t)∥2
+V + µ
+� t
+0
+∥Ayλ
+N(s)∥2
+H + 2β
+� t
+0
+∥|∇yλ
+N(s)||yλ
+N(s)|
+r−1
+2 ∥2
+Hds
+≤ C +
+
+
+
+
+
+C sup
+s∈[0,t]
+∥yλ
+N(s)∥2
+H
+� t
+0 ∥yλ
+N(s)∥2(r−1)
+V
+ds,
+for d = 2,
+C sup
+s∈[0,t]
+∥yλ
+N(s)∥3−r
+H
+� t
+0 ∥yλ
+N(s)∥3(r−1)
+V
+ds +
+� t
+0 ∥yλ
+N(s)∥6
+Vds,
+for d = 3,
+(A.11)
+where C = C
+�
+∥y0∥V, ∥f∥L2(0,T;H), ∥Φ(0)∥H
+�
+. Then one can use the similar techniques as in
+the proof [30, Theorem 2.1] to pass λ → 0 and then use Gronwall’s inequality to obtain
+∥yN(t)∥V ≤ C,
+for all t ∈ [0, T0],
+(A.12)
+where constant C is independent of N and T0 = T for d = 2 and T0 < T for d = 3. Hence for
+large N, we can choose N ≥ C so that BN(yN) = B(yN) and therefore yN = y is a solution
+of (1.4) with the regularity properties given in Theorems 1.3 and 1.4. So, yN satisfies (1.4)
+on the set
+EN = {t ∈ [0, T] : ∥yN(t)∥V ≤ N}.
+(A.13)
+By using Markov’s inequality, we have
+m([0, T]/EN) ≤ C
+N2,
+(A.14)
+where m is the Lebesgue measure. Since m([0, T]) = m(EN) + m([0, T]/EN), for large N,
+from (A.13) and (A.14), we conclude that m([0, T]) = m(EN) and y(·) satisfies (1.4) for
+a.e. t ∈ [0, T]. The case of y0 ∈ V ∩ D(Φ) in Theorem 1.4 can be completed by a density
+argument as in the proof of [30, Theorems 2.2 and 2.3].
+□
+Acknowledgments: The first author would like to thank Ministry of Education, Government
+of India - MHRD for financial assistance. K. Kinra would like to thank the Council of Scien-
+tific & Industrial Research (CSIR), India for financial assistance (File No. 09/143(0938)/2019-
+EMR-I). M. T. Mohan would like to thank the Department of Science and Technology (DST),
+India for Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award
+(IFA17-MA110).
+
+CBF EQUATIONS WITH POTENTIAL
+39
+Declarations:
+Ethical Approval: Not applicable.
+Competing interests:
+The authors declare no competing interests.
+Authors’ contributions:
+All authors have contributed equally.
+Funding:
+CSIR, India, 09/143(0938)/2019-EMR-I (K. Kinra), DST, India, IFA17-MA110
+(M. T. Mohan).
+References
+[1] F. Abergel and R. Temam, On some control problems in fluid mechanics, Theor. Comput. Fluid Dyn.,
+1 (1990), 303–325.
+[2] C. T. Anh and T.M. Nguyet, Time optimal control of the unsteady 3D Navier-Stokes-Voigt equations,
+Appl. Math. Optim., 79(2) (2019), 397–426.
+[3] S.N. Antontsev and H.B. de Oliveira, The Navier-Stokes problem modified by an absorption term, Appl.
+Anal., 89(12) (2010), 1805–1825.
+[4] J. Babutzka and P. C. Kunstmann, Lq-Helmholtz decomposition on periodic domains and applications
+to Navier-Stokes equations, J. Math. Fluid Mech., 20(3) (2018), 1093–1121.
+[5] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International
+Publishing, 1976.
+[6] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, 1993.
+[7] V. Barbu, The time optimal control of Navier-Stokes equations, Systems Control Lett., 30(2-3) (1997),
+93–100.
+[8] V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering Series,
+Springer, London, 2011.
+[9] V. Barbu, Controllability and Stabilization of Parabolic Equations, Progress in Nonlinear Differential
+Equations and their Applications Subseries in Control 90, Birkh¨auser/Springer, Cham, 2018.
+[10] V. Barbu, I. Lasiecka and R. Triggiani, Tangential Boundary Stabilization of Navier-Stokes Equations,
+Memoirs of the American Mathematical Society, 2006.
+[11] V. Barbu and C. Lefter, Internal stabilizability of the Navier-Stokes equations, System Control Lett.,
+48(3-4) (2003), 161–167.
+[12] V. Barbu and N. H. Pavel, Flow-invariant closed sets with respect to nonlinear semigroup flows, NoDEA
+Nonlinear Differential Equations Appl., 10(1) (2003), 57–72.
+[13] V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes
+equation, J. Math. Anal. Appl., 255(1) (2001), 281–307.
+[14] V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional
+controllers, Indiana Univ. Math. J., 53(5) (2004), 1443–1494.
+[15] T. Breiten, K. Kunisch and L. Pfeiffer, Feedback stabilization of the two-dimensional Navier–Stokes
+equations by value function approximation, Appl. Math. Optim., 80(3) (2019), 599–641.
+[16] H. Brezis, Operateurs Maximaux et Semi-groupes de Contractions das les Espaces de Hilbert, North
+Holland, New York, 1973.
+[17] Z. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with
+damping, J. Math. Anal. Appl., 343(2) (2008), 799–809.
+[18] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM Philadelphia, Philadel-
+phia (2013).
+[19] Z. Denkowski, S. Mig´orski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications,
+Kluwer Academic Publishers, Boston, MA, 2003.
+[20] R. Farwig, H. Kozono and H. Sohr, An Lq-approach to Stokes and Navier-Stokes equations in general
+domains, Acta Math., 195 (2005), 21–53.
+[21] B. P. W. Fernando, S. S. Sritharan and M. Xu, A simple proof of global solvability of 2-D Navier-Stokes
+equations in unbounded domains, Differential Integral Equations, 23(3–4) (2010), 223–235.
+[22] D. Fujiwara and H. Morimoto, An Lr-theorem of the Helmholtz decomposition of vector fields, J. Fac.
+Sci. Univ. Tokyo Sect. IA Math., 24(3) (1977), 685–700.
+
+40
+S. GAUTAM, K. KINRA AND M. T. MOHAN
+[23] A. V. Fursikov, Optimal Control of Distributed Systems: Theory and Applications, American Mathe-
+matical Society, Providence, Rhode Island, 2000.
+[24] T. Fukao, Variational inequality for the Stokes equations with constraint, Discrete Contin. Dyn. Syst.,
+Dynamical systems, differential equations and applications, 8th AIMS Conference. Suppl. Vol. I (2011),
+437–446.
+[25] M. Gokieli, N. Kenmochi and M. Niezg`odka, Variational inequalities of Navier-Stokes type with time
+dependent constraints, J. Math. Anal. Appl., 449(2) (2017), 1229–1247.
+[26] M. D. Gunzburger, Perspectives in Flow Control and Optimization, SIAM’s Advances in Design and
+Control Philadelphia, 2003.
+[27] K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer
+equations, J. Differential Equations, 263(11) (2017), 7141–7161.
+[28] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol I: Theory, Mathematics and its
+Applications, 419, Kluwer Academic Publishers, Dordrecht, 1997.
+[29] V. K. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast
+growing nonlinearities, Commun. Pure Appl. Anal., 11(5) (2012), 2037–2054.
+[30] A. I. Lefter, Navier-Stokes equations with potentials, Abstr. Appl. Anal., (2007), Art. ID 79406, 30 pp.
+[31] A. I. Lefter, Nonlinear feedback controllers for the Navier-Stokes equations, Nonlinear Anal., 71(1-2)
+(2009), 301–316.
+[32] S. Li and G. Wang, The time optimal control of the Boussinesq equations, Numer. Funct. Anal. Optim.,
+24(1–2) (2007), 163–180.
+[33] P. A. Markowich, E. S. Titi and S. Trabelsi, Continuous data assimilation for the three dimensional
+Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 29(4) (2016), 1292–1328.
+[34] M. T. Mohan, On the convective Brinkman-Forchheimer equations, Submitted.
+[35] M.
+T.
+Mohan,
+Stochastic
+convective
+Brinkman-Forchheimer
+equations,
+Submitted.
+https://arxiv.org/pdf/2007.09376.pdf.
+[36] M. T. Mohan, Well-posedness and asymptotic behavior of stochastic convective Brinkman–Forchheimer
+equations perturbed by pure jump noise, Stoch PDE: Anal. Comp., 10(2) (2022), 614–690.
+[37] M. T. Mohan, The time optimal control of two dimensional convective Brinkman-Forchheimer equations,
+Appl. Math. Optim., 84(3) (2021), 3295– 3338.
+[38] J. Peypouquet and S. Sorin, Evolution equations for maximal monotone operators: asymptotic analysis
+in continuous and discrete time, J. Convex Anal., 17(3-4) (2010), 1113–1163.
+[39] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipatives Parabolic
+PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.
+[40] J. C. Robinson and W. Sadowski, A local smoothness criterion for solutions of the 3D Navier–Stokes
+equations, Rend. Semin. Mat. Univ. Padova, 131 (2014), 159–178.
+[41] J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier–Stokes equations,
+classical theory, Cambridge University Press, Cambridge, UK, 2016.
+[42] M. R¨ockner and X. Zhang, Tamed 3D Navier-Stokes equation: existence, uniqueness and regularity,
+Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12(4) (2009), 525–549.
+[43] D. T. Son and L. T. Thuy, Time optimal control problem of the 3D Navier-Stokes-α equations, Numer.
+Funct. Anal. Optim., 43(6) (2022), 667–697.
+[44] S. S. Sritharan, An introduction to deterministic and stochastic control of viscous flow, in Optimal
+Control of Viscous Flow, SIAM, Philadelphia, 1998.
+[45] D.
+Stone
+and
+S.
+Zelik,
+The
+non-autonomous
+Navier-Stokes-Brinkmann-Forchhiemer
+equation
+with
+dirichlet
+boundary
+conditions:
+dissipativity,
+regularity
+and
+attractors,
+https://arxiv.org/pdf/2210.05580.pdf.
+[46] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, CBMS-NSF
+Regional Conference Series in Applied Mathematics, 1995.
+[47] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Vol.
+68, Applied Mathematical Sciences, Springer, 1997.
+[48] G. Wang, L. Wang, Y. Xu and Y. Zhang, Time Optimal Control of Evolution Equations, Springer, New
+York (2018).
+[49] Z. Zhang, X. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier-Stokes
+equations with damping, J. Math. Anal. Appl., 377(1) (2011), 414–419.
+
diff --git a/KdAzT4oBgHgl3EQfkP0Y/content/tmp_files/load_file.txt b/KdAzT4oBgHgl3EQfkP0Y/content/tmp_files/load_file.txt
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@@ -0,0 +1,1749 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf,len=1748
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='01527v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='OC]  4 Jan 2023  2D AND 3D CONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS  PERTURBED BY A SUBDIFFERENTIAL AND APPLICATIONS TO CONTROL  PROBLEMS  SAGAR GAUTAM1, KUSH KINRA2 AND MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN3*  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The following convective Brinkman-Forchheimer (CBF) equations (or damped  Navier-Stokes (NS) equations) with potential  ∂y  ∂t − µ∆y + (y · ∇)y + αy + β|y|r−1y + ∇p + Ψ(y) ∋ g, ∇ · y = 0,  in a d-dimensional torus is considered in this work, where d ∈ {2, 3}, µ, α, β > 0 and  r ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For d = 2 with r ∈ [1, ∞) and d = 3 with r ∈ [3, ∞) (2βµ ≥ 1 for d = r = 3), we  establish the existence of a unique global strong solution for the above multivalued problem  with the help of the abstract theory of m-accretive operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Moreover, we demonstrate  that the same results hold local in time for the case d = 3 with r ∈ [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We explored  the m-accretivity of the nonlinear as well as multivalued operators, Yosida approximations  and their properties, and several higher order energy estimates in the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For r ∈ [1, 3],  we quantize the NS nonlinearity (y · ∇)y to establish the existence and uniqueness results,  while for r ∈ [3, ∞) (2βµ ≥ 1 for r = 3), we handle the NS nonlinearity by the nonlinear  damping term |y|r−1y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Finally, we discuss the applications of the above developed theory  in feedback control problems like flow invariance, time optimal control and stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let Td =  �  R/LZ  �d be a d-dimensional torus (d = 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The convective  Brinkman-Forchheimer (CBF) equations describe the motion of incompressible fluid flows  in a saturated porous medium ([33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' With control applications in mind, we consider the  following CBF equations with potential (perturbed by a subdifferential, see Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1  below):  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∂y  ∂t − µ∆y + (y · ∇)y + αy + β|y|r−1y + ∇p + Ψ(y) ∋ g,  in Td × (0, ∞),  ∇ · y = 0,  in Td × (0, ∞),  y(0) = y0 in Td,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1)  1,2,3Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar High-  way, Roorkee, Uttarakhand 247667, INDIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  e-mail: Manil T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Mohan:  maniltmohan@ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='iitr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='in, maniltmohan@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  e-mail: Kush Kinra:  kkinra@ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='iitr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  e-mail: Sagar Gautam:  sagar_g@ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='iitr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Key words:  Convective Brinkman-Forchheimer equations, monotone operators, strong solution, stabi-  lization, feedback control, time optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Mathematics Subject Classification (2020): Primary 49J20, 49N35, 93D15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Secondary 35Q35, 76D03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  1  2  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  where y(x, t) : Td × (0, ∞) → Rd represents the velocity field at time t and position x,  p(x, t) : Td ×(0, ∞) → R denotes the pressure field, g(x, t) : Td ×(0, ∞) → Rd is an external  forcing and Ψ(·) ⊂ L2(Td) ×L2(Td) is a multivalued map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Moreover, y(·, ·), p(·, ·) and g(·, ·)  satisfies the following periodic conditions:  y(x + Lei, ·) = y(x, ·), p(x + Lei, ·) = p(x, ·) and g(x + Lei, ·) = g(x, ·),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2)  for every x ∈ Rd and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' , d, where {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' , ed} is the canonical basis of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The constant  µ > 0 denotes the Brinkman coefficient (effective viscosity), the positive constants α and  β represent the Darcy (permeability of porous medium) and Forchheimer (proportional to  the porosity of the material) coefficients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The absorption exponent r ∈ [1, ∞)  and r = 3 is known as the critical exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The critical homogeneous CBF equations ((1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1)  without potential, r = 3 and g = 0) have the same scaling as Navier-Stokes (NS) equations  only when α = 0 ([27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We refer the case r < 3 as subcritical and r > 3 as supercritical  (or fast growing nonlinearities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The model is accurate when the flow velocity is too large  for Darcy’s law to be valid, and apart from that the porosity is not too small ([33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' If one  considers (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1) without potential and if α = β = 0, then we obtain the classical NS equations,  and if α, β > 0, then it can be considered as damped NS equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' A discussion on NS  equations with potential can be accessed from [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Literature survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' In the literature, CBF equations are also known as tamed Navier-  Stokes equations or Navier-Stokes equations modified with an absorption term, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' [3, 42] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=',  and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The damping αy+β|y|r−1y arises from the resistance to the motion of  the flow, which describes several physical phenomena such as drag or friction effects, porous  media flow, some dissipative mechanisms, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' [17, 27, 33, 49] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  The continuous data assimilation problem for CBF model is described in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The global  solvability results for CBF model (for fast growing nonlinearities in 3D) can be accessed  from [3, 29, 33, 27, 34], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Similar to 3D Navier-Stokes equations, the existence of a unique  global (in time) weak solution of 3D CBF equations with r ∈ [1, 3) (for any β, µ > 0) and  r = 3 (for 2βµ < 1) is also an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  The theory of monotone operators is an important tool in the study of nonlinear operator  equations, we refer the readers to [5, 6, 19, 28], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' When the operator has  some kind of monotonicity properties, then one can pass the limit in the Galerkin and Faedo-  Galerkin approximations of the original equation, with a-priori estimates that are in general  weaker than those necessary in the compactness methods ([21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' In particular, monotone  operators are suitable tools for studying variational inequalities ([6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Local and global  solvability of NS equations with potential (or perturbed by a subdifferential) is established  in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Control of ordinary/partial differential equations associated with fluid flow motions have  numerous applications in science, engineering and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Behavior and control of turbu-  lent flows are some of the most difficult problems in fluid mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' By control of turbulent  flows, we meant to determine an optimal action which minimizes the turbulence inside the  flow, (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [1, 23, 26, 44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  One of the interesting control problem is the flow invariance  preserving feedback controllers for fluid flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The authors in [13] developed a procedure to  design feedback controllers that ensure the resultant dynamics of turbulence preserve some  prescribed physical constraints such as enstrophy, helicity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Flow invariance of controlled  CBF EQUATIONS WITH POTENTIAL  3  flux sets with respect to Navier-Stokes equations is discussed in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The existence prob-  lem of the variational inequality for Stokes and Navier-Stokes equations with constraints of  obstacle type is considered in [24, 25], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  An another interesting feedback control problem is the time optimal control problem,  where one finds a control of bang-bang type to reach a fixed state from an arbitrary state  in minimal time (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' [6, 48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' As far as the time optimal control of fluid flow models are  concerned, the time optimal control problem for 2D NS equations, Boussinesq equations, 3D  Navier-Stokes-Voigt equations, 2D CBF equations with r ∈ [1, 3], and 3D NS-α model is  considered in [7, 32, 2, 37, 43], respectively, and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Stabilization of NS equations is dealt to stabilize the equilibrium solution of NS equations  by using finite dimensional feedback controllers having support either in interior or on the  boundary of the domain (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' [8, 9, 10], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The internal stabilizability of NS equations  (with slip and non-slip Dirichlet boundary conditions) is developed in [11, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The author  in [31] discussed the feedback stabilization of NS equations preserving the invariance of a  given convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Recently the authors in [15] established feedback stabilization of 2D NS  equations by using the Taylor approximation of the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The main objective of this work is to establish the solvability results of  the inclusion problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1) and discuss their applications in the context of control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Since α is not playing a major role in this work, so we fix α = 0 in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let  us state the main results of this work for the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1) in an abstract framework (see  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We will prove these results in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us denote f = Pg  and Φ(·) = PΨ(·), where P is the Helmholtz-Hodge (or Leray) projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The functional  setting has been provided in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The following assumption is imposed on Φ(·) to  achieve our goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let Φ be a maximal monotone operator on H × H satisfying the following  hypothesis [30]:  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1) Φ = ∂ϕ, where ϕ : H → R := R ∪ {+∞} is a lower semicontinuous proper convex  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2) 0 ∈ D(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3) There exists two constants γ ≥ 0, ς ∈ (0, 1  µ) such that  (Ay, Φλ(y)) ≥ −γ(1 + ∥y∥2  H)  −  �  ς∥Φλ(y)∥2  H,  for d = 2 with r ∈ [1, ∞) and d = 3 with r ∈ [1, 5),  0,  for d = 3 with r ∈ [5, ∞),  for all λ > 0 and y ∈ D(A), where Φλ =  1  λ(I − (I + λΦ)−1) : H → H is the Yosida  approximation of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We observe from here that  Φλ(y) ∈ Φ((I + λΦ)−1)(y)),  for every y ∈ H and λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (1) The results of this work hold true if one replaces (1+∥y∥2  H) by (1+∥y∥2  V)  in (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (2) Using condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1), the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4) can be considered as CBF equations perturbed  by a subdifferential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  4  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let T > 0 and assume that Φ ⊂ H × H satisfies Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let y0 ∈  D(A) ∩ D(Φ) and f ∈ W1,1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For d = 2 with r ∈ [1, ∞) and d = 3 with r ∈ [3, ∞)  (2βµ ≥ 1 for r = 3), there exists a unique strong solution  y ∈ W1,∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' D(A)) ∩ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3)  such that in H  \uf8f1  \uf8f2  \uf8f3  dy(t)  dt  + µAy(t) + B(y(t)) + βC(y(t)) + Φ(y(t)) ∋ f(t),  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ (0, T),  y(0) = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4)  Furthermore, y is right differentiable, d+y  dt  is right continuous, and  d+y(t)  dt  + (µAy(t) + B(y(t)) + βC(y(t)) + Φ(y(t)) − f(t))0 = 0,  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5)  For d = 3 with r ∈ [1, 3] and 2βµ < 1 for r = 3 the solution y exists on some interval [0, T0),  where  T0 = T0  �  ∥y0∥V, ∥f∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  �  ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let T > 0 and assume that Φ ⊂ H × H satisfies Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let y0 ∈  D(A) ∩ D(Φ) and f ∈ W1,2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) (y0 ∈ V ∩ D(Φ) and f ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) for d = 2, 3 and  r ∈ [1, 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For d = 2 with r ∈ [1, ∞) and d = 3 with r ∈ [3, ∞), there exists a unique strong  solution  y ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V) ∩ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' D(A)) ∩ Lr+1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' �L3(r+1)) ∩ W1,2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V),  with dy  dt , B(y), C(y) ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) such that in H  \uf8f1  \uf8f2  \uf8f3  dy(t)  dt  + µAy(t) + B(y(t)) + βC(y(t)) + Φ(y(t)) ∋ f(t),  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ (0, T),  y(0) = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6)  For d = 3 with r ∈ [1, 3), there exists a time T0 = T0  �  ∥y0∥V, ∥f∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  �  ≤ T such that  the solution y exists on some interval [0, T0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Difficulties, approaches and novelties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The main concern for considering the CBF equa-  tions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1) in a d-dimensional torus is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' In the torus Td, the Helmholtz-Hodge  projection P and −∆ is commute ([41, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' So, the equality ([27, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1])  �  Td(−∆y(x)) · |y(x)|r−1y(x)dx  =  �  Td |∇y(x)|2|y(x)|r−1dx + 4  � r − 1  (r + 1)2  � �  Td |∇|y(x)|  r+1  2 |2dx,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7)  is quite useful in obtaining regularity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' It is also noticed in the literature that the  above equality may not be useful in domains other than the whole domain or a d-dimensional  torus (see [29, 34], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' for a detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Recently, the authors in [45] addressed  this regularity problem for Dirichlet’s boundary conditions and the well-posedness of CBF  equations with potential in bounded domains will be a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  The main difficulty with nonlinear terms arises when we multiply them by a generalized  function to get m-accretivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' In the literature for NS equations with potential (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' [30, 31])  or for feedback control problems (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' [13, 7]), V-quantization of the nonlinear term (y · ∇)y  is used to obtain the m-accretivity of the operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Whereas, for the supercritical CBF  CBF EQUATIONS WITH POTENTIAL  5  equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1) (that is, for r > 3), one can handle the NS nonlinearity (y · ∇)y by the  Forchheimer nonlinearity |y|r−1y (see steps (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='44), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Along with this  fact, the monotonicity of the nonlinear term |y|r−1y helps to obtain the m-accretivity of  the operators without using a quantization technique (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The same  results hold true for r = 3 with 2βµ ≥ 1 also without quantization, but for r ∈ [1, 3] (2βµ < 1  for r = 3), we need an �L4-quantization technique (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  The condition (Ay, Φλ(y)) ≥ −γ(1 + ∥y∥2  H) is considered in Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3) for  the case d = 3 with r ∈ [5, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' This is required to handle the term |(C(yλ), Φλ(yλ))| in  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3, while taking the inner product with Φλ(yλ) for the Yoisida approximated  stationary problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The Sobolev embedding of V ⊂ �Lp for any p ∈ [1, ∞) helps us to resolve  this problem in 2D, whereas in 3D, the embedding is true only for p ∈ [2, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Moreover,  for the supercritical case, by choosing F1(·) = µ(1 − δ1)A + β(1 − δ2)C(·) and F2(·) =  µδ1A + B(·) + βδ2C(·) + κI, for some δ1, δ2 ∈ (0, 1) and κ ≥ ̺ =  r−3  2µ(r−1)  �  2  βµ(r−1)  �  2  r−3, we used  the well-known perturbation theorem for nonlinear m-accretive operators ([5, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5,  Chapter II]) to show that the operator F1 + F2 = µA + B(·) + βC(·) + κI with the domain  D(A) is m-accretive in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  For NS equations with potential, the authors in [30] proved a result similar to Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4 by assuming that y0 ∈ V ∩ D(Φ) and f ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Under the same assumptions,  we are able to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4 for the case d = 2, 3 and r ∈ [1, 3] only (Appendix  A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For d = 2, 3 and r ∈ (3, ∞), we need y0 ∈ D(A) ∩ D(Φ) and f ∈ W1,2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) to  control the term  � T  0 ∥Φλ(y(t))∥2  Hdt (Step IV, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' In order to do this, we first  obtain the regularity estimates for  ��� d+yλ(·)  dt  ���  H and  � T  0  ��� dyλ(t)  dt  ���  2  Vdt (Step II, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1)  by taking the difference of Yosida approximated CBF equations (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74) below) at t + h  and t for h > 0 and t ∈ [0, T] and then using the monotonicity of Φλ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Due to the lack  of Gateaux derivative of Φλ(y(·)), one cannot differentiate the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74) and get the  required estimates by taking inner product with dyλ(·)  dt  in the resulting equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' This kind  of difficulty is not appearing in the case of NS equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Flow invraince preserving feedback controllers for 2D as well 3D NS equations with normal  cone as potential were considered in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The results obtained in the work [13] were global  for d = 2 and local for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' But the presence of the damping term |y|r−1y helps us  to obtain global results in 3D as well for supercritical CBF equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The author in [37]  discussed the time optimal control problem for 2D CBF equations with r ∈ [1, 3] by using a  V-quantization and m-accretivity of the nonlinear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 and the results  in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4 help us to study the time optimal control problem of CBF equations for  d = 2, 3 with r > 3 also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Moreover, the author in [31] examined the feedback stabilization  of 2D and 3D NS equations preserving the invariance of a given convex set by deducing  the existence of weak solutions (uniqueness only in 2D) for the NS system perturbed by a  subdifferential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Whereas, for the CBF equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4), one can address similar problems for  d = 2, 3 with r > 3 by establishing uniqueness results also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The rest of the paper is organized as follows: The next section is  devoted for the functional settings, definition and properties of linear, bilinear and nonlinear  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3 for the case r ∈ [3, ∞) (2βµ ≥ 1 for r = 3) is provided in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' In  order to do this, we apply the abstract theory of m-accretive operators available in [5, 6], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=',  6  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  in Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We prove the m-accretivity of the operator F(·) = µA + B(·) +  βC(·) + κI for some κ > 0 in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 by showing the monotonicity, demicontinuity  and coercivity of operator F(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Furthermore, in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3, we establish the maximal  monotonicity of the multivalued operator A(·), where A(·) := µA + B(·) + βC(·) + Φ(·) + κI,  by showing the range condition R(I + A) = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The primary tool in establishing the range  condition is the well-posedness of a Yosida approximated problem (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='33) below for yλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Then we establish the necessary stationary energy estimates and obtain uniform bounds of  the sequence {yλ}λ>0 (see Step II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' By passing to limit and applying the abstract theory  for maximal monotone operators (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' [5, 6], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' ), we finally deduce the m-accretivity of the  operator A(·) (see Step III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' A result similar to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3 for the Yosida approximated  problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74) is established in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  We first derive necessary higher order energy estimates to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4 in Section 4  (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then we prove some convergence results using the Banach-Alaoaglu  theorem and Aubin-Lions compactness lemma (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Lastly, we conclude  the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4 by using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4, and the uniqueness result is provided in  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  In Section 5, we discuss three applications of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4, namely, flow invariance  feedback controllers, time optimal control problem and feedback stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' A brief sketch  of proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4 is provided for the case r ∈ [1, 3] in Appendix A by using a  quantization of the NS nonlinearity (y·∇)y and the abstract theory of m-accretive operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Functional settings and Preliminaries  In this section, we provide the necessary functional setting needed to obtain the results of  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We consider the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1) on a d-dimensional torus Td =  �  R/LZ  �d (d = 2, 3),  with periodic boundary conditions and zero-mean value constraint for the functions, that is,  �  Td y(x)dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let  ˙C∞  p (Td;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Rd) denote the space of all infinitely differentiable func-  tions (Rd-valued) such that  �  Td y(x)dx = 0 and satisfy periodic boundary conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  The Sobolev space ˙Hk  p(Td) := ˙Hk  p(Td;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Rd) is the completion of ˙C∞  p (Td;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Rd) with respect to  the Hs norm  ∥y∥ ˙Hsp :=  \uf8eb  \uf8ed �  0≤|α|≤s  ∥Dαy∥2  L2(Td)  \uf8f6  \uf8f8  1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  The Sobolev space of periodic functions with zero mean ˙Hk  p(Td) is the same as [39, Proposition  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='39]  �  y : y =  �  k∈Zd  yke2πik·x/L, y0 = 0, ¯yk = y−k, ∥y∥ ˙Hs  f :=  �  k∈Zd  |k|2s|yk|2 < ∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  From [39, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='38], we infer that the norms ∥ · ∥ ˙Hsp and ∥ · ∥ ˙Hs  f are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let  us define  V := {y ∈ ˙C∞  p (Td;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Rd) : ∇ · y = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  The spaces H and �Lp are the closure of V in the Lebesgue spaces L2(Td;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Rd) and Lp(Td;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Rd)  for p ∈ (2, ∞), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The space V is the closure of V in the Sobolev space H1(Td;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  CBF EQUATIONS WITH POTENTIAL  7  The zero mean condition provides the well-known Poincar´e inequality,  λ1∥y∥2  H ≤ ∥y∥2  V,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1)  where λ1 = 4π2  L2 ([39, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then, we characterize the spaces H, �Lp and V with the  norms  ∥y∥2  H :=  �  Td |y(x)|2dx,  ∥y∥p  �Lp =  �  Td |y(x)|pdx and ∥y∥2  V :=  �  Td |∇y(x)|2dx,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let (·, ·) denote the inner product in the Hilbert space H and ⟨·, ·⟩ represent the  induced duality between the spaces V and its dual V′ as well as �Lp and its dual �Lp′, where  1  p + 1  p′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Note that H can be identified with its own dual H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The sum space V′ + �Lp′ is  well defined (see [20, Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Furthermore, we have  (V′ + �Lp′)′ = V ∩ �Lp and (V ∩ �Lp)′ = V′ + �Lp′,  where ∥y∥V∩�Lp = max{∥y∥V, ∥y∥�Lp}, which is equivalent to the norms ∥y∥V + ∥y∥�Lp and  �  ∥y∥2  V + ∥y∥2  �Lp, and  ∥y∥V′+�Lp′ = inf{∥y1∥V′ + ∥y2∥�Lp′ : y = y1 + y2, y1 ∈ V′ and y2 ∈ �Lp′}  = sup  �|⟨y1 + y2, f⟩|  ∥f∥V∩�Lp  : 0 ̸= f ∈ V ∩ �Lp  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Note that V ∩ �Lp and V′ + �Lp′ are Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Moreover, we have the continuous  embedding V ∩ �Lp ֒→ V ֒→ H ֒→ V′ ֒→ V′ + �Lp′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let Pp : Lp(Td) → �Lp, p ∈ [1, ∞) be the Helmholtz-Hodge (or Leray)  projection (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' [4, 22], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Note that Pp is a bounded linear operator and for p = 2, P := P2  is an orthogonal projection ([41, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We define the Stokes operator  Ay := −P∆y, y ∈ D(A) := V ∩ ˙H2  p(Td).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Note that D(A) can also be written as D(A) =  �  y ∈ ˙H2  p(Td) : ∇·y = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' It should be noted  that P and ∆ commutes in a torus ([41, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Note that for d ≤ 4, by Sobolev’s inequality, one has D(A) ⊂ H2 ⊂ Lp, for all  p ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Bilinear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us define the trilinear form b(·, ·, ·) : V × V × V → R by  b(y, z, w) =  �  Td(y(x) · ∇)z(x) · w(x)dx =  d  �  i,j=1  �  Td yi(x)∂zj(x)  ∂xi  wj(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  If y, z are such that the linear map b(y, z, ·) is continuous on V, the corresponding element  of V′ is denoted by B(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We also denote B(y) = B(y, y) = P[(y · ∇)y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' An integration  by parts yields  �b(y, z, w) = −b(y, w, z),  for all y, z, w ∈ z,  b(y, z, z) = 0,  for all y, z ∈ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2)  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We need the following estimates on the trilinear form b(·, ·, ·) in the sequel (see  [46, Chapter 2, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3]):  8  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  (i) For d = 2,  |b(y, z, w)| ≤ C ×  �  ∥y∥1/2  H ∥y∥1/2  V ∥z∥V∥w∥1/2  H ∥w∥1/2  V ,  for all y, z, w ∈ V,  ∥y∥1/2  H ∥y∥1/2  V ∥z∥1/2  V ∥Az∥1/2  H ∥w∥H,  for all y ∈ V, z ∈ D(A), w ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3)  (ii) For d = 3,  |b(y, z, w)| ≤ C ×  �  ∥y∥1/4  H ∥y∥3/4  V ∥z∥V∥w∥1/4  H ∥w∥3/4  V ,  for all y, z, w ∈ V,  ∥y∥V∥z∥1/2  V ∥Az∥1/2  H ∥w∥H,  for all y ∈ V, z ∈ D(A), w ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Nonlinear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us now consider the operator C(y) := P(|y|r−1y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' It is imme-  diate that ⟨C(y), y⟩ = ∥y∥r+1  �Lr+1 and the map C(·) : �Lr+1 → �L  r+1  r  is Gateaux differentiable  with Gateaux derivative  C′(y)z =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  P(z),  for r = 1,  �  P(|y|r−1z) + (r − 1)P  �  y  |y|3−r (y · z)  �  ,  if y ̸= 0,  0,  if y = 0,  for 1 < r < 3,  P(|y|r−1z) + (r − 1)P(y|y|r−3(y · z)),  for r ≥ 3,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5)  for all y, z ∈ �Lr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  From [36, Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4], we have  ⟨C(y) − C(z), y − z⟩ ≥ 1  2∥|y|  r−1  2 (y − z)∥2  H + 1  2∥|z|  r−1  2 (y − z)∥2  H  ≥  1  2r−1∥y − z∥r+1  �Lr+1 ≥ 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6)  for r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' In periodic domain (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' [36, Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5]), we have  ∥y∥r+1  �L3(r+1) ≤ C  �  Td |∇y(x)|2|y(x)|r−1dx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7)  for d = 3 and r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Also from [40, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2], we obtain  ∥y∥r+1  �Lp(r+1) = ∥|y|  r+1  2 ∥2  �L2p(Td) ≤ C  �  Td |∇|y|  r+1  2 |2dxC  �  Td |∇y(x)|2|y(x)|r−1dx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='8)  for d = 2 and for all p ∈ [2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3  We prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3 in this section for the case r ∈ [3, ∞) (2βµ ≥ 1 for r = 3) by using  the abstract theory available in the works [5, 6], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For d = 2, 3 with r > 3, define the operator G(·) : D(G) → H by  G(·) = µA + B(·) + βC(·),  CBF EQUATIONS WITH POTENTIAL  9  where D(G) = {y ∈ V ∩ �Lr+1 : Ay ∈ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then G + κI is m-accretive in H × H for some  κ ≥ ̺, where  ̺ =  r − 3  2µ(r − 1)  �  2  βµ(r − 1)  �  2  r−3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='9)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We shall first show that G + κI is a monotone operator for κ ≥ ̺ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then we  will show that G + κI is coercive and demicontinuous, which imply the m-accretivity of the  operator G + κI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The proof is divided into following steps:  Step I: G + κI is monotone for some κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We estimate ⟨Ay − Az, y − z⟩ by using an  integration by parts as  ⟨Ay − Az, y − z⟩ = ∥y − z∥2  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='10)  Note that ⟨B(y, y−z), y−z⟩ = 0 which implies along with H¨older’s and Young’s inequalities  that  |⟨B(y) − B(z), y − z⟩| = |⟨B(y − z, y − z), z⟩| ≤ µ  2 ∥y − z∥2  V + 1  2µ∥z(y − z)∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='11)  We take the term ∥z(y − z)∥2  H from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='11) and use H¨older’s and Young’s inequalities to  estimate it as (see [27] also)  �  Td |z(x)|2|y(x) − z(x)|2dx =  �  Td |z(x)|2|y(x) − z(x)|  4  r−1|y(x) − z(x)|  2(r−3)  r−1 dx  ≤  ��  Td |z(x)|r−1|y(x) − z(x)|2dx  �  2  r−1��  Td |y(x) − z(x)|2dx  � r−3  r−1  ≤ βµ∥|z|  r−1  2 (y − z)∥2  H + r − 3  r − 1  �  2  βµ(r − 1)  �  2  r−3  ∥y − z∥2  H,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='12)  for r > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='12) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='11), we find  |⟨B(y) − B(z), y − z⟩|  ≤ µ  2 ∥y − z∥2  V + β  2 ∥|z|  r−1  2 (y − z)∥2  H +  r − 3  2µ(r − 1)  �  2  βµ(r − 1)  �  2  r−3  ∥y − z∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='13)  From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6), we easily have  β⟨C(y) − C(z), y − z⟩ ≥ β  2 ∥|z|  r−1  2 (y − z)∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='14)  Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='13)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='14), we conclude that  ⟨(G + κI)(y) − (G + κI)(z), y − z⟩ ≥ µ  2 ∥y − z∥2  V + (κ − ̺)∥y − z∥2  H ≥ µ  2 ∥y − z∥2  V, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='15)  for κ ≥ ̺ and r > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Thus G + κI is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Step II: G + κI is demicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us take a sequence yn → y in V ∩ �Lr+1, so that  ∥yn − y∥V + ∥yn − y∥�Lr+1 → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For any z ∈ V ∩ �Lr+1, we consider  ⟨(F + κI)(yn) − (F + κI)(y), z⟩  = µ⟨A(yn) − A(y), z⟩ + ⟨B(yn) − B(y), z⟩ − β⟨C(yn) − C(y), z⟩ + κ⟨yn − y, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='16)  10  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  Note that  |µ⟨A(yn − y), z⟩ + κ⟨yn − y, z⟩| ≤ µ∥yn − y∥V∥z∥V + κ∥yn − y∥H∥z∥H → 0 as n → ∞,  since yn → y strongly in V ∩ �Lr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We estimate the term |⟨B(yn) − B(y), z⟩| by using the  H¨older’s and interpolation inequalities  |⟨B(yn) − B(y), z⟩| = |⟨B(yn, yn − y), z⟩ + ⟨B(yn − y, y), z⟩|  ≤ |⟨B(yn, z), yn − y⟩| + |⟨B(yn − y, z), y⟩|  ≤  �  ∥yn∥�L  2(r+1)  r−1 + ∥y∥�L  2(r+1)  r−1  �  ∥yn − y∥�Lr+1∥z∥V  ≤  �  ∥yn∥  r−3  r−1  H  ∥yn∥  2  r−1  �Lr+1 + ∥y∥  r−3  r−1  H  ∥y∥  2  r−1  �Lr+1  �  ∥yn − y∥�Lr+1∥z∥V  → 0,  as n → ∞,  since yn → y strongly in V ∩ �Lr+1 and yn, y ∈ V ∩ �Lr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We estimate the term |⟨C(yn) −  C(y), z⟩| using the Taylor’s formula ([18, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1]) as  |⟨C(yn) − C(y), z⟩| ≤ sup  0<θ<1  r∥(yn − y)|θyn + (1 − θ)y|r−1∥�L  r+1  r ∥z∥�Lr+1  ≤ r∥yn − y∥�Lr+1  �  ∥yn∥�Lr+1 + ∥y∥�Lr+1  �r−1∥z∥�Lr+1 → 0 as n → ∞,  since yn → y strongly in V ∩ �Lr+1 and yn, y ∈ V ∩ �Lr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From the above convergences, it is  immediate that ⟨(G+κI)(yn)−(G+κI)(y), z⟩ → 0, for all z ∈ V∩ �Lr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Hence the operator  G + κI : V ∩ �Lr+1 → V′ + �L  r+1  r  is demicontinuous and hence it is hemicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Step III: G + κI is coercive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We consider  ⟨(G + κI)(y), y⟩  ∥y∥V∩�Lr+1  =  µ∥y∥2  V + β∥y∥r+1  �Lr+1 + κ∥y∥2  H  �  ∥y∥2  V + ∥y∥2  �Lr+1  ≥  min{µ, β}  �  ∥y∥2  V + ∥y∥2  �Lr+1  �  − 1  �  ∥y∥2  V + ∥y∥2  �Lr+1  ,  where we have used the fact that x2 ≤ xr+1 + 1, for x ≥ 0 and r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Thus, we have  lim  ∥y∥V∩�Lr+1→∞  ⟨(G + κI)(y), y⟩  ∥y∥V∩�Lr+1  = ∞,  and it shows that the operator G + κI is coercive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Step IV: F(·) := G(·) + κI is m-accretive in H × H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us define an operator  F(y) = µAy + B(y) + βC(y) + κy,  where D(A) = {y ∈ V ∩ �Lr+1 : µAy + B(y) + βC(y) ∈ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Note that the space V ∩ �Lr+1 is  reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Since G+κI is monotone, hemicontinuous and coercive from V∩ �Lr+1 to V′ + �L  r+1  r ,  then by an application of [16, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7], we obtain that G+ κI is maximal monotone in  H with domain D(F) ⊇ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' In fact, we shall prove that F is m-accretive for κ sufficiently  large with D(F) = D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Let us consider the operators for some δ1, δ2 ∈ (0, 1) as  F1(·) = µ(1 − δ1)A + β(1 − δ2)C(·),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='17)  F2(·) = µδ1A + B(·) + βδ2C(·) + κI,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='18)  CBF EQUATIONS WITH POTENTIAL  11  where D(F1) = {y ∈ V ∩ �Lr+1 : F1(·) ∈ H} and D(F2) = {y ∈ V ∩ �Lr+1 : F2(·) ∈ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Taking  the inner product with y in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='17), we obtain  µ(1 − δ1)∥y∥2  V + β(1 − δ2)∥y∥r+1  �Lr+1 ≤ (F1(y), y) ≤ ∥F1(y)∥H∥y∥H,  so that  ∥y∥2  V ≤  1  µ(1 − δ1)∥F1(y)∥H∥y∥H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='19)  Taking the inner product with Ay in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='17) and using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7), we get  µ(1 − δ1)∥Ay∥2  H + β(1 − δ2)  �  ∥|y|  r−1  2 ∇y∥2  H + 4(r − 1)  (r + 1)2 ∥|∇|y|  r+1  2 |∥2  H  �  = (F1(y), Ay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Therefore, we have  ∥Ay∥H ≤  1  µ(1 − δ1)∥F1(y)∥H which implies D(F1) ⊆ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='20)  Moreover, using Sobolev’s inequality, we infer  ∥F1(y)∥H ≤ µ(1 − δ1)∥Ay∥H + Cβ(1 − δ2)∥y∥r  �L2r  ≤ µ(1 − δ1)∥Ay∥H + Cβ(1 − δ2)∥Ay∥r  H,  which gives D(F1) ⊇ D(A) and therefore D(A) = D(F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Taking the inner product with C(y)  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='17), we find  µ(1 − δ1)  �  ∥|y|  r−1  2 ∇y∥2  H + 4(r − 1)  (r + 1)2 ∥|∇|y|  r+1  2 |∥2  H  �  + β(1 − δ2)∥C(y)∥2  H = (F1(y), C(y)),  so that  ∥C(y)∥H ≤  1  β(1 − δ2)∥F1(y)∥H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='21)  For r > 3, we estimate ∥B(y)∥H using H¨older’s inequality as follows:  ∥B(y)∥2  H ≤  �  Td |y(x)|2|∇y(x)|2dx =  �  Td |y(x)|2|∇y(x)|  4  r−1|∇y(x)|  2(r−3)  r−1 dx  ≤ ∥|y|  r−1  2 ∇y∥  4  r−1  H  ∥y∥  2(r−3)  r−1  V  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='22)  Note that (C(y), Ay) =  �  Td(−∆y(x)) · |y(x)|r−1y(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Using the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='19) and the  equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='22), we find  ∥B(y)∥2  H ≤ [(C(y), Ay)]  2  r−1  �  1  µ(1 − δ1)∥F1(y)∥H∥y∥H  � r−3  r−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Therefore, we estimate ∥B(y)∥H as  ∥B(y)∥H ≤ ∥C(y)∥  1  r−1  H  ∥Ay∥  1  r−1  H  �  1  µ(1 − δ1)∥F1(y)∥H∥y∥H  �  r−3  2(r−1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='23)  Using the estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='20)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='21) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='23), then using Young’s inequality, we get  ∥B(y)∥H ≤  �  ∥F1(y)∥2  H  βµ(1 − δ1)(1 − δ2)  �  1  r−1�∥F1(y)∥H∥y∥H  µ(1 − δ1)  �  r−3  2(r−1)  12  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  =  1  �  µ(1 − δ1)  �  1  β(1 − δ2)  �  1  r−1  ∥F1(y)∥  r+1  2(r−1)  H  ∥y∥  r−3  2(r−1)  H  ≤  δ1  1 − δ1  ∥F1(y)∥H + Cδ1,δ2,µ,β∥y∥H,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='24)  where Cδ1,δ2,µ,β =  r−3  2(r−1)  �  1−δ1  µ  r−1  2  β(1−δ2)  �  2  r−3�  r+1  2δ1(r−1)  � r+1  r−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Now using the estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='20)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='21)  and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='24) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='18), we deduce  ∥F2(y)∥H ≤ µδ1∥Ay∥H + βδ2∥C(y)∥H + ∥B(y)∥H + κ∥y∥H  ≤  � 2δ1  1 − δ1  +  δ2  1 − δ2  �  ∥F1(y)∥H + (Cδ1,δ2,µ,β + κ)∥y∥H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Let us choose δ1 and δ2 in such a way that ρ =  2δ1  1−δ1 +  δ2  1−δ2 < 1, for example, one can choose  δ1 = 1  9, δ2 = 1  5, so that ρ = 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then by the well-known perturbation theorem for nonlinear  m-accretive operators ([5, Chapter II, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5]), we conclude that the operator F1 + F2  with the domain D(A) is m-accretive in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Since F1 + F2 = G + κI , the operator G + κI is  m-accretive in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For d = 2 with r ∈ [1, ∞) and d = 3 with r ∈ [1, 5], Sobolev’s embedding  yields V ⊂ �Lr+1, so that V ∩ �Lr+1 = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  For d = r = 3 and 2βµ ≥ 1, one can obtain global monotonicity of the operator  G(·) : V → V′ in the following way:  We estimate |⟨B(y − z, y − z), z⟩| using H¨older’s and Young’s inequalities as  |⟨B(y − z, y − z), z⟩| ≤ ∥z(y − z)∥H∥y − z∥V ≤ µ∥y − z∥2  V + 1  4µ∥z(y − z)∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='25)  Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='25), we obtain  ⟨G(y) − G(z), y − z⟩ ≥ 1  2  �  β − 1  2µ  �  ∥z(y − z)∥2  H ≥ 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='26)  provided 2βµ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Moreover, other properties like demicontinuity and coercivity can be  proved in similar way as r > 3 case (see the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let Φ ⊂ H × H be a maximal monotone operator satisfying Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Define the multivalued operator A : D(A) → H by  A(·) = µA + B(·) + βC(·) + Φ(·) + κI,  with the domain D(A) = {y ∈ H : ∅ ̸= A(y) ⊂ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then D(A) = D(A) ∩ D(Φ) and A is a  maximal monotone operator in H × H, where κ is as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Furthermore, the following estimates holds  ∥Aw∥2  H ≤ C(1 + ∥w∥2  H + ∥µAw + B(w) + βC(w) + Φλ(w)∥2  H)ϑ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='27)  for every w ∈ D(A), λ > 0 and  ∥Aw∥2  H ≤ C(1 + ∥w∥2  H + ∥µAw + B(w) + βC(w) + ξ∥2  H)ϑ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='28)  CBF EQUATIONS WITH POTENTIAL  13  for every w ∈ D(A) ∩ D(Φ) and ξ ∈ Φ(w), where  ϑ =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  r,  when d = 2 with r ∈ (3, ∞),  r+3  5−r,  when d = 3 with r ∈ (3, 5),  3,  when d = r = 3 with 2βµ ≥ 1,  1,  when d = 3 with r ∈ [5, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='29)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' It has been shown in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 that the operator F(·) = µA + B(·) + βC(·) + κI  is maximal monotone with domain D(F) = D(A) in H × H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Note that A = F + Φ implies  D(A) ∩ D(Φ) ⊆ D(A) and since A is the sum of two monotone operators, it is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  In order to prove A is maximal monotone, we need to show that  R(I + A) = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='30)  Step I: Well-posedness of the Yosida approximated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let f ∈ H be arbitrary  but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We approximate the inclusion problem  y + µAy + B(y) + βC(y) + Φ(y) + κy ∋ f,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='31)  by the equation  yλ + µAyλ + B(yλ) + βC(yλ) + Φλ(yλ) + κyλ = f,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='32)  where Φλ is the Yosida approximation of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' By the properties of Yosida approximation, Φλ  is demicontinuous and monotone (see [5, Chapter 2, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Therefore the sum  F(·) + Φλ(·) is maximal monotone (see [6, Chapter 2, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' This guarantees the  existence of a solution yλ ∈ D(A) for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let �κ = κ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='32) can be written as  µAyλ + B(yλ) + βC(yλ) + Φλ(yλ) + �κyλ = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='33)  We shall now prove the uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let yλ and zλ be two solutions of the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='33)  with the same data f and let wλ = yλ − zλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then we have  µAwλ + B(yλ) − B(zλ) + β(C(yλ) − C(zλ)) + Φλ(yλ) − Φλ(zλ) + �κwλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='34)  Taking the inner product with wλ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='34), we get  µ∥wλ∥2  V + (Φλ(yλ) − Φλ(zλ), wλ) + �κ∥wλ∥2  H  = −(B(yλ) − B(zλ), wλ) − β(C(yλ) − C(zλ), wλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='35)  By similar calculations as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6), we obtain  −(B(yλ) − B(zλ) − β(C(yλ) − C(zλ)), wλ) ≤ µ  2 ∥wλ∥2  V + η∥wλ∥2  H,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='36)  where ̺ =  r−3  2µ(r−1)  �  2  βµ(r−1)  �  2  r−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' By [6, Chapter 2, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3, part (i)], we know that Φλ  is monotone, so that (Φλ(yλ) − Φλ(zλ), wλ) ≥ 0 for any λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Therefore, we conclude from  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='35) that  µ  2∥wλ∥2  V + (�κ − ̺)∥wλ∥2  H ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Since ̺ < �κ, we get wλ = 0 and thus yλ = zλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Step II: Uniform bounds for yλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us take the inner product with yλ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='33) to get  µ∥yλ∥2  V + β(C(yλ), yλ) + (Φλ(yλ), yλ) + �κ∥yλ∥2  H = (f, yλ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='37)  14  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  since (B(yλ), yλ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' As the operator Φλ is monotone with 0 ∈ D(Φλ) = H, we infer  (Φλ(yλ), yλ) ≥ (Φλ(0), yλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='38)  By applying Young’s inequality and by of [6, Chapter 2, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3, part (ii)], we have  − (Φλ(0), yλ) ≤ ∥Φλ(0)∥H∥yλ∥H ≤ 1  �κ∥Φ(0)∥2  H + �κ  4∥yλ∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='39)  Then equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='37) yields  µ∥yλ∥2  V + �κ  2∥yλ∥2  H + β∥yλ∥r+1  �Lr+1 ≤ 1  �κ∥f∥2  H + 1  �κ∥Φ(0)∥2  H,  which gives  ∥yλ∥2  H + ∥yλ∥2  V + ∥yλ∥r+1  �Lr+1 ≤ C(1 + ∥f∥2  H),  for all λ > 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='40)  where the constant C = C(µ, β, �κ, ∥Φ(0)∥H) does not depend on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Taking the inner product of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='33) with Ayλ, we get  µ∥Ayλ∥2  H + (B(yλ), Ayλ) + β(C(yλ), Ayλ) + (Φλ(yλ), Ayλ) + �κ∥yλ∥2  V = (f, Ayλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='41)  By [47, Chapter VI, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' 404], we have (B(yλ), Ayλ) = 0 for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For d = 3, we consider  the cases r > 3 and r = 3 with 2βµ ≥ 1 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Case I: r > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From Cauchy-Schwarz and Young’s inequalities, we obtain  (f, Ayλ) ≤ ∥f∥H∥Ay∥H ≤ µ  4∥Ay∥2  H + 1  µ∥f∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='42)  We estimate |(B(yλ), Ayλ)| using H¨older’s, and Young’s inequalities as  |(B(yλ), Ayλ)| ≤ ∥|yλ||∇yλ|∥H∥Ayλ∥H ≤ µ  2 ∥Ayλ∥2  H + 1  2µ∥|yλ||∇yλ|∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='43)  We estimate the final term from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='43) using H¨older’s and Young’s inequalities as  �  Td |yλ(x)|2|∇yλ(x)|2dx =  �  Td |yλ(x)|2|∇yλ(x)|  4  r−1|∇yλ(x)|  2(r−3)  r−1 dx  ≤  ��  Td |yλ(x)|r−1|∇yλ(x)|2dx  �  2  r−1��  Td |∇yλ(x)|2dx  � r−3  r−1  ≤ βµ  ��  Td |yλ(x)|r−1|∇yλ(x)|2dx  �  + 2µ̺  ��  Td |∇yλ(x)|2dx  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='44)  where ̺ =  r−3  2µ(r−1)  �  2  βµ(r−1)  �  2  r−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7), we can write  (C(yλ), Ayλ) = ∥|∇yλ||yλ|  r−1  2 ∥2  H + 4  � r − 1  (r + 1)2  �  ∥|∇|yλ|  r+1  2 |∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='45)  Using the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3) of Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1, estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='40), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='42)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='43) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='45) in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='41), it yields for all λ > 0  µ  4 ∥Ayλ∥2  H + β  2 ∥|∇yλ||yλ|  r−1  2 ∥2  H + 4β  � r − 1  (r + 1)2  �  ∥|∇|yλ|  r+1  2 |∥2  H  CBF EQUATIONS WITH POTENTIAL  15  ≤ C(1 + ∥f∥2  H) +  �  ς∥Φλ(yλ)∥2  H,  for d = 2 with r ∈ (3, ∞) and d = 3 with r ∈ (3, 5),  0,  for d = 3 with r ∈ [5, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='46)  This completes the proof of energy estimates for d = 3 with r ∈ [5, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Case II: r = 3 with 2βµ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7) and Cauchy-Schwarz and Young’s inequalities, we  calculate  |(B(yλ), Ayλ)| ≤ ∥|yλ||∇yλ|∥H∥Ayλ∥H ≤ µ  2 ∥Ayλ∥2  H + 1  2µ∥|yλ||∇yλ|∥2  H,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='47)  (C(yλ), Ayλ) = ∥|yλ||∇yλ|∥2  H + 1  2∥|∇|yλ|2|∥2  H,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='48)  |(f, Ayλ)| ≤ ∥f∥H∥Ayλ∥H ≤ µ  8 ∥Ayλ∥2  H + 1  2µ∥f∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='49)  Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='40) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='47)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='49) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='41), we get  3µ  8 ∥Ayλ∥2  H +  �  β − 1  2µ  �  ∥|yλ||∇yλ|∥2  H + β  2 ∥|∇|yλ|2|∥2  H  ≤ C(1 + ∥f∥2  H) + ς∥Φλ(yλ)∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='50)  Estimate for ∥Φλ(yλ)∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Taking the inner product with Φλ(yλ) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='33), we have  ∥Φλ(yλ)∥2  H = (f, Φλ(yλ)) − (B(yλ), Φλ(yλ)) − β(C(yλ), Φλ(yλ)) − µ(Ayλ, Φλ(yλ))  − �κ(yλ, Φλ(yλ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='51)  Similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='39), we have  (yλ, Φλ(yλ)) ≥ −1  2(∥Φ(0)∥2  H + ∥yλ∥2  H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='52)  We calculate |(B(yλ), Φλ(yλ)| using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3), Agmon’s and Young’s inequalities as  |(B(yλ), Φλ(yλ)| = |b(yλ, yλ, Φλ(yλ))|  ≤ C  �  ∥yλ∥  1  2  H∥yλ∥V∥Ayλ∥  1  2  H∥Φλ(yλ)∥H,  for d = 2,  ∥yλ∥  3  2  V∥Ayλ∥  1  2  H∥Φλ(yλ)∥H,  for d = 3,  ≤ 1 − µς  8  ∥Φλ(yλ)∥2  H + µ(1 − µς)  8ς  ∥Ayλ∥2  H + C(1 + ∥f∥2  H)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='53)  For d = 3 with r ∈ (3, 5), by using the Cauchy-Schwarz, interpolation and Young’s  inequalities, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='40) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='46), we obtain  |(C(yλ), Φλ(yλ))| ≤ ∥C(yλ∥H∥Φλ(yλ)∥H ≤ ∥yλ∥r  �L2r∥Φλ(yλ)∥H  ≤ ∥yλ∥  r+3  4  �Lr+1∥yλ∥  3(r−1)  4  �L3(r+1)∥Φλ(yλ)∥H  ≤ C∥yλ∥  r+3  4  �Lr+1∥|yλ|  r−1  2 ∇yλ∥  3(r−1)  2(r+1)  H  ∥Φλ(yλ)∥H  ≤ C(1 + ∥f∥2  H)  r+3  4(r+1)  �2ς  β ∥Φλ(yλ)∥2  H + 2C  β (1 + ∥f∥2  H)  � 3(r−1)  4(r+1)  ∥Φλ(yλ)∥H  16  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  ≤ C  �  ∥Φλ(yλ)∥  5r−1  2(r+1)  H  (1 + ∥f∥2  H)  r+3  4(r+1) + ∥Φλ(yλ)∥H(1 + ∥f∥2  H)  r  r+1  �  ≤ 1 − µς  8β  ∥Φλ(yλ)∥2  H + C(1 + ∥f∥2  H)  r+3  5−r + C(1 + ∥f∥2  H)  2r  r+1  ≤ 1 − µς  8β  ∥Φλ(yλ)∥2  H + C(1 + ∥f∥2  H)  r+3  5−r ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='54)  where we have used the fact that r+3  5−r >  2r  r+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Using the Sobolev embedding (for d = 2), we  deduce  |(C(yλ), Φλ(yλ))| ≤ ∥C(yλ∥H∥Φλ(yλ)∥H ≤ ∥yλ∥r  �L2r∥Φλ(yλ)∥H  ≤ ∥yλ∥r  V∥Φλ(yλ)∥H ≤ C(1 + ∥f∥2  H)  r  2∥Φλ(yλ)∥H  ≤ 1 − µς  8β  ∥Φλ(yλ)∥2  H + C(1 + ∥f∥2  H)r,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='55)  for d = 2 with r ∈ (3, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Also, by the Cauchy-Schwarz and Young’s inequalities, we get  |(f, Φλ(yλ))| ≤  1  1 − µς ∥f∥2  H + 1 − µς  4  ∥Φλ(yλ)∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='56)  Using the estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='40) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='52)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='56) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='51), we arrive at  ς∥Φλ(yλ)∥2  H ≤ µ  4∥Ay∥2  H +  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  C(1 + ∥f∥2  H)r,  for d = 2 with r ∈ (3, ∞),  C(1 + ∥f∥2  H)  r+3  5−r ,  for d = 3 with r ∈ (3, 5),  C(1 + ∥f∥2  H)3,  for d = r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='57)  Uniform boundedness of sequqences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' It implies from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='46) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='57) that  µ  4 ∥Ayλ∥2  H + β  2 ∥|∇yλ||yλ|  r−1  2 ∥2  H + 4β  � r − 1  (r + 1)2  �  ∥|∇|yλ|  r+1  2 |∥2  H  ≤  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  C(1 + ∥f∥2  H)r,  for d = 2 with r ∈ (3, ∞),  C(1 + ∥f∥2  H)  r+3  5−r ,  for d = 3 with r ∈ (3, 5),  C(1 + ∥f∥2  H),  for d = 3 with r ∈ (5, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='58)  For d = r = 3 with 2βµ ≥ 1, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='57) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='50), we obtain  µ  8∥Ayλ∥2  H +  �  β − 1  2µ  �  ∥|yλ||∇yλ|∥2  H + β  2 ∥|∇|yλ|2|∥2  H ≤ C(1 + ∥f∥H)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='59)  Thus under Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 (condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3)), we have  ∥Ayλ∥H ≤ C, ∥|∇yλ||yλ|  r−1  2 ∥H ≤ C and ∥|∇|yλ|  r+1  2 |∥H ≤ C,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='60)  for d = 2, 3 with r ∈ (3, ∞) and d = r = 3 with 2βµ ≥ 1 for all yλ ∈ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Using  interpolation inequality and estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='8), we have  ∥C(yλ)∥H ≤ ∥yλ∥r  �L2r ≤ ∥yλ∥  r+3  4  �Lr+1∥yλ∥  3(r−1)  4  �L3(r+1) ≤ C,  for all yλ ∈ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Also, by using H¨older’s and Agmon’s inequalities, we obtain  ∥B(yλ)∥H ≤ ∥(yλ · ∇)yλ∥H ≤ ∥yλ∥V∥yλ∥  1− d  4  H  ∥A(yλ)∥  d  4  H ≤ C,  for all yλ ∈ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  CBF EQUATIONS WITH POTENTIAL  17  Now, the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='32) can be rewritten as  yλ + F(yλ) + Φλ(yλ) = f,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='61)  where F(·) = µA + B(·) + βC(·) + �κI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Hence from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='60), we conclude that  ∥F(yλ)∥H ≤ C and ∥Φλ(yλ)∥H ≤ C,  for all yλ ∈ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='62)  Step III: Convergence of yλ and proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='40), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='60) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='62),  and the Banach-Alaoglu theorem guarantee the existence of a weakly convergent subsequence  {yλj} of {yλ} such that as j → ∞  � yλj ⇀ y,  in V,  Ayλj ⇀ Ay,  in H,  �  Φλ(yλj) ⇀ f 1,  in H,  F(yλj) ⇀ f 2,  in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='63)  Since the embedding D(A) ֒→ V is compact, we get the following strong convergence also:  yλj → y in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='64)  Passing weak limit in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='61), we get  y + f 1 + f 2 = f in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='65)  In order to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='30), we need to show that f 2 = F(y) and f 1 ∈ Φ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For this, we  rewrite equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='61) for λ and �λ, subtract and then take the inner product with yλ − y�λ  to find  (Φλ(yλ) − Φ�λ(y�λ), yλ − y�λ) + ((F + I)(yλ) − (F + I)(y�λ), yλ − y�λ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='66)  By the monotonicity of F + I (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1), we conclude that  (Φλ(yλ) − Φ�λ(y�λ), yλ − y�λ) ≤ 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='67)  for all λ, �λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' By [6, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3, part (iv), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' 49] (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='63)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='64) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='67)), we  conclude that (y, f 1) ∈ Φ and  lim  λ,�λ→0  (Φλ(yλ) − Φ�λ(y�λ), yλ − y�λ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  This also implies from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='66) that  lim  λ,�λ→0  ((F + I)(yλ) − (F + I)(y�λ), yλ − y�λ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='68)  Since yλ → y, F(yλ) ⇀ f 2 in H (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='63)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='64)), F + I is maximal monotone (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='68) holds, then by [6, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' 49], we deduce that (y, y +  f 2) ∈ F + I, and this implies that F(y) = f 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Hence it follows that f ∈ y + F(y) + Φ(y),  as claimed in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  It also follows that y ∈ D(F) ∩ D(Φ) = D(A) ∩ D(Φ) and hence  D(A) = D(A) ∩ D(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Step IV: Proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='27) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='58), it implies that  ∥Ayλ∥2  H ≤ C(1 + ∥f∥2  H)ϑ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='69)  where ϑ is given as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For a fixed λ > 0 and w ∈ D(A), let  gλ = µAw + B(w) + βC(w) + Φλ(w) + �κw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='70)  Then  ∥gλ∥2  H ≤ 2�κ2∥w∥2  H + 2∥µAw + B(w) + βC(w) + Φλ(w)∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='71)  18  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  Analogous to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='69) (for the solution yλ of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='32) with f ∈ H), it yields from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='71) that the  solution w of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='70) with gλ ∈ H satiesfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Now, for w ∈ D(A)∩D(Φ) and ξ ∈ Φ(w),  let  g = µAw + B(w) + βC(w) + ξ + �κw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Since g ∈ H, we obtain a sequence {wλ}λ>0 ⊂ H such that wλ is a solution of  µAwλ + B(wλ) + βC(wλ) + Φλ(wλ) + �κwλ = g,  for all λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Then as similar to Step III, we get wλ → w in V and Awλ ⇀ Aw in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Now we calculate  the estimate ∥Awλ∥2  H as we calculate above and then passing the limit as λ → 0, we obtain  that  ∥Aw∥2  H ≤ C(1 + ∥w∥2  H + ∥µAw + B(w) + βC(w) + ξ∥2  H)ϑ,  where ϑ is defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='29) and this completes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3 and [6, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' 214-216],  the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4) has unique solution y ∈ W1,∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) satisfying the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let  ξ(t) ∈ Φ(y(t)) be such that  dy(t)  dt  + µAy(t) + B(y(t)) + βC(y(t)) + ξ(t) = f(t),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='72)  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Since f ∈ W1,1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H), so f is absolutely continuous and subsequently  f ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) and therefore f − dy  dt ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='72), we have  µAy(t) + B(y(t)) + βC(y(t)) + ξ(t) ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H),  and thus from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='28), we conclude that Ay ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Moreover, we have the Gelfand  triplet D(A) ⊂ V ⊂ H, and  y ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' D(A)) and dy(t)  dt  ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H),  which imply that y ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  A similar result holds for the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4), when one replaces Φ with the Yosida approxi-  mation Φλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let Φ ⊂ H × H satisfy Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let f ∈ W1,1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) and y0 ∈  D(A) ∩ D(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then there exists a unique strong solution  yλ ∈ W1,∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' D(A)) ∩ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='73)  to the problem  \uf8f1  \uf8f2  \uf8f3  dyλ(t)  dt  + µAyλ(t) + B(yλ(t)) + βC(yλ(t)) + Φλ(yλ(t)) = f(t),  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ (0, T),  yλ(0) = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74)  Furthermore, yλ is right differentiable, d+yλ  dt  is right continuous, and  d+yλ(t)  dt  + µAyλ(t) + B(yλ(t)) + βC(yλ(t)) + Φλ(yλ(t)) = f(t),  for all t ∈ [0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='75)  CBF EQUATIONS WITH POTENTIAL  19  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1, we know that the operator y �→ F(y) = µAy +B(y)+βC(y)+  κy is maximal monotone (for κ sufficiently large) in H × H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Since Φλ(·) is single-valued,  monotone and demicontinuous in H × H (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' [6, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3]), then by [6, Chapter 2,  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1] (see [5] also), then the sum F(·) + Φλ(·) = µA + B(·) + βC(·) + κI + Φλ(·) is  maximal monotone in H×H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Since f ∈ W1,1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) and y0 ∈ D(A)∩D(Φ), an application  of [6, Chapter 4, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='8] yields the existence of a unique solution yλ ∈ W1,∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) to  the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Arguing similarly as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3 and using the estimate  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='27), one can conclude the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='73)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4  The aim of this section is to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4 using the solvability results obtained  in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We first provide some uniform energy estimates for the solutions of the  problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Energy estimates for the solution of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3, we infer  that the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4) has a unique strong solution with the regularity given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Our  aim in this subsection is to obtain some energy estimates for the solution of the problem  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' In order to do this, we first obtain suitable energy estimates for the solution yλ(·) for  the approximate problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74), which also has a unique strong solution with the regularity  given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let yλ(·) be the unique strong solution of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74) obtained in  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then for f ∈ W1,2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) and y0 ∈ D(A) ∩ D(Φ), the solution yλ(·)  satisfies the following energy estimates:  sup  t∈[0,T]  ∥yλ(t)∥2  H + µ  � T  0  ∥yλ(t)∥2  Vdt + β  � T  0  ∥yλ(t)∥r+1  �Lr+1dt  ≤ C  �  ∥y0∥H, ∥f∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H), ∥Φ(0)∥H  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1)  where C is independent of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Furthermore, we have  sup  t∈[0,T]  ∥yλ(t)∥2  V + µ  � T  0  ∥Ayλ(t)∥2  Hdt + β  � T  0  ∥|∇yλ(t)||yλ(t)|  r−1  2 ∥Hdt  ≤ C  �  µ, β, T, ∥Ay0∥H, ϕ(y0), ∥Φ(y0)∥H, ∥Φ(0)∥H, ∥f(0)∥H, ∥f∥W1,2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2)  where C is independent of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2) in the following steps:  Step I: Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Taking the inner product with yλ(·) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74), we get for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ [0, T],  1  2  d  dt∥yλ(t)∥2  H + µ∥yλ(t)∥2  V + β∥yλ(t)∥r+1  �Lr+1 + (Φλ(yλ(t)), yλ(t)) = (f(t), yλ(t)),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3)  since (B(yλ), yλ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' By the monotonicity of Φλ(·), [6, Chapter 2, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3, part(ii)]  and using the Cauchy-Schwarz and Young’s inequalities, we have  (Φλ(yλ(·)), yλ(·)) ≥ (Φλ(0), yλ(·)) ≥ −∥Φ(0)∥2  H − 1  4∥yλ(·)∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4)  20  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  Using the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1), and Cauchy-Schwarz and Young’s inequalities in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3),  we deduce  ∥yλ(t)∥2  H + µ  � t  0  ∥yλ(s)∥2  Vds + 2β  � t  0  ∥yλ(s)∥r+1  �Lr+1ds  ≤ ∥y0∥2  H +  1  µλ1  � t  0  ∥f(s)∥2  Hds + t∥Φ(0)∥2  H,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5)  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Step II: Regularity estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' In order to obtain the energy estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2), we first need fur-  ther regularity estimates on the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' This is due to the (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3) assumption in Hypothesis  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We observe that yλ(·) satisfies for any h > 0  dyλ(t + h)  dt  + µAyλ(t + h) + B(yλ(t + h)) + βC(yλ(t + h)) + Φλ(yλ(t + h)) = f(t + h),  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then subtracting above equation from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74), and taking the inner  product with yλ(· + h) − yλ(·) and then using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6), we get for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ (0, T)  1  2  d  dt∥yλ(t + h) − yλ(t)∥2  H + µ∥yλ(t + h) − yλ(t)∥2  V + β  2 ∥|y(t)|  r−1  2 (y(t + h) − y(t))∥2  H  ≤ (f(t + h) − f(t), yλ(t + h) − yλ(t)) − (B(yλ(t + h)) − B(yλ(t)), yλ(t + h) − yλ(t)),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6)  where we have used the monotonicity property of the Yosida approximation Φλ(·) also, that  is, (Φλ(yλ(· + h)) − Φλ(yλ(·)), yλ(· + h) − yλ(·)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We consider the follwing cases:  For r > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='13) and the Cauchy-Schwarz inequality, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6) yields  1  2  d  dt∥yλ(t + h) − yλ(t)∥2  H + µ  2∥yλ(t + h) − yλ(t)∥2  V  ≤ 1  2∥f(t + h) − f(t)∥2  H + 1  2∥yλ(t + h) − yλ(t)∥2  H + ̺∥yλ(t + h) − yλ(t)∥2  H,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7)  or we can write  d  dt∥yλ(t + h) − yλ(t)∥2  H ≤ ∥f(t + h) − f(t)∥2  H + (2̺ + 1)∥yλ(t + h) − yλ(t)∥2  H,  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' By Gronwall’s inequality, we have  ∥yλ(t + h) − yλ(t)∥2  H ≤ e(2̺+1)t  �  ∥yλ(h) − yλ(0)∥2  H +  � t  0  ∥f(s + h) − f(s)∥2  Hds  �  ,  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' On dividing by h2 and then taking limit as h → 0, we obtain for all t ∈ [0, T]  ����  d+yλ(t)  dt  ����  2  H  ≤ e(2̺+1)T  �����  d+yλ(0)  dt  ����  2  H  +  � T  0  ����  df  dt (t)  ����  2  H  dt  �  ≤ Ce(2̺+1)T �  µ∥Ayλ(0)∥2  H + ∥B(yλ(0))∥2  H + β∥C(yλ(0))∥2  H + ∥Φλ(yλ(0))∥2  H  +∥f(0)∥2  H + ∥f∥W1,2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  �  ≤ C  �  µ, β, T, ∥Ay0∥H, ∥Φ(y0)∥H, ∥f(0)∥H, ∥f∥W1,2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='8)  CBF EQUATIONS WITH POTENTIAL  21  where we have used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='75) and the fact that f ∈ W1,2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) implies f ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Now  on integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7), we get  ∥yλ(t + h) − yλ(t)∥2  H + µ  � t  0  ∥yλ(s + h) − yλ(s)∥2  Vds  ≤ ∥yλ(h) − yλ(0)∥2  H + (2̺ + 1)  � t  0  ∥yλ(s + h) − yλ(s)∥2  Hds +  � t  0  ∥f(s + h) − f(s)∥2  Hds,  or we can write  µ  � t  0  ∥yλ(s + h) − yλ(s)∥2  Vds  ≤ ∥yλ(h) − yλ(0)∥2  H + (2̺ + 1)  � t  0  ∥yλ(s + h) − yλ(s)∥2  Hds +  � t  0  ∥f(s + h) − f(s)∥2  Hds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  On dividing both sides by h2 and then passing limit as h → 0, we get  � T  0  ����  dyλ(s)  ds  ����  2  V  ds ≤ C  �  µ, β, T, ∥Ay0∥H, ∥Φ(y0)∥H, ∥f(0)∥H, ∥f∥W1,2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='9)  For r = 3 with 2βµ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Once again by using Cauchy-Schwarz and Young’s inequalities,  we get  |(B(yλ(t + h)) − B(yλ(t)), yλ(t + h) − yλ(t))|  ≤ ∥yλ(t)(yλ(t + h) − yλ(t))∥H∥yλ(t + h) − yλ(t)∥V  ≤ 1  2β∥yλ(t + h) − yλ(t)∥2  V + β  2 ∥yλ(t)(yλ(t + h) − yλ(t))∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Thus we get from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6)  d  dt∥yλ(t + h) − yλ(t)∥2  H + 2  �  µ − 1  2β  �  ∥yλ(t + h) − yλ(t)∥2  V  ≤ ∥f(t + h) − f(t)∥2  H + ∥yλ(t + h) − yλ(t)∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Using the similar calculations as we have done in the case r > 3, we get the similar estimates  as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Taking the inner product with  dyλ  dt  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74) and then using the Cauchy-Schwarz and  Young’s inequalities, we get for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ (0, T)  ����  dyλ(t)  dt  ����  2  H  + µ  2  d  dt∥yλ(t)∥2  V +  β  r + 1  d  dt∥yλ(t)∥r+1  �Lr+1 +  �dyλ(t)  dt  , Φλ(yλ(t))  �  =  �  f(t), dyλ(t)  dt  �  +  �  B(yλ(t)), dyλ(t)  dt  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='10)  We calculate  �  B(yλ), dyλ  dt  �  by using the interpolation, H¨older’s and Young’s inequalities as  �  B(yλ), dyλ  dt  �  = −b  �  yλ, dyλ  dt , yλ  �  ≤ ∥yλ∥2  �L4  ����  dyλ  dt  ����  V  ≤ 1  2  ����  dyλ  dt  ����  2  V  + 1  2∥yλ∥  2(r+1)  r−1  �Lr+1 ∥yλ∥  2(r−3)  r−1  H  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  22  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  Therefore from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='10), it is immediate that  µ  2 ∥yλ(t)∥2  V +  β  r + 1∥yλ(t)∥r+1  �Lr+1 + 1  2  � t  0  ����  dyλ(s)  ds  ����  2  H  ds +  � t  0  �dyλ(s)  ds  , Φλ(yλ(s))  �  ds  ≤ µ  2 ∥y0∥2  V +  β  r + 1∥y0∥r+1  �Lr+1 + 1  2  � t  0  ∥f(s)∥2  Hds + 1  2  � t  0  ����  dyλ(s)  ds  ����  2  V  ds  + 1  2t  r−3  r−1 sup  s∈[0,t]  ∥yλ(s)∥  2(r−3)  r−1  H  �� t  0  ∥yλ(s)∥r+1  �Lr+1ds  �  2  r−1  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='11)  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1, we know that Φ = ∂ϕ, where ϕ : H → ¯R is a lower  semicontinuous proper convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then by an application of [6, Chapter 2, Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2] yields that the Yosida approximation Φλ is the Gateaux derivative of ϕλ, for all λ > 0,  that is, Φλ = ∇ϕλ, where ϕλ is the regularization of ϕ [6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' 64], given by  ϕλ(y) = inf  �∥y − z∥2  H  2λ  + ϕ(z) : z ∈ H  �  ,  for all y ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='12)  Moreover by a standard calculation, we have  d  ds[ϕλ(yλ(·))] =  �dyλ(·)  ds  , (∇ϕλ)(yλ(·))  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='13)  and  � t  0  �dyλ(s)  ds  , Φλ(yλ(s))  �  ds = ϕλ(yλ(t)) − ϕλ(y0),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='14)  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From [6, Chapter 2, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3], we infer that Jλ := (I + λΦ)−1 is  bounded on bounded subsets of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Furthermore, from [6, Chapter 2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2], we also  have  ϕ(Jλ(y)) ≤ ϕλ(y) ≤ ϕ(y),  for all λ > 0, y ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='15)  From [6, Chapter 2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1], we know that any proper lower semicontinuous convex  function is bounded from below by an affine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Therefore, there exists w ∈ H and  q ∈ R such that  ϕ(y) ≥ (y, w) + q,  for all y ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='16)  From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='15), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='16) and application of the Cauchy-Schwarz inequality yield  −ϕλ(yλ) ≤ −ϕ(Jλ(yλ)) ≤ −(Jλ(yλ), w) − q ≤ ∥Jλ(yλ)∥H∥w∥H + |q| ≤ C,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='17)  where C is independent of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Thus using ϕλ(y0) ≤ ϕ(y0) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='14), we deduce  −  � t  0  �dyλ(s)  ds  , Φλ(yλ(s))  �  ds ≤ C,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='18)  where the constant C depends on ϕ(y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Thus from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='18), we get for all  t ∈ [0, T]  µ  2 ∥yλ(t)∥2  V +  β  r + 1∥yλ(t)∥r+1  �Lr+1 + 1  2  � t  0  ����  dyλ(s)  ds  ����  2  H  ds ≤ C,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='19)  where C = C  �  µ, β, T, ∥Ay0∥H, ϕ(y0), ∥Φ(y0)∥H, ∥Φ(0)∥H, ∥f(0)∥H, ∥f∥W1,2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  CBF EQUATIONS WITH POTENTIAL  23  Step III: Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We take the inner product with Ayλ(·) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74) to obtain  1  2  d  dt∥yλ(t)∥2  V + µ∥Ayλ(t)∥2  H + β(C(yλ(t)), Ayλ(t))  = (f(t), Ayλ(t)) − (B(yλ(t)), Ayλ(t)) − (Φλ(yλ(t)), Ayλ(t)),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='20)  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' This yields  ∥yλ(t)∥2  V + 2µ  � t  0  ∥Ayλ(s)∥2  Hds + 2β  � t  0  (C(yλ(s)), Ayλ(s))ds  = ∥y0∥2  V + 2  � t  0  (f(s), Ayλ(s))ds − 2  � t  0  (B(yλ(s)), Ayλ(s))ds  − 2  � t  0  (Φλ(yλ(s)), Ayλ(s))ds,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='21)  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We consider the cases r > 3 and for d = r = 3 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Case I: For r > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From the equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7), we infer  (C(yλ), Ayλ) = ∥|∇yλ||yλ|  r−1  2 ∥2  H + 4  � r − 1  (r + 1)2  �  ∥|∇|yλ|  r+1  2 |∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='22)  From [47, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' 404], for d = 2, we have (B(yλ), Ayλ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For d = 3 with  r ∈ (3, ∞), we estimate |(B(yλ), Ayλ)| as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='43)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='44) as  |(B(yλ), Ayλ)| ≤ µ  2∥Ayλ∥2  H + β  2 ∥|∇yλ||yλ|  r−1  2 ∥2  H + ̺∥yλ∥2  V,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='23)  where ̺ =  r−3  2µ(r−1)  �  2  βµ(r−1)  �  2  r−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3) of Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='22)-  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='23), we obtain from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='21) that  ∥yλ(t)∥2  V + µ  2  � t  0  ∥Ayλ(s)∥2  Hds + 3β  2  � t  0  ∥|∇yλ(s)||yλ(s)|  r−1  2 ∥2  Hds  ≤ C +  �  ς  � t  0 ∥Φλ(yλ(s))∥2  Hds,  for d = 2 with r ∈ (3, ∞) and d = 3 with r ∈ (3, 5),  0,  for d = 3 with r ∈ [5, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='24)  where C = C(µ, β, T, ∥y0∥H, ∥f∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H), ∥Φ(0)∥H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' This completes the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2) for  d = 3 with r ∈ [5, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Case II: For d = r = 3 with 2βµ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We calculate  (C(yλ), Ayλ) = ∥|∇yλ||yλ|∥2  H + 1  2∥|∇|yλ|2|∥2  H,  |(B(yλ), Ayλ)| ≤ µ  2 ∥Ayλ∥2  H + 1  2µ∥|yλ||∇yλ|∥2  H,  |(f, Ayλ)| ≤ µ  8 ∥Ayλ∥2  H + 1  2µ∥f∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Using above estimates and Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='21), we obtain  ∥yλ(t)∥2  V + 3µ  4  � t  0  ∥Ayλ(s)∥2  Hds + 2  �  β − 1  2µ  � � t  0  ∥|∇yλ(s)||yλ(s)|∥2  Hds  24  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  ≤ C + ς  � t  0  ∥Φλ(yλ(s))∥2  Hds,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='25)  where constant C is same as given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Step IV: An estimate for  � t  0 ∥Φλ(yλ(s))∥2  Hds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us now find a bound for  � t  0 ∥Φλ(yλ(s))∥2  Hds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  For this, taking the inner product of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74) with Φλ(yλ(·)), we get for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ (0, T)  �dyλ(t)  ds  , Φλ(yλ(t))  �  + µ(Φλ(yλ(t)), Ayλ(t)) + (B(yλ(t)), Φλ(yλ(t))  + β(C(yλ(t)), Φλ(yλ(t))) + ∥Φλ(yλ(t))∥2  H = (f(t), Φλ(yλ(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='26)  Integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='26) and using the condition (H3) of Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='18), we obtain  (1 − µς)  � t  0  ∥Φλ(yλ(s))∥2  Hds ≤ C +  � t  0  (f(s), Φλ(yλ(s)))ds −  � t  0  (B(yλ(s)), Φλ(yλ(s))ds  − β  � t  0  (C(yλ(s)), Φλ(yλ(s)))ds,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='27)  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' A calculation similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='53) yields  ����  � t  0  (B(yλ(s)), Φλ(yλ(s))ds  ���� ≤ C + 1 − µς  8  � t  0  ∥Φλ(yλ(s))∥2  Hds  + µ(1 − µς)  8ς  � t  0  ∥Ayλ(s)∥2  Hds,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='28)  where we have used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='19) also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Using the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='19) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='24), we find  ����  � t  0  (C(yλ(s)), Φλ(yλ(s)))ds  ���� ≤ C  � t  0  ∥yλ(s)∥r  �L2r∥Φλ(yλ(s))∥Hds  ≤ C sup  s∈[0,t]  ∥yλ(s)∥  r+3  4  �Lr+1  � t  0  ∥yλ(s)∥  3(r−1)  4  �L3(r+1)∥Φλ(yλ(s))∥Hds  ≤ Ct  5−r  4(r+1)  �� t  0  ∥Φλ(yλ(s))∥2  Hds  � 1  2�� t  0  ∥yλ(s)∥r+1  �L3(r+1)ds  � 3(r−1)  4(r+1)  ≤ C  �  C + ς  � t  0  ∥Φλ(yλ(s))∥2  Hds  � 3(r−1)  4(r+1)�� t  0  ∥Φλ(yλ(s))∥2  Hds  � 1  2  ≤ 1 − µς  8β  � t  0  ∥Φλ(yλ(s))∥2  Hds + C,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='29)  for d = 3 and r ∈ (3, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We further calculate by using Sobolev’s embedding V ⊂ �L2r and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='19), for d = 2 with r ∈ (3, ∞) as  ����  � t  0  (C(yλ(s)), Φλ(yλ(s)))ds  ���� ≤ C  � t  0  ∥yλ(s)∥r  �L2r∥Φλ(yλ(s))∥Hds  ≤ C  � t  0  ∥yλ(s)∥2r  V ds + 1 − µς  8β  � t  0  ∥Φλ(yλ(s))∥2  Hds  ≤ C + 1 − µς  8β  � t  0  ∥Φλ(yλ(s))∥2  Hds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='30)  CBF EQUATIONS WITH POTENTIAL  25  Using the Cauchy-Schwarz and Young’s inequailities, we further have  ����  � t  0  (f(s), Φλ(yλ(s)))ds  ���� ≤ C  � t  0  ∥f(s)∥2  Hds + 1 − µς  4  � t  0  ∥Φλ(yλ(s))∥2  Hds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='31)  By using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='28)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='31), we conclude from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='27) that  (1 − µς)  � t  0  ∥Φλ(yλ(s))∥2  Hds ≤ C + C  � t  0  ∥f(s)∥2  Hds + 1 − µς  2  � t  0  ∥Φλ(yλ(s))∥2  Hds  + µ(1 − µς)  8ς  � t  0  ∥Ayλ(s)∥2  Hds,  or we can write  ς  � t  0  ∥Φλ(yλ(s))∥2  Hds ≤ C + C  � t  0  ∥f(s)∥2  Hds + µ  4  � t  0  ∥Ayλ(s)∥2  Hds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='32)  Thus from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='24), for d = 2, 3 with r > 3, we get  ∥yλ(t)∥2  V + µ  4  � t  0  ∥Ayλ(s)∥2  Hds + 3β  2  � t  0  ∥|∇yλ(s)||yλ(s)|  r−1  2 ∥2  Hds ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='33)  Also from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='25), for d = r = 3 with 2βµ ≥ 1, we find  ∥yλ(t)∥2  V + µ  2  � t  0  ∥Ayλ(s)∥2  Hds + 2  �  β − 1  2µ  � � t  0  ∥|∇yλ(s)||yλ(s)|∥2  Hds ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='34)  Combining the above estimates with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='32), we deduce  � t  0  ∥Φλ(yλ(s))∥2  Hds ≤ C,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='35)  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Passing to the limit as λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us now pass λ → 0 and obtain the energy estimates  for the solution of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The limit of the sequence (yλ)λ>0 satisfies the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈  (0, T) in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We prove (yλ) satisfies the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ (0, T) in H in the following  steps:  Step I: For d = 2 with r ∈ (3, ∞) and d = 3 with r ∈ (3, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From the Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4, we  have  yλ ∈ W1,∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' D(A)) ∩ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='36)  From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='33)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='35), we have uniform bounds for the sequences  (Ayλ)λ>0 and (Φλ(yλ))λ>0 in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Then from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4), we have the sequence (B(yλ))λ is bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H), since  � T  0  ∥B(yλ(t))∥2  Hdt ≤ C  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  T 1/2 sup  t∈[0,T]  (∥yλ(t)∥H∥yλ(t)∥2  V)  �� T  0 ∥Ayλ(t)∥2  Hdt  �1/2  ,  for d = 2,  T 1/2 sup  t∈[0,T]  ∥yλ(t)∥3  V  �� T  0 ∥Ayλ(t)∥2  Hdt  �1/2  ,  for d = 3,  26  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='37)  Moreover, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='19), we have  �  dyλ  dt  �  λ>0 is uniformly bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74)  and the energy estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='19), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='33)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='35) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='37), we find  � T  0  ∥C(yλ(t))∥2  Hdt ≤ C  � T  0  �����  dyλ(t)  dt  ����  2  H  + ∥Ayλ(t)∥2  H + ∥B(yλ(t))∥2  H + ∥Φλ(yλ(t))∥2  H  + ∥f(t)∥2  H  �  dt  ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Therefore the sequence (C(yλ))λ>0 is bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Thus by making the use of the  Banach-Alaoglu theorem, we infer  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  yλ  ∗⇀ y  in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V ∩ �Lr+1),  yλ ⇀ y  in Lr+1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' �L3(r+1)),  dyλ  dt ⇀ dy  dt  in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V),  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  Ayλ ⇀ Ay  in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H),  B(yλ) ⇀ ζ  in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H),  C(yλ) ⇀ ϑ  in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H),  Φλ(yλ) ⇀ φ  in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='38)  Since V ֒→ H ֒→ V′, the embedding of V ֒→ H is compact, and the fact that y ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V),  dy  dt ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) ֒→ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V′) imply  yλ → y in C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='39)  by an application of the Aubin-Lions compactness lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Since D(A) ֒→ V ֒→ H, (yλ)λ>0  is bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' D(A)) and  �  dyλ  dt  �  λ>0 is bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H), and the embedding  D(A) ֒→ V is compact, it implies once again from Aubin-Lions compactness lemma that  yλ → y in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='40)  From [6, Chapter 2, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4, part(i)], we know that (I + λΦ)−1 is nonexpansive, that  is, Lipschitz with Lipschitz constant 1 and from [6, Chapter 2, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3, part (iii)],  we have  � T  0  ∥(I + λΦ)−1yλ(t) − y(t)∥2  Hdt  ≤ 2  � T  0  ∥(I + λΦ)−1(yλ(t)) − (I + λΦ)−1y(t)∥2  Hdt + 2  � T  0  ∥(I + λΦ)−1y(t) − y(t)∥2  Hdt  ≤ 2  � T  0  ∥yλ(t) − y(t)∥2  Hdt + 2  � T  0  ∥(I + λΦ)−1y(t) − y(t)∥2  Hdt  → 0 as λ → 0,  so that (I + λΦ)−1(yλ) → y in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) and (I + λΦ)−1(yλ) → (I + λΦ)−1y, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  t ∈ (0, T) in H (along a subsequence, which is still denoted by the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From [6, Chapter  2, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1, part (i)] and [38, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7], we know that the maximal monotone  operator Φ is weak-strong and strong-weak closed in H × H, that is, if Φλ(yλ) ∈ Φ(I +  λΦ)−1(yλ), (I + λΦ)−1(yλ) → y in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) and Φλ(yλ) ⇀  φ in  L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H), then  φ ∈ Φ(y) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ (0, T) in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  CBF EQUATIONS WITH POTENTIAL  27  Convergence of Bilinear operator B(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We have from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4)  ∥(B(yλ) − B(y)∥H ≤ C ×  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∥yλ − y∥  1  2  H∥yλ − y∥  1  2  V∥yλ∥  1  2  V∥Ayλ∥  1  2  H  +∥y∥  1  2  H∥y∥  1  2  V∥yλ − y∥  1  2  V∥Ayλ − Ay∥  1  2  H,  for d = 2,  ∥yλ − y∥H∥yλ∥  1  2  V∥Ayλ∥  1  2  H  +∥y∥H∥yλ − y∥  1  2  V∥Ayλ − Ay∥  1  2  H,  for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  For d = 2, we calculate  � T  0  ∥B(yλ(t)) − B(y(t))∥2  Hdt  ≤ C  � T  0  ∥yλ(t) − y(t)∥H∥yλ(t) − y(t)∥V∥yλ(t)∥V∥Ayλ(t)∥Hdt  + C  � T  0  ∥y(t)∥H∥y(t)∥V∥yλ(t) − y(t)∥V∥Ayλ(t) − Ay(t)∥Hdt  ≤ C∥yλ − y∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)∥yλ∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='V)∥yλ − y∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='V)∥Ayλ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  + ∥y∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)∥y∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='V)∥yλ − y∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='V)∥Ayλ − Ay∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  → 0, as λ → 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='41)  where we have used the H¨older’s inequality, strong convergences (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='39)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='40) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  For d = 3, we estimate  � T  0  ∥B(yλ(t)) − B(y(t))∥2  Hdt ≤ C  � T  0  ∥yλ(t) − y(t)∥2  V∥yλ(t)∥V∥Ayλ(t)∥Hdt  + C  � T  0  ∥y(t)∥2  V∥yλ(t) − y(t)∥V∥Ayλ(t) − Ay(t)∥Hdt  ≤ ∥yλ − y∥2  L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='V)∥yλ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='V)∥Ayλ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  + ∥y∥2  L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='V)∥yλ − y∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='V)∥Ayλ − Ay∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  → 0 as λ → 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='42)  where we have used the H¨older’s inequality, strong convergences (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='39)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='40) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Convergence of nonlinear operator C(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' By using Taylor’s formula [18, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1] and  H¨older’s inequality, we have  � T  0  ∥C(yλ(t)) − C(y(t))∥  r+1  r  H  dt  ≤  � T  0  �� 1  0  ∥C′(θyλ(t) + (1 − θ)y(t))(yλ(t) − y(t))∥  r+1  r  H  dθ  �  dt  ≤ r  � T  0  �  ∥yλ(t)∥r−1  �L2r + ∥y(t)∥r−1  �L2r  � r+1  r ∥yλ(t) − y(t)∥  r+1  r  �L2r dt  ≤ C  �� T  0  ∥yλ(t)∥  (r−1)(r+1)  r  �L2r  ∥yλ(t) − y(t)∥  r+1  r  �L2r dt +  � T  0  ∥y(t)∥  (r−1)(r+1)  r  �L2r  ∥yλ(t) − y(t)∥  r+1  r  �L2r dt  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='43)  28  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  Using the interpolation in 2 ≤ 2r ≤ 3(r + 1) and H¨older’s inequalities, we obtain  � T  0  ∥yλ(t)∥  (r−1)(r+1)  r  �L2r  ∥yλ(t) − y(t)∥  r+1  r  �L2r dt  ≤  � T  0  ∥yλ(t)∥  (r+3)(r−1)(r+1)  r2(3r+1)  H  ∥yλ(t)∥  3(r−1)2(r+1)2  r2(3r+1)  �L3(r+1)  ∥yλ(t) − y(t)∥  (r+3)(r+1)  r2(3r+1)  H  × ∥y(t)λ − y(t)∥  3(r−1)(r+1)2  r2(3r+1)  �L3(r+1)  dt  ≤  �  ∥yλ∥  (r+3)(r−1)(r+1)  r2(3r+1)  L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  ��  sup  t∈[0,T]  ∥yλ(t) − y(t)∥  (r+3)(r+1)  r2(3r+1)  H  �  ×  � T  0  ∥yλ(t)∥  3(r−1)2(r+1)2  r2(3r+1)  �L3(r+1)  ∥yλ(t) − y(t)∥  3(r−1)(r+1)2  r2(3r+1)  �L3(+1)  dt  ≤  �  ∥yλ∥  (r+3)(r−1)(r+1)  r2(3r+1)  L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H)  ��  sup  t∈[0,T]  ∥yλ(t) − y(t)∥  (r+3)(r+1)  r2(3r+1)  H  �  × T  r(3r+1)  r+3 ∥yλ∥Lr+1(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='�L3(r+1))∥yλ − y∥Lr+1(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='�L3(r+1))  → 0 as λ → 0,  where we have used the H¨older’s inequality, strong convergence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='39) and energy estimates  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2), and for H¨older’s inequality, we use the exponents  3(r−1)2(r+1)  r2(3r+1)  + 3(r−1)(r+1)  r2(3r+1)  +  r+3  r(3r+1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Similarly,  � T  0  ∥yλ(t)∥  (r−1)(r+1)  r  �L2r  ∥yλ(t) − y(t)∥  r+1  r  �L2r dt → 0, as λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Hence from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='43), we conclude that  C(yλ) → C(y) strongly in L  r+1  r (0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='44)  Letting λ → 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74), we obtain that y satisfies the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Uniqueness of solution to the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us now prove that the solution ob-  tained by passing to the limit with λ → 0 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The solution for the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6) is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let y1(·) and y2(·) be two solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6) satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then we have  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ (0, T),  1  2  d  dt∥y1(t) − y2(t)∥2  H + µ∥y1(t) − y2(t)∥2  V + (B(y1(t)) − B(y2(t)), y1(t) − y2(t))  + β(C(y1(t)) − C(y2(t)), y1(t) − y2(t)) + (ξ1(t) − ξ2(t), y1(t) − y2(t)) = 0,  where ξj(·) ∈ Φ(yj(·)), for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Integrating the above equality and using the monotonicity  of Φ, we can write  ∥y1(t) − y2(t)∥2  H + 2µ  � t  0  ∥y1(s) − y2(s)∥2  Vds  CBF EQUATIONS WITH POTENTIAL  29  ≤ ∥y1(0) − y2(0)∥2  H − 2  � t  0  (B(y1(s)) − B(y2(s)), y1(s) − y2(s))ds  − 2β  � t  0  (C(y1(s)) − C(y2(s)), y1(s) − y2(s))ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='45)  Using calculations similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='12)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='13), we find  |(B(y1) − B(y2), y1 − y2)| ≤ µ  2 ∥y1 − y2∥2  V + β  2 ∥|y2|  r−1  2 (y1 − y2)∥2  H + ̺∥y1 − y2∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='46)  Also calculation similar to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6) gives  β(C(y1) − C(y2), y1 − y2) ≥ β  2 ∥|y2|  r−1  2 (y1 − y2)∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='47)  Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='46)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='47) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='45), we get  ∥y1(t) − y2(t)∥2  H + µ  � t  0  ∥y1(s) − y2(s)∥2  Vds  ≤ ∥y1(0) − y2(0)∥2  H + ̺  � t  0  ∥y1(s) − y2(s)∥2  Hds,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='48)  for all t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Applying Gronwall’s inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='48), we obtain for all t ∈ (0, T)  ∥y1(t) − y2(t)∥2  H ≤ ∥y1(0) − y2(0)∥2  He̺T ,  which proves the uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3, we know that the operator A(·) is m-  accretive in H × H for sufficiently large κ ≥ ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' So by using the abstract theory, we obtain  the regularity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3) given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Moreover, from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2, the solution  y ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' D(A)) ∩ Lr+1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' �L3(r+1)) ∩ W1,2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V)  satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6) in H for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' in t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The uniqueness of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6) follows from  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3 and this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Applications  We discuss some applications of the results obtained in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' These  include flow invariance preserving feedback controllers, a time optimal control problem and  stabilizing feedback controllers for 2D and 3D CBF equations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Flow invariance preserving feedback controllers ([13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us consider the following  controlled CBF equations:  \uf8f1  \uf8f2  \uf8f3  dy(t)  dt  + µAy(t) + B(y(t)) + βC(y(t)) = f(t) + U(t), t ∈ (0, T],  y(0) = y0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1)  where U(·) is distributed control acting on the system, f ∈ W1,1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) and y0 ∈ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Consider a closed and convex set K ⊂ H such that 0 ∈ K and  (I + λA)−1K ⊂ K, for all λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2)  30  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  Our aim is to search for a feedback control U = Ψ(y) such that y(t) ∈ K, for all t ∈ [0, T], if  y0 ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' That is, we have to find a feedback controller for which the set K is invariant with  respect to CBF flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We establish this by solving the following CBF inclusion problem:  dy(t)  dt  + µAy(t) + B(y(t)) + βC(y(t)) − f(t) + NK(y(t)) ∋ 0, t ∈ (0, T],  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3)  where NK(y) = {w ∈ H : (w, y − z) ≥ 0, for all z ∈ K} is the well-known Clark’s normal  cone to K at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We consider the indicator function IK : H → R ([6]) given by  IK(x) =  �  0,  if x ∈ K,  +∞,  if x /∈ K,  whose subdifferential is given by  ∂IK(x) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ∅,  if x /∈ K,  {0},  if x ∈ int(K),  NK(x) = {y ∈ H : (y, x − z) ≥ 0, for all z ∈ K},  if x ∈ ∂K,  where int(K) and ∂K denote the interior and boundary of K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then regulariza-  tion of IK is given by [6, Chapter 2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2]  (IK)λ(x) = 1  2λ∥x − PK(x)∥2  H,  and its Gateaux derivative  (∂IK)λ(x) = 1  λ(x − PK(x)),  where PK : L2(Td) → K is the projection operator of x onto K which is equal to the resolvent  (I + λ∂IK)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' It implies that the above derivative is equal to the Yosida approximation of  ∂IK, that is  (∂IK)λ(x) = 1  λ(x − (I + λ∂IK)−1(x)), for all x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  We observe that the multi valued operator Φ := NK is a maximal monotone operator with  0 ∈ D(Φ) = D(∂IK) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Also, the operator A is single-valued maximal monotone, and thus  from [5, Chapter IV, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1, part(iv)], we have  (Ay, (∂IK)λ(y)) ≥ 0,  for all y ∈ D(A), λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Thus the multi valued operator NK = ∂IK satisfies all the assumptions (H1)-(H3) of Hypoth-  esis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 and therefore we can apply the main result Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3 to the inclusion problem  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3) to determine a feedback controller U ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) with U(t) ∈ −NK(y(t)) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  t ∈ [0, T] which is given by  U(t) = − f(t) + µAy(t) + B(y(t)) + βC(y(t))  − (−f(t) + µAy(t) + B(y(t)) + βC(y(t)) + NK(y(t)))0,  for all t ∈ [0, T),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4)  where NK(y) is the H-valued normal cone to K at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Flow invariance for the estrophy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We consider the constraint set  K = {y ∈ V : ∥∇ × y∥H = ∥∇y∥H = ∥A  1  2y∥H ≤ ̟}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  CBF EQUATIONS WITH POTENTIAL  31  Let f be any arbitrary element of K such that y + λAy = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Taking the inner product with  Ay and using the Cauchy-Schwarz and Young’s inequalities, we obtain  ∥y∥2  V + λ∥Ay∥2  H ≤ 1  2∥f∥2  V + 1  2∥y∥2  V ⇒ ∥y∥V ≤ ∥f∥V,  for all λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Thus from the definition of K we have y ∈ K and this imply (I+λA)−1K ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  We will find a feedback control so that enstrophy of the system kept inside this constraint  set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The normal cone corresponding to the convex set K is  NK(y) =  \uf8f1  \uf8f2  \uf8f3  0,  if ∥∇y∥H < ̟,  �  λ>0  λAy,  if ∥∇y∥H = ̟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  The feedback control is given by  U(·) ∈ −NK(y(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  For ∥∇y∥H < ̟, that is, when the flow remain inside the constraint set K, we have  U(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For ∥∇y∥H = ̟,  U(t) = −λ0Ay(t), for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ [0, T],  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5)  for some λ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4), we have  U(t) = − f(t) + µAy(t) + B(y(t)) + βC(y(t)) + d+y(t)  dt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Taking the inner product with Ay and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5), we obtain  λ0 =  −1  ∥Ay∥2  H  �  (f, Ay) − µ∥Ay∥2  H − b(y, y, Ay) − (C(y), Ay)  �  ,  where we have used the fact that  �d+y(t)  dt  , Ay(t)  �  = d+  dt ∥∇y(t)∥2  H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Therefore the feedback control becomes  U(t) = −Ay(t)  ∥Ay(t)∥2  H  �  (f(t), Ay(t)) − µ∥Ay(t)∥2  H − b(y(t), y(t), Ay(t)) − (C(y(t)), Ay(t))  �  ,  for all t ∈ [0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Thus, for y0 ∈ D(A) ∩ K and f ∈ W1,1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H), and the feedback  control given above, the closed loop problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1) has a unique solution y ∈ W1,∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H)∩  L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' D(A)) which satisfies  y(t) ∈ K,  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  We refer the interested readers to [13] for some other important flow invariance problems  like localized dissipation, pointwise velocity constraints, pointwise vorticity contraint, helicity  invariance, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  32  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' A time optimal control problem ([7, 37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us discuss the following time optimal  control for CBF equations  \uf8f1  \uf8f2  \uf8f3  dy(t)  dt  + µAy(t) + B(y(t)) + βC(y(t)) = U(t), for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t > 0, in H,  y(0) = y0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6)  Let κ > 0 and we define the class of controls  Uκ = {U(·) ∈ L∞(R+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) : ∥U(t)∥H ≤ κ, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Let y0 and y1 be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' A control U(·) ∈ Uκ is said to be admissible if it  steers from the (initial state) y0 to the (target) y1 in a finite time T along the trajectory  y(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' y0, U(·)) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6) which starts from y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We assume that the class of all such controls  (admissible class) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let T(y0, y1) be the infimum of all such times and it is called  minimal time, that is,  T(y0, y1) := inf  T∈R+{T : y(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' y0, U(·)) = y1, U(·) ∈ Uκ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  A control U∗(·) such that y(T(y0, y1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' y0, U∗(·)) = y1 is called time optimal control and the  time T(y0, y1) is said to be optimal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The pair (y∗, U∗) is called the time optimal pair,  where y∗ = y(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' y0, U∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  We define a multivalued operator sgn : H → H by  sgny =  �  y  ∥y∥H,  if y ̸= 0,  {z ∈ H : ∥z∥H ≤ 1},  if y = 0,  which is the subdifferential of ∥y∥H and hence it is maximal monotone in H×H ([6, Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1, Chapter 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From [6, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4, Chapter 4], the Yosida approximation of B :=  κ sgn(·) is given by  Bλ(y) = 1  λ  �  y − (I + λB)−1y  �  =  �  κy  ∥y∥H,  if ∥y∥H ≥ λ,  κ  λy,  if ∥y∥H < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  From the above definition, we conclude that  (Ay, Bλ(y − y1)) ≥ 0,  for all y ∈ D(A), λ > 0,  and therefore all the assumptions of Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Thus we can apply the  existence and uniqueness result (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3) of strong solutions for the system  \uf8f1  \uf8f2  \uf8f3  dy(t)  dt  + µAy(t) + B(y(t)) + βC(y(t)) + κ(sgn(y(t) − y1)) ∋ 0, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t > 0,  y(0) = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7)  Then the feedback law U(t) ∈ −κ(sgn(y(t) − y1)), for t > 0, ensures the existence of an  admissible control U(·) ∈ Uκ for the system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6), under the assumption that  ∥µAy1 + B(y1) + βC(y1)∥H < κ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='8)  and  ∥y0 − y1∥H ≤ κ − ∥µAy1 + B(y1) + βC(y1)∥H  ̺  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='9)  CBF EQUATIONS WITH POTENTIAL  33  for y0, y1 ∈ D(A), where ̺ is given as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' In order to prove this, we show that the  system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7) has finite extinction property in H, that is, y(T) = y1 for some T > 0 (see [6,  section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3, Chapter 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us set z(·) = y(·) − y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then z(·) satisfies  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  dz(t)  dt  + µAz(t) + B(z(t) + y1) − B(y1) + β(C(z(t) + y1) − C(y1))  +κ sgn(z(t)) ∋ −(µAy1 + B(y1) + βC(y1)), for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t > 0,  z(0) = y0 − y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  We assume that there exists no T such that z(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Taking the inner product with  sgn(z(·)) (using a smooth approximation of sgn(z(·)) [6], one can justify), we get  1  2  d  dt∥z(t)∥2  H + µ∥z(t)∥2  V + κ∥z(t)∥H + (C(z(t) + y1) − C(y1), z(t))  = (B(y1) − B(z(t) + y1), z(t)) − (µAy1 + B(y1) + βC(y1), z(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='13), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6) and using a calculation similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='36), we obtain  1  2  d  dt∥z(t)∥2  H + µ  2 ∥z(t)∥2  V + κ∥z(t)∥H ≤ ̺∥z(t)∥2  H + ∥µAy1 + B(y1) + βC(y1)∥H∥z(t)∥H,  and we can rewrite  d  dt∥z(t)∥H + η ≤ ̺∥z(t)∥H,  where η = κ − ∥µAy1 + B(y1) + βC(y1)∥H > 0 and ̺ is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' By using variation of  constant formula, we get  e−̺t∥z(t)∥H ≤  �  ∥z(0)∥H − η  ̺  �  + η  ̺e−̺t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  This shows as t → ∞ we are getting contradiction to the assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' This implies  that z = z(t) has finite extinction property in time T > 0 and this proves the existence of  an admissible control U(·) ∈ Uκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Stabilizing feedback controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us consider the following controlled CBF equa-  tions:  \uf8f1  \uf8f2  \uf8f3  dy(t)  dt  + µAy(t) + B(y(t)) + βC(y(t)) = f e + U(t), for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t > 0,  y(0) = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='10)  Let ye ∈ D(A) be the steady-state (equilibrium) solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='10), that is, ye satisfies  µAye + B(ye) + βC(ye) = f e in Td,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='11)  whose solvability results are available in [35, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let K ⊂ H be a closed and  convex set with 0 ∈ K such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  We set z(·) = y(·) − ye, then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='10) becomes  \uf8f1  \uf8f2  \uf8f3  dz(t)  dt  + µAz(t) + B(z(t) + ye) − B(ye) + βC(z(t) + ye) − C(ye) = U(t), for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t > 0,  z(0) = y0 − ye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='12)  34  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  Let �B(z(·)) := B(z(·)+ye)−B(ye) and �C(z(·)) := C(z(·)+ye)−C(ye).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='12) becomes  \uf8f1  \uf8f2  \uf8f3  dz(t)  dt  + µAz(t) + �B(z(t)) + �C(z(t)) = U(t), for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t > 0,  z(0) = y0 − ye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='13)  Using Step IV in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1, it is clear that �B(·) and �C(·) map from D(A)  to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Therefore the operator  µA + �B(·) + β�C(·) + θI + ∂IK(·),  is m-accretive in H × H for θ > 0 is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let  Φ(w) := θw + ∂IK(w),  for all w ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Since D(∂IK) = K and K ⊂ H, then from [6, Chapter 2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3], we  can write for all w ∈ H  Φ(w) = ∂  �θ  2∥w∥2  H + IK(w)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Clearly 0 ∈ D(Φ) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Also, from [5, Chapter IV, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1, part (iv)], we have  (Φ(w), Aw) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Thus the Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Therefore, we can apply the existence and uniqueness  result (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3) for the inclusion problem  \uf8f1  \uf8f2  \uf8f3  dz(t)  dt  + µAz(t) + �B(z(t)) + �C(z(t)) + θz(t) + ∂IK(z(t)) ∋ 0, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t > 0,  z(0) = y0 − ye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='14)  We intend to find a feedback controller U(·) given by U(t) ∈ −θz(t) − ∂IK(z(t) which  statbilizes the equilibrium solution ye exponentially under the invariance condition that  y0 −ye ∈ K, then y(t) −ye ∈ K for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The stability part will be discussed in a future  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The case of d = 2, 3 and r ∈ [1, 3]  The case of d = 2, 3 and r ∈ [1, 3] is considered in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We quantize the Navier-  Stokes nonlinearity B(·) and prove monotonicity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The authors in [13] took a V-ball  for quantization, while we are taking an �L4-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Define the quantized nonlinearity as  BN(y) =  \uf8f1  \uf8f2  \uf8f3  B(y),  if ∥y∥�L4 ≤ N,  �  N  ∥y∥�L4  �4  B(y),  if ∥y∥�L4 > N,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1)  where N ∈ N∗ := N ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The operator BN(·) : V → V′ satisfies  |⟨BN(y) − BN(z), y − z⟩| ≤ µ  2∥y − z∥2  V + CN∥y − z∥2  H,  for all y, z ∈ V,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2)  where µ > 0 is the same as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  CBF EQUATIONS WITH POTENTIAL  35  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Without loss of generality, one may assume that ∥y∥�L4 ≤ ∥z∥�L4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Therefore, we need  to consider the following three cases:  Case I: ∥y∥�L4, ∥z∥�L4 ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2), H¨older’s, Ladyzhenskaya’s and Young’s inequalities,  we have  |⟨BN(y) − BN(z), y − z⟩|  = |⟨B(y) − B(z), y − z⟩| = |⟨B(y − z), z⟩|  ≤ C∥y − z∥�L4∥y − z∥V∥z∥�L4 ≤ CN∥y − z∥�L4∥y − z∥V  ≤ CN∥y − z∥  1− d  4  H  ∥y − z∥  1+ d  4  V  ≤ µ  2∥y − z∥2  V + CN∥y − z∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3)  Case II: ∥y∥�L4, ∥z∥�L4 > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let us first consider  ⟨BN(y) − BN(z), y − z⟩  =  ��  N  ∥y∥�L4  �4  B(y) −  �  N  ∥z∥�L4  �4  B(z), y − z  �  =  ��  N  ∥y∥�L4  �4  −  �  N  ∥z∥�L4  �4�  ⟨B(y), y − z⟩ +  �  N  ∥z∥�L4  �4  ⟨B(y) − B(z), y − z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4)  Now by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2) and H¨older’s inequality, we calculate  |⟨B(y), y − z⟩| = |⟨B(y, y − z), z⟩| ≤ ∥y∥�L4∥y − z∥V∥z∥�L4,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5)  By Taylor’s formula, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5), H¨older’s, Ladyzhenskaya’s and Young’s inequalities, we obtain  ����  �  N  ∥y∥�L4  �4  −  �  N  ∥z∥�L4  �4����|⟨B(y), y − z⟩|  ≤ 4  ��  N  ∥y∥�L4  �  +  �  N  ∥z∥�L4  ��3����  N  ∥y∥�L4  −  N  ∥z∥�L4  ����|⟨B(y, y − z), z⟩|  ≤ CN∥y − z∥�L4∥y − z∥V  ≤ µ  4 ∥y − z∥2  V + CN∥y − z∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  A calculation similar to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3) yields  �����  �  N  ∥z∥�L4  �4  ⟨B(y) − B(z), y − z⟩  ����� ≤ µ  4 ∥y − z∥2  V + CN∥y − z∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='6)  Combining the above estimates imply (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Case III: ∥y∥�L4 ≤ N and ∥z∥�L4 > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' One can rewrite  ⟨BN(y) − BN(z), y − z⟩  =  �  B(y) −  �  N  ∥z∥�L4  �4  B(z), y − z  �  =  �  1 −  �  N  ∥z∥�L4  �4�  ⟨B(y), y − z⟩ +  �  N  ∥z∥�L4  �4  ⟨B(y) − B(z), y − z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  36  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  As 1 −  �  N  ∥z∥�L4  �4  =  ∥z∥4  �L4−N4  ∥z∥4  �L4  ≤  ∥z∥4  �L4−∥y∥4  �L4  ∥z∥4  �L4  , one can use the estimates in the previous cases to  conclude (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  □  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For d = 2, 3 and 1 ≤ r ≤ 3, define the operator ΥN : D(ΥN) → H by  ΥN(·) := µA + BN(·) + βC(·),  with D(ΥN) = D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Moreover, there exists ηN > 0 such that ΥN + ηNI is m-accretive in  H × H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1, proof follows in a similar way as the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 with  some minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  □  One can prove the following result similar to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let N ∈ N∗ be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let Φ ⊂ H × H be a maximal monotone operator  satisfying Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Define the multi-valued operator AN : D(AN) → H by  AN(·) = µA + BN(·) + βC(·) + Φ(·) + ηNI  with domain D(AN) = {y ∈ H : ∅ ̸= AN(y) ⊂ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then D(AN) = D(A) ∩ D(Φ) and AN is  a maximal monotone operator in H × H, where ηN is as in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Moreover, there exists a constant C such that  ∥Aw∥2  H ≤ C(1 + ∥w∥2  H + ∥µAw + BN(w) + βC(w) + Φλ(w)∥2  H)3,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7)  for every w ∈ D(A) and for every λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Furthermore, we have  ∥Aw∥2  H ≤ C(1 + ∥w∥2  H + ∥µAw + BN(w) + βC(w) + ξ∥2  H)3,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='8)  for every w ∈ D(A) ∩ D(Φ) and for every ξ ∈ Φ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Let us now consider the following approximate equation:  \uf8f1  \uf8f2  \uf8f3  dyN(t)  dt  + µAyN(t) + BN(yN(t)) + βC(yN(t)) + Φ(yN(t)) ∋ f(t),  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ (0, T),  yN(0) = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Using Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3, one can establish the following results in a similar way as in the  proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let Φ ⊂ H × H satisfy Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let f ∈ W1,1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) and y0 ∈  D(A) ∩ D(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then there exists a unique strong solution  yN ∈ W1,∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' D(A)) ∩ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V)  to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Furthermore, yN is right differentiable, d+yN  dt  is right continuous, and  d+yN(t)  dt  + (µAyN(t) + BN(yN(t)) + βC(yN(t)) + Φ(yN(t)) − f(t))0 = 0,  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let Φ ⊂ H × H satisfy Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Let f ∈ W1,1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) and y0 ∈  D(A) ∩ D(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then there exists a unique strong solution  yλ  N ∈ W1,∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' D(A)) ∩ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V)  CBF EQUATIONS WITH POTENTIAL  37  to the problem  \uf8f1  \uf8f2  \uf8f3  dyλ  N(t)  dt  + µAyλ  N(t) + BN(yλ  N(t)) + βC(yλ  N(t)) + Φλ(yλ  N(t)) = f(t),  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ (0, T),  yλ  N(0) = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Furthermore, yλ  N is right differentiable, d+yλ  N  dt  is right continuous, and  d+yλ  N(t)  dt  + µAyλ  N(t) + BN(yλ  N(t)) + βC(yλ  N(t)) + Φλ(yλ  N(t)) = f(t),  for all t ∈ [0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' For the case d = 2, 3 and r ∈ [1, 3], calculations similar to  the energy estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1) yields  ∥yλ  N(t)∥2  H + µ  � t  0  ∥yλ  N(s)∥2  Vds + 2β  � t  0  ∥yλ  N(s)∥r+1  �Lr+1ds  ≤ ∥y0∥2  H +  1  µλ1  � t  0  ∥f(s)∥2  Hds + t∥Φ(0)∥2  H,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='9)  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' We take the inner product with Ayλ  N(·) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='74) to obtain  1  2  d  dt∥yλ  N(t)∥2  V + µ∥Ayλ  N(t)∥2  H + (BN(yλ  N(t)) + βC(yλ  N(t)) + Φλ(yλ  N(t)), Ayλ  N(t))  = (f(t), Ayλ  N(t)),  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='7), we infer that  (C(yλ  N), Ayλ  N) = ∥|∇yλ  N||yλ  N|  r−1  2 ∥2  H + 4  � r − 1  (r + 1)2  �  ∥|∇|yλ  N|  r+1  2 |∥2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Using [47, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' 404] and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4), we find  |(BN(yλ  N), Ayλ  N)| ≤  �  0,  for d = 2,  µ  4∥Ayλ  N∥2  H + C∥yλ  N∥6  V,  for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Using Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3), and the above estimates, we deduce  ∥yλ  N(t)∥2  V + µ  � t  0  ∥Ayλ  N(s)∥2  H + 2β  � t  0  ∥|∇yλ  N(s)||yλ  N(s)|  r−1  2 ∥2  Hds  ≤ ∥y0∥2  V + 2γ  � t  0  (1 + ∥yλ  N(s)∥2  H) + 2ς  � t  0  ∥Φλ(yλ  N(s))∥2  Hds + 2  µ  � t  0  ∥f(s)∥2  Hds  +  �  0,  for d = 2,  � t  0 ∥yλ  N(s)∥6  Vds,  for d = 3,  for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='5), we obtain  ∥yλ  N(t)∥2  V + µ  � t  0  ∥Ayλ  N(s)∥2  H + 2β  � t  0  ∥|∇yλ  N(s)||yλ  N(s)|  r−1  2 ∥2  Hds  ≤ C + ς  � t  0  ∥Φλ(yλ  N(s))∥2  Hds +  �  0,  for d = 2,  � t  0 ∥yλ  N(s)∥6  Vds,  for d = 3,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='10)  38  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  where C = C  �  ∥y0∥V, ∥f∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H), ∥Φ(0)∥H  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Calculation similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='29) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='32) yield  ����  � t  0  (C(yλ  N(s)), Φλ(yλ  N(s)))ds  ����  ≤ C  � t  0  ∥yλ  N(s)∥r  �L2r∥Φλ(yλ  N(s))∥Hds  ≤ 1 − µς  8β  � t  0  ∥Φλ(yλ  N(s))∥2  Hds + C sup  s∈[0,t]  ∥yλ  N(s)∥r(2−d)+d  H  � t  0  ∥yλ  N(s)∥(r−1)d  V  ds,  and  ς  � t  0  ∥Φλ(yλ  N(s))∥2  Hds ≤ C + C sup  s∈[0,t]  ∥yλ  N(s)∥r(2−d)+d  H  � t  0  ∥yλ  N(s)∥(r−1)d  V  ds  + C  � t  0  ∥f(s)∥2  Hds + µ  4  � t  0  ∥Ayλ  N(s)∥2  Hds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Therefore, from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='10) one can conclude  ∥yλ  N(t)∥2  V + µ  � t  0  ∥Ayλ  N(s)∥2  H + 2β  � t  0  ∥|∇yλ  N(s)||yλ  N(s)|  r−1  2 ∥2  Hds  ≤ C +  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  C sup  s∈[0,t]  ∥yλ  N(s)∥2  H  � t  0 ∥yλ  N(s)∥2(r−1)  V  ds,  for d = 2,  C sup  s∈[0,t]  ∥yλ  N(s)∥3−r  H  � t  0 ∥yλ  N(s)∥3(r−1)  V  ds +  � t  0 ∥yλ  N(s)∥6  Vds,  for d = 3,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='11)  where C = C  �  ∥y0∥V, ∥f∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='H), ∥Φ(0)∥H  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Then one can use the similar techniques as in  the proof [30, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='1] to pass λ → 0 and then use Gronwall’s inequality to obtain  ∥yN(t)∥V ≤ C,  for all t ∈ [0, T0],  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='12)  where constant C is independent of N and T0 = T for d = 2 and T0 < T for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Hence for  large N, we can choose N ≥ C so that BN(yN) = B(yN) and therefore yN = y is a solution  of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4) with the regularity properties given in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' So, yN satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4)  on the set  EN = {t ∈ [0, T] : ∥yN(t)∥V ≤ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='13)  By using Markov’s inequality, we have  m([0, T]/EN) ≤ C  N2,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='14)  where m is the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Since m([0, T]) = m(EN) + m([0, T]/EN), for large N,  from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='13) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='14), we conclude that m([0, T]) = m(EN) and y(·) satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4) for  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' The case of y0 ∈ V ∩ D(Φ) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='4 can be completed by a density  argument as in the proof of [30, Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  □  Acknowledgments: The first author would like to thank Ministry of Education, Government  of India - MHRD for financial assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Kinra would like to thank the Council of Scien-  tific & Industrial Research (CSIR), India for financial assistance (File No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' 09/143(0938)/2019-  EMR-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Mohan would like to thank the Department of Science and Technology (DST),  India for Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award  (IFA17-MA110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  CBF EQUATIONS WITH POTENTIAL  39  Declarations:  Ethical Approval: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Competing interests:  The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Authors’ contributions:  All authors have contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Funding:  CSIR, India, 09/143(0938)/2019-EMR-I (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Kinra), DST, India, IFA17-MA110  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Mohan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  References  [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Abergel and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Temam, On some control problems in fluid mechanics, Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Fluid Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=',  1 (1990), 303–325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Anh and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Nguyet, Time optimal control of the unsteady 3D Navier-Stokes-Voigt equations,  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 79(2) (2019), 397–426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Antontsev and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' de Oliveira, The Navier-Stokes problem modified by an absorption term, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 89(12) (2010), 1805–1825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Babutzka and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Kunstmann, Lq-Helmholtz decomposition on periodic domains and applications  to Navier-Stokes equations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 20(3) (2018), 1093–1121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [5] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International  Publishing, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [6] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [7] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Barbu, The time optimal control of Navier-Stokes equations, Systems Control Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 30(2-3) (1997),  93–100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [8] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering Series,  Springer, London, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [9] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Barbu, Controllability and Stabilization of Parabolic Equations, Progress in Nonlinear Differential  Equations and their Applications Subseries in Control 90, Birkh¨auser/Springer, Cham, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [10] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Barbu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Lasiecka and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Triggiani, Tangential Boundary Stabilization of Navier-Stokes Equations,  Memoirs of the American Mathematical Society, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [11] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Barbu and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Lefter, Internal stabilizability of the Navier-Stokes equations, System Control Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=',  48(3-4) (2003), 161–167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [12] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Barbu and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Pavel, Flow-invariant closed sets with respect to nonlinear semigroup flows, NoDEA  Nonlinear Differential Equations Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 10(1) (2003), 57–72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [13] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Barbu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes  equation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 255(1) (2001), 281–307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [14] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Barbu and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional  controllers, Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 53(5) (2004), 1443–1494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Breiten, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Kunisch and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Pfeiffer, Feedback stabilization of the two-dimensional Navier–Stokes  equations by value function approximation, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 80(3) (2019), 599–641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Brezis, Operateurs Maximaux et Semi-groupes de Contractions das les Espaces de Hilbert, North  Holland, New York, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [17] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Cai and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with  damping, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 343(2) (2008), 799–809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM Philadelphia, Philadel-  phia (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [19] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Denkowski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Mig´orski and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Papageorgiou, An Introduction to Nonlinear Analysis: Applications,  Kluwer Academic Publishers, Boston, MA, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Farwig, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Kozono and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Sohr, An Lq-approach to Stokes and Navier-Stokes equations in general  domains, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 195 (2005), 21–53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Fernando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Sritharan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Xu, A simple proof of global solvability of 2-D Navier-Stokes  equations in unbounded domains, Differential Integral Equations, 23(3–4) (2010), 223–235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [22] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Fujiwara and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Morimoto, An Lr-theorem of the Helmholtz decomposition of vector fields, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Tokyo Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' IA Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 24(3) (1977), 685–700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  40  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' GAUTAM, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' KINRA AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' MOHAN  [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Fursikov, Optimal Control of Distributed Systems: Theory and Applications, American Mathe-  matical Society, Providence, Rhode Island, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Fukao, Variational inequality for the Stokes equations with constraint, Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=',  Dynamical systems, differential equations and applications, 8th AIMS Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' I (2011),  437–446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Gokieli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Kenmochi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Niezg`odka, Variational inequalities of Navier-Stokes type with time  dependent constraints, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 449(2) (2017), 1229–1247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Gunzburger, Perspectives in Flow Control and Optimization, SIAM’s Advances in Design and  Control Philadelphia, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [27] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Hajduk and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer  equations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Differential Equations, 263(11) (2017), 7141–7161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Hu and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Papageorgiou, Handbook of Multivalued Analysis, Vol I: Theory, Mathematics and its  Applications, 419, Kluwer Academic Publishers, Dordrecht, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [29] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Kalantarov and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast  growing nonlinearities, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 11(5) (2012), 2037–2054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Lefter, Navier-Stokes equations with potentials, Abstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', (2007), Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' ID 79406, 30 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Lefter, Nonlinear feedback controllers for the Navier-Stokes equations, Nonlinear Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 71(1-2)  (2009), 301–316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Li and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Wang, The time optimal control of the Boussinesq equations, Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=',  24(1–2) (2007), 163–180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [33] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Markowich, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Titi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Trabelsi, Continuous data assimilation for the three dimensional  Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 29(4) (2016), 1292–1328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Mohan, On the convective Brinkman-Forchheimer equations, Submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Mohan,  Stochastic  convective  Brinkman-Forchheimer  equations,  Submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='org/pdf/2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='09376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Mohan, Well-posedness and asymptotic behavior of stochastic convective Brinkman–Forchheimer  equations perturbed by pure jump noise, Stoch PDE: Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 10(2) (2022), 614–690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Mohan, The time optimal control of two dimensional convective Brinkman-Forchheimer equations,  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 84(3) (2021), 3295– 3338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Peypouquet and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Sorin, Evolution equations for maximal monotone operators: asymptotic analysis  in continuous and discrete time, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Convex Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 17(3-4) (2010), 1113–1163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipatives Parabolic  PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [40] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Robinson and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Sadowski, A local smoothness criterion for solutions of the 3D Navier–Stokes  equations, Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Semin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Padova, 131 (2014), 159–178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Robinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Rodrigo and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Sadowski, The Three-Dimensional Navier–Stokes equations,  classical theory, Cambridge University Press, Cambridge, UK, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' R¨ockner and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Zhang, Tamed 3D Navier-Stokes equation: existence, uniqueness and regularity,  Infin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Dimens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Quantum Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 12(4) (2009), 525–549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [43] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Son and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Thuy, Time optimal control problem of the 3D Navier-Stokes-α equations, Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 43(6) (2022), 667–697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [44] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Sritharan, An introduction to deterministic and stochastic control of viscous flow, in Optimal  Control of Viscous Flow, SIAM, Philadelphia, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [45] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Stone  and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  Zelik,  The  non-autonomous  Navier-Stokes-Brinkmann-Forchhiemer  equation  with  dirichlet  boundary  conditions:  dissipativity,  regularity  and  attractors,  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='org/pdf/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='05580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [46] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, CBMS-NSF  Regional Conference Series in Applied Mathematics, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [47] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  68, Applied Mathematical Sciences, Springer, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [48] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Xu and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Zhang, Time Optimal Control of Evolution Equations, Springer, New  York (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content='  [49] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Wu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Lu, On the uniqueness of strong solution to the incompressible Navier-Stokes  equations with damping, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
+page_content=', 377(1) (2011), 414–419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfkP0Y/content/2301.01527v1.pdf'}
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+Simultaneous superadditivity of the direct and complementary channel capacities
+Satvik Singh1 and Sergii Strelchuk1
+1DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB30WA, UK
+Quantum communication channels differ from their classical counterparts because their capacities
+can be superadditive.
+The principle of monogamy of entanglement suggests that superadditive
+improvements in the transmission capacity of a channel should reduce the amount of information
+loss to the environment. We challenge this intuition by demonstrating that the coherent and private
+information of a channel and its complement can be simultaneously superadditive for arbitrarily
+many channel uses.
+To quantify the limits of this effect, we consider the notion of max (resp.
+total) private information of a channel, which represents the maximum (resp. sum) of the private
+information of the channel itself and its complement, and study its relationship with the coherent
+information of the individual direct and complementary channels. For a varying number of channel
+uses, we show that these quantities can obey different interleaving sequences of inequalities.
+Quantum channels have several intriguing properties
+that separate them from their classical counterparts. One
+of them is the ability to send information superadditively
+[1], whereby multiple uses of the same channel increase
+the amount of information that it can reliably trans-
+mit. Another one, namely superactivation [2], demon-
+strates how some channels, having initially no capacity
+to send information can regain it when combined with
+an equally useless zero-capacity channel. Both of these
+effects were introduced and subsequently studied in the
+context of having access to a channel NA→B where Alice,
+the sender, communicates with the receiver, Bob.
+One can define different kinds of capacities of N de-
+pending on the type of information being sent.
+The
+quantum capacity Q(N) of N quantifies the maximum
+rate at which Alice can send quantum information re-
+liably to Bob and can be expressed as a regulariza-
+tion of the channel’s coherent information:
+Q(N) =
+limk→∞ Q(k)(N), Q(k)(N) := Q(1)(N ⊗k)/k, Q(1)(N) :=
+supρA[S(B) − S(E)]. The optimization here is over all
+states ρA and the von Neumann entropies S(B) and S(E)
+are evaluated on NA→B(ρA) and N c
+A→E(ρA), respec-
+tively, where the complementary channel N c
+A→E models
+the nature of information loss to the environment (Eve)
+(its precise definition will be introduced shortly). Simi-
+larly, the classical capacity C(N) of N quantifies the max-
+imum rate at which Alice can send classical information
+reliably to Bob. In addition, if the information being sent
+is to be kept private from Eve, one obtains the private
+capacity P(N) of the channel. These capacities admit
+regularized expressions in terms of the channel’s Holevo
+information:
+C(N)
+=
+limk→∞ C(k)(N), C(k)(N)
+:=
+C(1)(N ⊗k)/k, C(1)(N) := supρXA I(X
+: B), and pri-
+vate information P(N) = limk→∞ P(k)(N), P(k)(N) :=
+P(1)(N ⊗k)/k, P(1)(N) := supρXA[I(X : B) − I(X : E)].
+The optimizations above are over all classical quantum
+states ρXA = �
+x px |x⟩⟨x|X ⊗ ρx
+A and the mutual infor-
+mation terms I(X : B) and I(X : E) are evaluated on
+NA→B(ρXA) and N c
+A→E(ρXA), respectively.
+Since the first demonstration [1], there have been a
+variety of results that illustrate striking superadditive
+behavior of quantum channel capacities [3–14]. For in-
+stance, the k-letter coherent and private information of
+a channel obey the inequality: Q(k)(N) ≤ P(k)(N) ∀k.
+However, for different numbers of channel uses, the coher-
+ent information of N can exceed its private information
+[7], making the former inequality valid only for a fixed k.
+The above effects were all demonstrated in the set-
+ting where Alice optimizes her data transmission rate
+to Bob while minimizing information ‘leakage’ to Eve,
+who behaves as a non-participating party during trans-
+mission.
+There are several results that investigate in-
+formation transmission problems under non-trivial be-
+haviour of the environment [15–17]. In one of the ear-
+liest works [17] of such kind, the authors investigated
+the capacities of quantum channels by allowing Eve to
+locally measure and communicate classical messages to
+Bob. Later, this was extended to allow a helper [15] – a
+benevolent third party – who can adjust the environment
+state. This enabled one to derive streamlined examples
+for super-additivity due to the extra abilities of the helper
+to adjust the state of the environment depending on the
+message being sent. For example, the so-called locking
+capacity [18] of a channel in this regime is superadditive,
+whereas in the absence of auxiliary resources the question
+is still open.
+The above scenarios supplement the direct channel
+NA→B with extra resources which are extrinsic to its
+definition. This precludes one from learning about the
+total capacity for information transmission by using the
+channel alone. One may observe that defining a direct
+channel NA→B from Alice to Bob fixes the behaviour of
+information loss to the environment (Eve) via the com-
+plementary channel N c
+A→E. Indeed, the Stinespring dila-
+tion theorem [19, 20] shows that there exists an isometry
+V : HA → HB ⊗HE such that N(X) = TrE(V XV †) and
+N c(X) = TrB(V XV †).
+When optimizing the communication rates, we natu-
+rally want to take full advantage of the superadditive
+properties of the channel. The monogamy of entangle-
+ment principle suggests that when the direct channel is
+superadditive, its complementary channel to the envi-
+ronment is likely to have constrained data transmission
+capabilities.
+Contrary to this intuition, we show that
+arXiv:2301.05142v1  [quant-ph]  12 Jan 2023
+
+2
+superadditivity can persist for an unbounded number of
+channel uses in the strongest possible sense in both the
+direct and complementary channels simultaneously.
+To demonstrate this surprising effect we consider two
+quantities: the max and total coherent information of N:
+Q(k)
+max(N) := max{Q(k)(N), Q(k)(N c)},
+Q(k)
+tot(N) := Q(k)(N) + Q(k)(N c).
+The max- and total Holevo and private information quan-
+tities can be defined similarly. The max and total quan-
+tum capacities can be obtained by taking k → ∞ limits:
+lim
+k→∞ Q(k)
+max(N) = max{Q(N), Q(N c)},
+lim
+k→∞ Q(k)
+tot(N) = Q(N) + Q(N c).
+Operationally, Bob and Eve are now placed on an equal
+footing and Alice simply wants to send information at the
+best rate possible regardless of who plays the role of the
+receiver. She can either use the direct channel NA→B or
+its complement N c
+A→E to do so, thus arriving at the max
+rate. In such a scenario, it is crucial to analyse the super-
+additive behaviour of both the direct and complementary
+channels together to determine which one has higher ca-
+pacity to transmit information. A similar quantity was
+introduced in [21] in the context of entanglement distil-
+lation. On the other hand, to our best knowledge, the
+expression for the total quantum capacity first appeared
+in [22], where it was used to bound the difference be-
+tween the quantum and private capacities of any channel
+N. Intuitively, being the sum of the direct and comple-
+mentary channel capacities, the total capacity quantifies
+the overall information transmission capability of N.
+The above setting should be distinguished from that of
+quantum broadcast channels [23, 24] where two (or more)
+recipients (Bobs) share a joint environment. As such, this
+model is not representative of the concepts of max and
+total information where the notion of the environment
+does not feature.
+We now briefly describe our results. For n, α ∈ N sat-
+isfying nα−2 > 8, we construct a channel N such that
+(Theorem 1):
+∀k ≤ n :
+Q(k+1)(N), Q(k+1)(N c) > P(k)
+max(N). (1)
+Thus, for any number k of channel uses, the max k-letter
+private information of N can be exceeded by the coher-
+ent information of both the direct and complementary
+channels by using just one extra copy of each of these
+channels. This means that the coherent and private in-
+formation quantities of N are curiously interleaved:
+Q(1)(N), Q(1)(N c) ≤ P(1)
+max(N) < Q(2)(N), Q(2)(N c)
+≤ P(2)
+max(N) < . . .
+Remarkably, by choosing n large enough, this phe-
+nomenon can be made to persist for arbitrarily many
+channel uses. Moreover, the parameter p can be tuned
+to boost the superadditivity of the direct channel rel-
+ative to its complement or vice versa.
+More precisely,
+when 1/3 ≤ p ≤ 1/2 − 1/nα−1 and nα−2 > 12, even
+though both the direct and complementary channels are
+still superadditive, Eq. (1) now only holds for N and not
+for N c. Thus, the superadditivity of the direct channel
+dominates that of its complement (Theorem 2). For even
+smaller values of p, this effect becomes extreme: even the
+total k−letter private information of the channel (below a
+certain threshold k value) can be exceeded by the coher-
+ent information of the direct channel alone, provided that
+it is used sufficiently many times j > ck (Theorem B.3):
+Q(j)(N) > P(k)(N) + P(k)(N c) = P(k)
+tot (N).
+In all the above cases, it is possible to precisely quantify
+the effects of superaddivity by computing lower bounds
+on the difference quantities such as Q(k+1)(N)−P(k)
+max(N)
+and Q(k+1)(N c) − P(k)(N c).
+Finally, it turns out that simultaneous superadditiv-
+ity of both the direct and complementary channels is a
+non-trivial phenomenon. We prove this by constructing
+a channel with superadditive quantum capacity whose
+complement has additive capacity.
+The main construction.– Our channels Nn,p,d are made
+of two building blocks: the erasure and ‘rocket’ channels.
+The d-dimensional erasure channel Ep,d : A → B with
+erasure parameter p ∈ [0, 1] takes a d-dimensional input
+and replaces it with an erasure flag |e⟩⟨e| (orthogonal to
+the input space) with probability p and does nothing to
+it otherwise: Ep,d(ρ) = (1 − p)ρ + p Tr(ρ) |e⟩⟨e|. Its com-
+plement is again of the erasure type: Ec
+p,d = E1−p,d. The
+capacities of Ep,d are well known: Q(Ep,d) = P(Ep,d) =
+max{(1 − 2p) log d, 0}, C(Ep,d) = (1 − p) log d.
+The d-dimensional rocket channel [25] Rd : A1 ⊗
+A2 → B takes two d-dimensional quantum systems
+(A1 and A2) as inputs and applies local random unitaries
+on each input [26] followed by a controlled phase coupling
+P = �
+i,j ωij |i⟩⟨i|A1 ⊗|j⟩⟨j|A2, where ω = ei2π/d. Finally,
+A2 is discarded and Bob gets A1 along with classical in-
+formation about which local unitaries were applied. For
+the complementary channel Rc
+d : A1 ⊗ A2 → E, A1 is
+discarded and Eve gets A2 along with the same classical
+information about the local unitaries. Since Rd dephases
+the input registers in a random basis unknown to Alice,
+it has little capacity to transmit information on its own:
+C(Rd) ≤ 2. The same argument applies to Rc
+d as well:
+C(Rc
+d) ≤ 2 (the proof of [25] goes through by swapping
+labels for Bob and Eve). However, when Alice and Bob
+already share a maximally entangled state (this can be
+achieved with probability 1−p by using the erasure Ep,d),
+it turns out that Bob can undo the random phase cou-
+pling, thus allowing Alice to send quantum information
+at rate log d [25]. More precisely, we have
+Q(1)(Rd ⊗ Ep,d) ≥ (1 − p) log d.
+(2)
+A slight modification of this argument can be used to
+
+3
+obtain the same result for Rc
+d as well:
+Q(1)(Rc
+d ⊗ Ep,d) ≥ (1 − p) log d.
+(3)
+An intuitive graphical proof of the above two claims
+(Eqs. (2) and (3)) is provided in Appendix C.
+We are now ready to introduce our main channels. For
+n, d ∈ N and p ∈ [0, 1], we define
+Nn,p,d := R⊗n
+d
+⊕ Ep,d,
+N c
+n,p,d = Rc ⊗n
+d
+⊕ E1−p,d.
+The direct sum construction [27] allows Alice to control
+which of the two channels is being applied at the out-
+set, so that the coherent and private information of such
+channels is just the maximum of its building blocks, see
+Lemma A.1.
+In our case, by using the channel k + 1
+times, Alice gains access to blocks of the form
+R⊗n
+d
+⊗ E⊗k
+p,d or Rc ⊗n
+d
+⊗ E⊗k
+1−p,d,
+for which the superadditivty effects observed in Eqs. (2),
+and (3) can boost information transmission rates for each
+successive channel use as long as k ≤ n. For p = 1/2,
+these effects are identical for both the direct and comple-
+mentary channels. The parameter p in Eqs. (2),(3) can
+be adjusted to enhance the superadditivity of either the
+direct channel or its complement relative to the other.
+We employ these ideas to prove our main results below.
+Theorem 1. Let p = 1/2, and n, α ∈ N be such that
+nα−2 > 8. Furthermore, let d = 2nα and N = Nn,p,d.
+Then, for all k ≤ n :
+Q(k+1)(N) − P(k)
+max(N) ≥ nα − 4n(k + 1)
+2k(k + 1)
+> 0,
+Q(k+1)(N c) − P(k)
+max(N) ≥ nα − 4n(k + 1)
+2k(k + 1)
+> 0,
+Proof. Using Lemma A.1, and the fact that for any chan-
+nel N, C(1)(N ⊗ Ep,d) = C(1)(N) + C(1)(Ep,d) [7, Lemma
+2], we get:
+P(k)(N) = 1
+k max
+0≤l≤k P(1)(R⊗nl
+d
+⊗ E⊗k−l
+1/2,d )
+≤ max
+�
+�
+�
+�
+�
+2n
+2n
+k + k−1
+k
+1
+2 log d
+0
+= 2n
+k + (k − 1)nα
+2k
+.
+Analogous reasoning yields a bound for P(k)(N c). Com-
+bining Lemma A.1 with Eq. (2) and the fact that
+Q(1)(N1 ⊗ N2) ≥ Q(1)(N1) + Q(1)(N2) yields a simple
+lower bound:
+Q(k+1)(N) ≥
+Q(1)(R⊗n
+d
+⊗ E⊗k
+1/2,d)
+k + 1
+≥
+knα
+2(k + 1)
+for k ≤ n. Swapping Eq. (2) with Eq. (3) shows that the
+same bound holds for Q(k+1)(N c) too. Thus,
+Q(k+1)(N) − P(k)
+max(N) ≥ nα − 4n(k + 1)
+2k(k + 1)
+> 0,
+where the latter inequality holds because nα−2 > 8.
+Clearly, the same bounds hold for N c as well.
+When p < 1/2, the superadditivity analysis of Nn,p,d
+becomes tedious. In Lemma B.1, we prove several capac-
+ity bounds which we use to show that the superadditivity
+of the direct channel dominates that of its complement:
+Theorem 2. Let p ∈ [0, 1], and n, α ∈ N be such that
+1/3 < p ≤ 1/2 − 1/nα−1 and nα−2 > 12. Furthermore,
+let d = 2nα and N = Nn,p,d. Then,
+Q(k+1)(N) − P(k)
+max(N) ≥ nα(1 − p) − 2n(k + 1)
+k(k + 1)
+=: fn,p,α(k) > 0,
+Q(k+1)(N c) − P(k)(N c) ≥ nαp − 2n(k + 1)
+k(k + 1)
+=: f c
+n,p,a(k) > 0,
+where the first bound holds for 2 ≤ k ≤ n and the second
+bound holds for 1 ≤ k ≤ n.
+The full proof is located in Appendix B.
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+5.5
+Figure 1: Plot of the log of the lower bounds fn,p,α(k) and f c
+n,p,α(k)
+on the difference quantities Q(k+1)(Nn,p,d) − P(k)
+max(Nn,p,d) and
+Q(k+1)(N c
+n,p,d) − P(k)(N c
+n,p,d), respectively (see Theorem 2). Here,
+n = 100, p = 0.4, and α = 3.
+The log lower bounds for the difference quantities in
+Theorem 2 are plotted in Figure 1.
+Note that in the
+setting of Theorem 2, the following is true:
+k − 1
+k
+≥
+2 + nαp
+(1 − p)(n + 1)nα−1
+=⇒ P(n+1)(N c) ≤ Q(k)(N),
+(4)
+
+4
+provided that k ≤ n (see Theorem B.2). For instance,
+when n = 100, p = 0.4, and α = 3, the LHS in Eq. (4)
+holds for k ≥ 3. In other words, the direct channel in this
+case is vastly more superadditive than its complement,
+since only the 3-letter coherent information of Nn,p,d suf-
+fices to beat the 101-letter private information of N c
+n,p,d.
+For small values of p, the coherent information of
+Nn,p,d can even beat the total private information
+for some uses of the channel.
+For example, when
+p = 0.09, n = 100, and α = 3, Q(101)(Nn,p,d) beats
+P(k)
+tot (Nn,p,d) for k ≤ 9 (see Figure 2). A detailed analysis
+of this phenomenon is given in Theorem B.3.
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+8.2
+8.4
+8.6
+8.8
+9
+9.2
+9.4
+9.6
+9.8
+10
+105
+Figure 2: Plot of the lower bound Qmax = L(n + 1) on
+Q(n+1)(Nn,p,d) and the upper bound Uk =
+�
+U ′(k)
+if k ≤ k0
+U ′′(k)
+otherwise
+on
+P(k)
+tot (Nn,p,d) for p = 0.09, n = 100, and α = 3. The lower and upper
+bound functions are defined in Eqs. (B9),(B11),(B12). Clearly, Qmax
+exceeds the total private capacity for at least 9 uses of the channel.
+Platypus construction.– We now turn to constructing
+a channel with superadditive quantum capacity whose
+complement has additive capacity. The d−dimensional
+platypus channel Md : A → B introduced recently in
+[28] is defined via the isometry:
+V : HA → HB ⊗ HE
+V |0⟩ =
+1
+√
+d − 1
+d−2
+�
+j=0
+|j⟩ |j⟩ ,
+V |i⟩ = |d − 1⟩ |i − 1⟩ ,
+i = 1, 2, . . . , d − 1,
+as Md(X)
+=
+TrE(V XV †).
+Here,
+A and B are
+d−dimensional while E is (d − 1)−dimensional.
+The
+channel and its complement satisfy [28, 29]:
+Q(Md) ≤ log
+�
+1 +
+1
+√
+d − 1
+�
+, P(Md) = C(Md) = 1,
+Q(1)(Mc
+d) = Q(Mc
+d) = P(Mc
+d) = C(Mc
+d)
+= log(d − 1).
+As d → ∞, Q(Md) → 0. However, when coupled with
+an erasure channel, Md can be shown to retain some
+quantum capacity as d → ∞ [28]:
+Q(1)(Md+1 ⊗ E1/2,d) ≥ 1
+2 + O
+� 1
+√
+d
+�
+.
+To turn these effects into superadditivity of a single
+channel, we again turn to a direct sum construction:
+Nd := Md+1 ⊕ E1/2,d,
+N c
+d := Mc
+d+1 ⊕ E1/2,d.
+By choosing a large enough d, we can ensure that
+Q(1)(Md+1 ⊗ E1/2,d) > 2 log
+�
+1 + 1/
+√
+d
+�
+≥ Q(1)(M⊗2
+d+1).
+In other words, Nd is superadditive for large enough d:
+Q(2)(Nd) = 1
+2 max{Q(1)(M⊗2
+d+1), Q(1)(Md+1 ⊗ E1/2,d)}
+> log
+�
+1 + 1
+√
+d
+�
+≥ Q(1)(Nd).
+On the other hand, for any k1, k2 ∈ N:
+Q(1)(Mc ⊗k1
+d+1 ⊗ E⊗k2
+1/2,d) ≤ C(1)(Mc ⊗k1
+d+1 ⊗ E⊗k2
+1/2,d)
+= C(1)(Mc ⊗k1
+d+1 ) + C(1)(E⊗k2
+1/2,d)
+= k1 log d + k2
+2 log d
+≤ (k1 + k2) log d.
+Thus, N c
+d has additive quantum capacity:
+∀k : Q(k)(N c
+d) = 1
+k max
+0≤l≤k Q(1)(Mc ⊗l
+d+1 ⊗ E⊗k−l
+1/2,d )
+= Q(k)(Mc
+d+1) = log d.
+Discussion.– We have investigated superadditive effects
+for the coherent and private information of a channel and
+its complement relative to each other. We showed that
+contrary to intuitive expectations, the following two cases
+are both possible:
+• The direct and complementary channels are simul-
+taneously superadditive,
+• The direct channel is superadditive while the com-
+plement is additive (and vice versa).
+It is also possible to construct examples where both the
+direct and complementary channels are entanglement-
+breaking and hence trivially have additive coherent and
+private information (equal to zero) [30].
+One interesting question to investigate further is the
+extent to which superadditivity of the coherent and pri-
+vate information quantities can be attained simultane-
+ously for the direct and complementary channels.
+Is
+
+5
+there a constraint akin to the monogamy of entangle-
+ment which would prohibit a maximum violation of ad-
+ditivity for both channels? Another intriguing question
+is whether one can superactivate total capacity.
+Finally, it would be enlightening to check whether or
+not the superadditivity results presented in this letter
+hold for the classical capacities as well. Our techniques
+do not apply directly in this case. The problem lies in
+the fact that the classical capacity of a direct sum chan-
+nel is not merely the maximum of the classical capacity
+of its components. This is because when using a direct
+sum channel, say N = ⊕n
+i=1N (i), unlike private or co-
+herent information, classical information can not only be
+sent through the individual blocks N (i), but can also be
+encoded in the choice of the blocks i = 1, 2, . . . , n. More
+precisely, for our channel of interest, say N = Rd ⊕ Ep,d,
+Lemma A.1 shows that
+C(1)(N) = log
+�
+2C(1)(Rd) + 2C(1)(Ep,d)�
+, and
+C(2)(N) = 1
+2 log
+�
+2C(1)(R⊗2
+d
+) + 2.2C(1)(Rd⊗Ep,d) + 2C(1)(E⊗2
+p,d)�
+.
+Since C(1)(N ⊗ Ep,d) = C(1)(N) + C(1)(Ep,d) for all chan-
+nels N [7, Lemma 2], it is clear that the question of
+superadditivity of C(1)(N) boils down to the question of
+superadditivity of C(1)(Rd), which is currently open.
+Acknowledgements. The authors would like to thank
+Nilanjana Datta for helpful discussions.
+Satvik Singh
+acknowledges support from the Cambridge Trust’s In-
+ternational Scholarship.
+Sergii Strelchuk acknowledges
+support from the Royal Society University Research Fel-
+lowship.
+♦
+[1] D. P. DiVincenzo, P. W. Shor, and J. A. Smolin, Quantum-channel capacity of very noisy channels, Physical Review A
+57, 830 (1998).
+[2] G. Smith and J. Yard, Quantum communication with zero-capacity channels, Science 321, 1812 (2008).
+[3] M. M. Wilde, Quantum information theory (Cambridge University Press, 2013).
+[4] E. Y. Zhu, Q. Zhuang, M.-H. Hsieh, and P. W. Shor, Superadditivity in trade-off capacities of quantum channels, IEEE
+transactions on information theory 65, 3973 (2018).
+[5] E. Y. Zhu, Q. Zhuang, and P. W. Shor, Superadditivity of the classical capacity with limited entanglement assistance,
+Physical review letters 119, 040503 (2017).
+[6] T. Cubitt, D. Elkouss, W. Matthews, M. Ozols, D. Pérez-García, and S. Strelchuk, Unbounded number of channel uses
+may be required to detect quantum capacity, Nature communications 6, 1 (2015).
+[7] D. Elkouss and S. Strelchuk, Superadditivity of private information for any number of uses of the channel, Phys. Rev. Lett.
+115, 040501 (2015).
+[8] D. Elkouss and S. Strelchuk, Nonconvexity of private capacity and classical environment-assisted capacity of a quantum
+channel, Physical Review A 94, 040301 (2016).
+[9] G. Smith and J. A. Smolin, Extensive nonadditivity of privacy, Physical Review Letters 103, 120503 (2009).
+[10] F. G. Brandao, J. Oppenheim, and S. Strelchuk, When does noise increase the quantum capacity?, Physical review letters
+108, 040501 (2012).
+[11] M. Shirokov and T. Shulman, On superactivation of zero-error capacities and reversibility of a quantum channel, Commu-
+nications in Mathematical Physics 335, 1159 (2015).
+[12] F. Leditzky, D. Leung, V. Siddhu, G. Smith, and J. A. Smolin, Generic nonadditivity of quantum capacity in simple
+channels, arXiv preprint arXiv:2202.08377 (2022).
+[13] D. Leung, K. Li, G. Smith, and J. A. Smolin, Maximal privacy without coherence, Physical review letters 113, 030502
+(2014).
+[14] K. Li, A. Winter, X. Zou, and G. Guo, Private capacity of quantum channels is not additive, Physical Review Letters 103,
+120501 (2009).
+[15] S. Karumanchi, S. Mancini, A. Winter, and D. Yang, Classical capacities of quantum channels with environment assistance,
+Problems of Information Transmission 52, 214 (2016).
+[16] S. K. Oskouei, S. Mancini, and A. Winter, Capacities of gaussian quantum channels with passive environment assistance,
+IEEE Transactions on Information Theory 68, 339 (2021).
+[17] A. Winter, On environment-assisted capacities of quantum channels, arXiv preprint quant-ph/0507045 (2005).
+[18] S. Guha, P. Hayden, H. Krovi, S. Lloyd, C. Lupo, J. H. Shapiro, M. Takeoka, and M. M. Wilde, Quantum enigma machines
+and the locking capacity of a quantum channel, Physical Review X 4, 011016 (2014).
+[19] W. F. Stinespring, Positive functions on c*-algebras, Proceedings of the American Mathematical Society 6, 211 (1955).
+[20] V. Paulsen, Completely bounded maps and operator algebras, 78 (Cambridge University Press, 2002).
+[21] S. Singh and N. Datta, Fully undistillable quantum states are separable, preprint arxiv:2207:05193 (2022).
+[22] C. Hirche and F. Leditzky, Bounding quantum capacities via partial orders and complementarity, in 2022 IEEE Interna-
+tional Symposium on Information Theory (ISIT) (2022) pp. 2219–2224.
+[23] J. Yard, P. Hayden, and I. Devetak, Quantum broadcast channels, IEEE Transactions on Information Theory 57, 7147
+(2011).
+[24] R. Laurenza and S. Pirandola, General bounds for sender-receiver capacities in multipoint quantum communications,
+Physical Review A 96, 032318 (2017).
+[25] G. Smith and J. A. Smolin, Extensive nonadditivity of privacy, Phys. Rev. Lett. 103, 120503 (2009).
+[26] It actually suffices to apply local unitaries from a unitary 2-design, such as the Clifford group [31].
+
+6
+[27] M. Fukuda and M. M. Wolf, Simplifying additivity problems using direct sum constructions, Journal of Mathematical
+Physics 48, 072101 (2007).
+[28] F. Leditzky, D. Leung, V. Siddhu, G. Smith, and J. A. Smolin, Generic nonadditivity of quantum capacity in simple
+channels, preprint arxiv:2202:08377 (2022).
+[29] F. Leditzky, D. Leung, V. Siddhu, G. Smith, and J. A. Smolin, The platypus of the quantum channel zoo, preprint
+arXiv:2202.08380 (2022).
+[30] A. Müller-Hermes and S. Singh, Bi-PPT channels are entanglement breaking, arXiv:2204.01685 (2022).
+[31] C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity
+estimation, Phys. Rev. A 80, 012304 (2009).
+[32] I. Nechita and S. Singh, A graphical calculus for integration over random diagonal unitary matrices, Linear Algebra and
+its Applications 613, 46 (2021).
+[33] C. J. Wood, J. D. Biamonte, and D. G. Cory, Tensor networks and graphical calculus for open quantum systems, Quantum
+Information & Computation 15, 759 (2015).
+[34] J. C. Bridgeman and C. T. Chubb, Hand-waving and interpretive dance: an introductory course on tensor networks,
+Journal of Physics A: Mathematical and Theoretical 50, 223001 (2017).
+Appendix A: Notation and the direct sum construction
+We denote quantum systems with capital letters like A and B. Each quantum system A is associated with a finite-
+dimensional complex Hilbert space HA. For a joint system AB or A ⊕ B, HAB = HA ⊗ HB or HA⊕B = HA ⊕ HB.
+Quantum states ρA on A are positive semi-definite operators acting on HA with unit trace.
+The von Neumann
+entropy of ρA is defined as S(A) := − Tr(ρA log2 ρA). For a bipartite state ρAB,the mutual information is defined as
+I(A : B) := S(A) + S(B) − S(AB).
+A quantum channel N : A → B (denoted NA→B) is a completely positive and trace-preserving linear map taking
+linear operators acting on HA to those acting on HB. Every channel NA→B admits a Stinespring isometry V : HA →
+HB ⊗ HE such that N(X) = TrE(V XV †). The complementary channel N c
+A→E is defined as N c(X) = TrB(V XV †).
+For two channels N (1) : A1 → B1, N (2) : A2 → B2, the direct sum N = N (1) ⊕ N (2) : A1 ⊕ A2 → B1 ⊕ B2 acts as
+N
+��
+XA1
+∗
+∗
+XA2
+��
+=
+�
+N (1)(XA1)
+0
+0
+N (2)(XA2)
+�
+.
+It should be clear that the complement Nc = N (1)
+c
+⊕ N (2)
+c
+.
+Lemma A.1. For quantum channels N (i)
+Ai→Bi for i = 1, 2, . . . n, let N := ⊕n
+i=1N (i). Then,
+[27,
+Proposition
+1]
+Q(1)(N) = max
+1≤i≤n Q(1)(N (i)),
+[7,
+Lemma
+1]
+P(1)(N) = max
+1≤i≤n P(1)(N (i)),
+[27,
+Proposition
+1]
+C(1)(N) = log
+n
+�
+i=1
+2C(1)(N (i)).
+Appendix B: Proofs
+Recall from the main text that for n, d ∈ N and p ∈ [0, 1], the channel Nn,p,d was defined as the direct sum
+Nn,p,d := R⊗n
+d
+⊕ Ep,d, where Rd and Ep,d are the rocket and erasure channels. We can derive the following bounds on
+the capacities of this channel.
+Lemma B.1. Let p ∈ [0, 1] and n, α ∈ N (α > 1) be such that 4/nα−1 ≤ p ≤ 1/2 − 1/nα−1. Let d = 2nα and
+k0(n, p, α) := 1 − p − 2/nα−1
+p
+.
+
+7
+Then, the following bounds hold:
+∀k ≤ k0 :
+P(k)(Nn,p,d) ≤ (1 − 2p)nα
+(B1)
+∀k > k0 :
+P(k)(Nn,p,d) ≤ 2n
+k + k − 1
+k
+(1 − p)nα
+(B2)
+∀k :
+P(k)(N c
+n,p,d) ≤ 2n
+k + k − 1
+k
+pnα
+(B3)
+∀k ≤ n :
+Q(k+1)(Nn,p,d) ≥
+k
+k + 1(1 − p)nα
+(B4)
+∀k ≤ n :
+Q(k+1)(N c
+n,p,d) ≥
+k
+k + 1pnα
+(B5)
+Proof. By using Lemma A.1 and the fact that C(1)(N ⊗ Ep,d) = C(1)(N) + C(1)(Ep,d) for all channels N, it is easy to
+arrive at the following bounds:
+P(k)(Nn,p,d) = 1
+k max
+0≤l≤k P(1)(R⊗nl
+d
+⊗ E⊗k−l
+p,d
+)
+≤ max
+�
+�
+�
+�
+�
+2n
+2n
+k + k−1
+k (1 − p)nα
+(1 − 2p)nα,
+(B6)
+P(k)(N c
+n,p,d) = 1
+k max
+0≤l≤k P(1)(Rc ⊗nl
+d
+⊗ E⊗k−l
+1−p,d)
+≤ max
+�
+2n
+2n
+k + k−1
+k pnα.
+Let’s first deal with the maximum in Eq. (B6). Clearly, (1 − 2p)nα ≥ 2n since p ≤ 1/2 − 1/nα−1. Moreover,
+2n
+k + k − 1
+k
+(1 − p)nα ≤ (1 − 2p)nα
+⇐⇒ 2
+k + (1 − 1
+k )(1 − p)nα−1 ≤ (1 − 2p)nα−1
+⇐⇒ 1
+k ((1 − p)nα−1 − 2) ≥ nα−1p
+⇐⇒ k ≤ (1 − p − 2/nα−1)
+p
+= k0.
+This gives us the first two bounds in Eqs. (B1) and (B2). To obtain the bound in Eq. (B3), observe that
+4/nα−1 ≤ p ⇐⇒ 2n ≤ 1
+2pnα.
+Hence, for k ≥ 2, we have k−1
+k pnα ≥ pnα/2 ≥ 2n. To prove Eq. (B4), note that for k ≤ n, Lemma A.1 shows
+Q(k+1)(Nn,p,d) ≥
+Q(1)(R⊗n
+d
+⊗ E⊗k
+p,d)
+k + 1
+≥
+k
+k + 1(1 − p)nα,
+where we have used the bound in Eq. (2) along with the fact that Q(1)(N1 ⊗N2) ≥ Q(1)(N1)+Q(1)(N2). An identical
+argument works to prove Eq. (B5).
+We can now prove Theorem 2 from the main text.
+Theorem B.1. Let p ∈ [0, 1], and n, α ∈ N be such that 1/3 < p ≤ 1/2 − 1/nα−1 and nα−2 > 12. Furthermore, let
+d = 2nα and N = Nn,p,d. Then,
+Q(k+1)(N) − P(k)
+max(N) ≥ nα(1 − p) − 2n(k + 1)
+k(k + 1)
+=: fn,p,α(k) > 0,
+(B7)
+Q(k+1)(N c) − P(k)(N c) ≥ nαp − 2n(k + 1)
+k(k + 1)
+=: f c
+n,p,a(k) > 0,
+(B8)
+
+8
+where the first bound holds for 2 ≤ k ≤ n and the second bound holds for 1 ≤ k ≤ n. Q(2)(N) > P(1)
+max(N)
+Proof. For 1/3 < p ≤ 1/2 − 1/nα−1, Lemma B.1 shows that
+P(k)
+max(Nn,p,d) ≤ 2n
+k + k − 1
+k
+(1 − p)nα =: Un,p,α(k)
+for 2 ≤ k ≤ n. Furthermore, for k ≤ n, we have
+Q(k+1)(Nn,p,d) ≥
+k
+k + 1(1 − p)nα =: Ln,p,α(k + 1).
+(B9)
+Now,
+L(k + 1) − U(k) =
+k
+k + 1(1 − p)nα − 2n
+k − k − 1
+k
+(1 − p)nα
+= nα(1 − p) − 2n(k + 1)
+k(k + 1)
+> 0,
+where the final inequality holds since nα−2 > 12 > 8, k ≤ n, and 1 − p ≥ 1/2:
+2(k + 1) ≤ 2(n + 1) ≤ 4n < nα−1
+2
+≤ nα−1(1 − p).
+This establishes Eq. (B7). To prove Eq. (B8), we again use Lemma B.1 to obtain (for k ≤ n) :
+P(k)(N c
+n,p,d) ≤ 2n
+k + k − 1
+k
+pnα =: U c
+n,p,α(k),
+(B10)
+Q(k+1)(N c
+n,p,d) ≥
+k
+k + 1pnα =: Lc
+n,p,α(k + 1).
+As before, the claim follows by noting that
+Lc(k + 1) − U c(k) =
+k
+k + 1pnα − 2n
+k − k − 1
+k
+pnα
+= nαp − 2n(k + 1)
+k(k + 1)
+> 0,
+where the final inequality is true because nα−2 > 12, k ≤ n, and p > 1/3:
+2(k + 1) ≤ 2(n + 1) ≤ 4n < nα−1
+3
+< nα−1p.
+Theorem B.2. In the setting of Theorem 2, for k ≤ n, the following implication holds:
+k − 1
+k
+≥
+2 + nαp
+(1 − p)(n + 1)nα−1
+=⇒ P(n+1)(N c) ≤ Q(k)(N).
+Proof. For k ≤ n, we have
+P(k+1)(N c
+n,p,d) ≤ U c
+n,p,α(k + 1),
+Q(k)(N c
+n,p,d) ≥ Ln,p,α(k),
+where U c and L are defined in Eq. (B10), (B9). The claim follows by noting that L(k) ≥ U c(n + 1) if and only if k
+satisfies the desired inequality.
+
+9
+Theorem B.3. Let p ∈ [0, 1] and n, α ∈ N (α > 1) be such that 4/nα−1 ≤ p ≤ 1/2 − 1/nα−1. Let d = 2nα and
+k0 := (1 − p − 2/nα−1)/p. Fix k ≤ k0 and define
+cn,p,α(k) :=
+(1 − p)k
+p − 2/nα−1 .
+Then, for c(k) < j ≤ n + 1 (provided such a j exists),
+Q(j)(Nn,p,d) > P(k)
+tot (Nn,p,d).
+Proof. Lemma B.1 shows that for k ≤ k0 :
+P(k)
+tot (Nn,p,d) ≤ (1 − 2p)nα + 2n
+k + k − 1
+k
+pnα =: U ′
+n,p,α(k),
+(B11)
+and Q(j)(Nn,p,d) ≥ j−1
+j (1 − p)nα =: Ln,p,α(j) for j ≤ n + 1. The result follows by noting that
+Ln,p,α(j) > U ′
+n,p,α(k) ⇐⇒ j >
+(1 − p)k
+p − 2/nα−1 .
+A similar reasoning as above can be applied when k > k0. In this case, Lemma B.1 shows that
+∀k > k0 :
+P(k)
+tot (Nn,p,d) ≤ 4n
+k + k − 1
+k
+nα =: U ′′
+n,α(k),
+(B12)
+and the lower bound for Q(j) remains the same: Q(j)(Nn,p,d) ≥ Ln,p,α(j) for j ≤ n + 1. Thus, we have
+Ln,p,α(j) > U ′′
+n,α(k) ⇐⇒ j − 1
+j
+> 4/nα−1 + k − 1
+k(1 − p)
+,
+provided such a j ≤ n + 1 exists.
+
+10
+Appendix C: Rocket channels
+In this section, we provide a graphical proof of the superadditive nature of the Rocket channel when used jointly
+with erasure (Eq. (2), (3)). The graphical presentation makes for a more intuitive and easily digestible argument. A
+quick summary of the necessary diagrammatic notation can be found in [32, Section 3]. For more detailed expositions,
+we refer the readers to [33, 34].
+Recall that the rocket channel Rd : A1 ⊗ A2 → B first applies local (independent) random unitaries U, V on
+inputs A1, A2 respectively, and then couples them via a controlled phase gate P = �
+i,j ωij |i⟩⟨i|A1 ⊗ |j⟩⟨j|A2, where
+ω = ei2π/d. Finally, A2 is discarded and Bob gets A1 along with classical information about which local unitaries were
+applied. Hence, each random instance of the Rocket channel acting on an input XA1A2 can be depicted as follows:
+U
+V
+P
+U †
+P †
+V †
+X
+,
+where we have not shown the classical knowledge about U, V that is also delivered to Bob.
+Note that the final
+output state will be obtained by taking an expectation over the random variables U, V . The complementary channel
+Rc
+d : A1 ⊗ A2 → E acts nearly identically, except that in the final step, A1 is discarded and Eve gets A2 along with
+the same classical information about which local unitaries were applied (which is not depicted below):
+U
+V
+P
+U †
+P †
+V †
+X
+.
+To transmit information by jointly using the Rocket channel (or its complement) with erasure, Alice first prepares
+a maximally entangled state and sends one half of it through the erasure channel Ep,d. This establishes maximal
+entanglement between her and the receiver with probability 1 − p. Using this shared entanglement (shown in red in
+Figures 3 and 4), Alice and Bob can communicate through Rd and Rc
+d at rate log d by following the steps described in
+Figures 3 and 4, respectively. With probability p, the erasure channel fails to distribute entanglement between Alice
+and the receiver, in which case the protocol fails. Hence, we get the following net rates of quantum communication:
+Q(1)(Rd ⊗ Ep,d) ≥ (1 − p) log d
+and
+Q(1)(Rc
+d ⊗ Ep,d) ≥ (1 − p) log d.
+
+11
+U
+V
+P
+U †
+P †
+V †
+U
+P
+U †
+P †
+V T
+¯V
+U
+P
+U †
+P †
+U
+P Γ
+U †
+P Γ†
+U
+U †
+Slide V, V† across the wire
+Undo VT
+Undo PΓ = P
+Undo U
+Figure 3: Visual depiction of how the Rocket channel Rd can be used to transmit information with the help of pre-shared entanglement. Alice
+and Bob start with a pre-shared maximally entangled state (shown in red). Alice locally prepares another maximally entangled state (shown in
+purple) and sends half of each of the entangled states through Rd to Bob as shown. Hence, only the top two dangling wires are in Alice’s possession
+while the bottom four are with Bob. Since Bob knows which local random unitaries U, V are applied during the transmission, he can use the
+pre-shared entanglement with Alice to undo the phase coupling operation as described. Here, P Γ is the partial transpose of P with respect to the
+second subsystem. The two parties finally end up sharing one maximally entangled state (shown in orange), which can be used to send quantum
+information at rate log d.
+
+12
+U
+V
+P
+U †
+P †
+V †
+P
+P †
+P Γ1
+P Γ1†
+Slide U, U† across the wire
+Undo UT
+Undo PΓ1 = P
+U T
+¯U
+V
+V †
+P
+P †
+V
+V †
+V
+V †
+V
+V †
+Undo V
+Figure 4:
+Visual depiction of how the complementary Rocket channel Rc
+d can be used to transmit information with the help of pre-shared
+entanglement.
+Alice and Bob start with a pre-shared maximally entangled state (shown in red).
+Alice locally prepares another maximally
+entangled state (shown in purple) and sends half of each of the entangled states through Rc
+d to Bob as shown.
+Hence, only the bottom two
+dangling wires are in Alice’s possession while the top four are with Bob. Since Bob knows which local random unitaries U, V are applied during
+the transmission, he can use the pre-shared entanglement with Alice to undo the phase coupling operation as described. Here, P Γ1 is the partial
+transpose of P with respect to the first subsystem. The two parties finally end up sharing one maximally entangled state (shown in orange), which
+can be used to send quantum information at rate log d.
+
diff --git a/NdE4T4oBgHgl3EQfjg0O/content/tmp_files/load_file.txt b/NdE4T4oBgHgl3EQfjg0O/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2611070e7951a9eb836fff06abb6f42e301c4563
--- /dev/null
+++ b/NdE4T4oBgHgl3EQfjg0O/content/tmp_files/load_file.txt
@@ -0,0 +1,459 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf,len=458
+page_content='Simultaneous superadditivity of the direct and complementary channel capacities  Satvik Singh1 and Sergii Strelchuk1  1DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB30WA, UK  Quantum communication channels differ from their classical counterparts because their capacities  can be superadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The principle of monogamy of entanglement suggests that superadditive  improvements in the transmission capacity of a channel should reduce the amount of information  loss to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' We challenge this intuition by demonstrating that the coherent and private  information of a channel and its complement can be simultaneously superadditive for arbitrarily  many channel uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  To quantify the limits of this effect, we consider the notion of max (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  total) private information of a channel, which represents the maximum (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' sum) of the private  information of the channel itself and its complement, and study its relationship with the coherent  information of the individual direct and complementary channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For a varying number of channel  uses, we show that these quantities can obey different interleaving sequences of inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Quantum channels have several intriguing properties  that separate them from their classical counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' One  of them is the ability to send information superadditively  [1], whereby multiple uses of the same channel increase  the amount of information that it can reliably trans-  mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Another one, namely superactivation [2], demon-  strates how some channels, having initially no capacity  to send information can regain it when combined with  an equally useless zero-capacity channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Both of these  effects were introduced and subsequently studied in the  context of having access to a channel NA→B where Alice,  the sender, communicates with the receiver, Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  One can define different kinds of capacities of N de-  pending on the type of information being sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The  quantum capacity Q(N) of N quantifies the maximum  rate at which Alice can send quantum information re-  liably to Bob and can be expressed as a regulariza-  tion of the channel’s coherent information:  Q(N) =  limk→∞ Q(k)(N), Q(k)(N) := Q(1)(N ⊗k)/k, Q(1)(N) :=  supρA[S(B) − S(E)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The optimization here is over all  states ρA and the von Neumann entropies S(B) and S(E)  are evaluated on NA→B(ρA) and N c  A→E(ρA), respec-  tively, where the complementary channel N c  A→E models  the nature of information loss to the environment (Eve)  (its precise definition will be introduced shortly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Simi-  larly, the classical capacity C(N) of N quantifies the max-  imum rate at which Alice can send classical information  reliably to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' In addition, if the information being sent  is to be kept private from Eve, one obtains the private  capacity P(N) of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' These capacities admit  regularized expressions in terms of the channel’s Holevo  information:  C(N)  =  limk→∞ C(k)(N), C(k)(N)  :=  C(1)(N ⊗k)/k, C(1)(N) := supρXA I(X  : B), and pri-  vate information P(N) = limk→∞ P(k)(N), P(k)(N) :=  P(1)(N ⊗k)/k, P(1)(N) := supρXA[I(X : B) − I(X : E)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The optimizations above are over all classical quantum  states ρXA = �  x px |x⟩⟨x|X ⊗ ρx  A and the mutual infor-  mation terms I(X : B) and I(X : E) are evaluated on  NA→B(ρXA) and N c  A→E(ρXA), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Since the first demonstration [1], there have been a  variety of results that illustrate striking superadditive  behavior of quantum channel capacities [3–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For in-  stance, the k-letter coherent and private information of  a channel obey the inequality: Q(k)(N) ≤ P(k)(N) ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  However, for different numbers of channel uses, the coher-  ent information of N can exceed its private information  [7], making the former inequality valid only for a fixed k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The above effects were all demonstrated in the set-  ting where Alice optimizes her data transmission rate  to Bob while minimizing information ‘leakage’ to Eve,  who behaves as a non-participating party during trans-  mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  There are several results that investigate in-  formation transmission problems under non-trivial be-  haviour of the environment [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' In one of the ear-  liest works [17] of such kind, the authors investigated  the capacities of quantum channels by allowing Eve to  locally measure and communicate classical messages to  Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Later, this was extended to allow a helper [15] – a  benevolent third party – who can adjust the environment  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' This enabled one to derive streamlined examples  for super-additivity due to the extra abilities of the helper  to adjust the state of the environment depending on the  message being sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For example, the so-called locking  capacity [18] of a channel in this regime is superadditive,  whereas in the absence of auxiliary resources the question  is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The above scenarios supplement the direct channel  NA→B with extra resources which are extrinsic to its  definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' This precludes one from learning about the  total capacity for information transmission by using the  channel alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' One may observe that defining a direct  channel NA→B from Alice to Bob fixes the behaviour of  information loss to the environment (Eve) via the com-  plementary channel N c  A→E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Indeed, the Stinespring dila-  tion theorem [19, 20] shows that there exists an isometry  V : HA → HB ⊗HE such that N(X) = TrE(V XV †) and  N c(X) = TrB(V XV †).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  When optimizing the communication rates, we natu-  rally want to take full advantage of the superadditive  properties of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The monogamy of entangle-  ment principle suggests that when the direct channel is  superadditive, its complementary channel to the envi-  ronment is likely to have constrained data transmission  capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Contrary to this intuition, we show that  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='05142v1  [quant-ph]  12 Jan 2023  2  superadditivity can persist for an unbounded number of  channel uses in the strongest possible sense in both the  direct and complementary channels simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  To demonstrate this surprising effect we consider two  quantities: the max and total coherent information of N:  Q(k)  max(N) := max{Q(k)(N), Q(k)(N c)},  Q(k)  tot(N) := Q(k)(N) + Q(k)(N c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The max- and total Holevo and private information quan-  tities can be defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The max and total quan-  tum capacities can be obtained by taking k → ∞ limits:  lim  k→∞ Q(k)  max(N) = max{Q(N), Q(N c)},  lim  k→∞ Q(k)  tot(N) = Q(N) + Q(N c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Operationally, Bob and Eve are now placed on an equal  footing and Alice simply wants to send information at the  best rate possible regardless of who plays the role of the  receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' She can either use the direct channel NA→B or  its complement N c  A→E to do so, thus arriving at the max  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' In such a scenario, it is crucial to analyse the super-  additive behaviour of both the direct and complementary  channels together to determine which one has higher ca-  pacity to transmit information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' A similar quantity was  introduced in [21] in the context of entanglement distil-  lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' On the other hand, to our best knowledge, the  expression for the total quantum capacity first appeared  in [22], where it was used to bound the difference be-  tween the quantum and private capacities of any channel  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Intuitively, being the sum of the direct and comple-  mentary channel capacities, the total capacity quantifies  the overall information transmission capability of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The above setting should be distinguished from that of  quantum broadcast channels [23, 24] where two (or more)  recipients (Bobs) share a joint environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' As such, this  model is not representative of the concepts of max and  total information where the notion of the environment  does not feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  We now briefly describe our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For n, α ∈ N sat-  isfying nα−2 > 8, we construct a channel N such that  (Theorem 1):  ∀k ≤ n :  Q(k+1)(N), Q(k+1)(N c) > P(k)  max(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (1)  Thus, for any number k of channel uses, the max k-letter  private information of N can be exceeded by the coher-  ent information of both the direct and complementary  channels by using just one extra copy of each of these  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' This means that the coherent and private in-  formation quantities of N are curiously interleaved:  Q(1)(N), Q(1)(N c) ≤ P(1)  max(N) < Q(2)(N), Q(2)(N c)  ≤ P(2)  max(N) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Remarkably, by choosing n large enough, this phe-  nomenon can be made to persist for arbitrarily many  channel uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Moreover, the parameter p can be tuned  to boost the superadditivity of the direct channel rel-  ative to its complement or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  More precisely,  when 1/3 ≤ p ≤ 1/2 − 1/nα−1 and nα−2 > 12, even  though both the direct and complementary channels are  still superadditive, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (1) now only holds for N and not  for N c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Thus, the superadditivity of the direct channel  dominates that of its complement (Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For even  smaller values of p, this effect becomes extreme: even the  total k−letter private information of the channel (below a  certain threshold k value) can be exceeded by the coher-  ent information of the direct channel alone, provided that  it is used sufficiently many times j > ck (Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='3):  Q(j)(N) > P(k)(N) + P(k)(N c) = P(k)  tot (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  In all the above cases, it is possible to precisely quantify  the effects of superaddivity by computing lower bounds  on the difference quantities such as Q(k+1)(N)−P(k)  max(N)  and Q(k+1)(N c) − P(k)(N c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Finally, it turns out that simultaneous superadditiv-  ity of both the direct and complementary channels is a  non-trivial phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' We prove this by constructing  a channel with superadditive quantum capacity whose  complement has additive capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The main construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='– Our channels Nn,p,d are made  of two building blocks: the erasure and ‘rocket’ channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The d-dimensional erasure channel Ep,d : A → B with  erasure parameter p ∈ [0, 1] takes a d-dimensional input  and replaces it with an erasure flag |e⟩⟨e| (orthogonal to  the input space) with probability p and does nothing to  it otherwise: Ep,d(ρ) = (1 − p)ρ + p Tr(ρ) |e⟩⟨e|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Its com-  plement is again of the erasure type: Ec  p,d = E1−p,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The  capacities of Ep,d are well known: Q(Ep,d) = P(Ep,d) =  max{(1 − 2p) log d, 0}, C(Ep,d) = (1 − p) log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The d-dimensional rocket channel [25] Rd : A1 ⊗  A2 → B takes two d-dimensional quantum systems  (A1 and A2) as inputs and applies local random unitaries  on each input [26] followed by a controlled phase coupling  P = �  i,j ωij |i⟩⟨i|A1 ⊗|j⟩⟨j|A2, where ω = ei2π/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Finally,  A2 is discarded and Bob gets A1 along with classical in-  formation about which local unitaries were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For  the complementary channel Rc  d : A1 ⊗ A2 → E, A1 is  discarded and Eve gets A2 along with the same classical  information about the local unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Since Rd dephases  the input registers in a random basis unknown to Alice,  it has little capacity to transmit information on its own:  C(Rd) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The same argument applies to Rc  d as well:  C(Rc  d) ≤ 2 (the proof of [25] goes through by swapping  labels for Bob and Eve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' However, when Alice and Bob  already share a maximally entangled state (this can be  achieved with probability 1−p by using the erasure Ep,d),  it turns out that Bob can undo the random phase cou-  pling, thus allowing Alice to send quantum information  at rate log d [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' More precisely, we have  Q(1)(Rd ⊗ Ep,d) ≥ (1 − p) log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  (2)  A slight modification of this argument can be used to  3  obtain the same result for Rc  d as well:  Q(1)(Rc  d ⊗ Ep,d) ≥ (1 − p) log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  (3)  An intuitive graphical proof of the above two claims  (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (2) and (3)) is provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  We are now ready to introduce our main channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For  n, d ∈ N and p ∈ [0, 1], we define  Nn,p,d := R⊗n  d  ⊕ Ep,d,  N c  n,p,d = Rc ⊗n  d  ⊕ E1−p,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The direct sum construction [27] allows Alice to control  which of the two channels is being applied at the out-  set, so that the coherent and private information of such  channels is just the maximum of its building blocks, see  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  In our case, by using the channel k + 1  times, Alice gains access to blocks of the form  R⊗n  d  ⊗ E⊗k  p,d or Rc ⊗n  d  ⊗ E⊗k  1−p,d,  for which the superadditivty effects observed in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (2),  and (3) can boost information transmission rates for each  successive channel use as long as k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For p = 1/2,  these effects are identical for both the direct and comple-  mentary channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The parameter p in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (2),(3) can  be adjusted to enhance the superadditivity of either the  direct channel or its complement relative to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  We employ these ideas to prove our main results below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Let p = 1/2, and n, α ∈ N be such that  nα−2 > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Furthermore, let d = 2nα and N = Nn,p,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Then, for all k ≤ n :  Q(k+1)(N) − P(k)  max(N) ≥ nα − 4n(k + 1)  2k(k + 1)  > 0,  Q(k+1)(N c) − P(k)  max(N) ≥ nα − 4n(k + 1)  2k(k + 1)  > 0,  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1, and the fact that for any chan-  nel N, C(1)(N ⊗ Ep,d) = C(1)(N) + C(1)(Ep,d) [7, Lemma  2], we get:  P(k)(N) = 1  k max  0≤l≤k P(1)(R⊗nl  d  ⊗ E⊗k−l  1/2,d )  ≤ max  �  �  �  �  �  2n  2n  k + k−1  k  1  2 log d  0  = 2n  k + (k − 1)nα  2k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Analogous reasoning yields a bound for P(k)(N c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Com-  bining Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (2) and the fact that  Q(1)(N1 ⊗ N2) ≥ Q(1)(N1) + Q(1)(N2) yields a simple  lower bound:  Q(k+1)(N) ≥  Q(1)(R⊗n  d  ⊗ E⊗k  1/2,d)  k + 1  ≥  knα  2(k + 1)  for k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Swapping Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (2) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (3) shows that the  same bound holds for Q(k+1)(N c) too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Thus,  Q(k+1)(N) − P(k)  max(N) ≥ nα − 4n(k + 1)  2k(k + 1)  > 0,  where the latter inequality holds because nα−2 > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Clearly, the same bounds hold for N c as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  When p < 1/2, the superadditivity analysis of Nn,p,d  becomes tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' In Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1, we prove several capac-  ity bounds which we use to show that the superadditivity  of the direct channel dominates that of its complement:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Let p ∈ [0, 1], and n, α ∈ N be such that  1/3 < p ≤ 1/2 − 1/nα−1 and nα−2 > 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Furthermore,  let d = 2nα and N = Nn,p,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Then,  Q(k+1)(N) − P(k)  max(N) ≥ nα(1 − p) − 2n(k + 1)  k(k + 1)  =: fn,p,α(k) > 0,  Q(k+1)(N c) − P(k)(N c) ≥ nαp − 2n(k + 1)  k(k + 1)  =: f c  n,p,a(k) > 0,  where the first bound holds for 2 ≤ k ≤ n and the second  bound holds for 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The full proof is located in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  0  10  20  30  40  50  60  70  80  90  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='5  5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='5  Figure 1: Plot of the log of the lower bounds fn,p,α(k) and f c  n,p,α(k)  on the difference quantities Q(k+1)(Nn,p,d) − P(k)  max(Nn,p,d) and  Q(k+1)(N c  n,p,d) − P(k)(N c  n,p,d), respectively (see Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Here,  n = 100, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='4, and α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The log lower bounds for the difference quantities in  Theorem 2 are plotted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Note that in the  setting of Theorem 2, the following is true:  k − 1  k  ≥  2 + nαp  (1 − p)(n + 1)nα−1  =⇒ P(n+1)(N c) ≤ Q(k)(N),  (4)  4  provided that k ≤ n (see Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For instance,  when n = 100, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='4, and α = 3, the LHS in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (4)  holds for k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' In other words, the direct channel in this  case is vastly more superadditive than its complement,  since only the 3-letter coherent information of Nn,p,d suf-  fices to beat the 101-letter private information of N c  n,p,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  For small values of p, the coherent information of  Nn,p,d can even beat the total private information  for some uses of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  For example, when  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='09, n = 100, and α = 3, Q(101)(Nn,p,d) beats  P(k)  tot (Nn,p,d) for k ≤ 9 (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' A detailed analysis  of this phenomenon is given in Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  0  5  10  15  20  25  30  35  40  45  50  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='6  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='8  9  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='2  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='4  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='6  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='8  10  105  Figure 2: Plot of the lower bound Qmax = L(n + 1) on  Q(n+1)(Nn,p,d) and the upper bound Uk =  �  U ′(k)  if k ≤ k0  U ′′(k)  otherwise  on  P(k)  tot (Nn,p,d) for p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='09, n = 100, and α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The lower and upper  bound functions are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (B9),(B11),(B12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Clearly, Qmax  exceeds the total private capacity for at least 9 uses of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Platypus construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='– We now turn to constructing  a channel with superadditive quantum capacity whose  complement has additive capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The d−dimensional  platypus channel Md : A → B introduced recently in  [28] is defined via the isometry:  V : HA → HB ⊗ HE  V |0⟩ =  1  √  d − 1  d−2  �  j=0  |j⟩ |j⟩ ,  V |i⟩ = |d − 1⟩ |i − 1⟩ ,  i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' , d − 1,  as Md(X)  =  TrE(V XV †).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Here,  A and B are  d−dimensional while E is (d − 1)−dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The  channel and its complement satisfy [28, 29]:  Q(Md) ≤ log  �  1 +  1  √  d − 1  �  , P(Md) = C(Md) = 1,  Q(1)(Mc  d) = Q(Mc  d) = P(Mc  d) = C(Mc  d)  = log(d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  As d → ∞, Q(Md) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' However, when coupled with  an erasure channel, Md can be shown to retain some  quantum capacity as d → ∞ [28]:  Q(1)(Md+1 ⊗ E1/2,d) ≥ 1  2 + O  � 1  √  d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  To turn these effects into superadditivity of a single  channel, we again turn to a direct sum construction:  Nd := Md+1 ⊕ E1/2,d,  N c  d := Mc  d+1 ⊕ E1/2,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  By choosing a large enough d, we can ensure that  Q(1)(Md+1 ⊗ E1/2,d) > 2 log  �  1 + 1/  √  d  �  ≥ Q(1)(M⊗2  d+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  In other words, Nd is superadditive for large enough d:  Q(2)(Nd) = 1  2 max{Q(1)(M⊗2  d+1), Q(1)(Md+1 ⊗ E1/2,d)}  > log  �  1 + 1  √  d  �  ≥ Q(1)(Nd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  On the other hand, for any k1, k2 ∈ N:  Q(1)(Mc ⊗k1  d+1 ⊗ E⊗k2  1/2,d) ≤ C(1)(Mc ⊗k1  d+1 ⊗ E⊗k2  1/2,d)  = C(1)(Mc ⊗k1  d+1 ) + C(1)(E⊗k2  1/2,d)  = k1 log d + k2  2 log d  ≤ (k1 + k2) log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Thus, N c  d has additive quantum capacity:  ∀k : Q(k)(N c  d) = 1  k max  0≤l≤k Q(1)(Mc ⊗l  d+1 ⊗ E⊗k−l  1/2,d )  = Q(k)(Mc  d+1) = log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='– We have investigated superadditive effects  for the coherent and private information of a channel and  its complement relative to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' We showed that  contrary to intuitive expectations, the following two cases  are both possible:  The direct and complementary channels are simul-  taneously superadditive,  The direct channel is superadditive while the com-  plement is additive (and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  It is also possible to construct examples where both the  direct and complementary channels are entanglement-  breaking and hence trivially have additive coherent and  private information (equal to zero) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  One interesting question to investigate further is the  extent to which superadditivity of the coherent and pri-  vate information quantities can be attained simultane-  ously for the direct and complementary channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Is  5  there a constraint akin to the monogamy of entangle-  ment which would prohibit a maximum violation of ad-  ditivity for both channels?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Another intriguing question  is whether one can superactivate total capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Finally, it would be enlightening to check whether or  not the superadditivity results presented in this letter  hold for the classical capacities as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Our techniques  do not apply directly in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The problem lies in  the fact that the classical capacity of a direct sum chan-  nel is not merely the maximum of the classical capacity  of its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' This is because when using a direct  sum channel, say N = ⊕n  i=1N (i), unlike private or co-  herent information, classical information can not only be  sent through the individual blocks N (i), but can also be  encoded in the choice of the blocks i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' More  precisely, for our channel of interest, say N = Rd ⊕ Ep,d,  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1 shows that  C(1)(N) = log  �  2C(1)(Rd) + 2C(1)(Ep,d)�  , and  C(2)(N) = 1  2 log  �  2C(1)(R⊗2  d  ) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='2C(1)(Rd⊗Ep,d) + 2C(1)(E⊗2  p,d)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Since C(1)(N ⊗ Ep,d) = C(1)(N) + C(1)(Ep,d) for all chan-  nels N [7, Lemma 2], it is clear that the question of  superadditivity of C(1)(N) boils down to the question of  superadditivity of C(1)(Rd), which is currently open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The authors would like to thank  Nilanjana Datta for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Satvik Singh  acknowledges support from the Cambridge Trust’s In-  ternational Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Sergii Strelchuk acknowledges  support from the Royal Society University Research Fel-  lowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  ♦  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' DiVincenzo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Shor, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smolin, Quantum-channel capacity of very noisy channels, Physical Review A  57, 830 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smith and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Yard, Quantum communication with zero-capacity channels, Science 321, 1812 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Wilde, Quantum information theory (Cambridge University Press, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Zhu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Zhuang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Hsieh, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Shor, Superadditivity in trade-off capacities of quantum channels, IEEE  transactions on information theory 65, 3973 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Zhu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Zhuang, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Shor, Superadditivity of the classical capacity with limited entanglement assistance,  Physical review letters 119, 040503 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Cubitt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Elkouss, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Matthews, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Ozols, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Pérez-García, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Strelchuk, Unbounded number of channel uses  may be required to detect quantum capacity, Nature communications 6, 1 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Elkouss and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Strelchuk, Superadditivity of private information for any number of uses of the channel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  115, 040501 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Elkouss and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Strelchuk, Nonconvexity of private capacity and classical environment-assisted capacity of a quantum  channel, Physical Review A 94, 040301 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smith and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smolin, Extensive nonadditivity of privacy, Physical Review Letters 103, 120503 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Brandao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Oppenheim, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Strelchuk, When does noise increase the quantum capacity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=', Physical review letters  108, 040501 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Shirokov and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Shulman, On superactivation of zero-error capacities and reversibility of a quantum channel, Commu-  nications in Mathematical Physics 335, 1159 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Leditzky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Leung, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Siddhu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smith, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smolin, Generic nonadditivity of quantum capacity in simple  channels, arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='08377 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Leung, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smith, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smolin, Maximal privacy without coherence, Physical review letters 113, 030502  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [14] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Winter, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Zou, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Guo, Private capacity of quantum channels is not additive, Physical Review Letters 103,  120501 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Karumanchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Mancini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Winter, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Yang, Classical capacities of quantum channels with environment assistance,  Problems of Information Transmission 52, 214 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Oskouei, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Mancini, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Winter, Capacities of gaussian quantum channels with passive environment assistance,  IEEE Transactions on Information Theory 68, 339 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Winter, On environment-assisted capacities of quantum channels, arXiv preprint quant-ph/0507045 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Guha, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Hayden, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Krovi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Lloyd, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Lupo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Shapiro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Takeoka, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Wilde, Quantum enigma machines  and the locking capacity of a quantum channel, Physical Review X 4, 011016 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [19] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Stinespring, Positive functions on c*-algebras, Proceedings of the American Mathematical Society 6, 211 (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [20] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Paulsen, Completely bounded maps and operator algebras, 78 (Cambridge University Press, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Singh and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Datta, Fully undistillable quantum states are separable, preprint arxiv:2207:05193 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Hirche and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Leditzky, Bounding quantum capacities via partial orders and complementarity, in 2022 IEEE Interna-  tional Symposium on Information Theory (ISIT) (2022) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' 2219–2224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Yard, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Hayden, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Devetak, Quantum broadcast channels, IEEE Transactions on Information Theory 57, 7147  (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Laurenza and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Pirandola, General bounds for sender-receiver capacities in multipoint quantum communications,  Physical Review A 96, 032318 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [25] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smith and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smolin, Extensive nonadditivity of privacy, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' 103, 120503 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [26] It actually suffices to apply local unitaries from a unitary 2-design, such as the Clifford group [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  6  [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Fukuda and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Wolf, Simplifying additivity problems using direct sum constructions, Journal of Mathematical  Physics 48, 072101 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [28] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Leditzky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Leung, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Siddhu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smith, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smolin, Generic nonadditivity of quantum capacity in simple  channels, preprint arxiv:2202:08377 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [29] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Leditzky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Leung, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Siddhu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smith, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Smolin, The platypus of the quantum channel zoo, preprint  arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='08380 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Müller-Hermes and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Singh, Bi-PPT channels are entanglement breaking, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='01685 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [31] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Dankert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Cleve, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Emerson, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Livine, Exact and approximate unitary 2-designs and their application to fidelity  estimation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' A 80, 012304 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [32] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Nechita and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Singh, A graphical calculus for integration over random diagonal unitary matrices, Linear Algebra and  its Applications 613, 46 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [33] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Wood, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Biamonte, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Cory, Tensor networks and graphical calculus for open quantum systems, Quantum  Information & Computation 15, 759 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  [34] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Bridgeman and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Chubb, Hand-waving and interpretive dance: an introductory course on tensor networks,  Journal of Physics A: Mathematical and Theoretical 50, 223001 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Appendix A: Notation and the direct sum construction  We denote quantum systems with capital letters like A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Each quantum system A is associated with a finite-  dimensional complex Hilbert space HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For a joint system AB or A ⊕ B, HAB = HA ⊗ HB or HA⊕B = HA ⊕ HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Quantum states ρA on A are positive semi-definite operators acting on HA with unit trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  The von Neumann  entropy of ρA is defined as S(A) := − Tr(ρA log2 ρA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For a bipartite state ρAB,the mutual information is defined as  I(A : B) := S(A) + S(B) − S(AB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  A quantum channel N : A → B (denoted NA→B) is a completely positive and trace-preserving linear map taking  linear operators acting on HA to those acting on HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Every channel NA→B admits a Stinespring isometry V : HA →  HB ⊗ HE such that N(X) = TrE(V XV †).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The complementary channel N c  A→E is defined as N c(X) = TrB(V XV †).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  For two channels N (1) : A1 → B1, N (2) : A2 → B2, the direct sum N = N (1) ⊕ N (2) : A1 ⊕ A2 → B1 ⊕ B2 acts as  N  ��  XA1  ∗  ∗  XA2  ��  =  �  N (1)(XA1)  0  0  N (2)(XA2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  It should be clear that the complement Nc = N (1)  c  ⊕ N (2)  c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For quantum channels N (i)  Ai→Bi for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' n, let N := ⊕n  i=1N (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Then,  [27,  Proposition  1]  Q(1)(N) = max  1≤i≤n Q(1)(N (i)),  [7,  Lemma  1]  P(1)(N) = max  1≤i≤n P(1)(N (i)),  [27,  Proposition  1]  C(1)(N) = log  n  �  i=1  2C(1)(N (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Appendix B: Proofs  Recall from the main text that for n, d ∈ N and p ∈ [0, 1], the channel Nn,p,d was defined as the direct sum  Nn,p,d := R⊗n  d  ⊕ Ep,d, where Rd and Ep,d are the rocket and erasure channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' We can derive the following bounds on  the capacities of this channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Let p ∈ [0, 1] and n, α ∈ N (α > 1) be such that 4/nα−1 ≤ p ≤ 1/2 − 1/nα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Let d = 2nα and  k0(n, p, α) := 1 − p − 2/nα−1  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  7  Then, the following bounds hold:  ∀k ≤ k0 :  P(k)(Nn,p,d) ≤ (1 − 2p)nα  (B1)  ∀k > k0 :  P(k)(Nn,p,d) ≤ 2n  k + k − 1  k  (1 − p)nα  (B2)  ∀k :  P(k)(N c  n,p,d) ≤ 2n  k + k − 1  k  pnα  (B3)  ∀k ≤ n :  Q(k+1)(Nn,p,d) ≥  k  k + 1(1 − p)nα  (B4)  ∀k ≤ n :  Q(k+1)(N c  n,p,d) ≥  k  k + 1pnα  (B5)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' By using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1 and the fact that C(1)(N ⊗ Ep,d) = C(1)(N) + C(1)(Ep,d) for all channels N, it is easy to  arrive at the following bounds:  P(k)(Nn,p,d) = 1  k max  0≤l≤k P(1)(R⊗nl  d  ⊗ E⊗k−l  p,d  )  ≤ max  �  �  �  �  �  2n  2n  k + k−1  k (1 − p)nα  (1 − 2p)nα,  (B6)  P(k)(N c  n,p,d) = 1  k max  0≤l≤k P(1)(Rc ⊗nl  d  ⊗ E⊗k−l  1−p,d)  ≤ max  �  2n  2n  k + k−1  k pnα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Let’s first deal with the maximum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (B6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Clearly, (1 − 2p)nα ≥ 2n since p ≤ 1/2 − 1/nα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Moreover,  2n  k + k − 1  k  (1 − p)nα ≤ (1 − 2p)nα  ⇐⇒ 2  k + (1 − 1  k )(1 − p)nα−1 ≤ (1 − 2p)nα−1  ⇐⇒ 1  k ((1 − p)nα−1 − 2) ≥ nα−1p  ⇐⇒ k ≤ (1 − p − 2/nα−1)  p  = k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  This gives us the first two bounds in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (B1) and (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' To obtain the bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (B3), observe that  4/nα−1 ≤ p ⇐⇒ 2n ≤ 1  2pnα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Hence, for k ≥ 2, we have k−1  k pnα ≥ pnα/2 ≥ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' To prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (B4), note that for k ≤ n, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1 shows  Q(k+1)(Nn,p,d) ≥  Q(1)(R⊗n  d  ⊗ E⊗k  p,d)  k + 1  ≥  k  k + 1(1 − p)nα,  where we have used the bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (2) along with the fact that Q(1)(N1 ⊗N2) ≥ Q(1)(N1)+Q(1)(N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' An identical  argument works to prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (B5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  We can now prove Theorem 2 from the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Let p ∈ [0, 1], and n, α ∈ N be such that 1/3 < p ≤ 1/2 − 1/nα−1 and nα−2 > 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Furthermore, let  d = 2nα and N = Nn,p,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Then,  Q(k+1)(N) − P(k)  max(N) ≥ nα(1 − p) − 2n(k + 1)  k(k + 1)  =: fn,p,α(k) > 0,  (B7)  Q(k+1)(N c) − P(k)(N c) ≥ nαp − 2n(k + 1)  k(k + 1)  =: f c  n,p,a(k) > 0,  (B8)  8  where the first bound holds for 2 ≤ k ≤ n and the second bound holds for 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Q(2)(N) > P(1)  max(N)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For 1/3 < p ≤ 1/2 − 1/nα−1, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1 shows that  P(k)  max(Nn,p,d) ≤ 2n  k + k − 1  k  (1 − p)nα =: Un,p,α(k)  for 2 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Furthermore, for k ≤ n, we have  Q(k+1)(Nn,p,d) ≥  k  k + 1(1 − p)nα =: Ln,p,α(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  (B9)  Now,  L(k + 1) − U(k) =  k  k + 1(1 − p)nα − 2n  k − k − 1  k  (1 − p)nα  = nα(1 − p) − 2n(k + 1)  k(k + 1)  > 0,  where the final inequality holds since nα−2 > 12 > 8, k ≤ n, and 1 − p ≥ 1/2:  2(k + 1) ≤ 2(n + 1) ≤ 4n < nα−1  2  ≤ nα−1(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  This establishes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (B7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' To prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (B8), we again use Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1 to obtain (for k ≤ n) :  P(k)(N c  n,p,d) ≤ 2n  k + k − 1  k  pnα =: U c  n,p,α(k),  (B10)  Q(k+1)(N c  n,p,d) ≥  k  k + 1pnα =: Lc  n,p,α(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  As before, the claim follows by noting that  Lc(k + 1) − U c(k) =  k  k + 1pnα − 2n  k − k − 1  k  pnα  = nαp − 2n(k + 1)  k(k + 1)  > 0,  where the final inequality is true because nα−2 > 12, k ≤ n, and p > 1/3:  2(k + 1) ≤ 2(n + 1) ≤ 4n < nα−1  3  < nα−1p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' In the setting of Theorem 2, for k ≤ n, the following implication holds:  k − 1  k  ≥  2 + nαp  (1 − p)(n + 1)nα−1  =⇒ P(n+1)(N c) ≤ Q(k)(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For k ≤ n, we have  P(k+1)(N c  n,p,d) ≤ U c  n,p,α(k + 1),  Q(k)(N c  n,p,d) ≥ Ln,p,α(k),  where U c and L are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (B10), (B9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The claim follows by noting that L(k) ≥ U c(n + 1) if and only if k  satisfies the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  9  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Let p ∈ [0, 1] and n, α ∈ N (α > 1) be such that 4/nα−1 ≤ p ≤ 1/2 − 1/nα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Let d = 2nα and  k0 := (1 − p − 2/nα−1)/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Fix k ≤ k0 and define  cn,p,α(k) :=  (1 − p)k  p − 2/nα−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Then, for c(k) < j ≤ n + 1 (provided such a j exists),  Q(j)(Nn,p,d) > P(k)  tot (Nn,p,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1 shows that for k ≤ k0 :  P(k)  tot (Nn,p,d) ≤ (1 − 2p)nα + 2n  k + k − 1  k  pnα =: U ′  n,p,α(k),  (B11)  and Q(j)(Nn,p,d) ≥ j−1  j (1 − p)nα =: Ln,p,α(j) for j ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The result follows by noting that  Ln,p,α(j) > U ′  n,p,α(k) ⇐⇒ j >  (1 − p)k  p − 2/nα−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  A similar reasoning as above can be applied when k > k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' In this case, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='1 shows that  ∀k > k0 :  P(k)  tot (Nn,p,d) ≤ 4n  k + k − 1  k  nα =: U ′′  n,α(k),  (B12)  and the lower bound for Q(j) remains the same: Q(j)(Nn,p,d) ≥ Ln,p,α(j) for j ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Thus, we have  Ln,p,α(j) > U ′′  n,α(k) ⇐⇒ j − 1  j  > 4/nα−1 + k − 1  k(1 − p)  ,  provided such a j ≤ n + 1 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  10  Appendix C: Rocket channels  In this section, we provide a graphical proof of the superadditive nature of the Rocket channel when used jointly  with erasure (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' (2), (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The graphical presentation makes for a more intuitive and easily digestible argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' A  quick summary of the necessary diagrammatic notation can be found in [32, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' For more detailed expositions,  we refer the readers to [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Recall that the rocket channel Rd : A1 ⊗ A2 → B first applies local (independent) random unitaries U, V on  inputs A1, A2 respectively, and then couples them via a controlled phase gate P = �  i,j ωij |i⟩⟨i|A1 ⊗ |j⟩⟨j|A2, where  ω = ei2π/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Finally, A2 is discarded and Bob gets A1 along with classical information about which local unitaries were  applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Hence, each random instance of the Rocket channel acting on an input XA1A2 can be depicted as follows:  U  V  P  U †  P †  V †  X  ,  where we have not shown the classical knowledge about U, V that is also delivered to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Note that the final  output state will be obtained by taking an expectation over the random variables U, V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The complementary channel  Rc  d : A1 ⊗ A2 → E acts nearly identically, except that in the final step, A1 is discarded and Eve gets A2 along with  the same classical information about which local unitaries were applied (which is not depicted below):  U  V  P  U †  P †  V †  X  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  To transmit information by jointly using the Rocket channel (or its complement) with erasure, Alice first prepares  a maximally entangled state and sends one half of it through the erasure channel Ep,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' This establishes maximal  entanglement between her and the receiver with probability 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Using this shared entanglement (shown in red in  Figures 3 and 4), Alice and Bob can communicate through Rd and Rc  d at rate log d by following the steps described in  Figures 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' With probability p, the erasure channel fails to distribute entanglement between Alice  and the receiver, in which case the protocol fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Hence, we get the following net rates of quantum communication:  Q(1)(Rd ⊗ Ep,d) ≥ (1 − p) log d  and  Q(1)(Rc  d ⊗ Ep,d) ≥ (1 − p) log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  11  U  V  P  U †  P †  V †  U  P  U †  P †  V T  ¯V  U  P  U †  P †  U  P Γ  U †  P Γ†  U  U †  Slide V, V† across the wire  Undo VT  Undo PΓ = P  Undo U  Figure 3: Visual depiction of how the Rocket channel Rd can be used to transmit information with the help of pre-shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Alice  and Bob start with a pre-shared maximally entangled state (shown in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Alice locally prepares another maximally entangled state (shown in  purple) and sends half of each of the entangled states through Rd to Bob as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Hence, only the top two dangling wires are in Alice’s possession  while the bottom four are with Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Since Bob knows which local random unitaries U, V are applied during the transmission, he can use the  pre-shared entanglement with Alice to undo the phase coupling operation as described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Here, P Γ is the partial transpose of P with respect to the  second subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The two parties finally end up sharing one maximally entangled state (shown in orange), which can be used to send quantum  information at rate log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  12  U  V  P  U †  P †  V †  P  P †  P Γ1  P Γ1†  Slide U, U† across the wire  Undo UT  Undo PΓ1 = P  U T  ¯U  V  V †  P  P †  V  V †  V  V †  V  V †  Undo V  Figure 4:  Visual depiction of how the complementary Rocket channel Rc  d can be used to transmit information with the help of pre-shared  entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Alice and Bob start with a pre-shared maximally entangled state (shown in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Alice locally prepares another maximally  entangled state (shown in purple) and sends half of each of the entangled states through Rc  d to Bob as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content='  Hence, only the bottom two  dangling wires are in Alice’s possession while the top four are with Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Since Bob knows which local random unitaries U, V are applied during  the transmission, he can use the pre-shared entanglement with Alice to undo the phase coupling operation as described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' Here, P Γ1 is the partial  transpose of P with respect to the first subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
+page_content=' The two parties finally end up sharing one maximally entangled state (shown in orange), which  can be used to send quantum information at rate log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQfjg0O/content/2301.05142v1.pdf'}
diff --git a/OdFPT4oBgHgl3EQfmzXU/content/tmp_files/2301.13127v1.pdf.txt b/OdFPT4oBgHgl3EQfmzXU/content/tmp_files/2301.13127v1.pdf.txt
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+arXiv:2301.13127v1  [hep-ph]  30 Jan 2023
+A modern approach to HZZ vertex and direct bounds on its anomalous couplings
+from LHC data
+A.I. Hern´andez-Ju´arez, G. Tavares-Velasco and A. Fern´andez-T´ellez
+Facultad de Ciencias F´ısico Matem´aticas, Benem´erita Universidad
+Aut´onoma de Puebla, Apartado Postal 1152, Puebla, Pue., M´exico.
+(Dated: January 31, 2023)
+We present an evaluation of the standard model one-loop contributions to the H∗Z∗Z∗ coupling,
+where all the three particles are taken off-shell. Our results are presented in terms of Passarino-
+Veltman scalar functions.
+We then use the most current CMS results to obtain bounds on the
+off-shell H∗ZZ coupling, which at the one-loop level can be parametrized in terms of two CP
+conserving and one CP violating form factors. The limits are studied in three different context,
+which are usually used to analyze the HZZ coupling. In general, such limits are of the order of
+10−2 − 10−4 for different energy intervals. The Standard Model (SM) one-loop contributions for the
+H∗ZZ case are revisited, whereas the HZZ∗ case is calculated for the first time. The imaginary
+part of these couplings is also studied for the first time.
+I.
+INTRODUCTION
+The observation of the Higgs boson at the CERN LHC [1, 2] was a clear evidence that the mechanism of electroweak
+gauge symmetry breaking is realized in nature as conjectured by the standard model (SM) of elementary particles.
+Up to now, the data collected at the LHC has confirmed that the properties of the Higgs particle are consistent with
+the SM predictions, though some of its couplings are yet to be measured, such as those to light fermions and its self
+couplings as well. It is expected that the LHC run 3 can explore hints of some anomalous Higgs couplings. Along
+these lines, the CMS collaboration has reported for the first time data on the off-shell H∗ZZ coupling via off-shell
+Higgs boson production (O-SHBP) pp → H∗ → ZZ [3]. To produce a pair of on-shell Z gauge bosons, it is required
+that the four-momentum of the off-shell Higgs boson is above the threshold ∥q∥ = 2mZ. According to the SM, 10%
+of events of V pair production at the LHC are due to the H∗V V coupling [4], which is statistically large enough to
+allow its measurement. Moreover, it has been found that the V V invariant mass kinematic distribution is sensitive to
+the off-shell H∗ZZ contribution, whereas the ratio of off-shell to on-shell production rates can be used to determine
+the Higgs decay width ΓH [5, 6].
+The phenomenological and experimental implications of the H∗ZZ coupling at the LCH and future colliders have
+been explored in the past and also very recently [7–15]. Furthermore, the off-shell HZZ∗ coupling has also several
+phenomenological implications, which have been studied through HZ production [16–34]. Thus, the HZZ coupling
+is worth studying, thereby requiring a highly precise determination of all its lowest order contributions. In particular,
+the search for any anomalous contribution to the H∗ZZ coupling will play an important role at particle colliders in
+the near future.
+Off-shell couplings have been of great interest in recent years [35–37] as they can develop an imaginary (absorptive)
+part due to the optical theorem, thereby giving rise to some effects on physical processes. Such a property has been
+studied for instance in the off-shell chromomagnetic and chromoeletric dipole moments of quarks [38, 39], where the
+gluon is off-shell, and also in the trilinear neutral gauge boson couplings [40, 41], which are non-vanishing when at
+least one of the three gauge bosons is off-shell [42, 43]. Also, the phenomenological implications of the absorptive part
+of the off-shell Higgs boson couplings have been brought to attention by many authors [18, 20, 22, 28], nonetheless, to
+our knowledge, a precise determination of the SM contribution has not been reported yet, which may stem from the
+fact that there has been some controversy on whether or not off-shell couplings represent valid observable quantities.
+We will dwell on this issue below. In particular, the study of the absorptive part of the H∗ZZ and HZZ∗ couplings
+may be relevant at present and future colliders as it can explain slight deviations on some physical observables. This
+is the case of the absorptive part of the ttg coupling, which has direct effects on top pair production at the LHC [37].
+The one-loop corrections to the HZZ coupling were calculated long ago in the SM [44, 45] and also recently [46].
+On the other hand, the new physics contributions have also been reported in several beyond the SM theories, such as
+the two-Higgs doublet model [47], the minimal Higgs triplet model (HTM) [48], the Higgs singlet model (HSM) [47]
+and the minimal supersymmetric standard model (MSSM) [36]. Nevertheless, some of those results are reported in a
+somewhat old-fashioned notation, which can lead to some confusion when a numerical calculation is worked out and a
+cross-check is required. Therefore, a new evaluation of the SM one-loop contributions to the off-shell H∗ZZ coupling
+with a more up-to-date notation for the analytical results is in order. The purpose of this work is to present such an
+evaluation, which could be suited to numerical calculations in view of the O-SHBP results.
+The rest of this work is organized as follows. In Section II we present the most general effective Lagrangian for the
+
+2
+HZZ coupling up to dimension-five operators, along with the most popular parametrizations used to study the effects
+of this coupling at particle colliders. Sec. III is devoted to describe the calculation of the SM one-loop contributions
+to the off-shell H∗ZZ and HZZ∗ couplings via the background field method, which in the Feynman-t’Hooft gauge
+yields identical results to those obtained via the Pinch technique. This method is known to give gauge-independent
+and finite off-shell Green’s functions. All the lengthy results are given in Appendix A in terms of Passarino-Veltman
+scalar functions. In Section IV we present the numerical analysis of the behavior of the H∗ZZ and HZZ∗ couplings
+as functions of the off-shell boson transfer momentum and we also obtain bounds on the anomalous terms of such
+couplings. Finally, in Section VI we present the conclusions and outlook.
+II.
+THEORETICAL FRAMEWORK
+Several parametrizations have been introduced in the literature to study the phenomenology of the HZZ coupling.
+It is thus worth presenting a review of the more popular of such parametrizations with the aim that our analysis can
+be straightforwardly compared with other studies. The effective Lagrangian up to dimension-five operators for the
+HZZ interaction can be written in the so-called Hagiwara basis [17, 18, 23] as follows
+L = g
+cW
+mZ
+�(1 − aZ)
+2
+HZµZµ +
+1
+2m2
+Z
+�
+bZHZµνZµν + cZ
+��
+∂µH
+�
+Zν −
+�
+∂νH
+�
+Zµ
+�
+Zµν + ˜bZHZµν ˜Zµν��
+,
+(1)
+where Zµν = ∂µZν − ∂νZµ and ˜Zµν = ǫµναβZαβ/2. In the SM aZ vanishes at the tree level, the CP conserving form
+factors bZ and cZ are induced up to one-loop level [45], and the CP violating form factor �bZ would arise up to the
+three-loop level [7]. Therefore, the bZ, cZ and �bZ form factors are absent at the tree-level in the SM but can receive
+anomalous contributions from new physics at the one-loop-level or higher orders of perturbation theory. In Fig. 1,
+we introduce the notation used throughout the rest of this work for the vertex function ΓZZH
+µν
+.
+Zµ(p1)
+Zν(p2)
+H(q)
+= i g
+cW mZΓZZH
+µν
+(p2
+1, p2
+2, q2)
+FIG. 1. Nomenclature for the HZZ coupling and the ΓZZH
+µν
+vertex function.
+We are interested in the coupling with an off-shell Higgs boson H∗ZZ. Nevertheless, for completeness we will also
+study the HZZ∗ coupling, which has been largely studied as it has several implications at particle colliders [16–33].
+Therefore, from Lagrangian (1) neglecting the scalar component of the Z bosons (we use ∂µZµ = 0) and considering
+the kinematics H∗ → ZZ(Z∗ → HZ) along with the notation of Fig. 1 we obtain the vertex function when both the
+Higgs boson and one Z gauge boson are off-shell:
+ΓZZH
+µν
+= hV
+1 (q2, p2
+1, m2
+Z)gµν + hV
+2 (q2, p2
+1, m2
+Z)
+m2
+Z
+p1νp2µ + hV
+3 (q2, p2
+1, m2
+Z)
+m2
+Z
+ǫµναβpα
+1 pβ
+2,
+(2)
+where V stands for the off-shell boson and the dependence of the hV
+i form factors have been written explicitly. The
+relation between the form factors hi and the parameters of Lagrangian (1) for the H∗ZZ (HZZ∗) coupling are
+hV
+1 (q2, p2
+1, m2
+Z) = 1 + aZ − bZ
+q2 − p2
+1 − p2
+2
+m2
+Z
++ cZ
+q2
+m2
+Z
+,
+(3)
+hV
+2 (q2, p2
+1, m2
+Z) = ±2
+�
+bZ − cZ
+�
+,
+(4)
+hV
+3 (q2, p2
+1, m2
+Z) = ±2�bZ.
+(5)
+For V = H (V = Z) the Z (H) boson is on-shell, and we have p2
+1 = m2
+Z (q2 = m2
+H). The structure of Eq. (2) is
+the same for three, two or one off-shell bosons provided that one assumes that the Z gauge bosons are coupled to
+
+3
+conserved currents. It is worth noticing that the basis used in the Lagrangian (1) is not unique. From the motion
+equations one has
+HZµ∂νZµν = 1
+2
+��
+∂µH
+�
+Zν −
+�
+∂νH
+�
+Zµ
+�
+Zµν + 1
+2ZµνZµν,
+(6)
+where a surface term has been dropped. Thus the HZZ coupling can be alternatively described as follows
+L = g
+cW
+mZ
+�(1 − aZ)
+2
+HZµZµ +
+1
+2m2
+Z
+�
+ˆbZHZµνZµν + ˆcZHZµ∂νZµν + ˜bZHZµν ˜Zµν��
+,
+(7)
+where the ˆbZ and ˆcZ form factors are given as ˆbZ = bZ − cZ and ˆcZ = 2cZ. In this notation, the form factors hV
+i of
+Eq. (2) are now given by
+h1(q2, p2
+1, m2
+Z) = 1 + aZ − ˆbZ
+q2 − p2
+1 − p2
+2
+m2
+Z
++ ˆcZ
+2
+p2
+1 + p2
+2
+m2
+Z
+,
+(8)
+h2(q2, p2
+1, m2
+Z) = ±2ˆbZ,
+(9)
+h3(q2, p2
+1, m2
+Z) = ±2�bZ.
+(10)
+The main difference between the basis of Lagrangians (1) and (7) is that in the latter the form factor hV
+2 is given in
+terms of only one parameter. Although one can find analytic expressions for the form factors hV
+i , it is not possible to
+identify the contribution from each form factors bZ and cZ. Both notations are used without any distinction in the
+literature, which may be confusing for a cross-check of the numerical results. Thus, to avoid a misleading analysis
+of the HZZ coupling we consider both bases. The notation of Lagrangian (7) is more suited for the purpose of this
+work as we can calculate the form factor h2 and then extract the exact SM contribution to the ˆbZ coefficient.
+Direct bounds on the anomalous couplings have been obtained using polarization observables of the Z gauge boson
+produced in the Z∗ → HZ decay at the LHC at √s = 14 TeV [28]. According to the notation of Lagrangian (7) such
+bounds read
+��Re
+�ˆbZ
+��� ⩽ 3.5 × 10−4,
+��Im
+�ˆbZ
+��� ⩽ 7.94 × 10−3,
+(11)
+��Re
+��bZ
+��� ⩽ 4.76 × 10−3,
+��Im
+��bZ
+��� ⩽ 6.64 × 10−3,
+(12)
+Other limits on the anomalous Higgs boson couplings have been obtained from the analysis of several processes at
+e+e− [18, 23], ep [25] and γe colliders [9]. Below we will discuss some parametrizations used by several authors in the
+study of the HZZ coupling and their relationship with the above parametrization.
+A.
+The LHC framework
+In the analysis of the CMS collaboration, the following parametrization is used for the scattering amplitude that
+describes the HZZ interaction [49] (according to the notation of Fig. 1):
+A(H → ZZ) ∼ 1
+v
+�
+aZZ
+1
++ κZZ
+1
+p2
+1 + κZZ
+2
+p2
+2
+�
+ΛZZ
+1
+�2
+�
+m2
+Zǫ∗
+1ǫ∗
+2 + aZZ
+2
+v f ∗(1)
+µν f ∗(2)µν + aZZ
+3
+v f ∗(1)
+µν
+˜f ∗(2)µν,
+(13)
+where we use the notation of Fig. 1, whereas f (i)µν = ǫµ
+i pν
+i − ǫνpµ
+i and ˜f (i)
+µν = ǫµνρσf (i)ρσ are the field and dual field
+strength tensor of the Zi gauge boson with polarization vector ǫi and four-momentum pi. In the SM aZZ
+1
+= 2 at the
+tree-level. We can rewrite the above equation in terms of the hV
+i form factors. In the Hagiwara basis (1) and using
+the kinematics for H → ZZ we find the following relations:
+h1(q2, p2
+1, m2
+Z) = aZZ
+1
+2
++ κZZ
+1
+p2
+1 + κZZ
+2
+p2
+2
+2
+�
+ΛZZ
+1
+�2
++ aZZ
+2
+2
+q2 − p2
+1 − p2
+2
+m2
+Z
+,
+(14)
+h2(q2, p2
+1, m2
+Z) = −aZZ
+2
+,
+(15)
+h3(q2, p2
+1, m2
+Z) = −aZZ
+3
+.
+(16)
+
+4
+Thus, we can identify
+1 + aZ = aZZ
+1
+2 ,
+(17)
+−bZ
+q2 − p2
+1 − p2
+2
+m2
+Z
++ cZ
+q2
+m2
+Z
+= κZZ
+1
+p2
+1 + κZZ
+2
+p2
+2
+2
+�
+ΛZZ
+1
+�2
++ aZZ
+2
+2
+q2 − p2
+1 − p2
+2
+m2
+Z
+,
+(18)
+bZ − cZ = −aZZ
+2
+2 ,
+(19)
+�bZ = −aZZ
+3
+2 ,
+(20)
+which yields the following relationship
+cZ
+p2
+1 + p2
+2
+m2
+Z
+= κZZ
+1
+p2
+1 + κZZ
+2
+p2
+2
+2
+�
+ΛZZ
+1
+�2
+,
+(21)
+which in the case of κZZ
+1
+= κZZ
+2
+and ΛZZ
+1
+≡ mZ gives
+cZ = κZZ
+1
+2 .
+(22)
+On the other hand, in the basis of Lagrangian (7) we can identify
+−ˆbZ
+q2 − p2
+1 − p2
+2
+m2
+Z
++ ˆcZ
+2
+p2
+1 + p2
+2
+m2
+Z
+= κZZ
+1
+p2
+1 + κZZ
+2
+p2
+2
+2
+�
+ΛZZ
+1
+�2
++ aZZ
+2
+2
+q2 − p2
+1 − p2
+2
+m2
+Z
+(23)
+ˆbZ = −aZZ
+2
+2 ,
+(24)
+whereas Eqs. (17) and (20) remain valid. For κZZ
+1
+= κZZ
+2
+and ΛZZ
+1
+≡ mZ, we obtain
+ˆcZ = κZZ
+1
+.
+(25)
+Therefore, the parametrization of the HZZ vertex in (13) is redundant since only one form factor kZZ
+i
+is necessary
+in both basis. In the rest of this paper we will consider the relations (17), (20) and (24).
+Indirect bounds on aZZ
+i
+coefficients have been obtained by the CMS collaboration [3, 50–52] through effective
+fractional cross sections fai, which allows one to minimize the uncertainties. In addition, this approach is independent
+of the coupling convention. The effective cross section ratios are defined as
+f ZZ
+ai
+=
+|aZZ
+i
+|2α(2e2µ)
+ii
+�
+j |aZZ
+j
+|2α(2e2µ)
+jj
+sign
+�aZZ
+i
+aZZ
+1
+�
+,
+(26)
+where the coefficients α(2e2µ)
+ii
+are the cross sections of the processes H → ZZ/Zγ∗/γ∗γ∗ → 2e2µ when aZZ
+i
+= 1.
+The corresponding numerical values, which can be obtained through MonteCarlo simulation and are normalized with
+respect to the coefficient α(2e2µ)
+11
+, are shown in Table I. In the CMS analysis the couplings are considered as real.
+Therefore, the relative phase between the couplings aZZ
+i
+is 0 or π. The current bounds from CMS [3] are shown in
+Table II.
+TABLE I. Anomalous couplings aZZ
+i
+, cross sections ratios f ZZ
+ai
+and coefficients αii/α11 considered in this work. We use the
+relationship aZZ
+i
+= aW W
+i
+and the value Λ1 = mZ for the case of the κZZ
+1
+coupling. The negative sign arises from the convention
+in Eq. (26) adopted in [53].
+aZZ
+i
+f ZZ
+ai
+αii/α11
+a3
+fa3
+0.153
+a2
+fa2
+0.361
+-k1
+fΛ1
+1.016
+
+5
+TABLE II. Allowed intervals at the 95% CL for the coupling parameters fai obtained by the CMS collaboration [3], through
+a combined analysis of off-shell and on-shell events, where two scenarios are considered: ΓH = ΓSM
+H =4.1 GeV, or ΓH left
+unconstrained. The sign of the relative phase between ai and a1 is absorbed into the definition of fai. [52].
+Parameter in
+units ×10−5
+Scenario
+Observed
+at 95% CL
+fa2
+ΓH = ΓSM
+H
+�
+− 32,514
+�
+ΓH unconstrained
+�
+− 38,503
+�
+fa3
+ΓH = ΓSM
+H
+�
+− 46,107
+�
+ΓH unconstrained
+�
+− 46,110
+�
+fΛ1
+ΓH = ΓSM
+H
+�
+− 11,46
+�
+ΓH unconstrained
+�
+− 10,47
+�
+The coupling fractions are also useful to study the limits on the anomalous couplings. They can be obtained from
+the cross sections ratios in Eq (26) as
+aZZ
+i
+aZZ
+j
+=
+�
+�
+�
+�|f ZZ
+ai |α2e2µ
+jj
+|f ZZ
+aj |α2e2µ
+ii
+sign
+�
+f ZZ
+ai f ZZ
+aj
+�
+.
+(27)
+B.
+The standard model effective field theory framework
+A recent approach well-suited for the analysis of anomalous couplings is offered by the Standard Model effective
+field theory (SMEFT). The corresponding Lagrangian up to dimension six includes 2499 operators Oi [54]
+LSMEFT = LSM +
+2499
+�
+i
+CiOi,
+(28)
+where the Wilson coefficients Ci along with the SM parameters make up the the parameter space of the SMEFT,
+whereas the operators Oi can be described via the Warsaw basis [55], the SILH basis [56] or the Higgs boson basis
+[57]. These bases are equivalent and their Wilson coefficients can be mapped onto each other. The Higgs basis is well
+suited to study the Higgs boson interactions at the LHC, though this is not always true: for instance, the parameter
+space of diboson production at the LHC is larger in the Higgs boson basis than in other bases. In the Higgs boson
+basis [12], the Lagrangian of the HZZ interaction can be written, after a redefinition of the couplings, as follows
+L = H
+υ
+��
+1 + δcz
+��
+g2
+L + g2
+Y
+�
+v2
+4
+ZµZµ + czz
+g2
+L + g2
+Y
+4
+ZµνZµν + cz□g2
+LZµ∂νZµν + ˜czz
+g2
+L + g2
+Y
+4
+Zµν ˜Zµν�
+,
+(29)
+where υ is the vacuum expectation value (VEV) of the Higgs field and gL, gY stand for the SU(2)L × UY (1) coupling
+constants, which in a more usual notation read gL = g and gY = g′. In this Lagrangian the Higgs boson couplings
+δcz, czz, cz□, and ˜czz are assumed to be real [12]. The Lagrangian (29) can be straightforwardly written in a more
+familiar form
+L =
+g
+2cW mZ
+H
+��
+1 + δcz
+�
+m2
+ZZµZµ + czz
+g2
+L + g2
+Y
+4
+ZµνZµν + cz□g2
+LZµ∂νZµν + ˜czz
+g2
+L + g2
+Y
+4
+Zµν ˜Zµν�
+,
+(30)
+which allows one to identify the following relations with the form factors of Lagrangian (7)
+δcz = aZ
+(31)
+czz =
+4
+g2
+L + g2
+Y
+ˆbZ,
+(32)
+cz□ = 1
+g2
+L
+ˆcZ,
+(33)
+˜czz =
+4
+g2
+L + g2
+Y
+˜bZ,
+(34)
+which agree with those relations reported in Ref. [52].
+
+6
+III.
+ANALYTICAL RESULTS
+We now turn to present our analytical results for the one-loop contributions to the HZZ coupling. Since we are
+considering that either the Z gauge boson or the Higgs boson H are off-shell, we need to address the problem of
+the gauge dependence of off-shell Green’s functions. In order to deal with this issue and obtain well-behaved vertex
+functions, there are two well-known approaches. The first one is the so-called pinch technique (PT) [58], which is a
+diagrammatic approach that allows one to systematically obtain gauge-independent and well-behaved off-shell Green’s
+functions by combining self-energy, vertex and box diagrams contributing to a physical process, which may turn into
+a very cumbersome task. Nevertheless, it was shown that the results obtained up to one-loop level via the PT are
+equivalent to those obtained by the application of the background field method (BGFM) when the Feynman-t’Hooft
+gauge is used, namely, when one sets the gauge-fixing parameter ξQ = 1 [59, 60]. In this work, we use the latter
+method to obtain gauge independent form factors since the calculation is easier to perform than in the PT method.
+For the analytical calculation we used the Mathematica package FeynArts [61], which allows one to obtain the
+complete set of Feynman diagrams and their corresponding invariant amplitudes via the background field method.
+We then used FeynCalc [62–64] to manipulate the amplitudes an obtain expressions in terms of Passarino-Veltman
+scalar functions, which can be numerically evaluated with the aid of the LoopTools [65] and Collier [66] packages.
+The contributing Feynman diagrams, which can be classified into three types, are shown in Figs. 2, 3, and 4. The
+Feynman diagram for the fermion loop contribution is shown Fig. 2, whereas the remaining contributions are those
+arising from Feynman diagrams with W gauge boson exchange (Fig. 3) and H − Z (HZ) boson exchange.
+Zµ(p1)
+Zν(p2)
+H(q)
+f
+f
+f
+FIG. 2. Feynman diagram for the one-loop fermion contribution (F) to the HZZ coupling.
+
+7
+Zµ(p1)
+Zν(p2)
+H(q)
+G±
+G±
+W ±
+(a)
+Zµ(p1)
+Zν(p2)
+H(q)
+W ±
+G±
+G±
+(b)
+Zµ(p1)
+Zν(p2)
+H(q)
+G±
+W ±
+G±
+(c)
+Zµ(p1)
+Zν(p2)
+H(q)
+W ±
+W ±
+G±
+(d)
+Zµ(p1)
+Zν(p2)
+H(q)
+W ±
+W ±
+G±
+(e)
+Zµ(p1)
+Zν(p2)
+H(q)
+G±
+W ±
+W ±
+(f)
+Zµ(p1)
+Zν(p2)
+H(q)
+W ±
+W ±
+W ±
+(g)
+Zµ(p1)
+Zν(p2)
+H(q)
+G±
+(h)
+Zµ(p1)
+Zν(p2)
+H(q)
+u−, u+
+(i)
+Zµ(p1)
+Zν(p2)
+H(q)
+G±
+(j)
+Zµ(p1)
+Zν(p2)
+H(q)
+W ±
+(k)
+Zµ(p1)
+Zν(p2)
+H(q)
+W ±
+G±
+(l)
+Zµ(p1)
+Zν(p2)
+H(q)
+W ±
+G±
+(m)
+FIG. 3. The same as in Fig. 2 but for the contribution of W gauge boson exchange (W).
+
+8
+Zµ(p1)
+Zν(p2)
+H(q)
+H
+H
+Z
+(a)
+Zµ(p1)
+Zν(p2)
+H(q)
+Z
+G0
+H
+(b)
+Zµ(p1)
+Zν(p2)
+H(q)
+G0
+Z
+H
+(c)
+Zµ(p1)
+Zν(p2)
+H(q)
+Z
+Z
+H
+(d)
+Zµ(p1)
+Zν(p2)
+H(q)
+G0
+G0
+H
+(e)
+Zµ(p1)
+Zν(p2)
+H(q)
+H
+H
+G0
+(f)
+Zµ(p1)
+Zν(p2)
+H(q)
+G0, H
+(g)
+Zµ(p1)
+Zν(p2)
+H(q)
+Z
+H
+(h)
+Zµ(p1)
+Zν(p2)
+H(q)
+Z
+H
+(i)
+FIG. 4. The same as in Fig. 2 but for the contribution of H − Z bosone exchange (HZ).
+Since the SM contributions to the anomalous couplings are the main aim of this work we will focus on form factor
+hV
+2 . Therefore, following the notation of Ref. [45] our results can be written as
+hV
+2 (q2, p2
+1, m2
+Z) = mZ
+g2
+4π2cW mW
+� �
+f
+Nfm2
+f
+�
+g2
+V fAV
+V f(q2, p2
+1, m2
+Z) + g2
+AfAV
+Af(q2, p2
+1, m2
+Z)
+�
+(35)
++ AV
+W (q2, p2
+1, m2
+Z) + AV
+ZH(q2, p2
+1, m2
+Z)
+�
+,
+where AV
+V f,Af, AV
+W and AV
+ZH stand for the fermion, W gauge boson and H − Z boson contributions. Here gV f,Af
+stand for the fermion couplings to fermion pairs:
+gV f = If
+2 − Qfs2
+W ,
+gAf = If
+2 ,
+(36)
+with If and Qf the fermion weak isospin and electric charge.
+Analytic expressions for the AV
+V f,Af, AV
+W and AV
+ZH functions in terms of Passarino-Veltman scalar functions are
+presented in Appendix A. We present explicit results for the contributions to the H∗ZZ (p2
+1 = m2
+Z) and HZZ∗
+(q2 = m2
+H) couplings. We note that for the H∗ZZ coupling, our results agree with those reported in Ref. [45], though
+there seems to be a difference of sign in the factors of all three-point scalar function C0. This stems from the fact
+that there is an additional minus sign in the definition of such functions in [45] as compared to the usual definition
+presented in Appendix A. We would like to emphasize that, to our knowledge, the results for the HZZ∗ (q2 = m2
+H)
+coupling have never been reported in the literature. Thus, we present a more comprehensive calculation, which could
+allow to asses the anomalous contributions to HZZ coupling in distinct scenarios. The Mathematica code for our
+analytical results is available for the interested reader [67]. We also include master formulas for the general case where
+the three particles are off-shell, which are too cumbersome to be presented in this work. Our expressions reported in
+Appendix A can be straightforwardly obtained from such master formulas.
+It is interesting to note that the HZZ form factors are gauge-dependent as expected, except for the fermion
+contribution, which evidently does not depend on the gauge-fixing parameter.
+On the other hand, all the three
+contributions are always free of ultraviolet divergences.
+
+9
+IV.
+NUMERICAL ANALYSIS
+We now turn to present the numerical analysis. For the evaluation of the Passarino-Veltman scalar functions we
+used the LoopTools package [65]. We first present the evaluation of the H∗ZZ and HZZ∗ form factors in an energy
+interval that allows for on-shell final bosons.
+A.
+H∗ZZ form factor
+We show in Figs. 5(a) and 5(b) the real and imaginary parts of the fermion (F), W gauge boson (W), H − Z
+bosons (HZ), and total contributions to the H∗ZZ form factor as functions of the Higgs boson transfer momentum
+∥q∥. As for the real part of hH
+2 , the total contribution is dominated by the W contribution, whereas the F and HZ
+contributions only are relevant at low energies, where they reach values of a comparable magnitude to those of the W
+contribution. From ∥q∥ = 300 GeV onwards, the F and HZ contributions have a magnitude of similar size but they
+are of opposite sign and tend to cancel each other out. It is also interesting to note that the fermion contribution is
+mainly dominated by the top quark and all other fermions give negligible contributions. As far as the imaginary part
+of the hH
+2 form factor is concerned, we observe that the W contribution is also the dominant one, whereas the F and
+HZ contributions are negligible as they are one order of magnitude smaller than the W contribution. As expected,
+the absorptive part of the F contribution is non-vanishing only above the threshold ∥q∥ = 2mt, where the two top
+quarks attached to the off-shell Higgs boson can be real. In general, the real and imaginary parts of hH
+2 are of similar
+size and their magnitudes are of the order of 10−2 − 10−3, nevertheless, at high energies the absorptive part can be
+larger than the real one. This behavior has also been observed in other off-shell coupling form factors [38–42].
+200
+400
+600
+800
+1000
+1200
+1400
+||q||
+−0.0020
+−0.0015
+−0.0010
+−0.0005
+0.0000
+0.0005
+0.0010
+0.0015
+Re[h H
+2 ]
+ZZH ∗
+F
+W
+HZ
+Total
+(a)
+200
+400
+600
+800
+1000
+1200
+1400
+||q||
+−0.005
+−0.004
+−0.003
+−0.002
+−0.001
+0.000
+0.001
+Im[h H
+2 ]
+ZZH ∗
+F
+W
+HZ
+Total
+(b)
+FIG. 5. One loop fermion (F), W gauge boson (W), HZ bosons (HZ) and total contributions to the real (left plot) and
+absorptive (right plot) parts of the form factor hH
+2 as functions of the Higgs boson transfer momentum ∥q∥.
+For illustration purposes, the values of the real and imaginary parts of the hH
+2 form factors at a few ∥q∥ values are
+presented in Table III, along with the values for ˆbZ, aZZ
+2
+and czz, which can be obtained from Eqs. (9), (15) and (32),
+respectively. In effective field theories the couplings are taken as constant and do not depend on ∥q∥, but our results
+can be useful to constraint the energy scale Λ of the model.
+
+10
+TABLE III. Total contributions to the hH
+2 form factor for a few values of ∥q∥. The respective values of ˆbZ, aZZ
+2
+and czz are
+also shown. All these results are in units of 10−3.
+��q
+��
+hH
+2
+ˆbZ
+aZZ
+2
+czz
+190
+−12.99 − 14.02 i
+−6.49 − 7.01 i
+12.99 + 14.02 i
+−48.16 − 51.98 i
+220
+−6.82 − 13.86 i
+−3.42 − 6.93 i
+6.82 + 13.86 i
+−25.3 − 51.4 i
+350
+0.09 − 7.77 i
+0.04 − 3.88 i
+−0.09 + 7.77 i
+0.35 − 28.8 i
+450
+1.02 − 5.24 i
+0.51 − 2.62 i
+−1.02 + 5.24 i
+3.81 − 19.43 i
+600
+1.11 − 3.31 i
+0.55 − 1.65 i
+−1.11 + 3.31 i
+4.12 − 12.29 i
+1000
+0.78 − 1.5 i
+0.39 − 0.75 i
+−0.78 + 1.5 i
+2.9 − 5.56 i
+1500
+0.52 − 0.79 i
+0.26 − 0.39 i
+−0.52 + 0.79 i
+1.92 − 2.95 i
+B.
+HZZ∗ form factor
+We now show in Fig. 6 the behavior of the real and imaginary parts of the partial and total contributions to the
+HZZ∗ form factor as functions of the off-shell Z gauge boson transfer momentum ∥p1∥. We observe that the hZ
+2 form
+factor has a similar behavior to that of hH
+2 . In both cases the W contribution dominates, but at low energies the F
+and HZ contributions are of similar magnitude than the W contribution, whereas at high energies they are negligible.
+However, there is a slight difference with the behavior of hH
+2 , since both the real part of hZ
+2 reach its larger value at
+a different energy value than in the hH
+2 case.
+400
+600
+800
+1000
+1200
+1400
+||p1 ||
+−0.0020
+−0.0015
+−0.0010
+−0.0005
+0.0000
+0.0005
+0.0010
+0.0015
+Re[h Z
+2 ]
+HZZ ∗
+F
+W
+HZ
+Total
+(a)
+400
+600
+800
+1000
+1200
+1400
+||p1 ||
+−0.005
+−0.004
+−0.003
+−0.002
+−0.001
+0.000
+0.001
+Im[h Z
+2 ]
+HZZ ∗
+F
+W
+HZ
+Total
+(b)
+FIG. 6. The same as in Fig. 5, but for the form factor hZ
+2 as a function of off-shell Z gauge boson transfer momentum ∥p1∥.
+In Table IV, we present numerical values for the real and absorptive parts of hZ
+2 at a few values of ∥p1∥. We also
+present the respective values for ˆbZ, aZZ
+2
+and czz. We can observe that at some energy values, the hZ
+2 form factor is
+larger than the hH
+2 one, reaching values of the order of 10−2 − 10−3. We note that in the case of both H∗ZZ and
+HZZ∗ couplings, the real and absorptive parts of the anomalous coupling ˆbZ are of the order of 10−3 − 10−4, which
+agrees with [28]. In the experimental side, the current bounds on ˆbZ are of the same order of magnitude, thus this
+anomalous coupling could be at the reach of measurement at the LHC in a near future.
+
+11
+TABLE IV. The same as in Table III, but for the form factor hZ
+2 for as a function of ∥p1∥.
+��p1
+��
+hZ
+2
+ˆbZ
+aZZ
+2
+czz
+220
+−4.84 − 15.16 i
+2.42 + 7.58 i
+4.84 + 15.16 i
+17.93 + 56.19 i
+350
+1.07 − 7.37 i
+−0.53 + 3.68 i
+−1.07 + 7.37 i
+−3.98 + 27.33 i
+450
+1.21 − 5.11 i
+−0.6 + 2.55 i
+−1.21 + 5.11 i
+−4.49 + 18.95 i
+600
+1.27 − 3.35 i
+−0.63 + 1.67 i
+−1.27 + 3.35 i
+−4.72 + 12.41 i
+1000
+0.92 − 1.46 i
+−0.46 + 0.73 i
+−0.92 + 1.46 i
+−3.43 + 5.43 i
+1500
+0.59 − 0.73 i
+−0.29 + 0.36 i
+−0.59 + 0.73 i
+−2.22 + 2.71 i
+V.
+BOUNDS ON THE ANOMALOUS COUPLINGS
+Limits on the real and absorptive parts of the anomalous couplings of the HZZ coupling have been set in the past
+[28], nonetheless, the energy dependence has not been considered. Also, the current CMS bounds [3] only consider
+constraints on the ratios of such couplings, which cannot be used to assess the corresponding contributions to physical
+observables [28]. Thus, individual constraints on each one of the couplings are necessary. With this aim, we proceed
+as follows. Firstly, by means of Eq. (27) and the values of Table I we use the constraints from Table II to obtain
+limits on the ratios aZZ
+i
+/aZZ
+2
+, which are presented in Table V, where we only consider the scenario with ΓH = ΓSM
+H
+as the bounds in the unconstrained scenario yield limits of similar size.
+TABLE V. Allowed intervals at 95 % CL for the ratios defined in Eq. (27), where we consider the case ΓH = ΓSM
+H
+and the
+limits of Table II.
+Ratio
+Allowed values
+aZZ
+3
+/aZZ
+2
+�
+− 1.84, 0.70
+�
+κZZ
+1
+/aZZ
+2
+�
+− 0.35, 0.18
+�
+Secondly, we use the constraints on aZZ
+i
+/aZZ
+2
+of Table V and use Eqs. (9) and (24), along with the numerical results
+from Sec. IV, to find the allowed areas of the real parts of aZZ
+3
+and κZZ
+1
+as functions of the Higgs boson transfer
+momentum. Also, since the results of Table III indicate that the real and absorptive parts of the H∗ZZ form factor
+are of similar size, we assume that the limits of Table V are also valid for the imaginary parts of aZZ
+3
+and κZZ
+1
+, which
+allows us to set constraints on Im
+�
+aZZ
+3
+�
+and Im
+�
+κZZ
+1
+�
+. For our calculations we use the numerical values of the form
+factor hH
+2 , but the same results are expected for hZ
+2 . Moreover, only energy regions where both Z bosons can be
+on-shell are considered in our analysis.
+We thus show in Figs. 7(a) and 7(b) the allowed areas on the planes ∥q∥ vs Re
+�
+aZZ
+3
+�
+and ∥q∥ vs Im
+�
+aZZ
+3
+�
+. We
+observe that for small values of ∥q∥, Re
+�
+aZZ
+3
+�
+is allowed to have values in the [−10−2, 10−2] interval, nevertheless,
+the length of such an interval shrinks considerably as the energy increases. Thus, at high energies, Re
+�
+aZZ
+3
+�
+is only
+allowed to have values of the order of 10−4. It is also worth noting that around ∥q∥ = 85 GeV, the allowed Re
+�
+aZZ
+3
+�
+area vanishes, which stems from the fact that in such a region the real part of aZZ
+2
+changes sign and Re
+�
+aZZ
+2
+�
+≈ 0
+(see Fig. 5(a)). Therefore, very small values of Re
+�
+aZZ
+3
+�
+are required to get aZZ
+3
+/aZZ
+2
+inside the allowed region. As
+for Im
+�
+aZZ
+3
+�
+, we observe that for small energies the allowed area is large but becomes smaller as the energy increases.
+However, in this case there is no abrupt vanishing of the allowed area as occurs for the real part around ∥q∥ ∼ 85
+GeV: it turns out that the absorptive part of aZZ
+2
+does not flips sign in the considered energy range.
+
+12
+(a)
+(b)
+FIG. 7. Allowed area at 95% CL for the the real (left plot) and absorptive (right plot) parts of aZZ
+3
+as functions of ∥q∥. These
+results are compatible with the bounds of Table V.
+We now show the allowed areas of the real and absorptive parts of κZZ
+1
+in Figs. 8(a) and 8(b), respectively, as
+functions of ∥q∥. The results are similar to those obtained for aZZ
+3
+. Nevertheless, in this case, the allowed upper
+values of both the real and imaginary parts of aZZ
+3
+are of the order of 10−3 at low energies and become smaller as the
+energy increases. In general, the constraints on the CP violating form factor aZZ
+3
+are less tighter than those on the
+CP conserving one kZZ
+1
+, although both can be of the same order of magnitude in some energy regions.
+(a)
+(b)
+FIG. 8. The same as in Fig. 7, but for the κZZ
+1
+form factor.
+Our constrains on the form factors aZZ
+3
+and κZZ
+1
+, which correspond to the LHC framework, can be translated
+into constraints on the form factors used in other parametrizations via the relations of Sec. II. In this way we can
+straightforwardly obtain the allowed regions for ˜bZ, ˆcZ, ˜czz and cz□, which are used by other authors in their analysis
+of the HZZ coupling. For instance, in the SMEFT the anomalous couplings do not depend on the off-shell boson
+transfer momenta [68] and the energy scale Λ, which is associated with the scale where the effective model remains
+
+13
+valid, has been absorbed in the definition of the Wilson coefficients ˜czz and cz□. Hence, our result may be used to
+set a bound on such energy scale. The corresponding bounds are presented in Tables VI and VII. We observe that
+the real and imaginary parts of the CP violating form factors ˜bZ and ˜czz can be of the order of 10−4 at low energies,
+however, in the SMEFT they can be as large as 10−3 for some values of ∥q∥. The current bounds on ˜bZ are of the
+order of 10−3 [28], therefore our limits are more stringent. On the other hand, the CP conserving from factor ˆcZ
+can be as large as 10−4 at low energies, which is two orders of magnitude smaller than previous results [18]. In the
+SMEFT, cz□ can be in general of the order of 10−3 − 10−4 and decreases at high energies.
+TABLE VI. Allowed intervals of the real and absorptive parts of the CP violating form factor of the HZZ coupling for some
+values of the transfer momentum. We consider three different schemes: The LHC framework (aZZ
+3
+), in a general effective
+Lagrangian approach (˜bZ) and the SMEFT (˜czz).
+��q
+��
+Re
+�
+aZZ
+3
+�
+Re
+�˜bZ
+�
+Re
+�
+˜czz
+�
+Im
+�
+aZZ
+3
+�
+Im
+�˜bZ
+�
+Im
+�
+˜czz
+�
+190
+�
+− 0.024, 0.009
+�
+�
+− 0.0045, 0.012
+�
+�
+− 0.033, 0.088
+�
+�
+− 0.026, 0.01
+�
+�
+− 0.005, 0.013
+�
+�
+− 0.037, 0.096
+�
+285
+�
+− 0.0029, 0.0011
+�
+�
+− 0.00055, 0.0014
+�
+�
+− 0.004, 0.01
+�
+�
+− 0.018, 0.0069
+�
+�
+− 0.0034, 0.009
+�
+�
+− 0.025, 0.066
+�
+400
+�
+− 0.00053, 0.0014
+�
+�
+− 0.0007, 0.00026
+�
+�
+− 0.0051, 0.0019
+�
+�
+− 0.012, 0.0044
+�
+�
+− 0.0022, 0.006
+�
+�
+− 0.016, 0.044
+�
+800
+�
+− 0.00069, 0.0018
+�
+�
+− 0.0009, 0.00034
+�
+�
+− 0.0066, 0.0025
+�
+�
+− 0.0039, 0.0015
+�
+�
+− 0.00075, 0.0019
+�
+�
+− 0.0055, 0.014
+�
+1500
+�
+− 0.00036, 0.00095
+�
+�
+− 0.00047, 0.00018
+�
+�
+− 0.0034, 0.0013
+�
+�
+− 0.0015, 0.00057
+�
+�
+− 0.00028, 0.00075
+�
+�
+− 0.002, 0.0055
+�
+TABLE VII. Allowed intervals for the real and absorptive parts of one of the CP conserving form factors of the HZZ coupling
+for a few values of the transfer momentum. We consider three different schemes: The LHC framework (κZZ
+1
+), in a general
+effective Lagrangian approach (ˆcZ) and the SMEFT (˜cz□). From Eq. (25) we note that κZZ
+1
+= ˆcZ.
+��q
+��
+Re
+�
+kZZ
+1
+� �
+Re
+�
+ˆcZ
+��
+Re
+�
+cz□
+�
+Im
+�
+kZZ
+1
+� �
+Im
+�
+ˆcZ
+��
+Im
+�
+cz□
+�
+190
+�
+− 0.0024, 0.0046
+�
+�
+− 0.0058, 0.011
+�
+�
+− 0.0026, 0.005
+�
+�
+− 0.0063, 0.012
+�
+285
+�
+− 0.00028, 0.00055
+�
+�
+− 0.00068, 0.0013
+�
+�
+− 0.0018, 0.0035
+�
+�
+− 0.0043, 0.0085
+�
+400
+�
+− 0.00027, 0.00014
+�
+�
+− 0.00065, 0.00034
+�
+�
+− 0.0012, 0.0023
+�
+�
+− 0.0029, 0.0055
+�
+800
+�
+− 0.00034, 0.00017
+�
+�
+− 0.00082, 0.00041
+�
+�
+− 0.00038, 0.00075
+�
+�
+− 0.00092, 0.0018
+�
+1500
+�
+− 0.00019, 0.0001
+�
+�
+− 0.00046, 0.00024
+�
+�
+− 0.00015, 0.00029
+�
+�
+− 0.00036, 0.0007
+�
+VI.
+CONCLUSIONS AND OUTLOOK
+Appendix A: Analytical results
+We present the analytical expression for the one-loop contributions to the HZZ form factor hV
+2 of Eq. (35) in terms
+of Passarino-Veltman scalar functions.
+The two- and three-point Passarino-Veltman scalar functions are defined as
+B0(r2, m2
+1, m2
+2) =
+1
+iπ2
+�
+dDk
+(k2 − m2
+1)((k + r)2 − m2
+2),
+(A1)
+C0(r2
+1, (r1 + r2)2, r2
+2, m2
+1, m2
+2, m2
+3) =
+1
+iπ2
+�
+dDk
+(k2 − m2
+1)((k + r1)2 − m2
+2)((k + r2)2 − m2
+3).
+(A2)
+We now introduce the following shorthand notation
+B0ij(r2) = B0
+�
+r2, m2
+i , m2
+j
+�
+,
+C0ijk
+�
+q2�
+= C0
+�
+m2
+Z, m2
+Z, q2, m2
+i , m2
+j, m2
+k
+�
+,
+C0ijk
+�
+p2
+1
+�
+= C0
+�
+m2
+H, m2
+Z, p2
+1, m2
+i , m2
+j, m2
+k
+�
+,
+(A3)
+It is useful to observe the following symmetry relations
+Bij(r2) = Bji(r2),
+Cijk
+�
+q2�
+= Ckji
+�
+q2�
+,
+(A4)
+
+14
+1.
+Contributions to the H∗ZZ coupling
+The contributions from fermion to the AH
+V f,Af functions read
+AH
+V f(q2, m2
+Z) =
+1
+q2 (q2 − 4m2
+Z) 2
+�
+4q2m2
+Z
+�
+B0ff
+�
+q2�
+− B0ff
+�
+m2
+Z
+�
+− 3
+�
++ 8m4
+Z
+�
+B0ff
+�
+q2�
+− B0ff
+�
+m2
+Z
+�
++ 2
+�
+−
+�
+q2 − 2m2
+Z
+� �
+−4m2
+f
+�
+q2 − 4m2
+Z
+�
+− 6q2m2
+Z − 4m4
+Z + q4�
+C0fff
+�
+q2�
++ 2q4�
+,
+(A5)
+AH
+Af(q2, m2
+Z) =
+1
+q2 (q2 − 4m2
+Z) 2
+� �
+q2 − 2m2
+Z
+� �
+4
+�
+q2 − m2
+Z
+� �
+B0ff
+�
+q2�
+− B0ff
+�
+m2
+Z
+� �
++
+�
+−2m2
+Z
+�
+8m2
+f + q2�
++ 4q2m2
+f + 4m4
+Z + q4�
+C0
+�
+q2, m2
+Z, m2
+Z, m2
+f, m2
+f, m2
+f
+�
++ 2
+�
+q2 − 4m2
+Z
+� ��
+. (A6)
+As for the contributions from W gauge boson (W), and H − Z boson (HZ) exchange, they are given by
+AH
+W (q2, m2
+Z) =
+1
+8q2�
+q2 − 4m2
+Z
+�2
+�
+2
+��
+1 − 2c2
+W
+�2m2
+Hm2
+Z
+�
+2m2
+Z + q2�
++ 2m2
+W
+��
+12c4
+W − 4c2
+W − 7
+�
+q2m2
+Z
++
+�
+24c4
+W − 8c2
+W + 2
+�
+m4
+Z + 2q4���
+B0WW
+�
+m2
+Z
+�
+− B0WW
+�
+q2��
+− 2
+��
+1 − 2c2
+W
+�2m2
+H
+�
+2m4
+Z
+�
+q2 − m2
+Z
+�
++ m2
+W
+�
+− 6q2m2
+Z + 8m4
+Z + q4��
++ 2m2
+W
+�
+4c2
+W
+�
+1 − 2c2
+W
+�
+q6 + 2
+�
+32c4
+W − 16c2
+W + 1
+�
+q4m2
+Z
+− 2
+�
+52c4
+W − 28c2
+W + 3
+�
+q2m4
+Z +
+�
+12c4
+W − 4c2
+W + 1
+�
+m2
+W
+�
+− 6q2m2
+Z + 8m4
+Z + q4�
++
+�
+− 24c4
+W + 8c2
+W − 2
+�
+m6
+Z
+��
+C0WWW
+�
+q2�
++
+�
+− 6q2m2
+Z + 8m4
+Z + q4��
+−
+�
+1 − 2c2
+W
+�2m2
+H
+− 2
+�
+12c4
+W − 4c2
+W + 1
+�
+m2
+W
+��
+,
+(A7)
+and
+AH
+ZH(q2, m2
+Z) =
+1
+8q2�
+q2 − 4m2
+Z
+�2
+�
+− 2
+�
+m2
+Z − m2
+H
+��
+q2 − 4m2
+Z
+��
+m2
+ZB0ZZ
+�
+0
+�
+− m2
+HB0HH
+�
+0
+��
++
+�
+6m4
+H
+�
+q2 − m2
+Z
+�
+− 9q2m2
+Hm2
+Z
+�
+B0HH
+�
+q2�
++ 2
+�
+8m4
+Z
+�
+m2
+H − q2�
+− q2m4
+H
++ 2m2
+Z
+�
+2q2m2
+H − m4
+H + q4��
+B0HZ
+�
+m2
+Z
+�
+−
+�
+2q2m4
+H + m2
+Z
+�
+3q2m2
+H − 2m4
+H + 4q4�
+− 18q2m4
+Z + 8m6
+Z
+�
+B0ZZ
+�
+q2�
++ m2
+H
+��
+− 2m4
+H
+�
+q2 − m2
+Z
+�
+− m2
+H
+�
+− 2q2m2
+Z + 4m4
+Z + q4�
+− 6q4m2
+Z + 28q2m4
+Z − 16m6
+Z
+�
+C0ZHZ
+�
+q2�
+− 3
+�
+2m4
+H
+�
+q2 − m2
+Z
+�
++ m2
+H
+�
+8m4
+Z − 8q2m2
+Z
+�
++ q2m2
+Z
+�
+2m2
+Z + q2��
+C0HZH
+�
+q2��
++
+�
+4m2
+Z − q2��
+2m2
+H
+�
+q2 − 4m2
+Z
+�
++ 2m4
+H + q2m2
+Z
+��
+.
+(A8)
+2.
+Contributions to the HZZ∗ coupling
+The corresponding contributions to the HZZ∗ coupling can be written as
+AZ
+V f(p2
+1, m2
+Z, m2
+H) =
+1
+(−2m2
+H (m2
+Z + p2
+1) + m4
+H + (m2
+Z − p2
+1) 2) 2
+��
+2
+�
+m4
+H
+�
+m2
+Z + p2
+1
+�
+− 2m2
+H
+�
+−4p2
+1m2
+Z + m4
+Z + p4
+1
+�
++
+�
+m2
+Z − p2
+1
+� 2 �
+m2
+Z + p2
+1
+� ��
+B0ff
+�
+m2
+H
+�
++
+�
+− 2m2
+Z
+�
+4p2
+1
+�
+m2
+H + m2
+Z
+�
++
+�
+m2
+H − m2
+Z
+� 2 − 5p4
+1
+��
+B0ff
+�
+m2
+Z
+�
++
+�
+− 2p2
+1
+�
+m2
+H
+�
+4m2
+Z − 2p2
+1
+�
++ m4
+H + 4p2
+1m2
+Z
+− 5m4
+Z + p4
+1
+��
+B0ff
+�
+p2
+1
+�
++
+� �
+m2
+H − m2
+Z − p2
+1
+� �
+− p4
+1
+�
+−4m2
+f + 3m2
+H + m2
+Z
+�
+− p2
+1
+�
+8m2
+f
+�
+m2
+H + m2
+Z
+�
+− 10m2
+Hm2
+Z − 3m4
+H + m4
+Z
+�
++
+�
+m2
+H − m2
+Z
+� 2 �
+4m2
+f − m2
+H + m2
+Z
+�
++ p6
+1
+��
+C0fff
+�
+p2
+1
+�
++ 2
+�
+− 3m4
+H
+�
+m2
+Z + p2
+1
+�
++ m2
+H
+�
+2p2
+1m2
+Z + 3m4
+Z + 3p4
+1
+�
++ m6
+H
+−
+�
+m2
+Z − p2
+1
+� 2 �
+m2
+Z + p2
+1
+� ��
+,
+(A9)
+
+15
+AZ
+Af(p2
+1, m2
+Z, m2
+H) =
+1
+(−2m2
+H (m2
+Z + p2
+1) + m4
+H + (m2
+Z − p2
+1) 2) 2
+�� �
+m2
+H − m2
+Z − p2
+1
+� �
+− 2m2
+H
+�
+m2
+Z + p2
+1
+�
++ 4m4
+H
+− 2
+�
+m2
+Z − p2
+1
+� 2��
+B0ff
+�
+m2
+H
+�
++
+�
+− 2
+�
+m2
+H − m2
+Z − p2
+1
+� �
+m2
+H
+�
+m2
+Z − 2p2
+1
+�
++ m4
+H + p2
+1m2
+Z
+− 2m4
+Z + p4
+1
+��
+B0ff
+�
+m2
+Z
+�
++
+�
+− 2
+�
+m2
+H − m2
+Z − p2
+1
+� �
+p2
+1
+�
+m2
+H + m2
+Z
+�
++
+�
+m2
+H − m2
+Z
+� 2
+− 2p4
+1
+��
+B0ff
+�
+p2
+1
+�
++
+� �
+m2
+H − m2
+Z − p2
+1
+� �
+− p4
+1
+�
+−4m2
+f + m2
+H + m2
+Z
+�
+− p2
+1
+�
+8m2
+f
+�
+m2
+H + m2
+Z
+�
+− 6m2
+Hm2
+Z + m4
+H + m4
+Z
+�
++
+�
+m2
+H − m2
+Z
+� 2 �
+4m2
+f + m2
+H + m2
+Z
+�
++ p6
+1
+��
+C0fff
+�
+p2
+1
+�
++ 2
+�
+− 3m4
+H
+�
+m2
+Z + p2
+1
+�
++ m2
+H
+�
+2p2
+1m2
+Z + 3m4
+Z + 3p4
+1
+�
++ m6
+H
+−
+�
+m2
+Z − p2
+1
+� 2 �
+m2
+Z + p2
+1
+� ��
+,
+(A10)
+AZ
+W (p2
+1, m2
+Z, m2
+H) =
+1
+8
+�
+− 2m2
+H
+�
+m2
+Z + p2
+1
+�
++ m4
+H +
+�
+m2
+Z − p2
+1
+�2�2
+��
+− m6
+H
+��
+1 − 2c2
+W
+�2�
+m2
+Z + p2
+1
+�
++ 8m2
+W
+�
+− 2m4
+H
+��
+12c4
+W − 4c2
+W − 7
+�
+m2
+W
+�
+m2
+Z + p2
+1
+�
+−
+�
+1 − 2c2
+W
+�2�
+− 4p2
+1m2
+Z + m4
+Z + p4
+1
+��
+− m2
+H
+�
+4m2
+W
+�
+16p2
+1c2
+W
+�
+3c2
+W − 1
+�
+m2
+Z +
+�
+− 12c4
+W + 4c2
+W + 1
+�
+m4
+Z + p4
+1
+�
+− 12c4
+W + 4c2
+W + 1
+��
++
+�
+1 − 2c2
+W
+�2�
+m2
+Z − p2
+1
+�2�
+m2
+Z + p2
+1
+��
+− 2
+�
+12c4
+W − 4c2
+W + 1
+�
+m2
+W
+�
+m2
+Z − p2
+1
+�2�
+m2
+Z + p2
+1
+��
+× B0WW
+�
+m2
+H
+�
++
+�
+2m4
+H
+�
+m2
+W
+��
+12c4
+W − 4c2
+W − 1
+�
+m2
+Z − 6p2
+1
+�
+−
+�
+1 − 2c2
+W
+�2m2
+Z
+�
+m2
+Z − 2p2
+1
+��
++ m2
+H
+�
+4m2
+W
+�
+8p2
+1c2
+W
+�
+3c2
+W − 1
+�
+m2
+Z − 2
+�
+6c4
+W − 2c2
+W + 1
+�
+m4
+Z + 3p4
+1
+�
++
+�
+1 − 2c2
+W
+�2m2
+Z
+�
+4p2
+1m2
+Z + m4
+Z − 5p4
+1
+��
++ m6
+H
+��
+1 − 2c2
+W
+�2m2
+Z + 4m2
+W
+�
++ 2m2
+W
+�
+m2
+Z − p2
+1
+��
+p2
+1
+�
+60c4
+W − 20c2
+W + 1
+�
+m2
+Z +
+�
+12c4
+W − 4c2
+W + 3
+�
+m4
+Z + 2p4
+1
+��
+B0WW
+�
+m2
+Z
+�
++
+�
+− 2m4
+H
+�
+m2
+W
+�
+p2
+1
+�
+− 12c4
+W + 4c2
+W + 1
+�
++ 6m2
+Z
+�
++ p2
+1
+�
+1 − 2c2
+W
+�2�
+p2
+1 − 2m2
+Z
+��
++ m2
+H
+�
+4m2
+W
+�
+8p2
+1c2
+W
+�
+3c2
+W − 1
+�
+m2
+Z − 2p4
+1
+�
+6c4
+W − 2c2
+W + 1
+�
++ 3m4
+Z
+�
++ p2
+1
+�
+1 − 2c2
+W
+�2�
+4p2
+1m2
+Z − 5m4
+Z + p4
+1
+��
++ m6
+H
+�
+p2
+1
+�
+1 − 2c2
+W
+�2 + 4m2
+W
+�
+− 2m2
+W
+�
+m2
+Z − p2
+1
+��
+p2
+1
+�
+60c4
+W − 20c2
+W + 1
+�
+m2
+Z + p4
+1
+�
+12c4
+W − 4c2
+W + 3
+�
++ 2m4
+Z
+�
+]B0WW
+�
+p2
+1
+�
++ 2
+�
+m6
+H
+�
+−
+�
+52c4
+W − 20c2
+W − 1
+�
+m2
+W
+�
+m2
+Z + p2
+1
+�
+− 2p2
+1
+�
+1 − 2c2
+W
+�2m2
+Z
++
+�
+− 24c4
+W + 8c2
+W − 2
+�
+m4
+W
+�
++ m4
+H
+�
+6
+�
+12c4
+W − 4c2
+W + 1
+�
+m4
+W
+�
+m2
+Z + p2
+1
+�
++ m2
+W
+�
+2p2
+1
+�
+4c4
+W − 4c2
+W − 1
+�
+m2
+Z +
+�
+84c4
+W − 36c2
+W + 3
+�
+m4
+Z + 3p4
+1
+�
+28c4
+W − 12c2
+W + 1
+��
++ p2
+1
+�
+1 − 2c2
+W
+�2m2
+Z
+�
+m2
+Z + p2
+1
+��
++ m2
+H
+�
+− 2
+�
+12c4
+W − 4c2
+W + 1
+�
+m4
+W
+�
+2p2
+1m2
+Z + 3m4
+Z + 3p4
+1
+�
+− m2
+W
+�
+m2
+Z + p2
+1
+��
+− 4p2
+1
+�
+36c4
+W − 16c2
+W + 3
+�
+m2
+Z +
+�
+60c4
+W − 28c2
+W + 5
+�
+m4
+Z
++ p4
+1
+�
+60c4
+W − 28c2
+W + 5
+��
++ p2
+1
+�
+1 − 2c2
+W
+�2m2
+Z
+�
+m2
+Z − p2
+1
+�2�
++
+�
+12c4
+W − 4c2
+W − 1
+�
+m8
+Hm2
+W
++ 2m2
+W
+�
+m2
+Z − p2
+1
+�2�
+− p2
+1
+�
+1 − 2c2
+W
+�2m2
+Z +
+�
+12c4
+W − 4c2
+W + 1
+�
+m2
+W
+�
+m2
+Z + p2
+1
+�
++
+�
+8c4
+W − 4c2
+W + 1
+�
+m4
+Z + p4
+1
+�
+8c4
+W − 4c2
+W + 1
+���
+C0WWW
+�
+p2
+1
+�
+−
+��
+1 − 2c2
+W
+�2m2
+H
++ 2
+�
+12c4
+W − 4c2
+W + 1
+�
+m2
+W
+��
+− 3m4
+H
+�
+m2
+Z + p2
+1
+�
++ m2
+H
+�
+2p2
+1m2
+Z + 3m4
+Z + 3p4
+1
+�
++ m6
+H
+−
+�
+m2
+Z − p2
+1
+�2�
+m2
+Z + p2
+1
+���
+,
+(A11)
+
+16
+and
+AZ
+ZH(q2, m2
+Z) =
+1
+16
+�
+− 2m2
+H
+�
+m2
+Z + p2
+1
+�
++ m4
+H +
+�
+m2
+Z − p2
+1
+�2�2
+�
+− 4
+�
+m2
+H − m2
+Z
+��
+− 2m2
+H
+�
+m2
+Z + p2
+1
+�
++ m4
+H
++
+�
+m2
+Z − p2
+1
+�2��
+m2
+HB0HH
+�
+0
+�
+− m2
+ZB0ZZ
+�
+0
+��
++ 3m2
+H
+�
+− m4
+H
+�
+7m2
+Z + 3p2
+1
+�
++ 2m2
+H
+�
+m4
+Z − p2
+1m2
+Z
+�
++ 4m6
+H +
+�
+m2
+Z − p2
+1
+�3�
+B0HH
+�
+m2
+H
+�
++
+�
+m6
+H
+�
+p2
+1 − 11m2
+Z
+�
++ 4m4
+H
+�
+p2
+1m2
+Z + 7m4
+Z + p4
+1
+�
+− m2
+H
+�
+7p2
+1m4
+Z + p4
+1m2
+Z + 7m6
+Z + p6
+1
+�
+− 4m8
+H − 2
+�
+3m2
+Z + p2
+1
+��
+m3
+Z − p2
+1mZ
+�2�
+B0ZZ
+�
+m2
+H
+�
+− 2
+�
+− 2m6
+H
+�
+m2
+Z + p2
+1
+�
++ m4
+H
+�
+6p2
+1m2
+Z − 7m4
+Z + p4
+1
+�
++ 2m2
+Hm2
+Z
+�
+− 9p2
+1m2
+Z + 8m4
+Z + p4
+1
+�
++ m8
+H
++ 2p6
+1m2
+Z + 6p2
+1m6
+Z − 8m8
+Z
+�
+B0HZ
+�
+m2
+Z
+�
+− 2
+�
+m6
+H
+�
+2p2
+1 − 6m2
+Z
+�
++ m4
+H
+�
+− 11p2
+1m2
+Z + 12m4
+Z − p4
+1
+�
+− 2m2
+H
+�
+− 3p2
+1m4
+Z − 3p4
+1m2
+Z + 5m6
+Z + p6
+1
+�
++ m8
+H + 3m2
+Z
+�
+m2
+Z − p2
+1
+��
+m2
+Z + p2
+1
+�2�
+B0HZ
+�
+p2
+1
+�
+− 6m4
+H
+�
+− 3m4
+H
+�
+2m2
+Z + p2
+1
+�
++ m2
+H
+�
+p2
+1m2
+Z + 6m4
+Z + 3p4
+1
+�
++ 2m6
+H − 2
+�
+m2
+Z − p2
+1
+�2�
+m2
+Z + p2
+1
+��
+× C0HHZ
+�
+p2
+1
+�
+− 2
+�
+m8
+H
+�
+4m2
+Z − 2p2
+1
+�
+− 2m6
+H
+�
+11m4
+Z + p4
+1
+�
++ m4
+H
+�
+5p2
+1m4
+Z − 2p4
+1m2
+Z + 12m6
+Z + p6
+1
+�
++ m2
+H
+�
+11m2
+Z + 5p2
+1
+��
+m3
+Z − p2
+1mZ
+�2 + 3m10
+H − 2
+�
+m3
+Z − p2
+1mZ
+�2�
+p2
+1m2
+Z + 4m4
+Z + p4
+1
+��
+C0ZZH
+�
+p2
+1
+�
+− 2
+�
+− m6
+H
+�
+13m2
+Z + 10p2
+1
+�
++ m4
+H
+�
+5p2
+1m2
+Z + 15m4
+Z + 8p4
+1
+�
++ m2
+H
+�
+8p2
+1m4
+Z + p4
+1m2
+Z − 7m6
+Z − 2p6
+1
+�
++ 4m8
+H + m2
+Z
+�
+m2
+Z − p2
+1
+�3��
+.
+(A12)
+[1] S. Chatrchyan et al. (CMS), Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC,
+Phys. Lett. B 716, 30 (2012), arXiv:1207.7235 [hep-ex].
+[2] G. Aad et al. (ATLAS), Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS
+detector at the LHC, Phys. Lett. B 716, 1 (2012), arXiv:1207.7214 [hep-ex].
+[3] A. Tumasyan et al. (CMS), First evidence for off-shell production of the Higgs boson and measurement of its width, (2022),
+arXiv:2202.06923 [hep-ex].
+[4] N. Kauer and G. Passarino, Inadequacy of zero-width approximation for a light Higgs boson signal, JHEP 08, 116,
+arXiv:1206.4803 [hep-ph].
+[5] F.
+Caola
+and
+K.
+Melnikov,
+Constraining
+the
+Higgs
+boson
+width
+with
+ZZ
+production
+at
+the
+LHC,
+Phys. Rev. D 88, 054024 (2013), arXiv:1307.4935 [hep-ph].
+[6] J. M. Campbell, R. K. Ellis, and C. Williams, Bounding the Higgs Width at the LHC Using Full Analytic Results for
+gg− > e−e+µ−µ+, JHEP 04, 060, arXiv:1311.3589 [hep-ph].
+[7] S. Bolognesi, Y. Gao, A. V. Gritsan, K. Melnikov, M. Schulze, N. V. Tran, and A. Whitbeck, On the spin and parity of a
+single-produced resonance at the LHC, Phys. Rev. D 86, 095031 (2012), arXiv:1208.4018 [hep-ph].
+[8] I.
+Anderson
+et
+al.,
+Constraining
+Anomalous
+HVV
+Interactions
+at
+Proton
+and
+Lepton
+Colliders,
+Phys. Rev. D 89, 035007 (2014), arXiv:1309.4819 [hep-ph].
+[9] B. S¸ahin, Search for the anomalous ZZH couplings at the CLIC, Mod. Phys. Lett. A 34, 1950299 (2019).
+[10] A. V. Gritsan, J. Roskes, U. Sarica, M. Schulze, M. Xiao, and Y. Zhou, New features in the JHU generator frame-
+work:
+constraining Higgs boson properties from on-shell and off-shell production, Phys. Rev. D 102, 056022 (2020),
+arXiv:2002.09888 [hep-ph].
+[11] D. Gon¸calves,
+T. Han, S. Ching Iris Leung, and H. Qin, Off-shell Higgs couplings in H∗
+→
+ZZ
+→
+ℓℓνν,
+Phys. Lett. B 817, 136329 (2021), arXiv:2012.05272 [hep-ph].
+[12] A. Azatov et al., Off-shell Higgs Interpretations Task Force: Models and Effective Field Theories Subgroup Report, (2022),
+arXiv:2203.02418 [hep-ph].
+[13] P. Sharma and A. Shivaji, Probing non-standard HV V (V = W, Z) couplings in single Higgs production at future electron-
+proton collider, (2022), arXiv:2207.03862 [hep-ph].
+[14] J. A. Aguilar-Saavedra, A. Bernal, J. A. Casas, and J. M. Moreno, Testing entanglement and Bell inequalities in H → ZZ,
+(2022), arXiv:2209.13441 [hep-ph].
+[15] A. T. Nguyen, D. T. Tran, and K. H. Phan, One-loop on-shell and off-shell decay H∗ → V V at future e−e− collider, in
+47th Vietnam Conference on Theoretical Physics (2022) arXiv:2209.13153 [hep-ph].
+[16] B. A. Kniehl, Radiative corrections for associated ZH production at future e+e− colliders, Z. Phys. C 55, 605 (1992).
+
+17
+[17] K. Hagiwara and M. L. Stong, Probing the scalar sector in e+ e- —> f anti-f H, Z. Phys. C 62, 99 (1994),
+arXiv:hep-ph/9309248.
+[18] K. Hagiwara, S. Ishihara, J. Kamoshita, and B. A. Kniehl, Prospects of measuring general Higgs couplings at e+ e- linear
+colliders, Eur. Phys. J. C 14, 457 (2000), arXiv:hep-ph/0002043.
+[19] B. A. Kniehl, Theoretical aspects of standard model Higgs boson physics at a future e+ e- linear collider,
+Int. J. Mod. Phys. A 17, 1457 (2002), arXiv:hep-ph/0112023.
+[20] S. S. Biswal, R. M. Godbole, R. K. Singh, and D. Choudhury, Signatures of anomalous VVH interactions at a linear
+collider, Phys. Rev. D 73, 035001 (2006), [Erratum: Phys.Rev.D 74, 039904 (2006)], arXiv:hep-ph/0509070.
+[21] D. Choudhury and Mamta, Anomalous Higgs Couplings at an e gamma Collider, Phys. Rev. D 74, 115019 (2006),
+arXiv:hep-ph/0608293.
+[22] R. M. Godbole, D. J. Miller, and M. M. Muhlleitner, Aspects of CP violation in the H ZZ coupling at the LHC,
+JHEP 12, 031, arXiv:0708.0458 [hep-ph].
+[23] S. Dutta, K. Hagiwara, and Y. Matsumoto, Measuring the Higgs-Vector boson Couplings at Linear e+e− Collider,
+Phys. Rev. D 78, 115016 (2008), arXiv:0808.0477 [hep-ph].
+[24] S. D. Rindani and P. Sharma, Angular distributions as a probe of anomalous ZZH and gammaZH interactions at a linear
+collider with polarized beams, Phys. Rev. D 79, 075007 (2009), arXiv:0901.2821 [hep-ph].
+[25] I.
+T.
+Cakir,
+O.
+Cakir,
+A.
+Senol,
+and
+A.
+T.
+Tasci,
+Probing
+Anomalous
+HZZ
+Couplings
+at
+the
+LHeC,
+Mod. Phys. Lett. A 28, 1350142 (2013), arXiv:1304.3616 [hep-ph].
+[26] K. Rao, S. D. Rindani, and P. Sarmah, Probing anomalous gauge-Higgs couplings using Z boson polarization at e+e−
+colliders, Nucl. Phys. B 950, 114840 (2020), arXiv:1904.06663 [hep-ph].
+[27] S. Kumar, P. Poulose, R. Rahaman, and R. K. Singh, Measuring Higgs self-couplings in the presence of VVH and VVHH
+at the ILC, Int. J. Mod. Phys. A 34, 1950094 (2019), arXiv:1905.06601 [hep-ph].
+[28] K. Rao, S. D. Rindani, and P. Sarmah, Study of anomalous gauge-Higgs couplings using Z boson polarization at LHC,
+Nucl. Phys. B 964, 115317 (2021), arXiv:2009.00980 [hep-ph].
+[29] L. Chen, G. Heinrich, S. P. Jones, M. Kerner, J. Klappert, and J. Schlenk, ZH production in gluon fusion: two-loop
+amplitudes with full top quark mass dependence, JHEP 03, 125, arXiv:2011.12325 [hep-ph].
+[30] W. Bizo´n, F. Caola, K. Melnikov, and R. R¨ontsch, Anomalous couplings in associated VH production with Higgs boson
+decay to massive b quarks at NNLO in QCD, Phys. Rev. D 105, 014023 (2022), arXiv:2106.06328 [hep-ph].
+[31] K. Rao, S. D. Rindani, P. Sarmah, and B. Singh, Polarized Z cross sections in Higgsstrahlung for the determination of
+anomalous ZZH couplings, (2022), arXiv:2202.10215 [hep-ph].
+[32] P. Bittar and G. Burdman, Form Factors in Higgs Couplings from Physics Beyond the Standard Model,
+(2022),
+arXiv:2204.07094 [hep-ph].
+[33] A. V. Gritsan et al., Snowmass White Paper: Prospects of CP-violation measurements with the Higgs boson at future
+experiments, (2022), arXiv:2205.07715 [hep-ex].
+[34] X. Chen, X. Guan, C.-Q. He, Z. Li, X. Liu, and Y.-Q. Ma, Complete two-loop electroweak corrections to e+e− → HZ,
+(2022), arXiv:2209.14953 [hep-ph].
+[35] U. Haisch and G. Koole, Off-shell Higgs production at the LHC as a probe of the trilinear Higgs coupling, JHEP 02, 030,
+arXiv:2111.12589 [hep-ph].
+[36] C. Englert, Y. Soreq, and M. Spannowsky, Off-Shell Higgs Coupling Measurements in BSM scenarios, JHEP 05, 145,
+arXiv:1410.5440 [hep-ph].
+[37] A. I. Hern´andez-Ju´arez, A. Moyotl, and G. Tavares-Velasco, Bounds on the absorptive parts of the chromomagnetic and
+chromoelectric dipole moments of the top quark from LHC data, (2021), arXiv:2109.09978 [hep-ph].
+[38] A. I. Hern´andez-Ju´arez, A. Moyotl, and G. Tavares-Velasco, New estimate of the chromomagnetic dipole moment of quarks
+in the standard model, Eur. Phys. J. Plus 136, 262 (2021), arXiv:2009.11955 [hep-ph].
+[39] A. I. Hern´andez-Ju´arez, G. Tavares-Velasco, and A. Moyotl, Chromomagnetic and chromoelectric dipole moments of quarks
+in the reduced 331 model, Chin. Phys. C 45, 113101 (2021), arXiv:2012.09883 [hep-ph].
+[40] A. I. Hern´andez-Ju´arez, A. Moyotl, and G. Tavares-Velasco, Contributions to ZZV ∗ (V = γ, Z, Z′) couplings from CP
+violating flavor changing couplings, Eur. Phys. J. C 81, 304 (2021), arXiv:2102.02197 [hep-ph].
+[41] A. I. Hern´andez-Ju´arez and G. Tavares-Velasco, Non-diagonal contributions to ZγV ∗ vertex and bounds on Ztq couplings,
+(2022), arXiv:2203.16819 [hep-ph].
+[42] G. J. Gounaris, J. Layssac, and F. M. Renard, New and standard physics contributions to anomalous Z and gamma
+selfcouplings, Phys. Rev. D 62, 073013 (2000), arXiv:hep-ph/0003143.
+[43] D.
+Choudhury,
+S.
+Dutta,
+S.
+Rakshit,
+and
+S.
+Rindani,
+Trilinear
+neutral
+gauge
+boson
+couplings,
+Int. J. Mod. Phys. A 16, 4891 (2001), arXiv:hep-ph/0011205.
+[44] J. Fleischer and F. Jegerlehner, Radiative Corrections to Higgs Decays in the Extended Weinberg-Salam Model,
+Phys. Rev. D 23, 2001 (1981).
+[45] B. A. Kniehl, Radiative corrections for H → ZZ in the standard model, Nucl. Phys. B 352, 1 (1991).
+[46] K. H. Phan and D. T. Tran, One-loop formulas for off-shell decay H∗ → ZZ in ’t Hooft-Veltman gauge and its applications,
+(2022), arXiv:2209.12410 [hep-ph].
+[47] S. Kanemura, M. Kikuchi, K. Sakurai, and K. Yagyu, Gauge invariant one-loop corrections to Higgs boson couplings in
+non-minimal Higgs models, Phys. Rev. D 96, 035014 (2017), arXiv:1705.05399 [hep-ph].
+[48] M. Aoki, S. Kanemura, M. Kikuchi, and K. Yagyu, Radiative corrections to the Higgs boson couplings in the triplet model,
+Phys. Rev. D 87, 015012 (2013), arXiv:1211.6029 [hep-ph].
+
+18
+[49] Y. Gao, A. V. Gritsan, Z. Guo, K. Melnikov, M. Schulze, and N. V. Tran, Spin Determination of Single-Produced Reso-
+nances at Hadron Colliders, Phys. Rev. D 81, 075022 (2010), arXiv:1001.3396 [hep-ph].
+[50] A. M. Sirunyan et al. (CMS), Constraints on anomalous Higgs boson couplings using production and decay information in
+the four-lepton final state, Phys. Lett. B 775, 1 (2017), arXiv:1707.00541 [hep-ex].
+[51] A. M. Sirunyan et al. (CMS), Measurements of the Higgs boson width and anomalous HV V couplings from on-shell and
+off-shell production in the four-lepton final state, Phys. Rev. D 99, 112003 (2019), arXiv:1901.00174 [hep-ex].
+[52] A. M. Sirunyan et al. (CMS), Constraints on anomalous Higgs boson couplings to vector bosons and fermions in its
+production and decay using the four-lepton final state, Phys. Rev. D 104, 052004 (2021), arXiv:2104.12152 [hep-ex].
+[53] V. Khachatryan et al. (CMS), Constraints on the spin-parity and anomalous HVV couplings of the Higgs boson in proton
+collisions at 7 and 8 TeV, Phys. Rev. D 92, 012004 (2015), arXiv:1411.3441 [hep-ex].
+[54] R. Alonso, E. E. Jenkins, A. V. Manohar, and M. Trott, Renormalization Group Evolution of the Standard Model Dimen-
+sion Six Operators III: Gauge Coupling Dependence and Phenomenology, JHEP 04, 159, arXiv:1312.2014 [hep-ph].
+[55] B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian,
+JHEP 10, 085, arXiv:1008.4884 [hep-ph].
+[56] R. Contino, M. Ghezzi, C. Grojean, M. Muhlleitner, and M. Spira, Effective Lagrangian for a light Higgs-like scalar,
+JHEP 07, 035, arXiv:1303.3876 [hep-ph].
+[57] D. de Florian et al. (LHC Higgs Cross Section Working Group), Handbook of LHC Higgs Cross Sections: 4. Deciphering
+the Nature of the Higgs Sector 2/2017, 10.23731/CYRM-2017-002 (2016), arXiv:1610.07922 [hep-ph].
+[58] J. M. Cornwall, Dynamical Mass Generation in Continuum QCD, Phys. Rev. D 26, 1453 (1982).
+[59] J.
+Papavassiliou,
+On
+the
+connection
+between
+the
+pinch
+technique
+and
+the
+background
+field
+method,
+Phys. Rev. D 51, 856 (1995), arXiv:hep-ph/9410385.
+[60] S. Hashimoto, J. Kodaira, Y. Yasui, and K. Sasaki, The Background field method: Alternative way of deriving the pinch
+technique’s results, Phys. Rev. D 50, 7066 (1994), arXiv:hep-ph/9406271.
+[61] T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140, 418 (2001),
+arXiv:hep-ph/0012260.
+[62] R. Mertig,
+M. Bohm, and A. Denner, FEYN CALC: Computer algebraic calculation of Feynman amplitudes,
+Comput. Phys. Commun. 64, 345 (1991).
+[63] V.
+Shtabovenko,
+R.
+Mertig,
+and
+F.
+Orellana,
+New
+Developments
+in
+FeynCalc
+9.0,
+Comput. Phys. Commun. 207, 432 (2016), arXiv:1601.01167 [hep-ph].
+[64] V.
+Shtabovenko,
+R.
+Mertig,
+and
+F.
+Orellana,
+FeynCalc
+9.3:
+New
+features
+and
+improvements,
+Comput. Phys. Commun. 256, 107478 (2020), arXiv:2001.04407 [hep-ph].
+[65] T.
+Hahn
+and
+M.
+Perez-Victoria,
+Automatized
+one
+loop
+calculations
+in
+four-dimensions
+and
+D-dimensions,
+Comput. Phys. Commun. 118, 153 (1999), arXiv:hep-ph/9807565.
+[66] A. Denner, S. Dittmaier, and L. Hofer, Collier: a fortran-based Complex One-Loop LIbrary in Extended Regularizations,
+Comput. Phys. Commun. 212, 220 (2017), arXiv:1604.06792 [hep-ph].
+[67] A. I. Hern´andez-Ju´arez, Mathematica code for hzz analysis, https://gitlab.com/fcfm-buap-rc-group/zzh-anomalous-couplings,
+accessed: 2022-11-22.
+[68] C. Degrande, N. Greiner, W. Kilian, O. Mattelaer, H. Mebane, T. Stelzer, S. Willenbrock, and C. Zhang, Effective Field
+Theory: A Modern Approach to Anomalous Couplings, Annals Phys. 335, 21 (2013), arXiv:1205.4231 [hep-ph].
+
+200
+400
+600
+800
+1000
+1200
+1400
+|q|
+−0.00010
+−0.00005
+0.00000
+0.00005
+0.00010
+0.00015
+h2
+
diff --git a/OdFPT4oBgHgl3EQfmzXU/content/tmp_files/load_file.txt b/OdFPT4oBgHgl3EQfmzXU/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..949a4742f644754204a3507a4fec1d62b0e3a7e4
--- /dev/null
+++ b/OdFPT4oBgHgl3EQfmzXU/content/tmp_files/load_file.txt
@@ -0,0 +1,2556 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf,len=2555
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='13127v1  [hep-ph]  30 Jan 2023  A modern approach to HZZ vertex and direct bounds on its anomalous couplings  from LHC data  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hern´andez-Ju´arez, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Tavares-Velasco and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Fern´andez-T´ellez  Facultad de Ciencias F´ısico Matem´aticas, Benem´erita Universidad  Aut´onoma de Puebla, Apartado Postal 1152, Puebla, Pue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=', M´exico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  (Dated: January 31, 2023)  We present an evaluation of the standard model one-loop contributions to the H∗Z∗Z∗ coupling,  where all the three particles are taken off-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Our results are presented in terms of Passarino-  Veltman scalar functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  We then use the most current CMS results to obtain bounds on the  off-shell H∗ZZ coupling, which at the one-loop level can be parametrized in terms of two CP  conserving and one CP violating form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The limits are studied in three different context,  which are usually used to analyze the HZZ coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In general, such limits are of the order of  10−2 − 10−4 for different energy intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The Standard Model (SM) one-loop contributions for the  H∗ZZ case are revisited, whereas the HZZ∗ case is calculated for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The imaginary  part of these couplings is also studied for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  INTRODUCTION  The observation of the Higgs boson at the CERN LHC [1, 2] was a clear evidence that the mechanism of electroweak  gauge symmetry breaking is realized in nature as conjectured by the standard model (SM) of elementary particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Up to now, the data collected at the LHC has confirmed that the properties of the Higgs particle are consistent with  the SM predictions, though some of its couplings are yet to be measured, such as those to light fermions and its self  couplings as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' It is expected that the LHC run 3 can explore hints of some anomalous Higgs couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Along  these lines, the CMS collaboration has reported for the first time data on the off-shell H∗ZZ coupling via off-shell  Higgs boson production (O-SHBP) pp → H∗ → ZZ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' To produce a pair of on-shell Z gauge bosons, it is required  that the four-momentum of the off-shell Higgs boson is above the threshold ∥q∥ = 2mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' According to the SM, 10%  of events of V pair production at the LHC are due to the H∗V V coupling [4], which is statistically large enough to  allow its measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Moreover, it has been found that the V V invariant mass kinematic distribution is sensitive to  the off-shell H∗ZZ contribution, whereas the ratio of off-shell to on-shell production rates can be used to determine  the Higgs decay width ΓH [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  The phenomenological and experimental implications of the H∗ZZ coupling at the LCH and future colliders have  been explored in the past and also very recently [7–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Furthermore, the off-shell HZZ∗ coupling has also several  phenomenological implications, which have been studied through HZ production [16–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Thus, the HZZ coupling  is worth studying, thereby requiring a highly precise determination of all its lowest order contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In particular,  the search for any anomalous contribution to the H∗ZZ coupling will play an important role at particle colliders in  the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Off-shell couplings have been of great interest in recent years [35–37] as they can develop an imaginary (absorptive)  part due to the optical theorem, thereby giving rise to some effects on physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Such a property has been  studied for instance in the off-shell chromomagnetic and chromoeletric dipole moments of quarks [38, 39], where the  gluon is off-shell, and also in the trilinear neutral gauge boson couplings [40, 41], which are non-vanishing when at  least one of the three gauge bosons is off-shell [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Also, the phenomenological implications of the absorptive part  of the off-shell Higgs boson couplings have been brought to attention by many authors [18, 20, 22, 28], nonetheless, to  our knowledge, a precise determination of the SM contribution has not been reported yet, which may stem from the  fact that there has been some controversy on whether or not off-shell couplings represent valid observable quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  We will dwell on this issue below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In particular, the study of the absorptive part of the H∗ZZ and HZZ∗ couplings  may be relevant at present and future colliders as it can explain slight deviations on some physical observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' This  is the case of the absorptive part of the ttg coupling, which has direct effects on top pair production at the LHC [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  The one-loop corrections to the HZZ coupling were calculated long ago in the SM [44, 45] and also recently [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  On the other hand, the new physics contributions have also been reported in several beyond the SM theories, such as  the two-Higgs doublet model [47], the minimal Higgs triplet model (HTM) [48], the Higgs singlet model (HSM) [47]  and the minimal supersymmetric standard model (MSSM) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Nevertheless, some of those results are reported in a  somewhat old-fashioned notation, which can lead to some confusion when a numerical calculation is worked out and a  cross-check is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Therefore, a new evaluation of the SM one-loop contributions to the off-shell H∗ZZ coupling  with a more up-to-date notation for the analytical results is in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The purpose of this work is to present such an  evaluation, which could be suited to numerical calculations in view of the O-SHBP results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  The rest of this work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In Section II we present the most general effective Lagrangian for the  2  HZZ coupling up to dimension-five operators, along with the most popular parametrizations used to study the effects  of this coupling at particle colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' III is devoted to describe the calculation of the SM one-loop contributions  to the off-shell H∗ZZ and HZZ∗ couplings via the background field method, which in the Feynman-t’Hooft gauge  yields identical results to those obtained via the Pinch technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' This method is known to give gauge-independent  and finite off-shell Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' All the lengthy results are given in Appendix A in terms of Passarino-Veltman  scalar functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In Section IV we present the numerical analysis of the behavior of the H∗ZZ and HZZ∗ couplings  as functions of the off-shell boson transfer momentum and we also obtain bounds on the anomalous terms of such  couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Finally, in Section VI we present the conclusions and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  THEORETICAL FRAMEWORK  Several parametrizations have been introduced in the literature to study the phenomenology of the HZZ coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  It is thus worth presenting a review of the more popular of such parametrizations with the aim that our analysis can  be straightforwardly compared with other studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The effective Lagrangian up to dimension-five operators for the  HZZ interaction can be written in the so-called Hagiwara basis [17, 18, 23] as follows  L = g  cW  mZ  �(1 − aZ)  2  HZµZµ +  1  2m2  Z  �  bZHZµνZµν + cZ  ��  ∂µH  �  Zν −  �  ∂νH  �  Zµ  �  Zµν + ˜bZHZµν ˜Zµν��  ,  (1)  where Zµν = ∂µZν − ∂νZµ and ˜Zµν = ǫµναβZαβ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In the SM aZ vanishes at the tree level, the CP conserving form  factors bZ and cZ are induced up to one-loop level [45], and the CP violating form factor �bZ would arise up to the  three-loop level [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Therefore, the bZ, cZ and �bZ form factors are absent at the tree-level in the SM but can receive  anomalous contributions from new physics at the one-loop-level or higher orders of perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 1,  we introduce the notation used throughout the rest of this work for the vertex function ΓZZH  µν  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Zµ(p1)  Zν(p2)  H(q)  = i g  cW mZΓZZH  µν  (p2  1, p2  2, q2)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Nomenclature for the HZZ coupling and the ΓZZH  µν  vertex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  We are interested in the coupling with an off-shell Higgs boson H∗ZZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Nevertheless, for completeness we will also  study the HZZ∗ coupling, which has been largely studied as it has several implications at particle colliders [16–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Therefore, from Lagrangian (1) neglecting the scalar component of the Z bosons (we use ∂µZµ = 0) and considering  the kinematics H∗ → ZZ(Z∗ → HZ) along with the notation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 1 we obtain the vertex function when both the  Higgs boson and one Z gauge boson are off-shell:  ΓZZH  µν  = hV  1 (q2, p2  1, m2  Z)gµν + hV  2 (q2, p2  1, m2  Z)  m2  Z  p1νp2µ + hV  3 (q2, p2  1, m2  Z)  m2  Z  ǫµναβpα  1 pβ  2,  (2)  where V stands for the off-shell boson and the dependence of the hV  i form factors have been written explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The  relation between the form factors hi and the parameters of Lagrangian (1) for the H∗ZZ (HZZ∗) coupling are  hV  1 (q2, p2  1, m2  Z) = 1 + aZ − bZ  q2 − p2  1 − p2  2  m2  Z  + cZ  q2  m2  Z  ,  (3)  hV  2 (q2, p2  1, m2  Z) = ±2  �  bZ − cZ  �  ,  (4)  hV  3 (q2, p2  1, m2  Z) = ±2�bZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  (5)  For V = H (V = Z) the Z (H) boson is on-shell, and we have p2  1 = m2  Z (q2 = m2  H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The structure of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (2) is  the same for three, two or one off-shell bosons provided that one assumes that the Z gauge bosons are coupled to  3  conserved currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' It is worth noticing that the basis used in the Lagrangian (1) is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' From the motion  equations one has  HZµ∂νZµν = 1  2  ��  ∂µH  �  Zν −  �  ∂νH  �  Zµ  �  Zµν + 1  2ZµνZµν,  (6)  where a surface term has been dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Thus the HZZ coupling can be alternatively described as follows  L = g  cW  mZ  �(1 − aZ)  2  HZµZµ +  1  2m2  Z  �  ˆbZHZµνZµν + ˆcZHZµ∂νZµν + ˜bZHZµν ˜Zµν��  ,  (7)  where the ˆbZ and ˆcZ form factors are given as ˆbZ = bZ − cZ and ˆcZ = 2cZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In this notation, the form factors hV  i of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (2) are now given by  h1(q2, p2  1, m2  Z) = 1 + aZ − ˆbZ  q2 − p2  1 − p2  2  m2  Z  + ˆcZ  2  p2  1 + p2  2  m2  Z  ,  (8)  h2(q2, p2  1, m2  Z) = ±2ˆbZ,  (9)  h3(q2, p2  1, m2  Z) = ±2�bZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  (10)  The main difference between the basis of Lagrangians (1) and (7) is that in the latter the form factor hV  2 is given in  terms of only one parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Although one can find analytic expressions for the form factors hV  i , it is not possible to  identify the contribution from each form factors bZ and cZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Both notations are used without any distinction in the  literature, which may be confusing for a cross-check of the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Thus, to avoid a misleading analysis  of the HZZ coupling we consider both bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The notation of Lagrangian (7) is more suited for the purpose of this  work as we can calculate the form factor h2 and then extract the exact SM contribution to the ˆbZ coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Direct bounds on the anomalous couplings have been obtained using polarization observables of the Z gauge boson  produced in the Z∗ → HZ decay at the LHC at √s = 14 TeV [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' According to the notation of Lagrangian (7) such  bounds read  ��Re  �ˆbZ  ��� ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='5 × 10−4,  ��Im  �ˆbZ  ��� ⩽ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='94 × 10−3,  (11)  ��Re  ��bZ  ��� ⩽ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='76 × 10−3,  ��Im  ��bZ  ��� ⩽ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='64 × 10−3,  (12)  Other limits on the anomalous Higgs boson couplings have been obtained from the analysis of several processes at  e+e− [18, 23], ep [25] and γe colliders [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Below we will discuss some parametrizations used by several authors in the  study of the HZZ coupling and their relationship with the above parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  The LHC framework  In the analysis of the CMS collaboration, the following parametrization is used for the scattering amplitude that  describes the HZZ interaction [49] (according to the notation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 1):  A(H → ZZ) ∼ 1  v  �  aZZ  1  + κZZ  1  p2  1 + κZZ  2  p2  2  �  ΛZZ  1  �2  �  m2  Zǫ∗  1ǫ∗  2 + aZZ  2  v f ∗(1)  µν f ∗(2)µν + aZZ  3  v f ∗(1)  µν  ˜f ∗(2)µν,  (13)  where we use the notation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 1, whereas f (i)µν = ǫµ  i pν  i − ǫνpµ  i and ˜f (i)  µν = ǫµνρσf (i)ρσ are the field and dual field  strength tensor of the Zi gauge boson with polarization vector ǫi and four-momentum pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In the SM aZZ  1  = 2 at the  tree-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We can rewrite the above equation in terms of the hV  i form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In the Hagiwara basis (1) and using  the kinematics for H → ZZ we find the following relations:  h1(q2, p2  1, m2  Z) = aZZ  1  2  + κZZ  1  p2  1 + κZZ  2  p2  2  2  �  ΛZZ  1  �2  + aZZ  2  2  q2 − p2  1 − p2  2  m2  Z  ,  (14)  h2(q2, p2  1, m2  Z) = −aZZ  2  ,  (15)  h3(q2, p2  1, m2  Z) = −aZZ  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  (16)  4  Thus, we can identify  1 + aZ = aZZ  1  2 ,  (17)  −bZ  q2 − p2  1 − p2  2  m2  Z  + cZ  q2  m2  Z  = κZZ  1  p2  1 + κZZ  2  p2  2  2  �  ΛZZ  1  �2  + aZZ  2  2  q2 − p2  1 − p2  2  m2  Z  ,  (18)  bZ − cZ = −aZZ  2  2 ,  (19)  �bZ = −aZZ  3  2 ,  (20)  which yields the following relationship  cZ  p2  1 + p2  2  m2  Z  = κZZ  1  p2  1 + κZZ  2  p2  2  2  �  ΛZZ  1  �2  ,  (21)  which in the case of κZZ  1  = κZZ  2  and ΛZZ  1  ≡ mZ gives  cZ = κZZ  1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  (22)  On the other hand, in the basis of Lagrangian (7) we can identify  −ˆbZ  q2 − p2  1 − p2  2  m2  Z  + ˆcZ  2  p2  1 + p2  2  m2  Z  = κZZ  1  p2  1 + κZZ  2  p2  2  2  �  ΛZZ  1  �2  + aZZ  2  2  q2 − p2  1 − p2  2  m2  Z  (23)  ˆbZ = −aZZ  2  2 ,  (24)  whereas Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (17) and (20) remain valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' For κZZ  1  = κZZ  2  and ΛZZ  1  ≡ mZ, we obtain  ˆcZ = κZZ  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  (25)  Therefore, the parametrization of the HZZ vertex in (13) is redundant since only one form factor kZZ  i  is necessary  in both basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In the rest of this paper we will consider the relations (17), (20) and (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Indirect bounds on aZZ  i  coefficients have been obtained by the CMS collaboration [3, 50–52] through effective  fractional cross sections fai, which allows one to minimize the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In addition, this approach is independent  of the coupling convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The effective cross section ratios are defined as  f ZZ  ai  =  |aZZ  i  |2α(2e2µ)  ii  �  j |aZZ  j  |2α(2e2µ)  jj  sign  �aZZ  i  aZZ  1  �  ,  (26)  where the coefficients α(2e2µ)  ii  are the cross sections of the processes H → ZZ/Zγ∗/γ∗γ∗ → 2e2µ when aZZ  i  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  The corresponding numerical values, which can be obtained through MonteCarlo simulation and are normalized with  respect to the coefficient α(2e2µ)  11  , are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In the CMS analysis the couplings are considered as real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Therefore, the relative phase between the couplings aZZ  i  is 0 or π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The current bounds from CMS [3] are shown in  Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Anomalous couplings aZZ  i  , cross sections ratios f ZZ  ai  and coefficients αii/α11 considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We use the  relationship aZZ  i  = aW W  i  and the value Λ1 = mZ for the case of the κZZ  1  coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The negative sign arises from the convention  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (26) adopted in [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  aZZ  i  f ZZ  ai  αii/α11  a3  fa3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='153  a2  fa2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='361  k1  fΛ1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='016  5  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Allowed intervals at the 95% CL for the coupling parameters fai obtained by the CMS collaboration [3], through  a combined analysis of off-shell and on-shell events, where two scenarios are considered: ΓH = ΓSM  H =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='1 GeV, or ΓH left  unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The sign of the relative phase between ai and a1 is absorbed into the definition of fai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Parameter in  units ×10−5  Scenario  Observed  at 95% CL  fa2  ΓH = ΓSM  H  �  − 32,514  �  ΓH unconstrained  �  − 38,503  �  fa3  ΓH = ΓSM  H  �  − 46,107  �  ΓH unconstrained  �  − 46,110  �  fΛ1  ΓH = ΓSM  H  �  − 11,46  �  ΓH unconstrained  �  − 10,47  �  The coupling fractions are also useful to study the limits on the anomalous couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' They can be obtained from  the cross sections ratios in Eq (26) as  aZZ  i  aZZ  j  =  �  �  �  �|f ZZ  ai |α2e2µ  jj  |f ZZ  aj |α2e2µ  ii  sign  �  f ZZ  ai f ZZ  aj  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  (27)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  The standard model effective field theory framework  A recent approach well-suited for the analysis of anomalous couplings is offered by the Standard Model effective  field theory (SMEFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The corresponding Lagrangian up to dimension six includes 2499 operators Oi [54]  LSMEFT = LSM +  2499  �  i  CiOi,  (28)  where the Wilson coefficients Ci along with the SM parameters make up the the parameter space of the SMEFT,  whereas the operators Oi can be described via the Warsaw basis [55], the SILH basis [56] or the Higgs boson basis  [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' These bases are equivalent and their Wilson coefficients can be mapped onto each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The Higgs basis is well  suited to study the Higgs boson interactions at the LHC, though this is not always true: for instance, the parameter  space of diboson production at the LHC is larger in the Higgs boson basis than in other bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In the Higgs boson  basis [12], the Lagrangian of the HZZ interaction can be written, after a redefinition of the couplings, as follows  L = H  υ  ��  1 + δcz  ��  g2  L + g2  Y  �  v2  4  ZµZµ + czz  g2  L + g2  Y  4  ZµνZµν + cz□g2  LZµ∂νZµν + ˜czz  g2  L + g2  Y  4  Zµν ˜Zµν�  ,  (29)  where υ is the vacuum expectation value (VEV) of the Higgs field and gL, gY stand for the SU(2)L × UY (1) coupling  constants, which in a more usual notation read gL = g and gY = g′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In this Lagrangian the Higgs boson couplings  δcz, czz, cz□, and ˜czz are assumed to be real [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The Lagrangian (29) can be straightforwardly written in a more  familiar form  L =  g  2cW mZ  H  ��  1 + δcz  �  m2  ZZµZµ + czz  g2  L + g2  Y  4  ZµνZµν + cz□g2  LZµ∂νZµν + ˜czz  g2  L + g2  Y  4  Zµν ˜Zµν�  ,  (30)  which allows one to identify the following relations with the form factors of Lagrangian (7)  δcz = aZ  (31)  czz =  4  g2  L + g2  Y  ˆbZ,  (32)  cz□ = 1  g2  L  ˆcZ,  (33)  ˜czz =  4  g2  L + g2  Y  ˜bZ,  (34)  which agree with those relations reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  6  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  ANALYTICAL RESULTS  We now turn to present our analytical results for the one-loop contributions to the HZZ coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Since we are  considering that either the Z gauge boson or the Higgs boson H are off-shell, we need to address the problem of  the gauge dependence of off-shell Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In order to deal with this issue and obtain well-behaved vertex  functions, there are two well-known approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The first one is the so-called pinch technique (PT) [58], which is a  diagrammatic approach that allows one to systematically obtain gauge-independent and well-behaved off-shell Green’s  functions by combining self-energy, vertex and box diagrams contributing to a physical process, which may turn into  a very cumbersome task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Nevertheless, it was shown that the results obtained up to one-loop level via the PT are  equivalent to those obtained by the application of the background field method (BGFM) when the Feynman-t’Hooft  gauge is used, namely, when one sets the gauge-fixing parameter ξQ = 1 [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In this work, we use the latter  method to obtain gauge independent form factors since the calculation is easier to perform than in the PT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  For the analytical calculation we used the Mathematica package FeynArts [61], which allows one to obtain the  complete set of Feynman diagrams and their corresponding invariant amplitudes via the background field method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  We then used FeynCalc [62–64] to manipulate the amplitudes an obtain expressions in terms of Passarino-Veltman  scalar functions, which can be numerically evaluated with the aid of the LoopTools [65] and Collier [66] packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  The contributing Feynman diagrams, which can be classified into three types, are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 2, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The  Feynman diagram for the fermion loop contribution is shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 2, whereas the remaining contributions are those  arising from Feynman diagrams with W gauge boson exchange (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 3) and H − Z (HZ) boson exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Zµ(p1)  Zν(p2)  H(q)  f  f  f  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Feynman diagram for the one-loop fermion contribution (F) to the HZZ coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  7  Zµ(p1)  Zν(p2)  H(q)  G±  G±  W ±  (a)  Zµ(p1)  Zν(p2)  H(q)  W ±  G±  G±  (b)  Zµ(p1)  Zν(p2)  H(q)  G±  W ±  G±  (c)  Zµ(p1)  Zν(p2)  H(q)  W ±  W ±  G±  (d)  Zµ(p1)  Zν(p2)  H(q)  W ±  W ±  G±  (e)  Zµ(p1)  Zν(p2)  H(q)  G±  W ±  W ±  (f)  Zµ(p1)  Zν(p2)  H(q)  W ±  W ±  W ±  (g)  Zµ(p1)  Zν(p2)  H(q)  G±  (h)  Zµ(p1)  Zν(p2)  H(q)  u−, u+  (i)  Zµ(p1)  Zν(p2)  H(q)  G±  (j)  Zµ(p1)  Zν(p2)  H(q)  W ±  (k)  Zµ(p1)  Zν(p2)  H(q)  W ±  G±  (l)  Zµ(p1)  Zν(p2)  H(q)  W ±  G±  (m)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 2 but for the contribution of W gauge boson exchange (W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  8  Zµ(p1)  Zν(p2)  H(q)  H  H  Z  (a)  Zµ(p1)  Zν(p2)  H(q)  Z  G0  H  (b)  Zµ(p1)  Zν(p2)  H(q)  G0  Z  H  (c)  Zµ(p1)  Zν(p2)  H(q)  Z  Z  H  (d)  Zµ(p1)  Zν(p2)  H(q)  G0  G0  H  (e)  Zµ(p1)  Zν(p2)  H(q)  H  H  G0  (f)  Zµ(p1)  Zν(p2)  H(q)  G0, H  (g)  Zµ(p1)  Zν(p2)  H(q)  Z  H  (h)  Zµ(p1)  Zν(p2)  H(q)  Z  H  (i)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 2 but for the contribution of H − Z bosone exchange (HZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Since the SM contributions to the anomalous couplings are the main aim of this work we will focus on form factor  hV  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Therefore, following the notation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' [45] our results can be written as  hV  2 (q2, p2  1, m2  Z) = mZ  g2  4π2cW mW  � �  f  Nfm2  f  �  g2  V fAV  V f(q2, p2  1, m2  Z) + g2  AfAV  Af(q2, p2  1, m2  Z)  �  (35)  + AV  W (q2, p2  1, m2  Z) + AV  ZH(q2, p2  1, m2  Z)  �  ,  where AV  V f,Af, AV  W and AV  ZH stand for the fermion, W gauge boson and H − Z boson contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Here gV f,Af  stand for the fermion couplings to fermion pairs:  gV f = If  2 − Qfs2  W ,  gAf = If  2 ,  (36)  with If and Qf the fermion weak isospin and electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Analytic expressions for the AV  V f,Af, AV  W and AV  ZH functions in terms of Passarino-Veltman scalar functions are  presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We present explicit results for the contributions to the H∗ZZ (p2  1 = m2  Z) and HZZ∗  (q2 = m2  H) couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We note that for the H∗ZZ coupling, our results agree with those reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' [45], though  there seems to be a difference of sign in the factors of all three-point scalar function C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' This stems from the fact  that there is an additional minus sign in the definition of such functions in [45] as compared to the usual definition  presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We would like to emphasize that, to our knowledge, the results for the HZZ∗ (q2 = m2  H)  coupling have never been reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Thus, we present a more comprehensive calculation, which could  allow to asses the anomalous contributions to HZZ coupling in distinct scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The Mathematica code for our  analytical results is available for the interested reader [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We also include master formulas for the general case where  the three particles are off-shell, which are too cumbersome to be presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Our expressions reported in  Appendix A can be straightforwardly obtained from such master formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  It is interesting to note that the HZZ form factors are gauge-dependent as expected, except for the fermion  contribution, which evidently does not depend on the gauge-fixing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  On the other hand, all the three  contributions are always free of ultraviolet divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  9  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  NUMERICAL ANALYSIS  We now turn to present the numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' For the evaluation of the Passarino-Veltman scalar functions we  used the LoopTools package [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We first present the evaluation of the H∗ZZ and HZZ∗ form factors in an energy  interval that allows for on-shell final bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  H∗ZZ form factor  We show in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 5(a) and 5(b) the real and imaginary parts of the fermion (F), W gauge boson (W), H − Z  bosons (HZ), and total contributions to the H∗ZZ form factor as functions of the Higgs boson transfer momentum  ∥q∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' As for the real part of hH  2 , the total contribution is dominated by the W contribution, whereas the F and HZ  contributions only are relevant at low energies, where they reach values of a comparable magnitude to those of the W  contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' From ∥q∥ = 300 GeV onwards, the F and HZ contributions have a magnitude of similar size but they  are of opposite sign and tend to cancel each other out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' It is also interesting to note that the fermion contribution is  mainly dominated by the top quark and all other fermions give negligible contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' As far as the imaginary part  of the hH  2 form factor is concerned, we observe that the W contribution is also the dominant one, whereas the F and  HZ contributions are negligible as they are one order of magnitude smaller than the W contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' As expected,  the absorptive part of the F contribution is non-vanishing only above the threshold ∥q∥ = 2mt, where the two top  quarks attached to the off-shell Higgs boson can be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In general, the real and imaginary parts of hH  2 are of similar  size and their magnitudes are of the order of 10−2 − 10−3, nevertheless, at high energies the absorptive part can be  larger than the real one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' This behavior has also been observed in other off-shell coupling form factors [38–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  200  400  600  800  1000  1200  1400  ||q||  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0020  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0015  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0010  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0015  Re[h H  2 ]  ZZH ∗  F  W  HZ  Total  (a)  200  400  600  800  1000  1200  1400  ||q||  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='005  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='003  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='002  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='001  Im[h H  2 ]  ZZH ∗  F  W  HZ  Total  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' One loop fermion (F), W gauge boson (W), HZ bosons (HZ) and total contributions to the real (left plot) and  absorptive (right plot) parts of the form factor hH  2 as functions of the Higgs boson transfer momentum ∥q∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  For illustration purposes, the values of the real and imaginary parts of the hH  2 form factors at a few ∥q∥ values are  presented in Table III, along with the values for ˆbZ, aZZ  2  and czz, which can be obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (9), (15) and (32),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In effective field theories the couplings are taken as constant and do not depend on ∥q∥, but our results  can be useful to constraint the energy scale Λ of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  10  TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Total contributions to the hH  2 form factor for a few values of ∥q∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The respective values of ˆbZ, aZZ  2  and czz are  also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' All these results are in units of 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  ��q  ��  hH  2  ˆbZ  aZZ  2  czz  190  −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='99 − 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='02 i  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='49 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='01 i  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='99 + 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='02 i  −48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='16 − 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='98 i  220  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='82 − 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='86 i  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='42 − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='93 i  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='82 + 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='86 i  −25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='3 − 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='4 i  350  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='09 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='77 i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='04 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='88 i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='09 + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='77 i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='35 − 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='8 i  450  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='02 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='24 i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='51 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='62 i  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='02 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='24 i  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='81 − 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='43 i  600  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='11 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='31 i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='55 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='65 i  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='11 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='31 i  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='12 − 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='29 i  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='78 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='5 i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='39 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='75 i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='78 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='5 i  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='9 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='56 i  1500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='52 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='79 i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='26 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='39 i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='52 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='79 i  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='92 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='95 i  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  HZZ∗ form factor  We now show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 6 the behavior of the real and imaginary parts of the partial and total contributions to the  HZZ∗ form factor as functions of the off-shell Z gauge boson transfer momentum ∥p1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We observe that the hZ  2 form  factor has a similar behavior to that of hH  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In both cases the W contribution dominates, but at low energies the F  and HZ contributions are of similar magnitude than the W contribution, whereas at high energies they are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  However, there is a slight difference with the behavior of hH  2 , since both the real part of hZ  2 reach its larger value at  a different energy value than in the hH  2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  400  600  800  1000  1200  1400  ||p1 ||  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0020  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0015  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0010  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0015  Re[h Z  2 ]  HZZ ∗  F  W  HZ  Total  (a)  400  600  800  1000  1200  1400  ||p1 ||  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='005  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='003  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='002  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='001  Im[h Z  2 ]  HZZ ∗  F  W  HZ  Total  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 5, but for the form factor hZ  2 as a function of off-shell Z gauge boson transfer momentum ∥p1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  In Table IV, we present numerical values for the real and absorptive parts of hZ  2 at a few values of ∥p1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We also  present the respective values for ˆbZ, aZZ  2  and czz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We can observe that at some energy values, the hZ  2 form factor is  larger than the hH  2 one, reaching values of the order of 10−2 − 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We note that in the case of both H∗ZZ and  HZZ∗ couplings, the real and absorptive parts of the anomalous coupling ˆbZ are of the order of 10−3 − 10−4, which  agrees with [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In the experimental side, the current bounds on ˆbZ are of the same order of magnitude, thus this  anomalous coupling could be at the reach of measurement at the LHC in a near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  11  TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The same as in Table III, but for the form factor hZ  2 for as a function of ∥p1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  ��p1  ��  hZ  2  ˆbZ  aZZ  2  czz  220  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='84 − 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='16 i  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='42 + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='58 i  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='84 + 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='16 i  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='93 + 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='19 i  350  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='07 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='37 i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='53 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='68 i  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='07 + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='37 i  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='98 + 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='33 i  450  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='21 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='11 i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='6 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='55 i  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
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+page_content='11 i  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='49 + 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='95 i  600  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='27 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='35 i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='63 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='67 i  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='27 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='35 i  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='72 + 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='41 i  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
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+page_content='46 i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
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+page_content='73 i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
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+page_content='46 i  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='43 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='43 i  1500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='59 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='73 i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='29 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='36 i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='59 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='73 i  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='22 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='71 i  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  BOUNDS ON THE ANOMALOUS COUPLINGS  Limits on the real and absorptive parts of the anomalous couplings of the HZZ coupling have been set in the past  [28], nonetheless, the energy dependence has not been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Also, the current CMS bounds [3] only consider  constraints on the ratios of such couplings, which cannot be used to assess the corresponding contributions to physical  observables [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Thus, individual constraints on each one of the couplings are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' With this aim, we proceed  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Firstly, by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (27) and the values of Table I we use the constraints from Table II to obtain  limits on the ratios aZZ  i  /aZZ  2  , which are presented in Table V, where we only consider the scenario with ΓH = ΓSM  H  as the bounds in the unconstrained scenario yield limits of similar size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Allowed intervals at 95 % CL for the ratios defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (27), where we consider the case ΓH = ΓSM  H  and the  limits of Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Ratio  Allowed values  aZZ  3  /aZZ  2  �  − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='84, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='70  �  κZZ  1  /aZZ  2  �  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='18  �  Secondly, we use the constraints on aZZ  i  /aZZ  2  of Table V and use Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (9) and (24), along with the numerical results  from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' IV, to find the allowed areas of the real parts of aZZ  3  and κZZ  1  as functions of the Higgs boson transfer  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Also, since the results of Table III indicate that the real and absorptive parts of the H∗ZZ form factor  are of similar size, we assume that the limits of Table V are also valid for the imaginary parts of aZZ  3  and κZZ  1  , which  allows us to set constraints on Im  �  aZZ  3  �  and Im  �  κZZ  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' For our calculations we use the numerical values of the form  factor hH  2 , but the same results are expected for hZ  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Moreover, only energy regions where both Z bosons can be  on-shell are considered in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  We thus show in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 7(a) and 7(b) the allowed areas on the planes ∥q∥ vs Re  �  aZZ  3  �  and ∥q∥ vs Im  �  aZZ  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We  observe that for small values of ∥q∥, Re  �  aZZ  3  �  is allowed to have values in the [−10−2, 10−2] interval, nevertheless,  the length of such an interval shrinks considerably as the energy increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Thus, at high energies, Re  �  aZZ  3  �  is only  allowed to have values of the order of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' It is also worth noting that around ∥q∥ = 85 GeV, the allowed Re  �  aZZ  3  �  area vanishes, which stems from the fact that in such a region the real part of aZZ  2  changes sign and Re  �  aZZ  2  �  ≈ 0  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 5(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Therefore, very small values of Re  �  aZZ  3  �  are required to get aZZ  3  /aZZ  2  inside the allowed region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' As  for Im  �  aZZ  3  �  , we observe that for small energies the allowed area is large but becomes smaller as the energy increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  However, in this case there is no abrupt vanishing of the allowed area as occurs for the real part around ∥q∥ ∼ 85  GeV: it turns out that the absorptive part of aZZ  2  does not flips sign in the considered energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  12  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Allowed area at 95% CL for the the real (left plot) and absorptive (right plot) parts of aZZ  3  as functions of ∥q∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' These  results are compatible with the bounds of Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  We now show the allowed areas of the real and absorptive parts of κZZ  1  in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 8(a) and 8(b), respectively, as  functions of ∥q∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The results are similar to those obtained for aZZ  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Nevertheless, in this case, the allowed upper  values of both the real and imaginary parts of aZZ  3  are of the order of 10−3 at low energies and become smaller as the  energy increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In general, the constraints on the CP violating form factor aZZ  3  are less tighter than those on the  CP conserving one kZZ  1  , although both can be of the same order of magnitude in some energy regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 7, but for the κZZ  1  form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Our constrains on the form factors aZZ  3  and κZZ  1  , which correspond to the LHC framework, can be translated  into constraints on the form factors used in other parametrizations via the relations of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In this way we can  straightforwardly obtain the allowed regions for ˜bZ, ˆcZ, ˜czz and cz□, which are used by other authors in their analysis  of the HZZ coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' For instance, in the SMEFT the anomalous couplings do not depend on the off-shell boson  transfer momenta [68] and the energy scale Λ, which is associated with the scale where the effective model remains  13  valid, has been absorbed in the definition of the Wilson coefficients ˜czz and cz□.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hence, our result may be used to  set a bound on such energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The corresponding bounds are presented in Tables VI and VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We observe that  the real and imaginary parts of the CP violating form factors ˜bZ and ˜czz can be of the order of 10−4 at low energies,  however, in the SMEFT they can be as large as 10−3 for some values of ∥q∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' The current bounds on ˜bZ are of the  order of 10−3 [28], therefore our limits are more stringent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' On the other hand, the CP conserving from factor ˆcZ  can be as large as 10−4 at low energies, which is two orders of magnitude smaller than previous results [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' In the  SMEFT, cz□ can be in general of the order of 10−3 − 10−4 and decreases at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Allowed intervals of the real and absorptive parts of the CP violating form factor of the HZZ coupling for some  values of the transfer momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We consider three different schemes: The LHC framework (aZZ  3  ), in a general effective  Lagrangian approach (˜bZ) and the SMEFT (˜czz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  ��q  ��  Re  �  aZZ  3  �  Re  �˜bZ  �  Re  �  ˜czz  �  Im  �  aZZ  3  �  Im  �˜bZ  �  Im  �  ˜czz  �  190  �  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='024, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='009  �  �  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0045, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='012  �  �  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='033, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='088  �  �  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='026, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='01  �  �  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='013  �  �  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='037, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='096  �  285  �  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0029, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0011  �  �  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
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+page_content=' Allowed intervals for the real and absorptive parts of one of the CP conserving form factors of the HZZ coupling  for a few values of the transfer momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' We consider three different schemes: The LHC framework (κZZ  1  ), in a general  effective Lagrangian approach (ˆcZ) and the SMEFT (˜cz□).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (25) we note that κZZ  1  = ˆcZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  ��q  ��  Re  �  kZZ  1  � �  Re  �  ˆcZ  ��  Re  �  cz□  �  Im  �  kZZ  1  � �  Im  �  ˆcZ  ��  Im  �  cz□  �  190  �  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
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+page_content='0007  �  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  CONCLUSIONS AND OUTLOOK  Appendix A: Analytical results  We present the analytical expression for the one-loop contributions to the HZZ form factor hV  2 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (35) in terms  of Passarino-Veltman scalar functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  The two- and three-point Passarino-Veltman scalar functions are defined as  B0(r2, m2  1, m2  2) =  1  iπ2  �  dDk  (k2 − m2  1)((k + r)2 − m2  2),  (A1)  C0(r2  1, (r1 + r2)2, r2  2, m2  1, m2  2, m2  3) =  1  iπ2  �  dDk  (k2 − m2  1)((k + r1)2 − m2  2)((k + r2)2 − m2  3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  (A2)  We now introduce the following shorthand notation  B0ij(r2) = B0  �  r2, m2  i , m2  j  �  ,  C0ijk  �  q2�  = C0  �  m2  Z, m2  Z, q2, m2  i , m2  j, m2  k  �  ,  C0ijk  �  p2  1  �  = C0  �  m2  H, m2  Z, p2  1, m2  i , m2  j, m2  k  �  ,  (A3)  It is useful to observe the following symmetry relations  Bij(r2) = Bji(r2),  Cijk  �  q2�  = Ckji  �  q2�  ,  (A4)  14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Contributions to the H∗ZZ coupling  The contributions from fermion to the AH  V f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='Af functions read  AH  V f(q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' m2  Z) =  1  q2 (q2 − 4m2  Z) 2  �  4q2m2  Z  �  B0ff  �  q2�  − B0ff  �  m2  Z  �  − 3  �  + 8m4  Z  �  B0ff  �  q2�  − B0ff  �  m2  Z  �  + 2  �  −  �  q2 − 2m2  Z  � �  −4m2  f  �  q2 − 4m2  Z  �  − 6q2m2  Z − 4m4  Z + q4�  C0fff  �  q2�  + 2q4�  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  (A5)  AH  Af(q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' m2  Z) =  1  q2 (q2 − 4m2  Z) 2  � �  q2 − 2m2  Z  � �  4  �  q2 − m2  Z  � �  B0ff  �  q2�  − B0ff  �  m2  Z  � �  +  �  −2m2  Z  �  8m2  f + q2�  + 4q2m2  f + 4m4  Z + q4�  C0  �  q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' m2  Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' m2  Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' m2  f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' m2  f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' m2  f  �  + 2  �  q2 − 4m2  Z  � ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (A6)  As for the contributions from W gauge boson (W),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' and H − Z boson (HZ) exchange,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' they are given by  AH  W (q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
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+page_content='�3��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  (A12)  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Chatrchyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (CMS), Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' B 716, 30 (2012), arXiv:1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='7235 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (ATLAS), Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS  detector at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' B 716, 1 (2012), arXiv:1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='7214 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Tumasyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (CMS), First evidence for off-shell production of the Higgs boson and measurement of its width, (2022),  arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='06923 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [4] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kauer and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Passarino, Inadequacy of zero-width approximation for a light Higgs boson signal, JHEP 08, 116,  arXiv:1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='4803 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Caola  and  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Melnikov,  Constraining  the  Higgs  boson  width  with  ZZ  production  at  the  LHC,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 88, 054024 (2013), arXiv:1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='4935 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Campbell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Ellis, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Williams, Bounding the Higgs Width at the LHC Using Full Analytic Results for  gg− > e−e+µ−µ+, JHEP 04, 060, arXiv:1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='3589 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Bolognesi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Gao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Gritsan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Melnikov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Schulze, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Tran, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Whitbeck, On the spin and parity of a  single-produced resonance at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 86, 095031 (2012), arXiv:1208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='4018 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [8] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Anderson  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=',  Constraining  Anomalous  HVV  Interactions  at  Proton  and  Lepton  Colliders,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 89, 035007 (2014), arXiv:1309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='4819 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [9] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' S¸ahin, Search for the anomalous ZZH couplings at the CLIC, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' A 34, 1950299 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Gritsan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Roskes, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Sarica, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Schulze, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Xiao, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Zhou, New features in the JHU generator frame-  work:  constraining Higgs boson properties from on-shell and off-shell production, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 102, 056022 (2020),  arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='09888 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [11] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Gon¸calves,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Han, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Ching Iris Leung, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Qin, Off-shell Higgs couplings in H∗  →  ZZ  →  ℓℓνν,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' B 817, 136329 (2021), arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='05272 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Azatov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=', Off-shell Higgs Interpretations Task Force: Models and Effective Field Theories Subgroup Report, (2022),  arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='02418 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Sharma and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Shivaji, Probing non-standard HV V (V = W, Z) couplings in single Higgs production at future electron-  proton collider, (2022), arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='03862 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Aguilar-Saavedra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Bernal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Casas, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Moreno, Testing entanglement and Bell inequalities in H → ZZ,  (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='13441 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Nguyen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Tran, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phan, One-loop on-shell and off-shell decay H∗ → V V at future e−e− collider, in  47th Vietnam Conference on Theoretical Physics (2022) arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='13153 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kniehl, Radiative corrections for associated ZH production at future e+e− colliders, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' C 55, 605 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  17  [17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hagiwara and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Stong, Probing the scalar sector in e+ e- —> f anti-f H, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' C 62, 99 (1994),  arXiv:hep-ph/9309248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [18] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hagiwara, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Ishihara, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kamoshita, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kniehl, Prospects of measuring general Higgs couplings at e+ e- linear  colliders, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' C 14, 457 (2000), arXiv:hep-ph/0002043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [19] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kniehl, Theoretical aspects of standard model Higgs boson physics at a future e+ e- linear collider,  Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' A 17, 1457 (2002), arXiv:hep-ph/0112023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Biswal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Godbole, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Singh, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Choudhury, Signatures of anomalous VVH interactions at a linear  collider, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 73, 035001 (2006), [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='D 74, 039904 (2006)], arXiv:hep-ph/0509070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Choudhury and Mamta, Anomalous Higgs Couplings at an e gamma Collider, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 74, 115019 (2006),  arXiv:hep-ph/0608293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Godbole, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Miller, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Muhlleitner, Aspects of CP violation in the H ZZ coupling at the LHC,  JHEP 12, 031, arXiv:0708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0458 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Dutta, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hagiwara, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Matsumoto, Measuring the Higgs-Vector boson Couplings at Linear e+e− Collider,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 78, 115016 (2008), arXiv:0808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0477 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rindani and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Sharma, Angular distributions as a probe of anomalous ZZH and gammaZH interactions at a linear  collider with polarized beams, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 79, 075007 (2009), arXiv:0901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='2821 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [25] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Cakir,  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Cakir,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Senol,  and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Tasci,  Probing  Anomalous  HZZ  Couplings  at  the  LHeC,  Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' A 28, 1350142 (2013), arXiv:1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='3616 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [26] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rindani, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Sarmah, Probing anomalous gauge-Higgs couplings using Z boson polarization at e+e−  colliders, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' B 950, 114840 (2020), arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='06663 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kumar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Poulose, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rahaman, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Singh, Measuring Higgs self-couplings in the presence of VVH and VVHH  at the ILC, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' A 34, 1950094 (2019), arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='06601 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [28] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rindani, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Sarmah, Study of anomalous gauge-Higgs couplings using Z boson polarization at LHC,  Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' B 964, 115317 (2021), arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='00980 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [29] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Heinrich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Jones, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kerner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Klappert, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Schlenk, ZH production in gluon fusion: two-loop  amplitudes with full top quark mass dependence, JHEP 03, 125, arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='12325 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [30] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Bizo´n, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Caola, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Melnikov, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' R¨ontsch, Anomalous couplings in associated VH production with Higgs boson  decay to massive b quarks at NNLO in QCD, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 105, 014023 (2022), arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='06328 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [31] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rindani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Sarmah, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Singh, Polarized Z cross sections in Higgsstrahlung for the determination of  anomalous ZZH couplings, (2022), arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='10215 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [32] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Bittar and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Burdman, Form Factors in Higgs Couplings from Physics Beyond the Standard Model,  (2022),  arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='07094 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Gritsan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=', Snowmass White Paper: Prospects of CP-violation measurements with the Higgs boson at future  experiments, (2022), arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='07715 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [34] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Guan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' He, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Liu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Ma, Complete two-loop electroweak corrections to e+e− → HZ,  (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='14953 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [35] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Haisch and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Koole, Off-shell Higgs production at the LHC as a probe of the trilinear Higgs coupling, JHEP 02, 030,  arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='12589 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [36] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Englert, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Soreq, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Spannowsky, Off-Shell Higgs Coupling Measurements in BSM scenarios, JHEP 05, 145,  arXiv:1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='5440 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hern´andez-Ju´arez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Moyotl, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Tavares-Velasco, Bounds on the absorptive parts of the chromomagnetic and  chromoelectric dipole moments of the top quark from LHC data, (2021), arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='09978 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hern´andez-Ju´arez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Moyotl, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Tavares-Velasco, New estimate of the chromomagnetic dipole moment of quarks  in the standard model, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Plus 136, 262 (2021), arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='11955 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [39] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hern´andez-Ju´arez, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Tavares-Velasco, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Moyotl, Chromomagnetic and chromoelectric dipole moments of quarks  in the reduced 331 model, Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' C 45, 113101 (2021), arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='09883 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hern´andez-Ju´arez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Moyotl, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Tavares-Velasco, Contributions to ZZV ∗ (V = γ, Z, Z′) couplings from CP  violating flavor changing couplings, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' C 81, 304 (2021), arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='02197 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hern´andez-Ju´arez and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Tavares-Velasco, Non-diagonal contributions to ZγV ∗ vertex and bounds on Ztq couplings,  (2022), arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='16819 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [42] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Gounaris, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Layssac, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Renard, New and standard physics contributions to anomalous Z and gamma  selfcouplings, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 62, 073013 (2000), arXiv:hep-ph/0003143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [43] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Choudhury,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Dutta,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Rakshit,  and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Rindani,  Trilinear  neutral  gauge  boson  couplings,  Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' A 16, 4891 (2001), arXiv:hep-ph/0011205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Fleischer and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Jegerlehner, Radiative Corrections to Higgs Decays in the Extended Weinberg-Salam Model,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 23, 2001 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [45] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kniehl, Radiative corrections for H → ZZ in the standard model, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' B 352, 1 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [46] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phan and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Tran, One-loop formulas for off-shell decay H∗ → ZZ in ’t Hooft-Veltman gauge and its applications,  (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='12410 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [47] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kanemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kikuchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Sakurai, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Yagyu, Gauge invariant one-loop corrections to Higgs boson couplings in  non-minimal Higgs models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 96, 035014 (2017), arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='05399 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Aoki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kanemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kikuchi, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Yagyu, Radiative corrections to the Higgs boson couplings in the triplet model,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 87, 015012 (2013), arXiv:1211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='6029 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  18  [49] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Gao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Gritsan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Guo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Melnikov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Schulze, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Tran, Spin Determination of Single-Produced Reso-  nances at Hadron Colliders, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 81, 075022 (2010), arXiv:1001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='3396 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [50] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (CMS), Constraints on anomalous Higgs boson couplings using production and decay information in  the four-lepton final state, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' B 775, 1 (2017), arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='00541 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [51] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (CMS), Measurements of the Higgs boson width and anomalous HV V couplings from on-shell and  off-shell production in the four-lepton final state, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 99, 112003 (2019), arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='00174 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [52] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (CMS), Constraints on anomalous Higgs boson couplings to vector bosons and fermions in its  production and decay using the four-lepton final state, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 104, 052004 (2021), arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='12152 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [53] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Khachatryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (CMS), Constraints on the spin-parity and anomalous HVV couplings of the Higgs boson in proton  collisions at 7 and 8 TeV, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 92, 012004 (2015), arXiv:1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='3441 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [54] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Alonso, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Jenkins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Manohar, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Trott, Renormalization Group Evolution of the Standard Model Dimen-  sion Six Operators III: Gauge Coupling Dependence and Phenomenology, JHEP 04, 159, arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='2014 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [55] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Grzadkowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Iskrzynski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Misiak, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rosiek, Dimension-Six Terms in the Standard Model Lagrangian,  JHEP 10, 085, arXiv:1008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='4884 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [56] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Contino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Ghezzi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Grojean, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Muhlleitner, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Spira, Effective Lagrangian for a light Higgs-like scalar,  JHEP 07, 035, arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='3876 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [57] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' de Florian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' (LHC Higgs Cross Section Working Group), Handbook of LHC Higgs Cross Sections: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Deciphering  the Nature of the Higgs Sector 2/2017, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='23731/CYRM-2017-002 (2016), arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='07922 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [58] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Cornwall, Dynamical Mass Generation in Continuum QCD, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 26, 1453 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [59] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Papavassiliou,  On  the  connection  between  the  pinch  technique  and  the  background  field  method,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 51, 856 (1995), arXiv:hep-ph/9410385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [60] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hashimoto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kodaira, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Yasui, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Sasaki, The Background field method: Alternative way of deriving the pinch  technique’s results, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' D 50, 7066 (1994), arXiv:hep-ph/9406271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [61] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 140, 418 (2001),  arXiv:hep-ph/0012260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [62] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Mertig,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Bohm, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Denner, FEYN CALC: Computer algebraic calculation of Feynman amplitudes,  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 64, 345 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [63] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Shtabovenko,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Mertig,  and  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Orellana,  New  Developments  in  FeynCalc  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='0,  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 207, 432 (2016), arXiv:1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='01167 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [64] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Shtabovenko,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Mertig,  and  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Orellana,  FeynCalc  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='3:  New  features  and  improvements,  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 256, 107478 (2020), arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='04407 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [65] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Hahn  and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  Perez-Victoria,  Automatized  one  loop  calculations  in  four-dimensions  and  D-dimensions,  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 118, 153 (1999), arXiv:hep-ph/9807565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [66] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Denner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Dittmaier, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hofer, Collier: a fortran-based Complex One-Loop LIbrary in Extended Regularizations,  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 212, 220 (2017), arXiv:1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='06792 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [67] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Hern´andez-Ju´arez, Mathematica code for hzz analysis, https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='com/fcfm-buap-rc-group/zzh-anomalous-couplings,  accessed: 2022-11-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  [68] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Degrande, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Greiner, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Kilian, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Mattelaer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Mebane, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Stelzer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Willenbrock, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' Zhang, Effective Field  Theory: A Modern Approach to Anomalous Couplings, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content=' 335, 21 (2013), arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='4231 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
+page_content='  200  400  600  800  1000  1200  1400  |q|  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf'}
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+Draft version January 11, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Updating the 56Ni Problem in Core-collapse Supernova Explosion
+Ryo Sawada
+1 and Yudai Suwa
+1, 2
+1Department of Earth Science and Astronomy, Graduate School of Arts and Sciences, The University of Tokyo, Tokyo 153-8902, Japan
+2Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto
+606-8502, Japan
+Submitted to ApJ
+ABSTRACT
+Details of the core-collapse supernova (CCSN) explosion mechanism still need to be fully understood.
+There is an increasing number of successful examples of reproducing explosions in multidimensional
+hydrodynamic simulations, but subsequent studies pointed out that the growth rates of the explosion
+energy ˙Eexpl of these simulations are insufficient to produce enough 56Ni to match observations. This
+issue is known as the ‘56Ni problem’ in CCSNe. Recently, however, some studies have suggested that
+this 56Ni problem is derived from the simplicity of the explosion model. In response, we investigate the
+effect of the explosion energy growth rate ˙Eexpl on the behavior of nucleosynthesis in CCSNe in a more
+realistic model. We employ the 1D Lagrangian hydrodynamic code, in which we take neutrino heating
+and cooling terms into account with the light-bulb approximation. We reiterate that, consistent with
+previous rebuttal studies, there is the 56Ni problem: Although 56Ni is synthesized to almost the same
+mass coordinate independent of ˙Eexpl, some of the innermost material in the low- ˙Eexpl model failed
+to escape, leading to a shift in the innermost mass coordinate of the ejecta to the outer positions.
+Comparing our results with observations, we find that while modern slow explosions can, in principle,
+reproduce observations of standard Type II SNe, this is not possible with stripped-envelope SNe. Our
+finding places a strong constraint on the explosion mechanism. There are significant differences in the
+progenitor structures and the explosion mechanism between Type II and stripped-envelope SNe.
+Keywords: (stars:) supernovae: general—hydrodynamics
+1. INTRODUCTION
+Radioisotope
+56Ni is an important product in su-
+pernova nucleosynthesis, which drives supernova (SN)
+brightness.
+56Ni decays into 56Co, and then into 56Fe.
+This nuclear decay chain powers the light curve of SNe,
+and thus, 56Ni masses of SNe have been estimated with
+reasonable accuracy from the light curve (see, e.g., Ar-
+Corresponding author: Ryo Sawada
+ryo@g.ecc.u-tokyo.ac.jp
+nett 1982; Hamuy 2003).1
+On the other hand, the
+amount of synthesized 56Ni is sensitive to the tempera-
+ture T, the density ρ, and the number of electrons per
+nucleon (electron fraction) Ye, i.e., explosion property
+and pre-SNe core structure (e.g., Woosley & Weaver
+1995; Thielemann et al. 1996; Woosley et al. 2002).
+These two factors, that is, the amount of 56Ni synthe-
+sis can be accurately estimated from observations and
+strongly reflect the explosion’s innermost nature, sug-
+gest the following. 56Ni is the best probe to constrain an
+1 For Type-II SNe, the tail luminosity provides 56Ni mass, assum-
+ing the complete trapping of γ-rays produced from the nuclear
+decay.
+For Type-I SNe, on the contrary, the peak luminosity
+has often been used, assuming that it should be equal to the
+instantaneous energy deposition rate by the nuclear decay, so-
+called Arnett rule (Arnett 1982). Core-collapse SNe in the latter
+category is called stripped-envelope SNe (SE-SNe; Smartt 2009).
+arXiv:2301.03610v1  [astro-ph.HE]  9 Jan 2023
+
+ID2
+Sawada & Suwa
+aspect of the SN explosion mechanism accurately (e.g.,
+Maeda & Tominaga 2009; Suwa & Tominaga 2015).
+Details of the explosion mechanism of core-collapse
+supernovae (CCSNe) are not yet fully understood. The
+most promising scenario is the delayed neutrino-driven
+explosion (Bethe & Wilson 1985). While this scenario
+had once not been reproduced by numerical simulations,
+the situation has brought substantial progress over a few
+decades. Now, there is an increasing number of success-
+ful examples of reproducing explosions in multidimen-
+sional hydrodynamic simulations, with a detailed neu-
+trino transport (see, e.g.,
+Lentz et al. 2015; Takiwaki
+et al. 2016; M¨uller et al. 2017; O’Connor & Couch 2018;
+Glas et al. 2019; Bollig et al. 2021; Burrows & Vartanyan
+2021; Bruen et al. 2022, and references therein).
+Al-
+though the details now depend on the numerical meth-
+ods and physical approximations employed in each sim-
+ulation, there seems to be a general understanding that
+the explosion succeeds by the growth of the hydrody-
+namic instability over a sufficient time. Indeed, most, if
+not all, of those state-of-the-art simulations, have shown
+a slow increase of explosion energy, and the growing rate
+of the explosion energy is typically ˙Eexpl = O(0.1) Bethe
+s−1 (1 Bethe≡ 1 × 1051 erg), especially for 3D simula-
+tions.
+However, recent several studies have shown that to
+reproduce the typical observed mass of 56Ni by the ex-
+plosive nucleosynthesis in the ejecta, the growth rate
+of the explosion energy of ˙Eexpl = O(1) Bethe s−1 is re-
+quired in several methods (Sawada & Maeda 2019; Suwa
+et al. 2019; Saito et al. 2022). Sawada & Maeda (2019)
+found the inverse-correlation between 56Ni yield and ex-
+plosion energy growth rate ˙Eexpl by 1D simulations with
+the simple thermal-bomb modeling and post-processing
+detailed-nucleosynthesis, and Suwa et al. (2019) also
+came to the same conclusion by conducting hydrody-
+namic simulations with an approximate neutrino heat-
+ing model that self-consistently follows core-collapse and
+shock-revival.
+Saito et al. (2022) also confirmed this
+trend, using the same method as Sawada & Maeda
+(2019), but modeled for individual objects to reduce ob-
+servational uncertainties.
+If these results are correct,
+the current multi-D simulations, which give explosion
+energy growth rates of ˙Eexpl = O(0.1) Bethe s−1, would
+be observationally unfavorable. We refer to this issue
+as the nickel mass problem (‘56Ni problem,’ hereafter)
+in this paper. However, this 56Ni problem is still under
+some debate.
+In particular, Imasheva et al. (2023) just recently
+pointed out the most obvious question to the 56Ni prob-
+lem. Imasheva et al. (2023) used the same method as
+Sawada & Maeda (2019), with simple thermal injection
+modeling and post-processing detailed nucleosynthesis,
+but scrutinized the treatment of initial conditions. In
+the recent slow explosion scenario, the pre-SN star ex-
+periences sufficient gravitational contraction just before
+the successful explosion.
+They found that the corre-
+lation between 56Ni yield and explosion energy growth
+rate ˙Eexpl is the result of ignoring this initial collapse.
+They argued that this correlation disappears when the
+initial collapse is included and also that further initial
+collapse inversely results in more 56Ni being synthesized
+in slower explosions.
+Their arguments also apply to Sawada & Maeda
+(2019) and Saito et al. (2022), but not to Suwa et al.
+(2019). Suwa et al. (2019) solved self-consistently the
+core collapse and shock revival with the light-bulb
+scheme and found this correlation even though they
+took into account the initial collapse phase. This result
+is inconsistent with the conclusion of Imasheva et al.
+(2023). Note that Suwa et al. (2019) performed no de-
+tailed nucleosynthesis calculations. Instead, they esti-
+mated the 56Ni amount simply by the temperature of
+hydrodynamic simulations. Therefore, we perform hy-
+drodynamic and detailed nucleosynthesis calculations in
+this study. This study aims to clarify the detailed pic-
+ture of how 56Ni synthesis occurs in the current CCSN
+explosion scenario. By clarifying this picture, we also
+expect to explain the origin of the differences between
+the two studies and, by extension, the cause of the 56Ni
+problem itself.
+In this paper, therefore, we simulate one-dimensional
+hydrodynamics in the light-bulb scheme as in Suwa et al.
+(2019), then perform detailed nucleosynthesis in a post-
+process manner. The goal of this study is to present a
+detailed picture of 56Ni nucleosynthesis in CCSNe with
+self-consistent explosion modeling. Furthermore, we aim
+to sort out the controversial 56Ni problem. In Section
+2, we describe our simulation methods, the progenitor
+models, and post-processing analysis. Our results are
+summarized in Section 3. In Section 4, we revisit the
+56Ni problem through a detailed comparison of our re-
+sults and observations, and discuss the uncertainties in-
+volved. We conclude in Section 5.
+2. SIMULATION METHODS
+Following the computational setup performed in Suwa
+et al. (2019), we employ a 1D Lagrangian Newtonian
+hydrodynamic code based on blcode.2 Basic equations
+under a spherically symmetric configuration, as we per-
+2 This code is a prototype code of SNEC (Morozova et al. 2015), and
+available from https://stellarcollapse.org
+
+Updating the 56Ni Problem in CCSNe
+3
+form in this paper, are given as follows:
+∂r
+∂Mr
+=
+1
+4πr2ρ ,
+(1)
+Dv
+Dt = −GMr
+r2
+− 4πr2 ∂P
+∂Mr
+,
+(2)
+Dϵ
+Dt = −P D
+Dt
+�
+1
+ρ
+�
++ H − C ,
+(3)
+where r is the radius, Mr is the mass coordinate, t
+is time, ρ is the density, v is the radial velocity, P is
+pressure, ϵ is the specific internal energy, and D/Dt ≡
+∂/∂t + vr∂/∂r is the Lagrangian time derivative. The
+artificial viscosity of Von Neumann & Richtmyer (1950)
+is employed to capture a shock. The system of equa-
+tions (1)-(3) is closed with the Helmholtz equation of
+state (Timmes & Swesty 2000), which describes the stel-
+lar plasma as a mixture of arbitrarily degenerate and
+relativistic electrons and positrons, black-body radia-
+tion, and ideal Boltzmann gases of a defined set of fully
+ionized nuclei, taking into account corrections for the
+Coulomb effects.
+In this work, neutrino heating and cooling are added
+by a light-bulb scheme. In the light-bulb scheme, neu-
+trino cooling is given as a function of temperature, and
+neutrino heating is a function of the radius with param-
+eterized neutrino luminosity. The heating term H and
+the cooling term C, terms in Equation (3) are assumed
+to be
+H = 1.544 × 1020 erg g−1 s−1
+×
+�
+Lνe
+1052MeV
+� �
+rνe
+100km
+�−2 �
+Tνe
+4.0MeV
+�2
+,
+(4)
+C = 1.399 × 1020 erg g−1 s−1 ×
+�
+T
+2.0MeV
+�6
+.
+(5)
+Here, we fix the neutrino temperature as Tνe = 4 MeV.
+We take into account these terms only in the post-shock
+regime. We modified the inner boundary conditions so
+that the innermost mass shell does not shrink within 50
+km from the center to mimic the existence of a proto-
+neutron star (PNS). Also, the light-bulb scheme in this
+study tends to overestimate the neutrino-driven wind
+from the PNS surface at the post-explosion phase be-
+cause it keeps giving a constant neutrino luminosity.
+Therefore, in this study, we consider the mass coordi-
+nate that experienced r < 200 km as the neutrino-driven
+wind and separate it from the ejecta.
+The numerical computational domain contains 1.5M⊙
+and uses a 1500 grid with a mass resolution of 10−3M⊙.
+We set the inter boundary at Ms/kb=4 − 0.5M⊙ for each
+density [g/cm3]
+mass radius [Msun]
+ 12.3M
+ 16.0M
+ 18.0M
+ 19.5M
+10-10
+10-5
+100
+105
+1010
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+104
+105
+106
+107
+108
+109
+ 1.2
+ 1.3
+ 1.4
+ 1.5
+ 1.6
+ 1.7
+ 1.8
+ 1.9
+ 2
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+ 4.5
+ 5
+ 5.5
+ 6
+density [g/cm3]
+entropy [kb/baryon]
+Mass Radius [Msun]
+Figure 1. Density structure as a function of the enclosed
+mass for the considered progenitors with MZAMS = 12.3M⊙
+(cyan line), 16.0M⊙ (blue line), 19.7M⊙ (red line), and
+21.0M⊙ (magenta line), and its details with the entropy per
+nucleon.
+pre-explosion star. Each mass coordinate captures the
+time evolution of the hydrodynamic quantities, so that
+nucleosynthesis calculations are performed as a post-
+processing analysis with this trajectory. We calculate a
+reaction network of 640 nuclear species with the torch
+code (Timmes 1999).
+The initial conditions adopted in this study are a sub-
+set of non-rotating stars with solar metallicity, which
+evolved from the main sequence to the onset of iron-core
+collapse, as published by Sukhbold et al. (2018). The
+physics of this set of progenitors was discussed in detail
+in this literature. Figure 1 shows the density structures
+
+4
+Sawada & Suwa
+ 1x109
+ 1x1010
+ 1x1011
+ 1.5
+ 1.6
+ 1.7
+ 1.8
+ 1.9
+ 2
+temperature [K]
+mass radius [Msun]
+-2x109
+-1.5x109
+-1x109
+-5x108
+ 0
+ 5x108
+ 1x109
+ 1.5x109
+ 2x109
+ 1.5
+ 1.6
+ 1.7
+ 1.8
+ 1.9
+ 2
+velocity [cm/s]
+mass radius [Msun]
+Figure 2. Time evolution of the velocity (top) and the tem-
+perature (bottom) as a function of the mass coordinate for
+model 16.0M⊙. In both panels, each snapshot time corre-
+sponds to approximately every 0.1 seconds from 0.5 seconds
+to 1.5 seconds from the start of the simulation. The gray
+line corresponds to T = 5 × 109 K.
+of the progenitor as a function of the enclosed mass, and
+its details with the entropy per nucleon.
+3. RESULT
+3.1. Overview of the explosion dynamics
+Figure 2 shows the time evolution of radial velocity
+and temperature as a function of the mass coordinate
+for the model 16.0M⊙. We first use an example of a
+model with MZAMS = 16.0M⊙ throughout this section.
+From the velocity figure, it can be seen that the shock
+begins to propagate outward from the point where the
+silicon/oxygen (Si–O) layer (≈ 1.62M⊙) accretes onto
+the shock wave, due to the rapid decrease in ram pres-
+sure (Marek & Janka 2009; Suwa et al. 2016). From the
+temperature figure, we can confirm that the post-shock
+ 1x107
+ 1x108
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+ 1.2
+radius [cm]
+time [sec]
+Figure 3. Radius evolution of Lagrangian mass shells with
+time for the explosion (Lν = 3 × 1052 erg s−1) and non-
+exploding model (Lν = 0 erg s−1) of 16.0M⊙.
+The thick
+black solid lines are the mass shells, spaced in steps of 0.1
+M⊙, and the thin gray solid/dashed lines are spaced in steps
+of 0.02 M⊙.
+The difference between the dotted and solid
+lines corresponds to the explosion and non-exploding models,
+respectively.
+The blue line marks the shock radius of the
+explosion model.
+temperature of the ejecta is spatially almost constant so
+we define the shock temperature as the temperature of
+the material just behind the shock wave.
+Figure 3 shows that the mass shell until the arrival of
+the shock in the explosion model is consistent with its
+behavior in the non-exploding model. In other words,
+we can confirm that the behavior of the mass shell up to
+the arrival of the shock is independent of the explosion
+detail. Thus, by overlaying the shock evolution on the
+trajectory of the mass shell in the non-exploding model,
+we can compare several models at once to see where the
+shock impacts each of the mass shells.
+In the following subsections, we present the results
+focusing on the effect of the explosion energy growth
+rate ˙Eexpl on 56Ni nucleosynthesis. The results are sum-
+marized in Table 1.
+These yields consist only of un-
+bound 56Ni by gravity as determined by a 10-second
+simulation. We first use an example of a model with
+MZAMS = 16.0M⊙ throughout Sections 3.2 and 3.3.
+3.2. Hydrodynamics and 56Ni Synthesis Region
+Figure 4 shows the time evolution of the shock for the
+models Lν = 3, 5, and 7 × 1052 erg s−1 and the trajec-
+tory of the mass shell in the unexploded model. In each
+model, the time evolutions of the shock are shown by col-
+ored lines for the range where the shock satisfies T9 > 5
+(T9 ≡ T/109 K), and by black dashed lines for the range
+where T9 < 5. We refer to the mass coordinate that can
+
+Updating the 56Ni Problem in CCSNe
+5
+ 1x107
+ 1x108
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+radius [cm]
+time [sec]
+Lnu=3e52 [erg/s]
+Lnu=5e52 [erg/s]
+Lnu=7e52 [erg/s]
+Lnu=9e52 [erg/s]
+Figure 4. The time evolution of the shock radius in models
+Lν = 3, 5, and 7×1052 erg s−1 with the mass shell trajectory
+in the unexploded model, on the time-radius plane.
+spread at the shock temperature of T9 ≈ 5 as MT9=5.
+We find that in all models MT9=5 is near the mass co-
+ordinate with the enclosed mass ≈ 1.65M⊙, which is
+indicated by the dotted line in Figure 4. More detailed
+values are given in Table 1, and this trend is almost
+universal, independent of the progenitor models. In all
+models, we find that MT9=5 is near the mass coordinate
+with an enclosed mass of approximately 1.65M⊙, as in-
+dicated by the dotted line in Figure 4. More detailed
+values are given in Table 1, and this trend is nearly uni-
+versal and independent of the progenitor models.
+Qualitatively, this can be understood by using a zero-
+order approximation to estimate the shock radius at
+which the shock temperature is T9 = 5. When apply-
+ing a simple fireball model in which the region behind
+the shock wave is uniform and dominated by radiation
+pressure (e.g., Woosley et al. 2002), we can estimate the
+following relation between the temperature T, the shock
+radius rsh and the explosion energy Eexpl as follows:
+Eexpl = 4π
+3 r3
+sh(t) aT 4 ,
+(6)
+where a is the radiation constant. Then with Eexpl(t) ≡
+1051 ergs, the radius with T9 = 5 (rT9=5) can be esti-
+mated as follows:
+rT9=5 ≈ 3.6 × 108(Eexpl/1051)1/3 cm .
+(7)
+This estimated radius is classically well-known (e.g.,
+Woosley et al. 2002; Nomoto et al. 2013). If we consider
+the time evolution of Eexpl(t) = ˙Eexpl·t, this classical ra-
+dius is satisfied with an adequate large ˙Eexpl. However,
+at the shock velocity Vsh = 109 cm s−1, it takes less than
+ 0
+ 1x1050
+ 2x1050
+ 3x1050
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+explosion energy [erg]
+time after bounce [sec]
+explosion energy Esim
+estimated energy Eapp
+Figure 5. Comparison of the explosion energy in the sim-
+ulation Esim (dashed line) with the estimated energy in the
+fireball approximation Eapp (solid line), which comes from
+Eq (8) in models Lν = 2, 3, and 5 × 1052 erg s−1.
+The
+horizontal axis is the post-bounce time.
+1 second to reach this radius. In other words, if the case
+of ˙Eexpl ≲ 1 Bethe s−1, it takes a few seconds to reach 1
+Bethe, and obviously, the radius of T9 = 5 will be small.
+In fact, from Figure 4, we can confirm that even in this
+simulation, the radius of T9 = 5 is reduced in the case of
+low- ˙Eexpl. But at the same time, the time evolution of
+the shock radius is also slower down for lower- ˙Eexpl mod-
+els, and the mass shell falls more inward due to collapse.
+Eventually, the ‘mass coordinate’ of MT9=5 seems to be
+approximately the same regardless of the ˙Eexpl.
+This result suggests a very interesting trend.
+56Ni
+is synthesized mainly by complete Si burning at T ≳
+5×109 K (see in detail in Appendix A and Woosley et al.
+1973). Thus, this hydrodynamical result suggests that
+the outermost mass coordinates, where 56Ni is primar-
+ily synthesized, are insensitive to the explosion energy
+growth rate ˙Eexpl. To confirm this trend in more detail,
+we next discuss the results of nucleosynthesis calcula-
+tions.
+Although this is not relevant to the main focus of
+this paper, we show in Figure 5 for reference the com-
+parison of the explosion energy in the simulation Esim
+with the estimated energy in the fireball approximation
+Eapp. The explosion energy Esim in the hydrodynamical
+simulation is defined as the integral of the sum of spe-
+cific internal, kinetic, and gravitational energies over all
+zones, in which it is positive. The estimated energy in
+the fireball approximation Eapp is given by the follow-
+ing equation using only the shock radius rsh and shock
+
+6
+Sawada & Suwa
+temperature T:
+Eapp = 4π
+3 r3
+sh(t) aT 4f(T9) ,
+(8)
+where f(T9) = 1 + (7/4) · T 2
+9 /(T 2
+9 + 5.3) is a correction
+term to account for both radiation pressure and non-
+degenerate electron-positron pairs (e.g., Freiburghaus
+et al. 1999). As Figure 5 shows, this simple estimation is
+able to reproduce the explosion energy of the simulation
+with good enough accuracy. This supports the validity
+of the above discussion, and also suggests that thermal
+energy is dominant in the early phases of the explosion.
+3.3. Nucleosynthesis: Distribution of 56Ni synthesis
+Figure 6 shows the abundance distribution as a func-
+tion of the mass coordinate, for Lν = 3, 5, and 7 × 1052
+erg s−1 with 16.0M⊙.
+As shown in Figure 6, focus-
+ing specifically on 56Ni, we can confirm that the outer-
+most mass radius, where 56Ni is primarily synthesized,
+is in a similar position independent of the explosion en-
+ergy growth rate ˙Eexpl. In Table 1, we show the out-
+ermost mass radius where 56Ni is largely synthesized
+(here, we define it as X(56Ni) > 0.5). However, at the
+same time, the innermost mass radius, which is gravita-
+tionally unbounded, depends strongly on the explosion
+energy growth rate ˙Eexpl.
+4. DISCUSSION: UPDATE ‘NI PROBLEM’
+Figure 7 shows the synthesized amount of 56Ni as a
+function of the explosion energy growth rate ˙Eexpl. It
+can be clearly seen that there is a decreasing trend of
+the synthesized amount of 56Ni toward decreasing ˙Eexpl.
+The reason for this trend is explained in section 3.3, but
+this figure tells us that the same trend is generally ob-
+served regardless of the mass and structure of the pro-
+genitor.
+For comparison with observations, in Figure 7, we
+adopted two typical values based on a recent systematic
+survey for more than 300 events of CCSNe; 0.07M⊙3 as
+the median estimated from stripe-envelope supernovae
+(SE-SNe) and 0.03M⊙ from Type-II SNe (Rodr´ıguez
+et al. 2021, 2022). Note that the figure plots the syn-
+thesized amount of 56Ni ; not all 56Ni can be ejected. In
+other words, the figure shows the maximum amount of
+3 The ∼ 0.07M⊙ is often adopted as a typical value obtained for
+well-studied nearby SNe is on average (e.g., SN 1987A, SN 1994I,
+SN 2002ap; Arnett et al. 1989; Iwamoto et al. 1994; Mazzali et al.
+2002) and we adopt this value in previous studies. However, in
+this study, we clearly mention here that we do not use 0.07M⊙
+in the context of the typical for nearby SNe because we con-
+sider observational constraints from recently updated large-scale
+observational data.
+56Ni that can be ejected by each CCSN model, and if
+the calculated mass of 56Ni is larger than the observed
+value, then the model can reproduce the observed value.
+First, compared to the median value of Type II super-
+novae 0.03M⊙, even a modern slow explosion ( ˙Eexpl ≲ 1
+Bethe s−1 ) provides enough amount of 56Ni to repro-
+duce the observations. On the other hand, compared
+to the SE-SNe median of 0.07M⊙, a very rapid explo-
+sion of ˙Eexpl≳ 2 Bethe s−1 is required to reproduce this
+value. This translates to a time scale of t ≲ 0.5 sec-
+onds to the typical explosion energy ∼ 1.0 Bethe, and
+this timescale is very difficult to reproduce with current
+multi-D self-consistent calculations.
+Here we discuss a few caveats in this problem as fol-
+lows.
+1. [Observation of Type-II SNe]
+The typical 56Ni mass of canonical-CCSNe has
+been extensively discussed by large-scale observa-
+tions in recent years. In particular, Type II SNe,
+when volume-limited, account for nearly ∼ 60%
+of the observed CCSNe (e.g., Li et al. 2011; Jones
+et al. 2021).
+Recently, Type II SNe have been
+found to have lower median nickel masses than SE-
+SNe (e.g., Anderson 2019), confirming that this is
+not due to observational bias (Ouchi et al. 2021).
+Furthermore, the observed kinetic energy is also
+found to have a lower median value than the clas-
+sical typical value (∼ 0.6 Bethe;
+Martinez et al.
+2022). These facts also support the possibility that
+the ‘slow’ explosion results in the current state-of-
+the-art simulations are relatively consistent with
+standard Type II SNe.
+However, nickel synthe-
+sis and explosion energy (MNi ≈ 0.03M⊙ and
+Eexpl ∼ 0.6 Bethe) still remain important bench-
+marks for multidimensional self-consistent simula-
+tions, and it should be checked whether they are
+truly achieved. And, another important point is
+that this is only a statement of the median. Ac-
+cording to Ouchi et al. (2021), the fitting function
+for the cumulative histogram for observed 56Ni
+masses of each CCSNe is f(x) = tanh (14.60 × x)
+4 as a variable of observed 56Ni masses.
+This
+roughly implies that more than 20% of the Type II
+supernovae synthesize 56Ni above 0.075M⊙. While
+0.03M⊙ is a somewhat explainable value, this
+value is challenging to reproduce in multi-D self-
+consistent simulations. So, we need to explain and
+4 This function is obtained by fitting non-linear least squares of
+observed reports of Type II SNe with a sample size of 115 events
+(Ouchi et al. 2021).
+
+Updating the 56Ni Problem in CCSNe
+7
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+ 1.45  1.5  1.55  1.6  1.65  1.7  1.75  1.8  1.85  1.9
+proto-NS
+56Ni
+16O
+28Si
+40Ca
+12C
+44Ti
+28Mg
+abundance
+mass radius[M]
+(a)
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+ 1.45  1.5  1.55  1.6  1.65  1.7  1.75  1.8  1.85  1.9
+proto-NS
+abundance
+mass radius[M]
+(b)
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+ 1.45  1.5  1.55  1.6  1.65  1.7  1.75  1.8  1.85  1.9
+proto-NS
+abundance
+mass radius[M]
+(c)
+Figure 6. Abundance distribution as a function of the enclosed mass Mr, for (a) Lν = 3 × 1052 erg s−1, (b) Lν = 5 × 1052 erg
+s−1, and (c) Lν = 7 × 1052 erg s−1. All the models here are with 16.0M⊙ of Sukhbold et al. (2018). In all panels, the vertical
+dotted grey line indicates the location of the mass shell with an enclosed mass 1.65M⊙.
+ 0
+ 0.02
+ 0.04
+ 0.06
+ 0.08
+ 0.1
+ 0.12
+ 0.14
+ 5x1050
+ 1x1051
+ 2x1051
+ 3x1051 4x1051
+median of SESNe
+median of typeII-SNe
+nickel mass [Msun]
+Edot [erg/sec]
+ 12.3M
+ 16.0M
+ 18.0M
+ 19.5M
+Figure 7. The amount of 56Ni as a function of the growth
+rate of the explosion energy, ˙Eexpl. The gray line indicates a
+typical value of 56Ni , 0.07M⊙.
+reproduce the high 56Ni objects that will exist to
+some extent.
+2. [Multidimensional effect]
+How the amount of synthesized 56Ni changes in a
+multi-D explosion model is one of the issues to be
+discussed. Since this model is a 1D model and ex-
+plodes only with thermal energy as shown in Fig-
+ure 5, the temperature should be higher than the
+multi-D model, especially considering the geomet-
+ric structure and the change to kinetic energy in
+the non-radial direction (Suwa et al. 2019). There-
+fore, we should note that the same ˙Eexpl in a multi-
+D model would have less 56Ni than in a 1D model.
+In fact, with the exception of particular model
+results (Bollig et al. 2021,
+discussed next), the
+multi-D self-consistent simulation has even more
+difficulty with 56Ni synthesis than the estimate of
+this study (e.g., Bruen et al. 2022).
+Therefore,
+for the same synthesis conditions, the 1D model
+gives a robust maximum limit on the volume and
+the amount of 56Ni synthesis. The additional 56Ni
+amount newly occurring due to the multi-D effect
+will be discussed next.
+3. [Additional mechanism to add 56Ni ]
+Another possibility for an additional 56Ni , and one
+of the most often cited candidates for a solution
+to this problem, is the ‘outflow’ from the PNS sur-
+face for several seconds of the post-explosion phase
+(e.g., Wongwathanarat et al. 2017; Witt et al.
+2021). Recent detailed simulations have predicted
+proton-rich ejecta in the post-explosion ‘outflow’
+(e.g., Bruenn et al. 2016).
+In particular, Bollig
+et al. (2021) have observed the downflow/outflow
+system that results in a smooth and efficient tran-
+sition from the incoming flow to the outgoing flow,
+with the outflow providing 56Ni to ≲ 0.05M⊙.
+However, we already found that the contribution
+of such replenishment is small for regular CCSNe
+explosions (Sawada & Suwa 2021). That is, this
+outflow system is part of an ‘energetic’ model of
+the state-of-the-art simulations that succeeds in
+producing sufficient amounts of 56Ni , and it is de-
+batable whether this outflow system contributes
+to canonical-CCSNe explosions.
+
+8
+Sawada & Suwa
+56Ni
+PNS
+New Picture of the 56Ni problem:
+by more realistic explosion model
+(light-bulb scheme)
+mass coordinate
+*(slow)-expl: ሶ𝐸expl ≲ 1.0 × 1051 erg s−1 , (rapid)-expl: ሶ𝐸expl > 1.0 × 1051 erg s−1
+innermost (ejectable) mass radius 
+is sensitive to explosion ሶ𝐸expl !!
+PNS
+Imasheva, Janka,& Weiss (2023) :
+by collapsed thermal bomb model
+(rapid)
+(slow)
+PNS
+Fixed, In case of thermal bomb.
+Sawada & Maeda (2019) :
+by uncollapsed thermal bomb model
+mass coordinate
+mass coordinate
+(rapid)
+(slow)
+(rapid)
+(slow)
+56Ni
+56Ni
+Fixed, In case of thermal bomb.
+56Ni synthesized mass 
+radius is insensitive !
+56Ni problem;
+56Ni mass depends on ሶ𝐸expl
+We repropose the 56Ni problem
+Figure 8.
+Schematic picture to the ‘56Ni problem in CCSNe’ as suggested by the the previous studies and this study, respec-
+tively. At first, Sawada & Maeda (2019) raised the 56Ni problem because the 56Ni synthesized region varies with the growth rate
+of the explosion energy ˙Eexpl when an un-collapsed progenitor is used. When taking into account that the progenitor collapses
+just before the explosion, the 56Ni synthesized region becomes insensitive to ˙Eexpl, and thus, Imasheva et al. (2023) proposed
+a disappearance of the 56Ni problem. However, this study re-proposes the 56Ni problem on the grounds that while the 56Ni
+synthesized region is insensitive to ˙Eexpl, the ejectable innermost mass radius depends on the ˙Eexpl, as calculated using the
+light-bulb scheme in which the PNS masses is determined self-consistently.
+4. [Comparison to Imasheva et al. (2023) ]
+Finally, we compare our results with those of Ima-
+sheva et al. (2023) who just recently pointed out
+the most obvious doubts about the ‘56Ni prob-
+lems’. Figure 8 is a schematic comparison of our
+results with theirs.
+Their argument is that the
+correlation between
+˙Eexpl
+and 56Ni disappears
+when the initial collapse is included, and that fur-
+ther initial collapse inversely results in more 56Ni
+being synthesized in slower explosions.
+Noting
+that the innermost ejecta radius is fixed in the
+thermal bomb model, their argument is consis-
+tent with the present results where 56Ni is syn-
+thesized to almost the same mass coordinate in-
+dependent of ˙Eexpl, shown in Figure 4. We also
+confirm that 56Ni is synthesized slightly more out-
+wardly in models with slower initial collapse (i.e.,
+the MZAMS = 19.5M⊙ model).
+The difference
+from their study is the treatment of the inner-
+most mass coordinate of the ejecta, i.e., their inner
+boundary condition. Our explosion model deter-
+mines the innermost mass coordinate of the ejecta
+self-consistently.
+We then found that the inner-
+most material that could be ejected in the high-
+˙Eexpl model could not achieve the escape condi-
+tion in the low- ˙Eexpl model, leading to moving
+the innermost mass coordinate of the ejecta out-
+ward. In fact, for low ˙Eexpl models, Imasheva et al.
+(2023) themselves had mentioned the possibility
+that some of the innermost material may be unable
+to achieve escape conditions, remain gravitation-
+ally bound, and thus not contribute to the yield,
+and we confirmed this in this paper. Although our
+results are consistent with theirs, we confirm that
+56Ni
+problems reappear because the innermost
+ejecta radius shifts depending on the intensity of
+the ˙Eexpl.
+We conclude that the modern slow explosion ( ˙Eexpl ≲
+1 Bethe s−1 ) can reproduce the observations of a stan-
+dard Type II supernova. However, this is only a state-
+ment of a principal possibility. How much 56Ni can be
+synthesized is an important benchmark for multidimen-
+sional self-consistent simulations, and it should be con-
+firmed whether the median value for a standard Type II
+supernova (≈ 0.03M⊙) is indeed achieved.
+On the other hand, the 56Ni problem clearly exists in
+the explosion mechanism of SE-SNe, that is, the modern
+slow explosions cannot reproduce the SE-SNe observa-
+tions. As a simple and straightforward solution that sat-
+isfies the 56Ni problem without fine-tuning, we conclude
+that the SE-SNe favors active explosions in the early
+stages of shock revival ( ˙Eexpl ≳ 2 Bethe s−1).
+Since
+such high explosion energies are probably inconsistent
+with the standard explosion mechanism, the 56Ni prob-
+lem may require a different explosion mechanism for the
+SE-SNe. Anderson (2019) had already suggested from
+observations, but our results once again imply signifi-
+cant differences in the progenitor structures and/or the
+explosion mechanism between type II and SE-SNe.
+5. SUMMARY
+
+Updating the 56Ni Problem in CCSNe
+9
+In this paper, we investigated the effect of the explo-
+sion energy growth rate ˙Eexpl
+on the behavior of 56Ni
+nucleosynthesis in CCSNe. For numerical simulations,
+we employed the 1D Lagrangian hydrodynamic code in
+which neutrino heating and cooling terms are taken into
+account by the light-bulb approximation.
+The initial
+conditions are taken from Sukhbold et al. (2018), which
+have MZAMS = 12.3, 16.0, 18.0, and 19.5M⊙.
+Our first purpose was to present a detailed picture of
+56Ni nucleosynthesis in CCSNe with self-consistent ex-
+plosion modeling.
+We found that 56Ni is synthesized
+up to the almost same mass coordinate independent
+of ˙Eexpl. We also found that in the low- ˙Eexpl model,
+some of the innermost material that was ejected in the
+high- ˙Eexpl model failed to achieve the escape condition,
+leading to moving the innermost mass coordinate of the
+ejecta to the outer positions. This means that while the
+56Ni nucleosynthesis volume is insensitive to the nature
+of the explosion, the ejected amount of 56Ni is highly
+dependent on how much of the innermost PNS surface
+region is ejectable.
+Furthermore, our other goal was to sort out the re-
+cent controversial 56Ni problem. We found that there is
+a decreasing trend of the synthesized amount of 56Ni
+toward decreasing
+˙Eexpl.
+Compared to observations,
+we found that the modern slow explosion ( ˙Eexpl ≲ 1
+Bethe s−1 ) can reproduce the observations of a stan-
+dard Type II supernova in a principal. However, this
+does not mean that the 56Ni problem has been solved,
+and the 56Ni synthesis (MNi ≈ 0.03M⊙) still remains an
+important benchmark for multi-D self-consistent simu-
+lations.
+It should be checked whether they are truly
+achieved. And extremely important are the comparison
+results with SE-SNe. We found that the median value
+of SE-SNe (MNi ≈ 0.07M⊙) is challenging to reproduce
+in the modern slow explosion ( ˙Eexpl ≲ 1 Bethe s−1 ).
+As a simple and straightforward solution that satisfies
+the amount of 56Ni without fine-tuning, the SE-SNe fa-
+vors active explosions in the early stages of shock re-
+vival ( ˙Eexpl ≳ 2 Bethe s−1). Thus, we conclude that
+there are significant differences in the progenitor struc-
+tures and/or the explosion mechanism between type II
+and SE-SNe.
+Software: blcode, torch(Timmes 1999)
+ACKNOWLEDGMENTS
+The
+work
+has
+been
+supported
+by
+Japan
+Soci-
+ety for the Promotion of Science (JSPS) KAKENHI
+grants 21J00825, 21K13964 (RS), 18H05437, 20H00174,
+20H01904, and 22H04571 (YS).
+REFERENCES
+Anderson, J. P. 2019, A&A, 628, A7,
+doi: 10.1051/0004-6361/201935027
+Arnett, W. D. 1982, ApJ, 253, 785, doi: 10.1086/159681
+Arnett, W. D., Bahcall, J. N., Kirshner, R. P., & Woosley,
+S. E. 1989, ARA&A, 27, 629,
+doi: 10.1146/annurev.aa.27.090189.003213
+Bethe, H. A., & Wilson, J. R. 1985, ApJ, 295, 14,
+doi: 10.1086/163343
+Bodansky, D., Clayton, D. D., & Fowler, W. A. 1968,
+ApJS, 16, 299, doi: 10.1086/190176
+Bollig, R., Yadav, N., Kresse, D., et al. 2021, ApJ, 915, 28,
+doi: 10.3847/1538-4357/abf82e
+Bruen, S. W., Sieverding, A., Lentz, E. J., et al. 2022,
+arXiv e-prints, arXiv:2211.12675.
+https://arxiv.org/abs/2211.12675
+Bruenn, S. W., Lentz, E. J., Hix, W. R., et al. 2016, ApJ,
+818, 123, doi: 10.3847/0004-637X/818/2/123
+Burrows, A., & Vartanyan, D. 2021, Nature, 589, 29,
+doi: 10.1038/s41586-020-03059-w
+Clifford, F. E., & Tayler, R. J. 1965, MmRAS, 69, 21
+Fowler, W. A., & Hoyle, F. 1964, ApJS, 9, 201,
+doi: 10.1086/190103
+Freiburghaus, C., Rembges, J. F., Rauscher, T., et al. 1999,
+ApJ, 516, 381, doi: 10.1086/307072
+Glas, R., Just, O., Janka, H. T., & Obergaulinger, M. 2019,
+ApJ, 873, 45, doi: 10.3847/1538-4357/ab0423
+Hamuy, M. 2003, ApJ, 582, 905, doi: 10.1086/344689
+Hix, W. R., & Thielemann, F.-K. 1999, ApJ, 511, 862,
+doi: 10.1086/306692
+Imasheva, L., Janka, H.-T., & Weiss, A. 2023, MNRAS,
+518, 1818, doi: 10.1093/mnras/stac3239
+Iwamoto, K., Nomoto, K., H¨oflich, P., et al. 1994, ApJL,
+437, L115, doi: 10.1086/187696
+Jerkstrand, A., Timmes, F. X., Magkotsios, G., et al. 2015,
+ApJ, 807, 110, doi: 10.1088/0004-637X/807/1/110
+Jones, D. O., Foley, R. J., Narayan, G., et al. 2021, ApJ,
+908, 143, doi: 10.3847/1538-4357/abd7f5
+Lentz, E. J., Bruenn, S. W., Hix, W. R., et al. 2015, The
+Astrophysical Journal, 807, L31,
+doi: 10.1088/2041-8205/807/2/L31
+
+10
+Sawada & Suwa
+Table 1. Summary of simulations
+MZAMS
+Ms/kb=4
+a
+ξs/kb=4
+b
+Lνe
+˙Eexpl
+c
+MPNS
+d
+MT9=5
+e
+Mr(X56Ni > 0.5) f
+M56Ni
+g
+[M⊙]
+[M⊙]
+[1052erg s−1]
+[1051erg s−1]
+[M⊙]
+[M⊙]
+[M⊙]
+[M⊙]
+12.3
+1.462
+0.837
+1.4
+0.46
+1.464
+1.514
+1.523
+0.049
+2
+0.60
+1.464
+1.509
+1.516
+0.047
+3
+1.04
+1.458
+1.507
+1.512
+0.049
+4
+1.75
+1.449
+1.507
+1.512
+0.057
+5
+2.88
+1.427
+1.510
+1.514
+0.078
+16.0
+1.618
+0.700
+2
+0.48
+1.617
+1.655
+1.659
+0.039
+3
+0.82
+1.615
+1.651
+1.655
+0.039
+4
+1.35
+1.605
+1.650
+1.653
+0.045
+5
+2.39
+1.582
+1.653
+1.671
+0.068
+6
+3.47
+1.561
+1.651
+1.656
+0.087
+7
+4.62
+1.536
+1.650
+1.655
+0.109
+18.0
+1.535
+0.793
+2
+0.83
+1.537
+1.593
+1.607
+0.062
+3
+1.06
+1.537
+1.593
+1.600
+0.059
+4
+1.49
+1.535
+1.593
+1.599
+0.060
+5
+2.12
+1.530
+1.593
+1.600
+0.065
+6
+3.10
+1.514
+1.595
+1.602
+0.081
+7
+4.23
+1.514
+1.608
+1.615
+0.112
+19.5
+1.714
+0.762
+2
+0.70
+1.912
+1.962
+1.970
+0.054
+3
+1.22
+1.720
+1.806
+1.816
+0.090
+4
+1.64
+1.717
+1.801
+1.811
+0.089
+5
+2.10
+1.714
+1.800
+1.810
+0.092
+6
+2.80
+1.709
+1.801
+1.810
+0.097
+7
+3.69
+1.691
+1.801
+1.810
+0.114
+a
+Mass Coordinate with s = 4kB baryon−1.
+b
+Compactness parameter of Ms/kb=4
+which is defined as
+(Ms/kb=4/M⊙)/(Rs/kb=4/1000km).
+c
+Neutrino luminosity.
+d
+PNS mass which we define as the mass coordinate that expe-
+rienced r < 200 km as the neutrino-driven wind and separate it
+from the ejecta.
+e
+Mass Coordinate of shock at a time when the post-shock tem-
+perature is T = 5 × 109 K.
+f
+Outermost mass radius where the 56Ni is X56Ni > 0.5 synthe-
+sized.
+g
+56Ni mass computed in a post-process with the 640-isotope.
+Li, W., Leaman, J., Chornock, R., et al. 2011, MNRAS,
+412, 1441, doi: 10.1111/j.1365-2966.2011.18160.x
+Maeda, K., & Tominaga, N. 2009, Monthly Notices of the
+Royal Astronomical Society, 394, 1317,
+doi: 10.1111/j.1365-2966.2009.14460.x
+Magkotsios, G., Timmes, F. X., Hungerford, A. L., et al.
+2010, ApJS, 191, 66, doi: 10.1088/0067-0049/191/1/66
+Magkotsios, G., Timmes, F. X., & Wiescher, M. 2011, ApJ,
+741, 78, doi: 10.1088/0004-637X/741/2/78
+Marek, A., & Janka, H. T. 2009, ApJ, 694, 664,
+doi: 10.1088/0004-637X/694/1/664
+Martinez, L., Bersten, M. C., Anderson, J. P., et al. 2022,
+A&A, 660, A41, doi: 10.1051/0004-6361/202142076
+Mazzali, P. A., Deng, J., Maeda, K., et al. 2002, ApJL, 572,
+L61, doi: 10.1086/341504
+Morozova, V., Piro, A. L., Renzo, M., et al. 2015, ApJ, 814,
+63, doi: 10.1088/0004-637X/814/1/63
+M¨uller, B., Melson, T., Heger, A., & Janka, H.-T. 2017,
+MNRAS, 472, 491, doi: 10.1093/mnras/stx1962
+Nomoto, K., Kobayashi, C., & Tominaga, N. 2013,
+ARA&A, 51, 457,
+doi: 10.1146/annurev-astro-082812-140956
+
+Updating the 56Ni Problem in CCSNe
+11
+O’Connor, E. P., & Couch, S. M. 2018, ApJ, 865, 81,
+doi: 10.3847/1538-4357/aadcf7
+Ouchi, R., Maeda, K., Anderson, J. P., & Sawada, R. 2021,
+ApJ, 922, 141, doi: 10.3847/1538-4357/ac2306
+Rodr´ıguez, ´O., Maoz, D., & Nakar, E. 2022, arXiv e-prints,
+arXiv:2209.05552. https://arxiv.org/abs/2209.05552
+Rodr´ıguez, ´O., Meza, N., Pineda-Garc´ıa, J., & Ramirez, M.
+2021, MNRAS, 505, 1742, doi: 10.1093/mnras/stab1335
+Saito, S., Tanaka, M., Sawada, R., & Moriya, T. J. 2022,
+ApJ, 931, 153, doi: 10.3847/1538-4357/ac6bec
+Sawada, R., & Maeda, K. 2019, ApJ, 886, 47,
+doi: 10.3847/1538-4357/ab4da3
+Sawada, R., & Suwa, Y. 2021, ApJ, 908, 6,
+doi: 10.3847/1538-4357/abd476
+Seitenzahl, I. R., Timmes, F. X., Marin-Lafl`eche, A., et al.
+2008, ApJL, 685, L129, doi: 10.1086/592501
+Smartt, S. J. 2009, ARA&A, 47, 63,
+doi: 10.1146/annurev-astro-082708-101737
+Sukhbold, T., Woosley, S. E., & Heger, A. 2018, ApJ, 860,
+93, doi: 10.3847/1538-4357/aac2da
+Suwa, Y., & Tominaga, N. 2015, Monthly Notices of the
+Royal Astronomical Society, 451, 282,
+doi: 10.1093/mnras/stv901
+Suwa, Y., Tominaga, N., & Maeda, K. 2019, MNRAS, 483,
+3607, doi: 10.1093/mnras/sty3309
+Suwa, Y., Yamada, S., Takiwaki, T., & Kotake, K. 2016,
+ApJ, 816, 43, doi: 10.3847/0004-637X/816/1/43
+Takiwaki, T., Kotake, K., & Suwa, Y. 2016, Monthly
+Notices of the Royal Astronomical Society, 461, L112,
+doi: 10.1093/mnrasl/slw105
+Thielemann, F.-K., Nomoto, K., & Hashimoto, M.-A. 1996,
+ApJ, 460, 408, doi: 10.1086/176980
+Timmes, F. X. 1999, ApJS, 124, 241, doi: 10.1086/313257
+Timmes, F. X., & Swesty, F. D. 2000, ApJS, 126, 501,
+doi: 10.1086/313304
+Von Neumann, J., & Richtmyer, R. D. 1950, Journal of
+Applied Physics, 21, 232, doi: 10.1063/1.1699639
+Witt, M., Psaltis, A., Yasin, H., et al. 2021, ApJ, 921, 19,
+doi: 10.3847/1538-4357/ac1a6d
+Wongwathanarat, A., Janka, H.-T., M¨uller, E., Pllumbi, E.,
+& Wanajo, S. 2017, ApJ, 842, 13,
+doi: 10.3847/1538-4357/aa72de
+Woosley, S. E., Arnett, W. D., & Clayton, D. D. 1973,
+ApJS, 26, 231, doi: 10.1086/190282
+Woosley, S. E., Heger, A., & Weaver, T. A. 2002, Reviews
+of Modern Physics, 74, 1015,
+doi: 10.1103/RevModPhys.74.1015
+Woosley, S. E., & Weaver, T. A. 1995, ApJS, 101, 181,
+doi: 10.1086/192237
+
+12
+Sawada & Suwa
+APPENDIX
+A.
+56Ni SYNTHESIS CONDITION
+Here, we briefly review the general features of 56Ni nucleosynthesis in CCSNe, which is classically well enough
+understood (e.g., Bodansky et al. 1968; Woosley et al. 1973; Hix & Thielemann 1999). To illustrate the condition of
+56Ni synthesis, for simplicity, we use parameterized adiabatic expansion profiles. Let us focus on a certain Lagrangian
+particle. Assuming that a passing shock wave heats and compresses the particle to peak temperature T0 and peak
+density ρ0, and then it expands and cools in a constant T 3/ρ track, we can write the temperature and density evolution
+(Woosley et al. 1973) as
+T(t) = T0 exp (−t/3τdyn.),
+ρ(t) = ρ0 exp (−t/τdyn.) ,
+(A1)
+where the expansion timescale of the ejecta τ is the following (Fowler & Hoyle 1964);
+τdyn. = (24πGρ0)−1/2 ≈ 446/ρ1/2
+0
+sec.
+(A2)
+This expansion trajectory reproduces the general temperature and density trajectories in spherically symmetric or
+multi-dimensional CCSNe explosions, especially in the explosive nucleosynthesis region (e.g., Magkotsios et al. 2010,
+2011; Jerkstrand et al. 2015). We calculate a reaction network of 640 nuclear species with torch code (Timmes 1999),
+with the initial composition X(28Si) = 1. We then check the behavior of 56Ni nucleosynthesis using expansion profiles
+with peak temperature T0 and peak density ρ0 as parameters.
+Figure 9 shows the abundance of synthesized 56Ni as a function of the peak temperature T0 in the adiabatic expansion
+profile. When the peak temperature is T9 ≳ 5 (T9 ≡ T/109 K), we can see that a sizable amount of 56Ni is synthesized,
+almost independent of the peak density ρ0. In this temperature region, 28Si is wholly depleted at the same time
+as satisfying 56Ni synthesis, so this explosive nucleosynthesis is referred to as ‘complete Si burning’. This depletion
+of 28Si occurs when the lifetime over which 28Si burns up is much shorter than the given expansion time scale, i.e.,
+τ(28Si) ≪ τdyn.. According to Woosley et al. (1973), to estimate this condition quantitatively, we define the depletion
+point of 28Si as when its mass fraction falls below about 0.005 (i.e., Xf(28Si) < 0.005). Then, complete Si combustion
+is estimated to occur under the following initial conditions;
+T9 ≳ 5.0 ×
+�
+ρ0
+106 g cm−3
+�1/68
+(A3)
+This implies that an initial temperature must exceed T9 ≈ 5 to promote the depletion of 28Si in order for the complete
+silicon burning to occur at a reasonable density. This estimate is roughly consistent with our numerical result in Figure
+9.
+In complete Si burning, the forward and reverse reactions in the main reaction channel proceed much faster than
+the time scale of the change in thermal conditions. The compositions thus basically follow the Nuclear Statistical
+Equilibrium (NSE), except for the slow triple-alpha process. Then, as the temperature decreases rapidly, the reaction
+rates decrease and the abundance pattern ‘freezes-out’ (Hix & Thielemann 1999). In NSE, the composition ratio is
+determined to minimize the Helmholtz free energy (F = (U − Q) − TS) for the nuclide mass fraction (Clifford &
+Tayler 1965). When the temperature is not too high (T9 < 10), the binding energy per nucleon (BEN) Q = B/A
+roughly determines the abundance pattern of NSE. In environments where the number of protons and neutrons is
+almost equal (Ye ≈ 0.5), the most abundant isotope in NSE is 56Ni. Therefore, the main synthesis process of 56Ni in
+the SNe explosion is not a two-/three-body reaction, but a ‘freezes-out’ of the NSE abundance pattern that occurred
+via the ‘photodisintegration-rearrangement reactions’ associated with the photodisintegration of 28Si.
+Next, Figure 10 shows what the 56Ni synthesis would be in a different environment for Ye with different peak
+temperatures. We can clearly see that while 56Ni synthesis in the proton-rich environment (Ye > 0.5) is relatively the
+same as that in the Ye ≈ 0.5 environment, its synthesis is quite suppressed in the neutron-rich environment (Ye < 0.5).
+As explained by Seitenzahl et al. (2008), Ye dependence of nickel nucleosynthesis can be understood qualitatively by
+focusing on the BEN, Q = B/A, and the Helmholtz free energy F.
+
+Updating the 56Ni Problem in CCSNe
+13
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+ 2x109
+ 3x109
+ 4x109
+ 5x109
+ 6x109
+ 7x109
+ 8x109
+ 9x109
+complete Si-burning
+56Ni(solid)
+28Si(dashed)
+abundance
+peak temperature [K]
+ ρ = 3.0*107 [g/cc]
+ ρ = 1.0*107 [g/cc]
+ ρ = 3.0*106 [g/cc]
+ ρ = 1.0*106 [g/cc]
+Figure 9. The abundance of synthesized 56Ni as a function of peak temperature T0 with the peak density ρ0 at Ye = 0.5,
+calculated using the adiabatic expansion profile of Eq. (A1).
+electron fraction Ye
+peak tempareture [K]
+ 0.46
+ 0.48
+ 0.5
+ 0.52
+ 0.54
+ 2x109  3x109  4x109  5x109  6x109  7x109  8x109  9x109
+10-3
+10-2
+10-1
+100
+abundance of 56Ni
+Figure 10. The abundance of synthesized 56Ni in the peak temperature T0 and the initial electron fraction Ye plane at the
+peak density ρ0 = 107 g cm−3, calculated using the adiabatic expansion profile of Eq. (A1).
+Figure 11 (left) illustrates the BEN of various Ni-isotopes with blue crosses, where the number of protons per nucleon
+(Yp = Z/A, which corresponds to Ye) on the horizontal axis. To analyze stability, we consider a mixed composition
+of 56Ni plus an appropriate number of free protons or free neutrons (56Ni +nucleons) to match any Ye. The mean
+BEN of the mixed composition is calculated as ¯Q = Σi Qini/nB, where Qi is the binding energy of the nucleus i and
+Qneut = Qprot = 0. By comparing the BENs for each Ye, we find that the single Ni-isotope is significantly more stable
+than the 56Ni +nucleons mixture in the neutron-rich region (Ye < 0.5), but not in the proton-rich region (Ye ≥ 0.5).
+While this picture is somewhat modified when entropy is taken into account, this simple discussion shows that 56Ni
+synthesis is suppressed in the neutron-rich region.
+Figure 11 (right) shows Helmholtz free energies (F = (U −Q)−TS) for a single composition of only Ni isotopes and,
+as before, for a mixed composition of 56Ni plus an appropriate number of free protons or free neutrons. Helmholtz free
+energy was calculated using Helmholtz EoS (Timmes & Swesty 2000) with T = 5×109 K and ρ = 107 g cm−3 fixed. In
+an environment with the same T and ρ, the internal energy can be considered to be almost constant, U ≈ Const. The
+BEN term, Q, then contributes to the Helmholtz free energy on the order of 1 MeV/nuclear ≈ O(1018) erg g−1. For
+the TS term, the entropy is increased by the free nucleons in the 56Ni + nucleon mixture composition compared to the
+single Ni-isotope composition. It thus works toward lowering the Helmholtz free energy on the order of O(1017) erg
+
+14
+Sawada & Suwa
+ 7.6
+ 7.8
+ 8
+ 8.2
+ 8.4
+ 8.6
+ 8.8
+ 9
+ 0.4
+ 0.42
+ 0.44
+ 0.46
+ 0.48
+ 0.5
+ 0.52
+ 0.54
+ 0.56
+56Ni
+Binding Energy per Nucleon [MeV]
+Yp = Z/A  (=Ye:electron fraction)
+ANi
+56Ni+nucleons
+-9.2
+-9
+-8.8
+-8.6
+-8.4
+-8.2
+-8
+ 0.4
+ 0.42
+ 0.44
+ 0.46
+ 0.48
+ 0.5
+ 0.52
+ 0.54
+ 0.56
+56Ni
+Helmholtz free energy [1018 erg/g]
+ F = (U - Q) - TS
+Yp = Z/A  (=Ye:electron fraction)
+ANi
+56Ni+nucleons
+Figure 11. Binding energy per nucleon (left) and the Helmholtz free energy per nucleon (right) with the number of protons
+per nucleon Yp = Z/A (corresponds to Ye, and 28/Ye corresponds to the mass number A of each Ni isotope) on the horizontal
+axis. The blue crosses show the binding energy per nucleon (BEN) of the Ni-isotope, and the red circles indicate the mean BEN
+of a mixed composition, which is mixed with 56Ni and an appropriate number of free protons or free neutrons for any given Ye
+(56Ni +nucleons). The mean BEN of the mixed composition is derived by ¯Q = Σi Qini/nB. The Helmholtz free energies were
+calculated using Helmholtz EoS (Timmes & Swesty 2000) with T = 5 × 109 K and ρ = 107 g cm−3 fixed.
+g−1. Therefore, in the proton-rich region (Ye ≥ 0.5), the difference in BEN between the 56Ni +nucleon mixture and
+the single Ni-isotope is ∆Q ≲ O(1017) erg g−1, which is not much larger than the correction by entropy enhancement.
+As a result, the correction for the entropy enhancement due to the free nucleon makes the 56Ni +nucleon mixture more
+stable. On the other hand, in the neutron-rich region (Ye < 0.5), the difference in BEN is ∆Q ∼ O(1018) erg g−1,
+which is sufficiently larger than the effect of the correction by entropy enhancement, and thus the single Ni-isotope
+composition is more stable, as predicted from the BEN behavior.
+To summarize the above discussion, in the temperatures region where T9 ≳ 5 are attained behind the shock, NSE is
+achieved except for the slow triple-alpha process. Thus, the abundance pattern depends only on Ye and the entropy,
+and is dominated by iron-group elements (e.g., Hix & Thielemann 1999). The most abundant element in this region
+of complete Si burning with alpha-rich freeze-out is 56Ni , provided Ye ≳ 0.5. The 56Ni synthesis conditions in CCSNe
+explosion are an environment where the peak temperature T9 ≳ 5 is experienced by the shock and the electron fraction
+of the material is Ye ≳ 0.5.
+
diff --git a/StE2T4oBgHgl3EQfCAbz/content/tmp_files/load_file.txt b/StE2T4oBgHgl3EQfCAbz/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..dcadce4f8a24fa2499ac87da2de10686f4a9a3c6
--- /dev/null
+++ b/StE2T4oBgHgl3EQfCAbz/content/tmp_files/load_file.txt
@@ -0,0 +1,1061 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf,len=1060
+page_content='Draft version January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2023  Typeset using LATEX twocolumn style in AASTeX631  Updating the 56Ni Problem in Core-collapse Supernova Explosion  Ryo Sawada  1 and Yudai Suwa  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2  1Department of Earth Science and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Graduate School of Arts and Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Tokyo 153-8902,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Japan  2Center for Gravitational Physics and Quantum Information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Yukawa Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Kyoto University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Kyoto  606-8502,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Japan  Submitted to ApJ  ABSTRACT  Details of the core-collapse supernova (CCSN) explosion mechanism still need to be fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  There is an increasing number of successful examples of reproducing explosions in multidimensional  hydrodynamic simulations, but subsequent studies pointed out that the growth rates of the explosion  energy ˙Eexpl of these simulations are insufficient to produce enough 56Ni to match observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' This  issue is known as the ‘56Ni problem’ in CCSNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Recently, however, some studies have suggested that  this 56Ni problem is derived from the simplicity of the explosion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In response, we investigate the  effect of the explosion energy growth rate ˙Eexpl on the behavior of nucleosynthesis in CCSNe in a more  realistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We employ the 1D Lagrangian hydrodynamic code, in which we take neutrino heating  and cooling terms into account with the light-bulb approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We reiterate that, consistent with  previous rebuttal studies, there is the 56Ni problem: Although 56Ni is synthesized to almost the same  mass coordinate independent of ˙Eexpl, some of the innermost material in the low- ˙Eexpl model failed  to escape, leading to a shift in the innermost mass coordinate of the ejecta to the outer positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Comparing our results with observations, we find that while modern slow explosions can, in principle,  reproduce observations of standard Type II SNe, this is not possible with stripped-envelope SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Our  finding places a strong constraint on the explosion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' There are significant differences in the  progenitor structures and the explosion mechanism between Type II and stripped-envelope SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Keywords: (stars:) supernovae: general—hydrodynamics  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' INTRODUCTION  Radioisotope  56Ni is an important product in su-  pernova nucleosynthesis, which drives supernova (SN)  brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  56Ni decays into 56Co, and then into 56Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  This nuclear decay chain powers the light curve of SNe,  and thus, 56Ni masses of SNe have been estimated with  reasonable accuracy from the light curve (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Ar-  Corresponding author: Ryo Sawada  ryo@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='ecc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='jp  nett 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Hamuy 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1  On the other hand, the  amount of synthesized 56Ni is sensitive to the tempera-  ture T, the density ρ, and the number of electrons per  nucleon (electron fraction) Ye, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', explosion property  and pre-SNe core structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Woosley & Weaver  1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Thielemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Woosley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  These two factors, that is, the amount of 56Ni synthe-  sis can be accurately estimated from observations and  strongly reflect the explosion’s innermost nature, sug-  gest the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 56Ni is the best probe to constrain an  1 For Type-II SNe, the tail luminosity provides 56Ni mass, assum-  ing the complete trapping of γ-rays produced from the nuclear  decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  For Type-I SNe, on the contrary, the peak luminosity  has often been used, assuming that it should be equal to the  instantaneous energy deposition rate by the nuclear decay, so-  called Arnett rule (Arnett 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Core-collapse SNe in the latter  category is called stripped-envelope SNe (SE-SNe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Smartt 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='03610v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='HE]  9 Jan 2023  ID2  Sawada & Suwa  aspect of the SN explosion mechanism accurately (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=',  Maeda & Tominaga 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Suwa & Tominaga 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Details of the explosion mechanism of core-collapse  supernovae (CCSNe) are not yet fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The  most promising scenario is the delayed neutrino-driven  explosion (Bethe & Wilson 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' While this scenario  had once not been reproduced by numerical simulations,  the situation has brought substantial progress over a few  decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Now, there is an increasing number of success-  ful examples of reproducing explosions in multidimen-  sional hydrodynamic simulations, with a detailed neu-  trino transport (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=',  Lentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Takiwaki  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' O’Connor & Couch 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Glas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Bollig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Burrows & Vartanyan  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Bruen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2022, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Al-  though the details now depend on the numerical meth-  ods and physical approximations employed in each sim-  ulation, there seems to be a general understanding that  the explosion succeeds by the growth of the hydrody-  namic instability over a sufficient time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Indeed, most, if  not all, of those state-of-the-art simulations, have shown  a slow increase of explosion energy, and the growing rate  of the explosion energy is typically ˙Eexpl = O(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1) Bethe  s−1 (1 Bethe≡ 1 × 1051 erg), especially for 3D simula-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  However, recent several studies have shown that to  reproduce the typical observed mass of 56Ni by the ex-  plosive nucleosynthesis in the ejecta, the growth rate  of the explosion energy of ˙Eexpl = O(1) Bethe s−1 is re-  quired in several methods (Sawada & Maeda 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Suwa  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Sawada & Maeda (2019)  found the inverse-correlation between 56Ni yield and ex-  plosion energy growth rate ˙Eexpl by 1D simulations with  the simple thermal-bomb modeling and post-processing  detailed-nucleosynthesis, and Suwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2019) also  came to the same conclusion by conducting hydrody-  namic simulations with an approximate neutrino heat-  ing model that self-consistently follows core-collapse and  shock-revival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2022) also confirmed this  trend, using the same method as Sawada & Maeda  (2019), but modeled for individual objects to reduce ob-  servational uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  If these results are correct,  the current multi-D simulations, which give explosion  energy growth rates of ˙Eexpl = O(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1) Bethe s−1, would  be observationally unfavorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We refer to this issue  as the nickel mass problem (‘56Ni problem,’ hereafter)  in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' However, this 56Ni problem is still under  some debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  In particular, Imasheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2023) just recently  pointed out the most obvious question to the 56Ni prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Imasheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2023) used the same method as  Sawada & Maeda (2019), with simple thermal injection  modeling and post-processing detailed nucleosynthesis,  but scrutinized the treatment of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In  the recent slow explosion scenario, the pre-SN star ex-  periences sufficient gravitational contraction just before  the successful explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  They found that the corre-  lation between 56Ni yield and explosion energy growth  rate ˙Eexpl is the result of ignoring this initial collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  They argued that this correlation disappears when the  initial collapse is included and also that further initial  collapse inversely results in more 56Ni being synthesized  in slower explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Their arguments also apply to Sawada & Maeda  (2019) and Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2022), but not to Suwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Suwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2019) solved self-consistently the  core collapse and shock revival with the light-bulb  scheme and found this correlation even though they  took into account the initial collapse phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' This result  is inconsistent with the conclusion of Imasheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Note that Suwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2019) performed no de-  tailed nucleosynthesis calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Instead, they esti-  mated the 56Ni amount simply by the temperature of  hydrodynamic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Therefore, we perform hy-  drodynamic and detailed nucleosynthesis calculations in  this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' This study aims to clarify the detailed pic-  ture of how 56Ni synthesis occurs in the current CCSN  explosion scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' By clarifying this picture, we also  expect to explain the origin of the differences between  the two studies and, by extension, the cause of the 56Ni  problem itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  In this paper, therefore, we simulate one-dimensional  hydrodynamics in the light-bulb scheme as in Suwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  (2019), then perform detailed nucleosynthesis in a post-  process manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The goal of this study is to present a  detailed picture of 56Ni nucleosynthesis in CCSNe with  self-consistent explosion modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Furthermore, we aim  to sort out the controversial 56Ni problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In Section  2, we describe our simulation methods, the progenitor  models, and post-processing analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Our results are  summarized in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In Section 4, we revisit the  56Ni problem through a detailed comparison of our re-  sults and observations, and discuss the uncertainties in-  volved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We conclude in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' SIMULATION METHODS  Following the computational setup performed in Suwa  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2019), we employ a 1D Lagrangian Newtonian  hydrodynamic code based on blcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='2 Basic equations  under a spherically symmetric configuration, as we per-  2 This code is a prototype code of SNEC (Morozova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2015), and  available from https://stellarcollapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='org  Updating the 56Ni Problem in CCSNe  3  form in this paper, are given as follows:  ∂r  ∂Mr  =  1  4πr2ρ ,  (1)  Dv  Dt = −GMr  r2  − 4πr2 ∂P  ∂Mr  ,  (2)  Dϵ  Dt = −P D  Dt  �  1  ρ  �  + H − C ,  (3)  where r is the radius, Mr is the mass coordinate, t  is time, ρ is the density, v is the radial velocity, P is  pressure, ϵ is the specific internal energy, and D/Dt ≡  ∂/∂t + vr∂/∂r is the Lagrangian time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The  artificial viscosity of Von Neumann & Richtmyer (1950)  is employed to capture a shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The system of equa-  tions (1)-(3) is closed with the Helmholtz equation of  state (Timmes & Swesty 2000), which describes the stel-  lar plasma as a mixture of arbitrarily degenerate and  relativistic electrons and positrons, black-body radia-  tion, and ideal Boltzmann gases of a defined set of fully  ionized nuclei, taking into account corrections for the  Coulomb effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  In this work, neutrino heating and cooling are added  by a light-bulb scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In the light-bulb scheme, neu-  trino cooling is given as a function of temperature, and  neutrino heating is a function of the radius with param-  eterized neutrino luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The heating term H and  the cooling term C, terms in Equation (3) are assumed  to be  H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='544 × 1020 erg g−1 s−1  ×  �  Lνe  1052MeV  � �  rνe  100km  �−2 �  Tνe  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0MeV  �2  ,  (4)  C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='399 × 1020 erg g−1 s−1 ×  �  T  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0MeV  �6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  (5)  Here, we fix the neutrino temperature as Tνe = 4 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  We take into account these terms only in the post-shock  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We modified the inner boundary conditions so  that the innermost mass shell does not shrink within 50  km from the center to mimic the existence of a proto-  neutron star (PNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Also, the light-bulb scheme in this  study tends to overestimate the neutrino-driven wind  from the PNS surface at the post-explosion phase be-  cause it keeps giving a constant neutrino luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Therefore, in this study, we consider the mass coordi-  nate that experienced r < 200 km as the neutrino-driven  wind and separate it from the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  The numerical computational domain contains 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5M⊙  and uses a 1500 grid with a mass resolution of 10−3M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  We set the inter boundary at Ms/kb=4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5M⊙ for each  density [g/cm3]  mass radius [Msun]  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3M  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5M  10-10  10-5  100  105  1010  2  4  6  8  10  12  14  16  104  105  106  107  108  109  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='9  2  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  6  density [g/cm3]  entropy [kb/baryon]  Mass Radius [Msun]  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Density structure as a function of the enclosed  mass for the considered progenitors with MZAMS = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3M⊙  (cyan line), 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M⊙ (blue line), 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='7M⊙ (red line), and  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M⊙ (magenta line), and its details with the entropy per  nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  pre-explosion star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Each mass coordinate captures the  time evolution of the hydrodynamic quantities, so that  nucleosynthesis calculations are performed as a post-  processing analysis with this trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We calculate a  reaction network of 640 nuclear species with the torch  code (Timmes 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  The initial conditions adopted in this study are a sub-  set of non-rotating stars with solar metallicity, which  evolved from the main sequence to the onset of iron-core  collapse, as published by Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The  physics of this set of progenitors was discussed in detail  in this literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Figure 1 shows the density structures  4  Sawada & Suwa  1x109  1x1010  1x1011  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='9  2  temperature [K]  mass radius [Msun]  2x109  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5x109  1x109  5x108  0  5x108  1x109  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5x109  2x109  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='9  2  velocity [cm/s]  mass radius [Msun]  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Time evolution of the velocity (top) and the tem-  perature (bottom) as a function of the mass coordinate for  model 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In both panels, each snapshot time corre-  sponds to approximately every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1 seconds from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5 seconds  to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5 seconds from the start of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The gray  line corresponds to T = 5 × 109 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  of the progenitor as a function of the enclosed mass, and  its details with the entropy per nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' RESULT  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Overview of the explosion dynamics  Figure 2 shows the time evolution of radial velocity  and temperature as a function of the mass coordinate  for the model 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We first use an example of a  model with MZAMS = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M⊙ throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  From the velocity figure, it can be seen that the shock  begins to propagate outward from the point where the  silicon/oxygen (Si–O) layer (≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='62M⊙) accretes onto  the shock wave, due to the rapid decrease in ram pres-  sure (Marek & Janka 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Suwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' From the  temperature figure, we can confirm that the post-shock  1x107  1x108  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='2  radius [cm]  time [sec]  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Radius evolution of Lagrangian mass shells with  time for the explosion (Lν = 3 × 1052 erg s−1) and non-  exploding model (Lν = 0 erg s−1) of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  The thick  black solid lines are the mass shells, spaced in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1  M⊙, and the thin gray solid/dashed lines are spaced in steps  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='02 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  The difference between the dotted and solid  lines corresponds to the explosion and non-exploding models,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  The blue line marks the shock radius of the  explosion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  temperature of the ejecta is spatially almost constant so  we define the shock temperature as the temperature of  the material just behind the shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Figure 3 shows that the mass shell until the arrival of  the shock in the explosion model is consistent with its  behavior in the non-exploding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In other words,  we can confirm that the behavior of the mass shell up to  the arrival of the shock is independent of the explosion  detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Thus, by overlaying the shock evolution on the  trajectory of the mass shell in the non-exploding model,  we can compare several models at once to see where the  shock impacts each of the mass shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  In the following subsections, we present the results  focusing on the effect of the explosion energy growth  rate ˙Eexpl on 56Ni nucleosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The results are sum-  marized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  These yields consist only of un-  bound 56Ni by gravity as determined by a 10-second  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We first use an example of a model with  MZAMS = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M⊙ throughout Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Hydrodynamics and 56Ni Synthesis Region  Figure 4 shows the time evolution of the shock for the  models Lν = 3, 5, and 7 × 1052 erg s−1 and the trajec-  tory of the mass shell in the unexploded model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In each  model, the time evolutions of the shock are shown by col-  ored lines for the range where the shock satisfies T9 > 5  (T9 ≡ T/109 K), and by black dashed lines for the range  where T9 < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We refer to the mass coordinate that can  Updating the 56Ni Problem in CCSNe  5  1x107  1x108  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='7  radius [cm]  time [sec]  Lnu=3e52 [erg/s]  Lnu=5e52 [erg/s]  Lnu=7e52 [erg/s]  Lnu=9e52 [erg/s]  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The time evolution of the shock radius in models  Lν = 3, 5, and 7×1052 erg s−1 with the mass shell trajectory  in the unexploded model, on the time-radius plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  spread at the shock temperature of T9 ≈ 5 as MT9=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  We find that in all models MT9=5 is near the mass co-  ordinate with the enclosed mass ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='65M⊙, which is  indicated by the dotted line in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' More detailed  values are given in Table 1, and this trend is almost  universal, independent of the progenitor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In all  models, we find that MT9=5 is near the mass coordinate  with an enclosed mass of approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='65M⊙, as in-  dicated by the dotted line in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' More detailed  values are given in Table 1, and this trend is nearly uni-  versal and independent of the progenitor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Qualitatively, this can be understood by using a zero-  order approximation to estimate the shock radius at  which the shock temperature is T9 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' When apply-  ing a simple fireball model in which the region behind  the shock wave is uniform and dominated by radiation  pressure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Woosley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2002), we can estimate the  following relation between the temperature T, the shock  radius rsh and the explosion energy Eexpl as follows:  Eexpl = 4π  3 r3  sh(t) aT 4 ,  (6)  where a is the radiation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Then with Eexpl(t) ≡  1051 ergs, the radius with T9 = 5 (rT9=5) can be esti-  mated as follows:  rT9=5 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6 × 108(Eexpl/1051)1/3 cm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  (7)  This estimated radius is classically well-known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=',  Woosley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' If we consider  the time evolution of Eexpl(t) = ˙Eexpl·t, this classical ra-  dius is satisfied with an adequate large ˙Eexpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' However,  at the shock velocity Vsh = 109 cm s−1, it takes less than  0  1x1050  2x1050  3x1050  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='8  1  explosion energy [erg]  time after bounce [sec]  explosion energy Esim  estimated energy Eapp  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Comparison of the explosion energy in the sim-  ulation Esim (dashed line) with the estimated energy in the  fireball approximation Eapp (solid line), which comes from  Eq (8) in models Lν = 2, 3, and 5 × 1052 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  The  horizontal axis is the post-bounce time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  1 second to reach this radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In other words, if the case  of ˙Eexpl ≲ 1 Bethe s−1, it takes a few seconds to reach 1  Bethe, and obviously, the radius of T9 = 5 will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  In fact, from Figure 4, we can confirm that even in this  simulation, the radius of T9 = 5 is reduced in the case of  low- ˙Eexpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' But at the same time, the time evolution of  the shock radius is also slower down for lower- ˙Eexpl mod-  els, and the mass shell falls more inward due to collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Eventually, the ‘mass coordinate’ of MT9=5 seems to be  approximately the same regardless of the ˙Eexpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  This result suggests a very interesting trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  56Ni  is synthesized mainly by complete Si burning at T ≳  5×109 K (see in detail in Appendix A and Woosley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Thus, this hydrodynamical result suggests that  the outermost mass coordinates, where 56Ni is primar-  ily synthesized, are insensitive to the explosion energy  growth rate ˙Eexpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' To confirm this trend in more detail,  we next discuss the results of nucleosynthesis calcula-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Although this is not relevant to the main focus of  this paper, we show in Figure 5 for reference the com-  parison of the explosion energy in the simulation Esim  with the estimated energy in the fireball approximation  Eapp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The explosion energy Esim in the hydrodynamical  simulation is defined as the integral of the sum of spe-  cific internal, kinetic, and gravitational energies over all  zones, in which it is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The estimated energy in  the fireball approximation Eapp is given by the follow-  ing equation using only the shock radius rsh and shock  6  Sawada & Suwa  temperature T:  Eapp = 4π  3 r3  sh(t) aT 4f(T9) ,  (8)  where f(T9) = 1 + (7/4) · T 2  9 /(T 2  9 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3) is a correction  term to account for both radiation pressure and non-  degenerate electron-positron pairs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Freiburghaus  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' As Figure 5 shows, this simple estimation is  able to reproduce the explosion energy of the simulation  with good enough accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' This supports the validity  of the above discussion, and also suggests that thermal  energy is dominant in the early phases of the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Nucleosynthesis: Distribution of 56Ni synthesis  Figure 6 shows the abundance distribution as a func-  tion of the mass coordinate, for Lν = 3, 5, and 7 × 1052  erg s−1 with 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  As shown in Figure 6, focus-  ing specifically on 56Ni, we can confirm that the outer-  most mass radius, where 56Ni is primarily synthesized,  is in a similar position independent of the explosion en-  ergy growth rate ˙Eexpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In Table 1, we show the out-  ermost mass radius where 56Ni is largely synthesized  (here, we define it as X(56Ni) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' However, at the  same time, the innermost mass radius, which is gravita-  tionally unbounded, depends strongly on the explosion  energy growth rate ˙Eexpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' DISCUSSION: UPDATE ‘NI PROBLEM’  Figure 7 shows the synthesized amount of 56Ni as a  function of the explosion energy growth rate ˙Eexpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' It  can be clearly seen that there is a decreasing trend of  the synthesized amount of 56Ni toward decreasing ˙Eexpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  The reason for this trend is explained in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3, but  this figure tells us that the same trend is generally ob-  served regardless of the mass and structure of the pro-  genitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  For comparison with observations, in Figure 7, we  adopted two typical values based on a recent systematic  survey for more than 300 events of CCSNe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='07M⊙3 as  the median estimated from stripe-envelope supernovae  (SE-SNe) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='03M⊙ from Type-II SNe (Rodr´ıguez  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Note that the figure plots the syn-  thesized amount of 56Ni ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' not all 56Ni can be ejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In  other words, the figure shows the maximum amount of  3 The ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='07M⊙ is often adopted as a typical value obtained for  well-studied nearby SNe is on average (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', SN 1987A, SN 1994I,  SN 2002ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Arnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Iwamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Mazzali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  2002) and we adopt this value in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' However, in  this study, we clearly mention here that we do not use 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='07M⊙  in the context of the typical for nearby SNe because we con-  sider observational constraints from recently updated large-scale  observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  56Ni that can be ejected by each CCSN model, and if  the calculated mass of 56Ni is larger than the observed  value, then the model can reproduce the observed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  First, compared to the median value of Type II super-  novae 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='03M⊙, even a modern slow explosion ( ˙Eexpl ≲ 1  Bethe s−1 ) provides enough amount of 56Ni to repro-  duce the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' On the other hand, compared  to the SE-SNe median of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='07M⊙, a very rapid explo-  sion of ˙Eexpl≳ 2 Bethe s−1 is required to reproduce this  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' This translates to a time scale of t ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5 sec-  onds to the typical explosion energy ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0 Bethe, and  this timescale is very difficult to reproduce with current  multi-D self-consistent calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Here we discuss a few caveats in this problem as fol-  lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' [Observation of Type-II SNe]  The typical 56Ni mass of canonical-CCSNe has  been extensively discussed by large-scale observa-  tions in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In particular, Type II SNe,  when volume-limited, account for nearly ∼ 60%  of the observed CCSNe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Jones  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Recently, Type II SNe have been  found to have lower median nickel masses than SE-  SNe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Anderson 2019), confirming that this is  not due to observational bias (Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Furthermore, the observed kinetic energy is also  found to have a lower median value than the clas-  sical typical value (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6 Bethe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' These facts also support the possibility that  the ‘slow’ explosion results in the current state-of-  the-art simulations are relatively consistent with  standard Type II SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  However, nickel synthe-  sis and explosion energy (MNi ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='03M⊙ and  Eexpl ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6 Bethe) still remain important bench-  marks for multidimensional self-consistent simula-  tions, and it should be checked whether they are  truly achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' And, another important point is  that this is only a statement of the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Ac-  cording to Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2021), the fitting function  for the cumulative histogram for observed 56Ni  masses of each CCSNe is f(x) = tanh (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='60 × x)  4 as a variable of observed 56Ni masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  This  roughly implies that more than 20% of the Type II  supernovae synthesize 56Ni above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='075M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' While  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='03M⊙ is a somewhat explainable value, this  value is challenging to reproduce in multi-D self-  consistent simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' So, we need to explain and  4 This function is obtained by fitting non-linear least squares of  observed reports of Type II SNe with a sample size of 115 events  (Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Updating the 56Ni Problem in CCSNe  7  10-5  10-4  10-3  10-2  10-1  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='9  proto-NS  56Ni  16O  28Si  40Ca  12C  44Ti  28Mg  abundance  mass radius[M]  (a)  10-5  10-4  10-3  10-2  10-1  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='9  proto-NS  abundance  mass radius[M]  (b)  10-5  10-4  10-3  10-2  10-1  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='9  proto-NS  abundance  mass radius[M]  (c)  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Abundance distribution as a function of the enclosed mass Mr, for (a) Lν = 3 × 1052 erg s−1, (b) Lν = 5 × 1052 erg  s−1, and (c) Lν = 7 × 1052 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' All the models here are with 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M⊙ of Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In all panels, the vertical  dotted grey line indicates the location of the mass shell with an enclosed mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='65M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='14  5x1050  1x1051  2x1051  3x1051 4x1051  median of SESNe  median of typeII-SNe  nickel mass [Msun]  Edot [erg/sec]  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3M  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0M  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5M  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The amount of 56Ni as a function of the growth  rate of the explosion energy, ˙Eexpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The gray line indicates a  typical value of 56Ni , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='07M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  reproduce the high 56Ni objects that will exist to  some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' [Multidimensional effect]  How the amount of synthesized 56Ni changes in a  multi-D explosion model is one of the issues to be  discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Since this model is a 1D model and ex-  plodes only with thermal energy as shown in Fig-  ure 5, the temperature should be higher than the  multi-D model, especially considering the geomet-  ric structure and the change to kinetic energy in  the non-radial direction (Suwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' There-  fore, we should note that the same ˙Eexpl in a multi-  D model would have less 56Ni than in a 1D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  In fact, with the exception of particular model  results (Bollig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2021,  discussed next), the  multi-D self-consistent simulation has even more  difficulty with 56Ni synthesis than the estimate of  this study (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Bruen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Therefore,  for the same synthesis conditions, the 1D model  gives a robust maximum limit on the volume and  the amount of 56Ni synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The additional 56Ni  amount newly occurring due to the multi-D effect  will be discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' [Additional mechanism to add 56Ni ]  Another possibility for an additional 56Ni , and one  of the most often cited candidates for a solution  to this problem, is the ‘outflow’ from the PNS sur-  face for several seconds of the post-explosion phase  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Wongwathanarat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Witt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Recent detailed simulations have predicted  proton-rich ejecta in the post-explosion ‘outflow’  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Bruenn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  In particular, Bollig  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2021) have observed the downflow/outflow  system that results in a smooth and efficient tran-  sition from the incoming flow to the outgoing flow,  with the outflow providing 56Ni to ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='05M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  However, we already found that the contribution  of such replenishment is small for regular CCSNe  explosions (Sawada & Suwa 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' That is, this  outflow system is part of an ‘energetic’ model of  the state-of-the-art simulations that succeeds in  producing sufficient amounts of 56Ni , and it is de-  batable whether this outflow system contributes  to canonical-CCSNe explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  8  Sawada & Suwa  56Ni  PNS  New Picture of the 56Ni problem:  by more realistic explosion model  (light-bulb scheme)  mass coordinate  (slow)-expl: ሶ𝐸expl ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0 × 1051 erg s−1 , (rapid)-expl: ሶ𝐸expl > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0 × 1051 erg s−1  innermost (ejectable) mass radius  is sensitive to explosion ሶ𝐸expl !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  PNS  Imasheva, Janka,& Weiss (2023) :  by collapsed thermal bomb model  (rapid)  (slow)  PNS  Fixed, In case of thermal bomb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Sawada & Maeda (2019) :  by uncollapsed thermal bomb model  mass coordinate  mass coordinate  (rapid)  (slow)  (rapid)  (slow)  56Ni  56Ni  Fixed, In case of thermal bomb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  56Ni synthesized mass  radius is insensitive !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  56Ni problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  56Ni mass depends on ሶ𝐸expl  We repropose the 56Ni problem  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Schematic picture to the ‘56Ni problem in CCSNe’ as suggested by the the previous studies and this study, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' At first, Sawada & Maeda (2019) raised the 56Ni problem because the 56Ni synthesized region varies with the growth rate  of the explosion energy ˙Eexpl when an un-collapsed progenitor is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' When taking into account that the progenitor collapses  just before the explosion, the 56Ni synthesized region becomes insensitive to ˙Eexpl, and thus, Imasheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2023) proposed  a disappearance of the 56Ni problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' However, this study re-proposes the 56Ni problem on the grounds that while the 56Ni  synthesized region is insensitive to ˙Eexpl, the ejectable innermost mass radius depends on the ˙Eexpl, as calculated using the  light-bulb scheme in which the PNS masses is determined self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' [Comparison to Imasheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2023) ]  Finally, we compare our results with those of Ima-  sheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2023) who just recently pointed out  the most obvious doubts about the ‘56Ni prob-  lems’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Figure 8 is a schematic comparison of our  results with theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Their argument is that the  correlation between  ˙Eexpl  and 56Ni disappears  when the initial collapse is included, and that fur-  ther initial collapse inversely results in more 56Ni  being synthesized in slower explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Noting  that the innermost ejecta radius is fixed in the  thermal bomb model, their argument is consis-  tent with the present results where 56Ni is syn-  thesized to almost the same mass coordinate in-  dependent of ˙Eexpl, shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We also  confirm that 56Ni is synthesized slightly more out-  wardly in models with slower initial collapse (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=',  the MZAMS = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5M⊙ model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  The difference  from their study is the treatment of the inner-  most mass coordinate of the ejecta, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', their inner  boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Our explosion model deter-  mines the innermost mass coordinate of the ejecta  self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  We then found that the inner-  most material that could be ejected in the high-  ˙Eexpl model could not achieve the escape condi-  tion in the low- ˙Eexpl model, leading to moving  the innermost mass coordinate of the ejecta out-  ward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In fact, for low ˙Eexpl models, Imasheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  (2023) themselves had mentioned the possibility  that some of the innermost material may be unable  to achieve escape conditions, remain gravitation-  ally bound, and thus not contribute to the yield,  and we confirmed this in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Although our  results are consistent with theirs, we confirm that  56Ni  problems reappear because the innermost  ejecta radius shifts depending on the intensity of  the ˙Eexpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  We conclude that the modern slow explosion ( ˙Eexpl ≲  1 Bethe s−1 ) can reproduce the observations of a stan-  dard Type II supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' However, this is only a state-  ment of a principal possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' How much 56Ni can be  synthesized is an important benchmark for multidimen-  sional self-consistent simulations, and it should be con-  firmed whether the median value for a standard Type II  supernova (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='03M⊙) is indeed achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  On the other hand, the 56Ni problem clearly exists in  the explosion mechanism of SE-SNe, that is, the modern  slow explosions cannot reproduce the SE-SNe observa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' As a simple and straightforward solution that sat-  isfies the 56Ni problem without fine-tuning, we conclude  that the SE-SNe favors active explosions in the early  stages of shock revival ( ˙Eexpl ≳ 2 Bethe s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Since  such high explosion energies are probably inconsistent  with the standard explosion mechanism, the 56Ni prob-  lem may require a different explosion mechanism for the  SE-SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Anderson (2019) had already suggested from  observations, but our results once again imply signifi-  cant differences in the progenitor structures and/or the  explosion mechanism between type II and SE-SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' SUMMARY  Updating the 56Ni Problem in CCSNe  9  In this paper, we investigated the effect of the explo-  sion energy growth rate ˙Eexpl  on the behavior of 56Ni  nucleosynthesis in CCSNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' For numerical simulations,  we employed the 1D Lagrangian hydrodynamic code in  which neutrino heating and cooling terms are taken into  account by the light-bulb approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  The initial  conditions are taken from Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2018), which  have MZAMS = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0, and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Our first purpose was to present a detailed picture of  56Ni nucleosynthesis in CCSNe with self-consistent ex-  plosion modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  We found that 56Ni is synthesized  up to the almost same mass coordinate independent  of ˙Eexpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We also found that in the low- ˙Eexpl model,  some of the innermost material that was ejected in the  high- ˙Eexpl model failed to achieve the escape condition,  leading to moving the innermost mass coordinate of the  ejecta to the outer positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' This means that while the  56Ni nucleosynthesis volume is insensitive to the nature  of the explosion, the ejected amount of 56Ni is highly  dependent on how much of the innermost PNS surface  region is ejectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Furthermore, our other goal was to sort out the re-  cent controversial 56Ni problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We found that there is  a decreasing trend of the synthesized amount of 56Ni  toward decreasing  ˙Eexpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Compared to observations,  we found that the modern slow explosion ( ˙Eexpl ≲ 1  Bethe s−1 ) can reproduce the observations of a stan-  dard Type II supernova in a principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' However, this  does not mean that the 56Ni problem has been solved,  and the 56Ni synthesis (MNi ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='03M⊙) still remains an  important benchmark for multi-D self-consistent simu-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  It should be checked whether they are truly  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' And extremely important are the comparison  results with SE-SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We found that the median value  of SE-SNe (MNi ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='07M⊙) is challenging to reproduce  in the modern slow explosion ( ˙Eexpl ≲ 1 Bethe s−1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  As a simple and straightforward solution that satisfies  the amount of 56Ni without fine-tuning, the SE-SNe fa-  vors active explosions in the early stages of shock re-  vival ( ˙Eexpl ≳ 2 Bethe s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Thus, we conclude that  there are significant differences in the progenitor struc-  tures and/or the explosion mechanism between type II  and SE-SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Software: blcode, torch(Timmes 1999)  ACKNOWLEDGMENTS  The  work  has  been  supported  by  Japan  Soci-  ety for the Promotion of Science (JSPS) KAKENHI  grants 21J00825, 21K13964 (RS), 18H05437, 20H00174,  20H01904, and 22H04571 (YS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  REFERENCES  Anderson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2019, A&A, 628, A7,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1051/0004-6361/201935027  Arnett, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1982, ApJ, 253, 785, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/159681  Arnett, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Bahcall, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Kirshner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Woosley,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1989, ARA&A, 27, 629,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='aa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='090189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='003213  Bethe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Wilson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1985, ApJ, 295, 14,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/163343  Bodansky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Clayton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Fowler, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1968,  ApJS, 16, 299, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/190176  Bollig, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Yadav, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Kresse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2021, ApJ, 915, 28,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/1538-4357/abf82e  Bruen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Sieverding, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Lentz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2022,  arXiv e-prints, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='12675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='12675  Bruenn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Lentz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Hix, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2016, ApJ,  818, 123, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/0004-637X/818/2/123  Burrows, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Vartanyan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2021, Nature, 589, 29,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1038/s41586-020-03059-w  Clifford, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Tayler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1965, MmRAS, 69, 21  Fowler, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Hoyle, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1964, ApJS, 9, 201,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/190103  Freiburghaus, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Rembges, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Rauscher, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1999,  ApJ, 516, 381, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/307072  Glas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Just, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Obergaulinger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2019,  ApJ, 873, 45, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/1538-4357/ab0423  Hamuy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2003, ApJ, 582, 905, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/344689  Hix, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Thielemann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1999, ApJ, 511, 862,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/306692  Imasheva, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Weiss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2023, MNRAS,  518, 1818, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1093/mnras/stac3239  Iwamoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', H¨oflich, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1994, ApJL,  437, L115, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/187696  Jerkstrand, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Timmes, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Magkotsios, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2015,  ApJ, 807, 110, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1088/0004-637X/807/1/110  Jones, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Foley, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Narayan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2021, ApJ,  908, 143, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/1538-4357/abd7f5  Lentz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Bruenn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Hix, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2015, The  Astrophysical Journal, 807, L31,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  2010, ApJS, 191, 66, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1088/0067-0049/191/1/66  Magkotsios, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Timmes, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Wiescher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2011, ApJ,  741, 78, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1088/0004-637X/741/2/78  Marek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2009, ApJ, 694, 664,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1088/0004-637X/694/1/664  Martinez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Bersten, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Anderson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2022,  A&A, 660, A41, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1051/0004-6361/202142076  Mazzali, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Deng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Maeda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2002, ApJL, 572,  L61, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/341504  Morozova, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Piro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Renzo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2015, ApJ, 814,  63, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1088/0004-637X/814/1/63  M¨uller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Melson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Heger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2017,  MNRAS, 472, 491, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1093/mnras/stx1962  Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Kobayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Tominaga, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2013,  ARA&A, 51, 457,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1146/annurev-astro-082812-140956  Updating the 56Ni Problem in CCSNe  11  O’Connor, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Couch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2018, ApJ, 865, 81,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/1538-4357/aadcf7  Ouchi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Maeda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Anderson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Sawada, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2021,  ApJ, 922, 141, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/1538-4357/ac2306  Rodr´ıguez, ´O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Maoz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Nakar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2022, arXiv e-prints,  arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='05552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='org/abs/2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='05552  Rodr´ıguez, ´O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Meza, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Pineda-Garc´ıa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Ramirez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  2021, MNRAS, 505, 1742, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1093/mnras/stab1335  Saito, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Tanaka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Sawada, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Moriya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2022,  ApJ, 931, 153, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/1538-4357/ac6bec  Sawada, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Maeda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2019, ApJ, 886, 47,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/1538-4357/ab4da3  Sawada, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Suwa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2021, ApJ, 908, 6,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/1538-4357/abd476  Seitenzahl, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Timmes, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Marin-Lafl`eche, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  2008, ApJL, 685, L129, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/592501  Smartt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2009, ARA&A, 47, 63,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1146/annurev-astro-082708-101737  Sukhbold, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Heger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2018, ApJ, 860,  93, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/1538-4357/aac2da  Suwa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Tominaga, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2015, Monthly Notices of the  Royal Astronomical Society, 451, 282,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1093/mnras/stv901  Suwa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Tominaga, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Maeda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2019, MNRAS, 483,  3607, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1093/mnras/sty3309  Suwa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Yamada, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Takiwaki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Kotake, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2016,  ApJ, 816, 43, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/0004-637X/816/1/43  Takiwaki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Kotake, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Suwa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2016, Monthly  Notices of the Royal Astronomical Society, 461, L112,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1093/mnrasl/slw105  Thielemann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Hashimoto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1996,  ApJ, 460, 408, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/176980  Timmes, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1999, ApJS, 124, 241, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/313257  Timmes, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Swesty, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2000, ApJS, 126, 501,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/313304  Von Neumann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Richtmyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1950, Journal of  Applied Physics, 21, 232, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1699639  Witt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Psaltis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Yasin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2021, ApJ, 921, 19,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/1538-4357/ac1a6d  Wongwathanarat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', M¨uller, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Pllumbi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=',  & Wanajo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2017, ApJ, 842, 13,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='3847/1538-4357/aa72de  Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Arnett, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Clayton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1973,  ApJS, 26, 231, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/190282  Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Heger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Weaver, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2002, Reviews  of Modern Physics, 74, 1015,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1015  Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', & Weaver, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1995, ApJS, 101, 181,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='1086/192237  12  Sawada & Suwa  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  56Ni SYNTHESIS CONDITION  Here, we briefly review the general features of 56Ni nucleosynthesis in CCSNe, which is classically well enough  understood (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Bodansky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Woosley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Hix & Thielemann 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' To illustrate the condition of  56Ni synthesis, for simplicity, we use parameterized adiabatic expansion profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Let us focus on a certain Lagrangian  particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Assuming that a passing shock wave heats and compresses the particle to peak temperature T0 and peak  density ρ0, and then it expands and cools in a constant T 3/ρ track, we can write the temperature and density evolution  (Woosley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 1973) as  T(t) = T0 exp (−t/3τdyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' ),  ρ(t) = ρ0 exp (−t/τdyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=') ,  (A1)  where the expansion timescale of the ejecta τ is the following (Fowler & Hoyle 1964);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  τdyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' = (24πGρ0)−1/2 ≈ 446/ρ1/2  0  sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  (A2)  This expansion trajectory reproduces the general temperature and density trajectories in spherically symmetric or  multi-dimensional CCSNe explosions, especially in the explosive nucleosynthesis region (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Magkotsios et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2010,  2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Jerkstrand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We calculate a reaction network of 640 nuclear species with torch code (Timmes 1999),  with the initial composition X(28Si) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We then check the behavior of 56Ni nucleosynthesis using expansion profiles  with peak temperature T0 and peak density ρ0 as parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Figure 9 shows the abundance of synthesized 56Ni as a function of the peak temperature T0 in the adiabatic expansion  profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' When the peak temperature is T9 ≳ 5 (T9 ≡ T/109 K), we can see that a sizable amount of 56Ni is synthesized,  almost independent of the peak density ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In this temperature region, 28Si is wholly depleted at the same time  as satisfying 56Ni synthesis, so this explosive nucleosynthesis is referred to as ‘complete Si burning’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' This depletion  of 28Si occurs when the lifetime over which 28Si burns up is much shorter than the given expansion time scale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=',  τ(28Si) ≪ τdyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='. According to Woosley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (1973), to estimate this condition quantitatively, we define the depletion  point of 28Si as when its mass fraction falls below about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='005 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Xf(28Si) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Then, complete Si combustion  is estimated to occur under the following initial conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  T9 ≳ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0 ×  �  ρ0  106 g cm−3  �1/68  (A3)  This implies that an initial temperature must exceed T9 ≈ 5 to promote the depletion of 28Si in order for the complete  silicon burning to occur at a reasonable density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' This estimate is roughly consistent with our numerical result in Figure  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  In complete Si burning, the forward and reverse reactions in the main reaction channel proceed much faster than  the time scale of the change in thermal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The compositions thus basically follow the Nuclear Statistical  Equilibrium (NSE), except for the slow triple-alpha process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Then, as the temperature decreases rapidly, the reaction  rates decrease and the abundance pattern ‘freezes-out’ (Hix & Thielemann 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In NSE, the composition ratio is  determined to minimize the Helmholtz free energy (F = (U − Q) − TS) for the nuclide mass fraction (Clifford &  Tayler 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' When the temperature is not too high (T9 < 10), the binding energy per nucleon (BEN) Q = B/A  roughly determines the abundance pattern of NSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In environments where the number of protons and neutrons is  almost equal (Ye ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5), the most abundant isotope in NSE is 56Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Therefore, the main synthesis process of 56Ni in  the SNe explosion is not a two-/three-body reaction, but a ‘freezes-out’ of the NSE abundance pattern that occurred  via the ‘photodisintegration-rearrangement reactions’ associated with the photodisintegration of 28Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Next, Figure 10 shows what the 56Ni synthesis would be in a different environment for Ye with different peak  temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' We can clearly see that while 56Ni synthesis in the proton-rich environment (Ye > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5) is relatively the  same as that in the Ye ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5 environment, its synthesis is quite suppressed in the neutron-rich environment (Ye < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  As explained by Seitenzahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (2008), Ye dependence of nickel nucleosynthesis can be understood qualitatively by  focusing on the BEN, Q = B/A, and the Helmholtz free energy F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Updating the 56Ni Problem in CCSNe  13  10-5  10-4  10-3  10-2  10-1  100  2x109  3x109  4x109  5x109  6x109  7x109  8x109  9x109  complete Si-burning  56Ni(solid)  28Si(dashed)  abundance  peak temperature [K]  ρ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0*107 [g/cc]  ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0*107 [g/cc]  ρ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0*106 [g/cc]  ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='0*106 [g/cc]  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The abundance of synthesized 56Ni as a function of peak temperature T0 with the peak density ρ0 at Ye = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5,  calculated using the adiabatic expansion profile of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  electron fraction Ye  peak tempareture [K]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='54  2x109  3x109  4x109  5x109  6x109  7x109  8x109  9x109  10-3  10-2  10-1  100  abundance of 56Ni  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The abundance of synthesized 56Ni in the peak temperature T0 and the initial electron fraction Ye plane at the  peak density ρ0 = 107 g cm−3, calculated using the adiabatic expansion profile of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Figure 11 (left) illustrates the BEN of various Ni-isotopes with blue crosses, where the number of protons per nucleon  (Yp = Z/A, which corresponds to Ye) on the horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' To analyze stability, we consider a mixed composition  of 56Ni plus an appropriate number of free protons or free neutrons (56Ni +nucleons) to match any Ye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The mean  BEN of the mixed composition is calculated as ¯Q = Σi Qini/nB, where Qi is the binding energy of the nucleus i and  Qneut = Qprot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' By comparing the BENs for each Ye, we find that the single Ni-isotope is significantly more stable  than the 56Ni +nucleons mixture in the neutron-rich region (Ye < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5), but not in the proton-rich region (Ye ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  While this picture is somewhat modified when entropy is taken into account, this simple discussion shows that 56Ni  synthesis is suppressed in the neutron-rich region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  Figure 11 (right) shows Helmholtz free energies (F = (U −Q)−TS) for a single composition of only Ni isotopes and,  as before, for a mixed composition of 56Ni plus an appropriate number of free protons or free neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Helmholtz free  energy was calculated using Helmholtz EoS (Timmes & Swesty 2000) with T = 5×109 K and ρ = 107 g cm−3 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' In  an environment with the same T and ρ, the internal energy can be considered to be almost constant, U ≈ Const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The  BEN term, Q, then contributes to the Helmholtz free energy on the order of 1 MeV/nuclear ≈ O(1018) erg g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' For  the TS term, the entropy is increased by the free nucleons in the 56Ni + nucleon mixture composition compared to the  single Ni-isotope composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' It thus works toward lowering the Helmholtz free energy on the order of O(1017) erg  14  Sawada & Suwa  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='8  8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='8  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='56  56Ni  Binding Energy per Nucleon [MeV]  Yp = Z/A  (=Ye:electron fraction)  ANi  56Ni+nucleons  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='2  9  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='6  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='2  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='56  56Ni  Helmholtz free energy [1018 erg/g]  F = (U - Q) - TS  Yp = Z/A  (=Ye:electron fraction)  ANi  56Ni+nucleons  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Binding energy per nucleon (left) and the Helmholtz free energy per nucleon (right) with the number of protons  per nucleon Yp = Z/A (corresponds to Ye, and 28/Ye corresponds to the mass number A of each Ni isotope) on the horizontal  axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The blue crosses show the binding energy per nucleon (BEN) of the Ni-isotope, and the red circles indicate the mean BEN  of a mixed composition, which is mixed with 56Ni and an appropriate number of free protons or free neutrons for any given Ye  (56Ni +nucleons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The mean BEN of the mixed composition is derived by ¯Q = Σi Qini/nB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The Helmholtz free energies were  calculated using Helmholtz EoS (Timmes & Swesty 2000) with T = 5 × 109 K and ρ = 107 g cm−3 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Therefore, in the proton-rich region (Ye ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5), the difference in BEN between the 56Ni +nucleon mixture and  the single Ni-isotope is ∆Q ≲ O(1017) erg g−1, which is not much larger than the correction by entropy enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  As a result, the correction for the entropy enhancement due to the free nucleon makes the 56Ni +nucleon mixture more  stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' On the other hand, in the neutron-rich region (Ye < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5), the difference in BEN is ∆Q ∼ O(1018) erg g−1,  which is sufficiently larger than the effect of the correction by entropy enhancement, and thus the single Ni-isotope  composition is more stable, as predicted from the BEN behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='  To summarize the above discussion, in the temperatures region where T9 ≳ 5 are attained behind the shock, NSE is  achieved except for the slow triple-alpha process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' Thus, the abundance pattern depends only on Ye and the entropy,  and is dominated by iron-group elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=', Hix & Thielemann 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The most abundant element in this region  of complete Si burning with alpha-rich freeze-out is 56Ni , provided Ye ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content=' The 56Ni synthesis conditions in CCSNe  explosion are an environment where the peak temperature T9 ≳ 5 is experienced by the shock and the electron fraction  of the material is Ye ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf'}
diff --git a/TNE0T4oBgHgl3EQflAGw/content/tmp_files/2301.02481v1.pdf.txt b/TNE0T4oBgHgl3EQflAGw/content/tmp_files/2301.02481v1.pdf.txt
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@@ -0,0 +1,2133 @@
+Distal and non-symmetrical crack nucleation in
+delamination of plates via dimensionally-reduced
+peridynamics
+R. Cavuotoa, A. Cutoloa, K. Dayalb,c, L. Deserid,e,b,f,g,∗, M. Fraldia,h,∗
+aDepartment of Structures for Engineering and Architecture, University of Naples, Italy
+bDepartment of Civil and Environmental Engineering, Carnegie Mellon University, USA
+cCenter for Nonlinear Analysis, Department of Mathematical Sciences, CMU, USA
+dDepartment of Civil, Environmental and Mechanical Engineering, University of Trento,
+Italy
+eDepartment of Civil and Environmental Engineering, Pittsburgh University, PA, USA
+fDepartment of Nanomedicine, Houston Methodist Hospital, Houston TX, USA
+gDepartment of Mechanical Engineering, Carnegie Mellon University, Pittsburgh PA, USA
+hD´epartment de Physique, Ecole Normale Sup´erieure, Paris, France
+Abstract
+Exploiting the framework of peridynamics, a dimensionally-reduced formula-
+tion for plates is developed that allows for the through-thickness nucleation
+and growth of fracture surfaces, enabling the treatment of delamination in a
+lower-dimensional model.
+Delamination fracture nucleation and propagation
+are treated by choosing the kinematics to be composed of an absolutely con-
+tinuous part and a zone where jumps in the displacements are allowed. This
+assumption allows the explicit derivation of the dimensionally-reduced elastic
+energy, which shows a hierarchy of terms characterising the stored energy in a
+the plane element. An interpretation of the various terms of the reduced energy
+is shown by means of the simplest paradigm of bond-based peridynamics. A
+striking feature of the reduced energy is that, despite the small-displacement
+assumption, there is a coupling between the membrane and bending terms.
+Semi-analytical solutions for simplified settings are obtained through a mini-
+mization procedure, and a range of nonstandard behaviors such as distal crack
+nucleation and curved crack path are captured by the model. Finally, the con-
+vergence of the proposed peridynamic reduced model to a local elastic theory
+for vanishing nonlocal lengthscale is determined, giving a local cohesive model
+for fracture.
+Full article available at https://doi.org/10.1016/j.jmps.2022.105189.
+Keywords:
+crack onset, peridynamics, plates, delamination
+∗Corresponding authors.
+Email addresses: luca.deseri@unitn.it (L. Deseri), fraldi@unina.it (M. Fraldi)
+Preprint submitted to Journal of the Mechanics and Physics of Solids
+arXiv:2301.02481v1  [physics.app-ph]  6 Jan 2023
+
+1. Introduction
+Delamination is a mode of failure that is typical of thin plate and shell
+structures in which the thickness is much smaller than the other two dimen-
+sions. This mode of failure is characterised by a fracture in which the crack
+front propagates within the plane of the structure, resulting in the structure
+being broken up into layers. Composite laminates, which naturally present a
+weak plane at the interface between different materials, are especially vulnera-
+ble to delamination, but it can occur in microstructured and homogeneous thin
+structures as well [1, 2].
+This mode of fracture has been studied through a variety of approaches.
+Leading approaches include Cohesive Zone Models (CZM) [3–8] and the ex-
+tended finite element methods (XFEM) [9, 10]. More recently, the phase-field
+technique for fracture has been used to model debonding in laminates [11, 12].
+These models, however, treat the thin structure as a fully three-dimensional
+body and treat delamination explicitly.
+In a distinct approach to the modeling of thin plates without accounting
+for delamination, an established procedure with a long history is to derive two-
+dimensional formulations for plates or films based on systematically reducing
+the three-dimensional theory using that the thickness is much smaller than
+the other dimensions. This has been studied in a variety of settings, and are
+particularly attractive as they can lead to faster computational algorithms with
+good convergence properties while still capturing the key physical phenomena
+[13–18].
+The aim of this work is to develop an approach that combines the advan-
+tages of dimensionally-reduced models while also accounting for delamination.
+That is, we aim to derive a two-dimensional model of plates that allows for
+delamination failure. Our approach is based on using peridynamics [19, 20], a
+nonlocal theory that models continuum bodies as a collection of infinitesimal
+material particles that interact through long-range forces, rather than the typi-
+cal contact tractions. In contrast to local theories of continuum mechanics that
+rely on the definition of strain, consequently constraining the displacement to
+have sufficient regularity, peridynamics works directly with the displacement
+and does not require regularity a priori.
+This makes it attractive to model
+damage, damage-fracture transition [21–26], and dynamic phenomena such as
+impact and blasts [27–30].
+Papers dealing with two-dimensional peridynamic bodies can be divided into
+two main categories based on the approach: (1) full 3D numerical simulations
+[31–33], and (2) 2D reduced models [34–43]. While the former approach has
+been extensively used to treat delamination explicitly [44–46], existing reduced
+formulation only account for crack propagation with the crack tip oriented nor-
+mal to the plane and thus cannot capture phenomena such as delamination.
+In this work, a reduced formulation of bond-based peridynamics, tailored to
+account for through-thickness delamination in thin plates characterized by a
+single material and no preexisting weak interface, is introduced. As a first step,
+in Section 3, the displacement field is additively decomposed into its absolutely
+2
+
+continuous part and its jump to account for delamination fracture nucleation
+and propagation. That is, the natural function space for the displacement field
+is the space of functions of Special Bounded Variations (SBV), e.g. [47, 48].
+Further assumptions on both parts of the displacement field lead to a reduced
+form of the peridynamics energy in Section 3.4. The reduction procedure gen-
+erates a hierarchy of terms characterising the strain energy stored inside the
+two-dimensional continuum element. A striking feature of the reduced energy
+is that, despite the small displacement assumption, there is a coupling between
+the membrane and bending terms. The hierarchy of the resulting functional
+allows for a consistent variational approach, enabling the displacement fields to
+be obtained by a minimization procedure.
+Semi-analytical solutions for test cases are then obtained in Section 4. The
+tests are performed on a thin cantilever plate, modeled with the proposed re-
+duced formulation. In the first case, such a plate undergoes an imposed upward
+vertical displacement of the upper part of the free edge and a downward vertical
+displacement of the lower edge in a symmetrical manner - much like a peeling
+test. The model shows that variation of the nonlocal interaction lengthscale
+δ, also called the horizon, induces different behaviors, namely distal or prox-
+imal damage nucleation. In the second case, an asymmetry is introduced by
+imposing the vertical upward displacement at various points of the upper edge
+of the plate, leading to non-symmetric crack propagation. In the third case,
+Mode-II fracture or sliding delamination is studied. In order to explore the cou-
+pling between bending and membrane terms in the reduced formulation (which
+is geometrically linear) in a local setting, in Section 5, we examine the conver-
+gence of the proposed model to a local theory when the nonlocal interaction
+scale δ tends to zero. By enforcing a condition of bounded and non-vanishing
+energy, the scaling of the displacement field with δ is established; this, in turn,
+determines the scaling of all the terms in the energy, thereby allowing for the
+localization of the nonlocal model, leading to a reduced local formulation. The
+reduced local formulation has a cohesive structure, due to some terms of the
+energy associated with the jump part of the displacement surviving the limit
+operation.
+Organization. In Section 2, the constitutive framework of bond-based peridy-
+namics is summarized.
+Section 3 sets up the theoretical framework for the
+reduced formulation, and the reduced elastic energy density of a peridynamic
+plate is obtained and interpreted. The theoretical model is then implemented
+in a variational setting to be tested in simple loading conditions in Section
+4. Lastly, in Section 5, the behavior of the reduced formulation proposed for
+vanishing horizon is investigated.
+2. Formulation of the peridynamics model
+Bond-based peridynamics models a continuum body B as a collection of
+material particles interacting with one another, in pairs, through bonds. The
+3
+
+equation for the static equilibrium of the body [19, 49] can be written as the
+integrodifferential equation
+�
+H
+f(x, x′, u, u′) dVx′ + b(x) = 0 ,
+(1)
+where: f, called the pairwise force field, is the force exerted between material
+particle x and x′, it can depend on the displacements u of such particles and
+hosts all the constitutive information of the model; b is the vector of external
+body force; H represents the so-called family of x and is the set of all the
+material points that are within a characteristic distance from it, called horizon,
+δ. In bond-based peridynamics, the balances of linear and angular momentum
+require the pairwise force field, f, to be anti-symmetric with respect to particles
+switch [19]
+f(x, x′, u, u′) = −f(x′, x, u′, u) .
+(2)
+Constitutive relations for the definition of f have been proposed by many au-
+thors [19, 50], among which one of the simplest is the standard linear elastic
+perfectly brittle relation revisited by Zhou [51],
+f = µ c s |ξ|2
+σ ξ ,
+(3)
+where c is called the bond constant (a positive scalar quantity), ξ and η are
+the relative position and displacement of the particles respectively, and s is the
+stretch of the bond. Lastly, σ = σ(ξ) is a function ensuring integrability of
+(3).
+Additional conditions on σ(ξ) are necessary to bound the stiffness and
+the energy respectively of the PD model to finite positive values. The effect
+of σ(ξ) on the PD model, equation (3), is depicted in Figure 1. The function
+µ = µ(ξ, η) in equation (3) is a history-dependent scalar-valued function (also
+called failure parameter) which enforces bond breakage under tension only:
+µ(ξ, η) =
+� 1
+for s < scr
+0
+otherwise
+,
+(4)
+where scr is a critical threshold for the bond elongation. Form (3) admits a
+potential, called pairwise potential function:
+ω(ξ, η) =
+�
+f(ξ, η) · dη =
+�
+ωel = 1
+2c s2 |ξ|4
+σ
+for s < scr
+ωcr = 1
+2c s2
+cr
+|ξ|4
+σ
+otherwise
+.
+(5)
+2.1. A microstructural interpretation of peridynamics
+The present study proposes a dimensionally reduced formulation of peridy-
+namic plates with a particular focus on through-thickness fracture propagation.
+Since the model, developed in Section 3, shows unconventional scalings of the en-
+ergy terms due to its nonlocal (peridynamic) nature, in the present preliminary
+4
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+Figure 1: Upper left: normalized stiffness of a bond for different σ; kr = |f|/|η|, δ is the hori-
+zon and c3 the bond constant for the case of σ = |ξ|3 which recovers the original formulation
+of Silling [19]. Upper right: normalized bond force for different σ; fr is the modulus of the
+force as a function of r (the normalized relative distance) that is applied to bonds under an
+imposed uniform stretch ϵ. Below: the bond energy ωr (energy per unit volume squared) for
+increasing bond length and imposed uniform stretch.
+section a possible micro-structural interpretation of peridynamics, functional to
+the mechanical characterization of our model, is discussed.
+The possibility to pass from the continuum to the discrete level is crucial in
+problems involving the transition from elastic to dissipative phenomena such
+as damage and fracture. Nevertheless, the exact equivalence between a given
+peridynamic model and a corresponding microstructure is usually not trivial to
+find. One possible interpretation, available from purely energetic arguments,
+can be given by means of a discrete structure, see Appendix B. To stress such
+observations, we build a simple numerical example in which a structure made
+of interconnected linear elastic elements leads to a densely packed truss ensem-
+ble. Despite one would assume that the asymptotic behavior for an increasing
+number of micro-beams of the structure tends to that of a standard local contin-
+uum, we demonstrate that –for a prescribed topology– significant discrepancies
+in terms of displacements emerges between discrete and homogenised local con-
+tinuum. Figure 2 depicts the case of a cantilever beam (1 meter long and 0.4
+meters thick), built by assembling trusses in a net-like pattern as displayed in
+the inset of such figure. These trusses are connecting each material point of
+the body with all the others that satisfy a relative distance requirement. The
+parameters involved, namely axial stiffness of the beams and horizon length, are
+5
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+x/L
+v0/vmat
+Figure 2: (Left) Cantilever beam characterized by a net-like micro-structure and subjected
+to an imposed displacement at its free end. (Right) The normalized deformed shapes (vmat
+maximum vertical displacement of structured material) according to peridynamic (PD) and
+local theory.
+calibrated in such a way that the behavior under tension reproduces that of an
+ideal homogenized continuous local beam. In Figure 2 the deflection of the struc-
+tured beam when subjected to a vertical force applied at one end is compared
+with local theories (in red Euler-Bernoulli and Timoshenko beam overlapping
+one onto the other) and with the predictions of peridynamics (PD, in blue).
+Failing of local theories to correctly characterize the bending behavior for the
+example introduced above, can be ascribed to the intricate internal structure of
+the beam.
+3. A dimensionally-reduced model for thin plates
+In order to develop the analytical calculations necessary for the formulation
+of a dimensionally reduced model for plates, certain assumptions are made on
+both the kinematics of the plate and on the constitutive relation of the bond-
+based peridynamic continua.
+3.1. Kinematics
+Since the fracture of a material can be seen as the nucleation and growth
+of a discontinuity in its displacement field, one can additively partition the
+kinematics into a continuous part, accounting for elastic deformations, and a
+jump part, accounting for the displacements due to the delamination, namely:
+u(x) = ua(x) + uJ(x) ,
+(6)
+where the index a indicates the absolutely continuous part while the index J
+denotes the jump part. In this sense, it can be said that u is a function in the
+space of Special Bounded Variations (SBV) [47, 48].
+We now restrict ourselves to the study of thin bodies, B, characterized by a
+constant thickness H. Given a region of the three-dimensional euclidean space
+E3, and a Cartesian reference frame (O, x1, x2, x3), Figure 3, the absolutely
+6
+
+continuous part of the displacement is approximated by a polynomial expansion
+as follows
+ua(x) ≈ A(x1, x2) + B(x1, x2)x3 + · · ·
+.
+(7)
+It is important to highlight that the choice of the reduction plane (the expansion
+point in the expansion above) can have effects on the hierarchical distribution
+of terms in the reduced formulation [35]. In the sequel the midplane of the plate
+is chosen to perform the dimension reduction, so as to align with classical local
+elastic reduced formulations.
+O
+x3
+x1
+x2
+x
+y
+H (x)
+H (y)
+u = ua + uj
+h(x1,x2)
+j(x1,x2)
+Figure 3: Kinematics of a plate of thickness H undergoing through-thickness fracture prop-
+agation, here also referred as delamination. Each point of the reference configuration lying
+on possible delamination surfaces to be determined jumps to a new position specified by the
+vector j(x1, x2).
+The jump part of the displacement field will be represented in a rather general
+way as
+uJ(x) = j(x1, x2) · Θ(x3 − h(x1, x2)),
+(8)
+where the h(x1, x2) can be regarded as the crack surface, defining the surface on
+which a displacement discontinuity may arise, while Θ is the Heaviside function;
+lastly, j(x1, x2) is the vector function defining the jump itself. It is worth noting
+that mixed-mode fracture processes are allowed by the ansatz made above on uJ.
+Due to the kinematic split imposed on equation (6), the relative displacement
+field now reads as follows:
+η = ηa + ηJ.
+(9)
+3.2. Damage
+It is important to highlight that in nonlocal theories a discontinuity in the
+displacement field does not necessarily mean fracture nucleation/propagation, as
+particles that are already separated by a finite distance can very well withstand a
+jump in their relative displacement. In PD, what ensures the effective occurrence
+of damage is the µ function (4), which represents the failure criterion for the
+bonds. Indeed, the state of interaction can be determined by means of equation
+(4), which enforces a critical stretch condition (s < scr) [19, 52, 53]. In many
+other cases available in the literature, instead of a critical elongation criterion,
+7
+
+an energy-based one is employed [54–56]. Such a criterion relates the breakage
+of a bond to the attainment of a threshold in the stored energy, called critical
+bond energy ωcr.
+Both the critical stretch and critical energy are typically
+evaluated by means of an energy comparison with the standard local theory of
+fracture mechanics. In particular, the PD energy necessary for the growth of a
+new surface in the body, defined as the energy required to break all the bonds
+which pass through that particular surface (Figure 4), is imposed to be equal to
+the critical energy release rate of Griffith theory [57], an operation that ensures
+the recovery of the Griffith theory in the limit of small horizon [25, 58, 59].
+Figure 4: Computation of the total energy necessary to break all the bonds connecting the
+material point x with those x′ on the other side of the fracture surface h(x).
+In this fashion, for the case of the critical stretch, one obtains [34, 60]
+scr =
+�
+Gc
+β(H, σ) δ ,
+(10)
+where Gc is the critical energy release rate and β is a scalar function of the
+shape of the family H and on σ(|ξ|).
+The transition from damage to fracture is therefore naturally tracked by the
+failure mechanism of PD.
+3.3. Lagrangian formulation
+Under certain conditions [61], the solution of the equilibrium problem of the
+nonlocal PD body coincides with the stationary points of the following functional
+[62–64]:
+L = −Eel + W,
+(11)
+where Eel is the elastic energy, and W is the work of the external loads. By
+explicitly expressing the various terms, equation (11) becomes
+L[u] = −1
+2
+�
+B
+�
+H
+ω(ξ, η) dV′dV +
+�
+B
+b · u dV ,
+(12)
+8
+
+where ω is the energy density defined in (5), while B and H are the continuum
+body and the family of a point, respectively. We here define the decomposition
+of B through a Cartesian product as B = Bα × B3, where
+Bα := {(x1, x2), ∀x ∈ B}
+and B3 := {x3, ∀x ∈ B} .
+Accordingly, one can define H = Hα × H3, where
+Hα :=
+�
+(x′
+1, x′
+2), ∀x′ ∈ B :
+�
+(x′
+1 − x1)2 + (x′
+2 − x2)2 ≤ δ
+�
+,
+H3 :=
+�
+x′
+3, ∀x′ ∈ B :
+�
+(x′
+1 − x1)2 + (x′
+2 − x2)2 ≤ δ
+�
+.
+Since in the present study x3 is chosen as the out-of-plane coordinate (see Figure
+3), its value ranges in between {−H/2, H/2}, H being the plate thickness.
+In view of the previous Cartesian products, one can now write
+L[u] =
+�
+Bα
+�
+�−1
+2
+�
+Hα
+�
+H3
+�
+B3
+ω(ξ, η) dx3dx′
+3dSα +
+�
+B3
+b · u dx3
+�
+� dSα.
+(13)
+Performing the integrations through the thickness of (13) allows one to obtain
+the reduced form of the total Lagrangian of the plate. In particular, the first
+addend in parenthesis of equation (13), which is the elastic energy per unit
+surface ΛE, becomes:
+ΛE = 1
+2
+�
+Hα
+�
+H3
+�
+B3
+ω(ξ, η) dx3dx′
+3dSα =
+= 1
+2
+�
+Hα
+H
+2
+�
+− H
+2
+H
+2
+�
+− H
+2
+ω(ξ, η)dx3dx′
+3dSα =
+�
+Hα
+ωreddSα ,
+(14)
+where ωred is the reduced form of the pairwise potential function ω.
+Similarly, we refer to the result of the through-thickness integration of the work
+of the external loads (second addend in parenthesis in equation (13)) as ΛW.
+All the functionals involved in (13) are nonlocal, as the unknown function u
+is evaluated at different points of the body. An equivalent form of the Euler-
+Lagrange equation for nonlocal functionals is now necessary to find the station-
+ary points of (13). The search for stationary points within the interior of the
+domain of the functional (or its minimization) has been investigated in [65]. For
+the particular case of static and elastic PD nonlocal functional [62–64] one has
+that the following implication holds:
+min
+u L → 2∂ΛE
+∂q − ∂ΛW
+∂q
+= 0 ,
+(15)
+9
+
+where the q is the vector of the unknown functions of the problem, which because
+of eqs. (6) to (8) reads as follows:
+q = {h(x1, x2), j(x1, x2), A(x1, x2), B(x1, x2)} .
+In order to retrieve equation (15), condition (2) must be enforced on the results
+of [62–64].
+3.4. Hierarchical form of the reduced pairwise potential function
+We here retrieve an explicit form of the reduced pairwise potential function,
+ωred =
+H
+2
+�
+− H
+2
+H
+2
+�
+− H
+2
+ω(ξ, η) dx3dx′
+3 ,
+(16)
+for a bond-based peridynamic body. For the linear elastic case, the influence
+of the function σ appearing in (3) on the overall behavior has been indirectly
+investigated in Bobaru at al. [53]. There, the authors have shown how the shape
+of the micromodulus function has indeed consequences on the overall behavior
+of the material, albeit this does not influence the rate of convergence for a
+vanishing horizon. The result is critical to this work, where the consequences
+of a localization procedure on the reduced peridynamic model will be explored
+(section 5) with the objective of retrieving a local reduced formulation for plates.
+For the purpose of simplifying the calculations, drawing on the results discussed
+above [53], condition σ = 1 is enforced in the sequel. Neglecting the failure
+parameter (denoted by µ in equation (4)) allows the evaluation of the reduced
+form of the energy for the fully elastic case, i.e. when the load has yet to break
+any bond. If c is the bond constant, and φ is the ratio between the in-plane
+component of the horizon and the thickness, then
+ωred
+c
+= H2 p1(φ, ua) + H4 p2(φ, ua) + H6 p3(φ, ua)
+�
+��
+�
+ωred,a
++ωred,J(H, φ, ua, uJ) ,
+(17)
+where
+p1(φ, ua) =(2φ − φ2)(x′
+1 − x1)2 (A1(x′
+1) − A1(x1))2 /4 ;
+p2(φ, ua) = { 2φ3(3φ − 4)(A3(x′
+1) − A3(x1))2+
+(x′
+1 − x1)2φ3(3φ − 4)(B1(x′
+1)2 + B1(x1)2)+
+(x′
+1 − x1)2(−2φ + 3φ2 − φ4)(B1(x′
+1) − B1(x1))2+
+2φ3(3φ − 4)(x′
+1 − x1)(A1(x′
+1) − A1(x1))(B3(x′
+1) + B3(x1))+
+2φ3(3φ − 4)(x′
+1 − x1)(A3(x′
+1) − A3(x1))(B1(x′
+1) + B1(x1))
+�
+/48 ;
+p3(φ, ua) ={
+�
+20 − 45φ + 72φ2 − 80φ3�
+(B3(x′
+1) − B3(x1))2 +
+(20 − 45φ − 72φ2 + 80φ3)(B3(x′
+1)B3(x1))
+�
+/1440 ;
+10
+
+while ωred,J, an implicit function of φ, H and the unknown fields, denotes the
+part of the reduced energy associated with the jump field. As shown in equation
+(17) the part of the reduced energy associated with the continuous displacements
+is henceforth denoted by ωred,a.
+In the case of a through-thickness horizon equal to the whole thickness of the
+thin element, the physical condition of isotropic interaction is assumed. In such
+a case:
+p1(1, u) =(x′
+1 − x1)2 (A1(x′
+1) − A1(x1))2 /4 ;
+(18)
+p2(1, u) = { − 2(x′
+1 − x1)(A3(x′
+1) − A3(x1))(B1(x′
+1) + B1(x1))+
+(19)
+− 2(A3(x′
+1) − A3(x1))2 − (x′
+1 − x1)2(B1(x′
+1)2 + B1(x1)2)+
+− 2(x′
+1 − x1)(A1(x′
+1) − A1(x1))(B3(x′
+1) + B3(x1))}/48 ;
+p3(1, u) ={10B3(x′
+1)B3(x1) + 7B3(x′
+1)2 + 7B3(x1)2}/1440 ;
+(20)
+and:
+ωred,J
+c
+= r0(uJ) + H r1(ua, uJ) + H2 r2(uJ)+
+(21)
+H3 r3(ua, uJ) + H4 r4(ua, uJ) + H5 r5(ua, uJ)
+,
+where the ri functions are reported in Appendix A.
+For simplicity, equation (17) has been specialized for the plane strain case.
+The variables x1 and x′
+1 represent respectively the in-plane component of the
+position vector for particle x and x′. Furthermore, we have used A = {A1, A3},
+B = {B1, B3} and j = {0, j3}. The latter limits the kinematics to that of a
+pure Mode-I fracture. It is possible to see how the dimension reduction of the
+pairwise potential function generates a hierarchy of terms characterizing the
+strain energy stored inside the planar element.
+In Appendix
+B, following the microstructural interpretation given in section
+2.1 and through the definition of a paradigmatic case of discrete peridynamics,
+a simple tool for the physical interpretation of the various terms in the reduced
+energy of our peridynamic continuum is presented. Thanks to the paradigmatic
+case, it is possible to give an immediate physical interpretation to the terms of
+(17) that are scaling with the square p1 and the fourth power p2 of the thickness,
+that is the former are membrane terms and the latter bending.
+From an analysis of the expression of p1 (see eq. (18)), since A1(x1) is the
+in-plane component of the displacement field for the points on the reduction
+plane, it can be confirmed that the terms scaling with H2 of ωred,a can be
+regarded as purely membrane. In the higher-order term, on the contrary, such
+as p2 (equation 19) which multiplies H4, one can assess the presence of purely
+bending contributions (for example, those depending solely on B1(x1)2), but also
+of mixed ones. The mixed terms introduce the coupling of membrane behavior
+and bending behavior. This is a unique feature of the nonlocal formulation.
+Indeed, coupling between the membrane and bending behaviors is a feature not
+easily recoverable in local theories, as shown in Appendix C. In Section 5 it is
+shown that the coupling is lost when the peridynamic model is localized, namely
+the reduced energy is evaluated in the limit of vanishing horizon δ.
+11
+
+Lastly, the terms scaling with H6 are higher order ones depending only on
+B3(x1), which is the nonlocal equivalent of a strain deformation through the
+thickness ∂x3(ua · e3), where e3 is the unit vector normal to the plane (x1, x2).
+We note that in order to recover the kinematics of the Kirchhoff plate theory
+B3(x1) must be null.
+The contribution of the jump part of the displacement field to the reduced en-
+ergy, reflected in ωred,J, is more scattered. We see contributions of the jump field
+to both membrane, mixed and bending-related quantities. Here, again, coupling
+occurs between the different fields of the jump part of the displacement and the
+continuous part. The highest order term in the thickness (H) is determined
+by the order of the truncation in the Taylor expansion of the continuous part
+of the displacement. By retaining only terms up to the first order in x3, the
+highest power becomes 6. This particular choice was made in order to check
+the convergence of the nonlocal model, which will be done in the last section of
+this work.
+4. Model implementation and applications
+Under the aforementioned conditions, the solution for the Euler-Lagrange
+system of equations (15) of the PD model was achieved by using a Galerkin
+approach, resulting in a system of the kind
+�
+Bα
+�
+2∂ΛE
+∂q − ∂ΛW
+∂q
+�
+δq = 0 ,
+(22)
+which can be solved iteratively.
+A Mathematica code has then been developed in order to test the model under
+different loading conditions.
+We here present displacement-induced tests for
+symmetric and non-symmetric load distributions.
+The results have shown that the reduced model is capable of reproducing both
+traditional and unconventional mechanical behavior such as distal crack nucle-
+ation and loss of symmetry in the crack pattern.
+Mechanical and geometric quantities
+Value
+Young’s Modulus [MPa]
+5000
+Critical surface energy Gc [J/m2]
+5.3
+Thickness over length (H/L)
+1/25
+Nonlocal parameters
+Value
+Horizon(δ)/Length(L)
+1/5
+Bond constant c [N/mm6]
+7.7x106
+Critical stretch scr [-]
+0.2x10−4
+Table 1: Parameters of the local equivalent material and geometry of the plate (up); nonlocal
+parameter of the peridynamic bond-based reduced model (down).
+12
+
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.
+0.2
+0.4
+0.6
+0.8
+1.
+u/H
+u
+(x 10-4)
+L
+x1
+x2
+x3
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+x3
+x1
+Crack nucleation
+- 0.6
+- 0.4
+- 0.2
+0.0
+0.2
+0.4
+0.6
+u=0.163 x 10-4H
+u=0.195 x 10-4H
+u=0.228 x 10-4H
+0.04
+0.04
+0.06
+0.06
+0.06
+0.08
+0.1
+0.12
+0.12
+0.14
+0.14
+0.05
+0.05
+0.1
+0.15
+0.2
+0.2
+0.2
+0.25
+0.25
+0.25
+0.3
+0.3
+0.3
+0.35
+0.35
+0.35
+0.35
+0.4
+0.4
+0.4
+0.45
+0.45
+0.45
+- 0.6
+- 0.4
+- 0.2
+0.0
+0.2
+0.4
+0.6
+0.04
+0.04
+0.06
+0.06
+0.06
+0.08
+0.08
+0.08
+0.1
+0.1
+0.12
+0.12
+0.14
+0.14
+- 0.6
+- 0.4
+- 0.2
+0.0
+0.2
+0.4
+0.6
+x3
+x1
+“Distal” crack nucleation
+Figure 5: Results of the analysis of a peridynamic cantilever plate subjected to two opposing
+vertical displacements at the upper and lower edges using the present formulation. In the
+upper part: on the left, the evolution of the force-displacement response and the schematic
+representation of the test carried out, while on the right, the deformed shape corresponding
+to an imposed displacement of u/H = 0.195 × 10−4 amplified by a factor of 100. In the lower
+part: level curves representing the displacements - normalized with respect to the imposed
+one - showing the evolution of crack surface h(x), represented by the dashed blue line, as the
+imposed displacements at the edges increase.
+4.1. Displacement-induced peeling test
+The peridynamic reduced formulation proposed above is used to study the
+case of displacement-controlled test inducing through-thickness fracture of a
+cantilever plate. As shown in Figure 5, the plate is loaded by the application
+of a pair of vertical displacements to the upper and lower part of the free edge.
+The initial geometry necessitates neither an a priori crack nor a notch in order
+to develop a crack. This is due to the damage being implemented at the con-
+stitutive level in the peridynamic theory and to the kinematic assumptions on
+the damage-fracture transition taken before.
+The test has been carried out until a final vertical displacement of around H/600,
+a quantity which is sufficient for the development of fracture for the chosen
+elastic and critical parameters (see Table 1). In Figure 5 (above), in blue, the
+normalized force vs displacement plot is presented, while in red is the fraction
+of bonds that have yet to break near the loaded area. The latter has been used
+13
+
+to investigate the propagation of damage before and during fracture growth. In
+particular, both damage and fracture surfaces first develop at a distal section
+from the plate edge (loci of the applied load) as shown in Figure 5 (below), and
+then propagate in both directions, as observed, e.g., in laminated paper [18].
+This unusual response is obtained for a significant nonlocal character of the peri-
+dynamic continuum, i.e. horizon larger than the thickness of the plate. In fact,
+by reducing this parameter the interaction becomes more local and a different
+response is obtained where the crack nucleates closer to the free edge, ultimately
+reaching it in the limit of vanishing horizon which is a typical result of standard
+local continuum theories. Figure 6 shows the results of a parametric analysis of
+δ ringing from a value of twice the thickness H down to approximately zero, the
+value at which the fracture is nucleating and propagating from the cross-section
+at the free edge (where the load is applied). When in a peridynamic discrete
+δ=H/2
+A
+A’
+A
+A’
+A
+A’
+0
+0.0005
+0.001
+0.0015
+0.
+0.2
+0.4
+0.6
+0.8
+1.
+u/H
+0
+0.0005
+0.001
+0.0015
+0.
+0.2
+0.4
+0.6
+0.8
+1.
+u/H
+0
+0.0005
+0.001
+0.0015
+0.
+0.2
+0.4
+0.6
+0.8
+1.
+u/H
+δ>H
+δ   0
+Figure 6: Qualitative behavior of crack nucleation and propagation for a peridynamic reduced
+cantilever plate subjected to a displacement-induced test for a decreasing horizon. The plates
+are clamped at cross-section AA’ and loaded at the opposite edge.
+body or continuum the horizon is reduced, the number of total interactions,
+i.e. the bonds, of a point is also reduced. As a consequence, the behavior of
+the structure becomes less cohesive; this feature is clearly shown in the force-
+displacement plots of the various cases depicted in Figure 6. Surprisingly, the
+cohesive trait is not completely lost in the local case as is shown in the next Sec-
+tion. Finally, the red lines in the force plots are the relative number of broken
+14
+
+bonds, thus they represent the total damage in the zone of the load application.
+Along with the loss in cohesiveness, a reduction in the number of bonds is due
+to affect the overall stiffness of the plate. We hence display the result of a com-
+parison of the plate behavior for different horizon sizes, given a constant overall
+stiffness. This condition can be obtained by increasing the bond constant c of
+the peridynamic model as δ decreases; having in mind the paradigmatic micro-
+structure of a peridynamic discrete body, an increase in c is achieved by thick-
+ening each beam that represents a bond, see Figure 7 (see Appendix B). In the
+same figure, the comparison shows that the nonlocal micro-structure is capable
+of absorbing more energy, displaying thus superior toughness when compared
+with the cases of smaller horizons, which are in this sense more brittle. The
+relationship between horizon size and total dissipated energy seems to be less
+than linear as, from our study, an increase of four times the volume of inter-
+action has brought about an increase of total energy dissipated by 1.5 times.
+It is worth mentioning here that contour plots and force displacements plots
+0
+0.0005
+0.001
+0.0015
+0.
+0.2
+0.4
+0.6
+0.8
+1.
+u/H
+fv /fmax
+Figure 7: Comparison of the force-displacement response of nonlocal plates with different
+horizon sizes but equal overall stiffness (on the left). On the right, is the equivalent micro-
+structure for the peridynamic body in the different cases; the increase in bond stiffness is
+achieved by thickening the cross-section of each beam.
+all depict early-stage crack propagation phenomenon in the peridynamic plate,
+that is the damaging onset, the nucleation of fracturing embryos and the crack
+advancement in the very close regions.
+4.2. Non-symmetric load distribution inducing a through-thickness crack
+Starting from the previous case of a symmetrically loaded plate, we here
+explore the effects of an asymmetry in the application of the loads on the crack
+surface of a cantilever plate. The non-symmetric loading condition is achieved
+by pulling the upper edge in multiple points while the lower edge of the plate
+is still pulled from a single one, see Figure 8 on the right. The parameters used
+for the simulation are c = 7618N/mm6 (bond constant), δ = H/2 (the horizon),
+15
+
+scr = 0.02036 (critical elongation of a single bond) and the test has been carried
+out until a final vertical displacement of approximately H/20.
+As far as the crack path is concerned, the non-symmetric load distribution in-
+duces an unexpected non-symmetric crack trajectory which nucleates and prop-
+agates from the external section towards the center of the plate, see Figure 8.
+This sensitivity of crack path to even slight loss of symmetry in the prescribed
+boundary conditions is not typically achieved in thin structures obeying Saint
+Venant’s principle.
+0.00
+0.01
+0.02
+0.03
+0.04
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+u/H
+0.05
+0.1
+0.15
+0.2
+0.25
+- 0.6
+- 0.4
+- 0.2
+0.0
+0.2
+0.4
+0.6
+Figure 8: Non-symmetrical load distribution leading to a loss of symmetry of the crack path.
+On the left, the force-displacement plot (in blue) normalized with respect to the peak force,
+and the number of intact bonds is in red. The latter is representative of damage evolution.
+On the right, the crack surface (blue dotted line) in the sample loaded by the non-symmetrical
+distribution of forces and displacements expressed in terms of level set curves.
+4.3. Mode-II fracture propagation
+Nonlocal peridynamic plates can show loss of continuity through the thick-
+ness due to the action of an external couple. To show this, a clamped peridy-
+namic plate is loaded through the application of two opposing forces, applied at
+the upper and lower edge of the free end of the plate, with a growing inclination
+(see Figure 9). The mechanical and geometrical parameters chosen for the sim-
+ulation are the same as the previous case shown in Section 4.2. Upon reaching
+a condition of forces almost horizontal (zero inclination, Figure 9 on the right)
+the characteristic distal nucleation shown for the previous case of opposing ver-
+tical forces and Mode-I failure is lost and a more “classical” crack growth is
+exhibited with nucleation occurring at the free-end section in a Mode-II fash-
+ion. Nonetheless, in the latter case, the evolution of the crack is not continuous
+and, at a later stage, a more distal crack nucleates far from the first.
+As a last observation it is useful to highlight that from the various examples
+presented above it emerges a complex and rich interaction between the applied
+loads and the displacement field. This makes it very hard to substitute a spe-
+cific load distribution with a possible static equivalent, such as the resultant,
+to be applied to a dimensionally reduced plate. The need to follow fractur-
+ing/delamination processes makes forces with overall vanishing resultants as
+16
+
+relevant as not vanishing ones, the nonzero force and couple resultants being
+thus not the sole effective loads to be considered. Indeed, the two opposite forces
+of the first example, applied at the same free surface of the plate along the same
+vertical direction, give zero global resultant but are however very relevant for
+delamination, consistently with the classical peeling tests.
+a)
+b)
+0.1
+0.2
+0.3
+0.4
+0.4
+0.4
+0.5
+0.5
+0.5
+0.6
+0.6
+0.7
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+“Distal” nucleation
+Classical nucleation
+u
+L
+x3x3
+x2
+x1
+x3
+x1
+x3
+x1
+u
+L
+x3x3
+x2
+x1
+0.3
+0.4
+0.5
+0.6
+0.7
+0.7
+0.8
+0.9
+0.9
+-0.6
+-0.4
+- 0.2
+0.0
+0.2
+0.4
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+x3
+x1
+Figure 9: A nonlocal peridynamic plate loaded with a couple obtained by two opposing forces
+of increasing inclination. In the upper part of the figure a schematic representation of the loads
+and the plate is reported for both the case of a) inclination of the forces of 30° and b) inclination
+of the forces close to zero. In the central part of the figure are reported the displacements
+(normalized with respect to the maximum displacement imposed, i.e. u/H = 0.01) induced
+in the plate by the forces for the two cases, displaying different crack nucleation and growth
+(blue dotted line). Lastly, in the lower part of the Figure, it is depicted the deformed shape
+of the nonlocal plate corresponding to an imposed horizontal displacement of u/H = 0.003
+(with an amplification of 40 times).
+17
+
+5. Convergence to a local elastic model
+Convergence of the proposed PD model to local elasticity is assured for
+the continuous part of the displacement field only [62, 66–68].
+Nonetheless,
+with appropriate scaling of the jump field functions, convergence for vanishing
+horizon leads to a bounded form of the energy.
+By applying (9) to (5), the first term of (12), which is the elastic energy of a
+bond-based PD body, becomes
+1
+4
+c
+σ(ξ)µ
+�
+B
+�
+H
+�
+(ξ · ηa)2 + 2(ξ · ηa)(ξ · ηJ) + (ξ · ηJ)2�
+dV ′dV .
+(23)
+To ensure convergence for vanishing nonlocality, i.e. δ → 0, the scaling of each
+term must be checked.
+5.1. Peridynamic parameter evaluation
+Isotropic homogeneous linear elastic materials in the bond-based peridy-
+namic theory are characterized by one single constant, called the bond constant
+c. This is due to the fact that the value of the Poisson’s ratio ν for a bond-based
+material is fixed to 1/4 or 1/3 depending on the dimension of the problem, leav-
+ing only one parameter tunable. The constant is typically defined by means of
+an energetic equivalence with standard local elastic material. It has huge effects
+on the value of this constant under what conditions this equivalence is imposed,
+i.e. isotropic expansion, pure elongation or even shear. The energy obtained
+after the convergence to the local model is, in fact, affected by the choice of
+the bond constant to the point that certain terms can converge to classical ones
+typical of local theories while others may not. For example, if one were to make
+the choice of imposing equivalence of the stretching energy in the peridynamic
+model and in the local elastic one, only first-order terms in H of the localized
+PD model would converge while quadratic, cubic and higher-order ones would
+not.
+Commonly, for the evaluation of the bond constant through energy equiv-
+alence with local continua, the choice of isotropic expansion is made for the
+deformation map. This choice is not expected to make all the terms converge to
+the classical ones, but it can give a general idea of the possibilities of the model
+obtained by the convergence. The energy density for a bond-based PD linear
+elastic material under isotropic expansion (η = αξ) is defined as
+WP D = 1
+2cα2
+�
+H
+|ξ|4−bdV = 1
+2cα2 γ(H, b, d) δ4−b+d,
+(24)
+where γ is a scalar function which depends on the shape of the family H, the
+dimension of the problem d, and the parameter b which comes from the choice
+of σ(|ξ|) = |ξ|b. Likewise, the energy of an isotropic expanding linear elastic
+material in classic local elasticity is defined as
+WCL = 1
+2α2 I · E[I] = 1
+2α2
+3E
+1 − 2ν ,
+(25)
+18
+
+where I is the identity tensor, while E is the fourth-order elasticity tensor, E is
+Young’s modulus of the local elastic material and ν is Poisson’s ratio.
+By enforcing equivalence between the energies (24) and (25) one recovers
+c =
+3E
+γ(b) δ4−b+d(1 − 2ν).
+(26)
+In the case of a spherical horizon, one obtains
+c = 15 E
+56 δ6 .
+(27)
+According to (26), the scaling of the bond constant is then defined as c ∼ δb−4−d.
+5.2. Displacement scaling
+The continuous part of the energy (the first term of equation (23)) is found
+to be scaling as
+�
+B
+�
+H
+c
+|ξ|b (ξ · ηa)2dVdV′ ∼ δ−(4−b+d)−b+2(1+m)+d = δ2(m−1),
+(28)
+since the scaling of c is defined in (27), and the other terms scale as follow: |ξ| ∼
+δ; |η| ∼ δm; V ′ ∼ δd. For the integral term to stay bounded and nonvanishing
+one requires m = 1.
+Hence, ηa ∼ δ1, which means that
+ηa ≈ ∇xA(x1, x2) ξ + ∇xB(x1, x2)ξ x3 + B(x1, x2) ξ · e3 + · · ·
+(29)
+defines the scaling of the shape functions, since ξ ∼ δ1. In particular, no scaling
+is required
+∇xA(x1, x2) ∼ δ0 ,
+∇xB(x1, x2) ∼ δ0 ,
+B(x1, x2) ∼ δ0.
+(30)
+In a similar fashion, the second term of (23) must follow the following scaling:
+�
+B
+�
+H
+c
+|ξ|b (ξ · ηJ)(ξ · ηa)dVdV′ ∼ δ−(4−b+d)−b+(2+m+n)+d = δ−2+m+n.
+(31)
+In order for the energy to stay bounded the scaling of the jump part of the
+displacement field (defined by n) must fulfill the condition of n ≥ 1, since from
+(28) m = 1. Accordingly, from the last term of the energy one obtains:
+�
+B
+�
+H
+c
+|ξ|b (ξ · ηJ)2dVdV′ ∼ δ−(4−b+d)−b+2(1+n)+d = δ2(n−1),
+(32)
+which gives the redundant condition: n ≥ 1. In view of (8) and for vanishing
+nonlocality one can approximate the relative jump displacement ηJ as
+ηJ ≈ ∇ξu′
+J|ξ=0 · ξ,
+(33)
+19
+
+where u′
+J is the displacement of the particle x′. If we call p the scaling of ∇ξu′
+J
+then by virtue of (32), p ≥ 0. Though, since
+∇x u′
+J|ξ=0 = ∇xj Θ(x3 − h(x1, x2)) + j ⊗ (e3 − ∇xh(x1, x2)) φ(x3 − h(x1, x2)),
+(34)
+where φ is the Dirac delta distribution, one can easily assess that in order for
+h(x1, x2) and the energy to be bounded, the following scaling must hold
+∇xj ∼ δ0 ,
+j ∼ δ0 ,
+∇xh ∼ δ0.
+(35)
+5.3. The scaling of the failure criterion
+Alongside the energy, also the damage criterion (s < scr) scales as δ → 0.
+The scaling of the critical stretch scr is defined by equation (10), so scr ∼ δ−1/2.
+The scaling of the stretch s, on the other hand, can be obtained by employing
+equations (33) and (34)
+s =
+ξ · ∇ξu′
+J|ξ=0 · ξ
+|ξ|2
+= ∇ξu′
+J|ξ=0 : ξ ⊗ ξ
+|ξ|2 .
+(36)
+The second tensor in the double dot product1 is a quantity that scales as δ0
+whereas the first tensor harbors a singularity, the Dirac’s Delta function φ,
+which for x3 = h(x1, x2) makes the stretch infinite. Hence, whenever on the
+crack surface, the criterion is immediately not satisfied. Finally:
+s < scr →
+�
+False
+for x3 = h(x1, x2)
+True
+otherwise
+(37)
+5.4. Localized energy in plane strain
+localization of the PD non-local model has been obtained by means of a
+limit operation, for vanishing δ, on the PD non-local elastic energy. The local-
+ized energy obtained in this way is composed of a part entirely defined by the
+continuous part of the displacement field, the term (28), and a part composed
+by mix and purely jump terms
+Elocal = Elocal,a + Elocal,J,
+(38)
+where for the assumption of continuous displacement field (7) truncated at first
+order in x3, and plane strain
+Elocal,a =HE
+� 3
+56A′
+1(x1)2 + 5
+84B3(x1)A′
+1(x1) +
+5
+168A′
+3(x1)2+
+5
+84B1(x1)A′
+3(x1) +
+5
+168B1(x1)2 + 3
+56B3(x1)2
+�
++
+H3E
+� 1
+224B′
+1(x1)2 + 5B′
+3(x1)2
+2016
+�
+.
+(39)
+1Given two second-order tensors, A and B, we mean by double dot product the operation
+A:BT = Tr(AB) = AijBji.
+20
+
+Here A = {A1, A3}, B = {B1, B3} and the primes indicates derivative with
+respect to x1. Meanwhile,
+Elocal,J =HE
+� 5
+168B1(x1)j′
+3(x1) +
+5
+336j′
+3(x1)2 +
+5
+168j′
+3(x1)A′
+3(x1)
+�
++
+(40)
+H2E
+� 5
+672j′
+3(x1)B′
+3(x1)
+�
++
+E
+� 3
+28j3(x1)B3(x1) − 5
+84j′
+3(x1)B1(x1)h(x1) −
+5
+168h(x1)j′
+3(x1)2+
+− 5
+84h′(x1)j3(x1)B1(x1) − 5
+84j3(x1)j′
+3(x1)h′(x1) +
+− 5
+84j′
+3(x1)h(x1)A′
+3(x1) − 5
+84j3(x1)h′(x1)A′
+3(x1)+
+5
+84j3(x1)A′
+1(x1) + 5
+84h(x1)j3(x1)B′
+1(x1)+
+− 5
+168j′
+3(x1)h(x1)2B′
+3(x1) − 5
+84j3(x1)h(x1)h′(x1)B′
+3(x1)
+�
+,
+where the hypothesis of purely Mode-I crack development j = {0, j3} has been
+considered.
+Equation (38) basically represents a material with cohesive constitutive law due
+to the energy associated with a jump in the displacement.
+The localized formulation of the nonlocal model introduced in section 3 can
+now be obtained by writing the Lagrangian for a local plate where the internal
+energy is that of equations (39-40) and then minimizing it.
+5.5. Kirchhoff-like plate under Mode-I fracture
+The kinematics of a Kirchhoff plate is readily recovered by imposing
+A1(x1) → 0,
+B3(x1) → 0,
+B1(x1) → −A′
+3(x1),
+(41)
+such that
+ua(x1, x3) = {−A′
+3(x1)x3, A3(x1)}.
+(42)
+Mode I delamination is achieved by choosing the jump function and its derivative
+in the following way:
+j1(x1) → 0,
+j′
+1(x1) → 0,
+(43)
+such that
+uJ[x1, x3] = {0, j3(x1) Θ(x3 − h(x1))},
+(44)
+where Θ is the Heaviside function. Under these conditions the terms of localized
+energy become
+Elocal,a =
+1
+224EH3A′′
+3(x1)2,
+(45)
+21
+
+and
+Elocal,J =E
+� 5
+84j3(x1)j′
+3(x1)h′(x1) +
+5
+168h(x1)j′
+3(x1)2+
+(46)
+− 5
+84j3(x1)h(x1)A′′
+3(x1) −
+5
+336j′
+3(x1)2
+�
+,
+respectively. It is worth mentioning that the limiting local energy for the con-
+tinuous part of the displacement is a quantity resembling the classical result
+for Kirchhoff plates: Elocal,a
+=
+H3E
+12(1−ν2)A′′
+3(x1)2, where for obvious reasons a
+Poisson ratio of 1/4 has to be considered.
+6. Conclusions
+In the present paper, a reduced model for the explicit study of though-
+thickness fracture nucleation and propagation in thin structures is put forward.
+The model is obtained by making a hypothesis on the kinematics of the thin
+element, which is assumed as the sum of a continuous part and a jump part. A
+particular choice of these fields is made which expresses the dependence on the
+out-of-plane variable explicitly, thus making the integration through the thick-
+ness feasible. The resulting reduced model retains information on the loss of
+continuity of the material through the functions defining the jump. The pro-
+posed model has distinguished weak planes/surfaces, yet it leads to the recovery
+of both horizontal and deviated crack patterns across the thickness of the plate.
+The dimension reduction procedure generates a hierarchy of terms in the elastic
+energy stored inside the plate. A mechanical interpretation of those terms is
+possible (especially for the part of the energy associated with the continuous
+part of the displacement) and is proposed through the definition of a simple
+paradigm of peridynamic structure. It is found that the hierarchical form of
+the energy shows the coupling of membrane and bending behavior despite the
+formulation being expressed in a linear setting.
+The reduced model is then tested in a symmetric displacement-induced delam-
+ination test for a cantilever plate and a broad range of qualitative responses
+are obtained for a varying horizon.
+In particular, for a horizon larger than
+the height of the plate
+distal nucleation is observed, whereas in the limit of
+vanishing horizon a classic result of linear fracture mechanics is recovered with
+the propagation of the fracture starting from the loaded cross-section of the
+plate. Apart from the crack path development, different horizons have proven
+to greatly influence the force-displacement response of the structure, leading to
+superior toughness and energy dissipation in the nonlocal model with a greater
+horizon and a more brittle behavior in the case of a smaller horizon.
+A non-symmetrical displacement-induced test is also performed and a relevant
+sensitivity of the peridynamic plate emerges from the simulation where a curved
+crack path characterizes the response at failure of the thin nonlocal element.
+To further investigate the local limit of the model, localization of the nonlocal
+22
+
+reduced formulation is performed. Firstly, the convergence of the nonlocal en-
+ergy to a finite and non-vanishing local equivalent is assessed. The localized
+reduced model shows a cohesive nature, which is expressed by the fact that en-
+ergy can be stored by the part of the energy associated with the discontinuous
+displacement field when a fracture is propagating.
+CRediT authorship contribution statement
+R. Cavuoto: Developed the theory, performed the calculations and compu-
+tations, wrote and edited the manuscript. A. Cutolo: Performed the computa-
+tions, wrote and edited the manuscript. K. Dayal: Developed the theory, wrote
+and edited the manuscript, supervised the whole work. M. Fraldi: Developed
+the theory, wrote and edited the manuscript, supervised the whole work. L.
+Deseri: Developed the theory, wrote and edited the manuscript, supervised
+the whole work.
+Declaration of Competing Interest
+The authors declare that they have no known competing financial interests or
+personal relationships that could have appeared to influence the work reported
+in this paper.
+Acknowledgements
+LD, AC and MF gratefully acknowledge the support of the Italian Min-
+istry of Research (MIUR) through the grants PRIN-20177TTP3S and PON
+“Stream”-ARS01 01182. LD also gratefully thanks the support of the Euro-
+pean Commission through (i) FET Open “Boheme” grant no.
+863179, and
+(ii) LIFE GREEN VULCAN LIFE19 ENV/IT/000213, and (iii) ERC-ADG-
+2021-101052956-BEYOND. Kaushik Dayal thanks Army Research Office (MURI
+W911NF-19-1-0245), Office of Naval Research (N00014-18-1-2528), and Na-
+tional Science Foundation (DMREF 2118945, DMS 2108784) for financial sup-
+port.
+Appendix A. Expanded form of the reduced energy
+The hierarchical form of the reduced energy is reported here again for the reader:
+ωred = ωred,a + ωred,J
+(A.1)
+where in the case of φ = 1, specializes to
+ωred,a = H2 p1(φ, ua) + H4 p2(φ, ua) + H6 p3(φ, ua) ;
+ωred,J = r0(uJ) + H r1(ua, uJ) + H2 r2(uJ)+
+H3 r3(ua, uJ) + H4 r4(ua, uJ) + H5 r5(ua, uJ)
+,
+23
+
+where the pi functions are those specified by eqs. (18) to (20). On the other
+hand, the coefficients of the reduced energy associated with the jump part of
+the displacements are:
+r0 = −1
+6j3(x′
+1)j3(x1)h(x′
+1)h(x1)
+�
+2h(x′
+1)2 − 3h(x′
+1)h(x1) + 2h(x1)2�
+;
+r1 = −1
+3j3(x′
+1)A3(x′
+1)h(x′
+1)3 − 1
+6j3(x′
+1)2h(x′
+1)3 + 1
+3j3(x′
+1)h(x′
+1)3A3(x1)+
+1
+2j3(x′
+1)h(x′
+1)2(x′
+1 − x)A1(x1) + 1
+2j3(x1)A1(x′
+1)(x − x′
+1)h(x1)2+
+1
+3j3(x1)A3(x′
+1)h(x1)3 + 1
+6j3(x′
+1)j3(x1)h(x′
+1)3 + 1
+6j3(x′
+1)j3(x1)h(x1)3+
+1
+3j3(x1)(x′
+1 − x)h(x1)3B1(x1) − 1
+4j3(x′
+1)h(x′
+1)4B3(x′
+1)+
+−1
+4j3(x1)h(x1)4B3(x1) + 1
+2j3(x′
+1)A1(x′
+1)h(x′
+1)2(x − x′
+1) +
+1
+2j3(x1)(x′
+1 − x)A1(x1)h(x1)2 + 1
+3j3(x′
+1)h(x′
+1)3(x − x′
+1)B1(x′
+1)+
+− 1
+6j3(x1)2 − 1
+3j3(x1)A3(x1)h(x1)3h(x1)3 ;
+r2 = −1
+8j3(x′
+1)j3(x1)
+�
+h(x′
+1)2 + h(x1)2�
+;
+r3 = − 1
+12j3(x′
+1)A3(x′
+1)h(x′
+1) − 1
+24j3(x′
+1)2h(x′
+1) + 1
+12j3(x′
+1)h(x′
+1)A3(x1)+
+− 1
+12j3(x1)A3(x1)h(x1) + 1
+8j3(x′
+1)A1(x′
+1)(x′
+1 − x)+
+1
+8j3(x1)(x − x′
+1)A1(x1) + 1
+12j3(x1)(x′
+1 − x)B1(x′
+1)h(x1)+
+1
+12j3(x′
+1)h(x′
+1)(x − x′
+1)B1(x1) − 1
+12j3(x′
+1)h(x′
+1)2B3(x1)+
+− 1
+24j3(x′
+1)h(x′
+1)2B3(x′
+1) − 1
+24j3(x1)2h(x1) − 1
+24j3(x1)h(x1)2B3(x1)+
+1
+8j3(x1)A1(x′
+1)(x′
+1 − x) + 1
+12j3(x1)A3(x′
+1)h(x1) − 1
+12j3(x1)B3(x′
+1)h(x1)2+
+1
+8j3(x′
+1)(x − x′
+1)A1(x1) + 1
+24j3(x′
+1)j3(x1)h(x1) + 1
+24j3(x′
+1)j3(x1)h(x′
+1) ;
+r4 = 1
+12j3(x′
+1)A3(x′
+1) − 1
+12j3(x′
+1)A3(x1) − 1
+12j3(x1)A3(x′
+1)+
+1
+24j3(x′
+1)(x′
+1 − x)B1(x1) + 1
+24j3(x1)(x − x′
+1)B1(x′
+1)+
+1
+12j3(x1)A3(x1) + 1
+24j3(x1)2 + 1
+24j3(x′
+1)(x′
+1 − x)B1(x′
+1)+
+− 1
+96j3(x′
+1)j3(x1) + j3(x′
+1)2
+24
++ 1
+24j3(x1)(x − x′
+1)B1(x1) ;
+r5 = 1
+48j3(x1)B3(x′
+1) + 1
+48j3(x′
+1)B3(x1) +
+5
+192j3(x′
+1)B3(x′
+1) +
+5
+192j3(x1)B3(x1).
+24
+
+Appendix B. A micro-structural interpretation for bond-based PD
+The structure of bond-based peridynamic constitutive equation (3) lends
+itself to an intuitive and simple physical interpretation. In fact, one can imag-
+ine the body under consideration to be uniformly divided into blocks and to
+substitute each block with a node embodying the mass of that specific block2.
+Then, massless connections can be introduced between nodes to represent their
+interactions. These connections must reflect how pairs of particles exert forces
+onto one another in the PD formulation.
+In order to do so, one can look at how the energy is stored between pairs of
+interacting volumes (or areas for a two-dimensional problem), Vi and Vj, of the
+PD continuum model. Taking into account equation (5), by means of the mean
+value theorem for integrals, one has
+Ei,j =
+�
+Vi
+�
+Vj
+c
+σ
+(ξ · η)2
+4
+dVidVj = c
+σ∗ (ξ∗ · η∗)2 ViVj
+4
+= c
+σ∗
+ViVj
+4
+|ξ∗|2 |η∗|2 cos2 β ,
+(B.1)
+where ξ∗ and η∗ are the relative position vector and relative displacement vector
+for a pair of points (x∗
+i , x∗
+j), belonging to the volumes Vi and Vj, for which
+the theorem holds, and β is the angle between them.
+In the limit of very
+small volumes Vi and Vj it is legitimate to assume the ξ and η functions to be
+constant between the volumes of interest and approximate the (x∗
+i , x∗
+j) with the
+mid-points of each volume.
+The right term of equation (B.1) resembles the energy of a truss tilted by the
+angle between ξ and a horizontal axis e1, and subjected to the edge displacement
+|η|. In this fashion, the stiffness of the bond representing the interaction between
+volumes would be
+ki,j = 1
+2
+c
+σ∗ ViVj|ξ∗|2 ,
+(B.2)
+where ξ∗ is now the relative position vector between mid-points of volumes Vi
+and Vj. Equation (B.2) implies that the stiffness is proportional to some power
+n of the distance between particles ki,j ∼ ln
+i,j (recalling that σ∗ = σ(|ξ∗|)).
+Performing this type of substitution for all the pairs of nodes transforms the
+continuum peridynamic body into an intricate reticular beam. The simplest
+paradigm of structure that can be built by following the PD approximation
+presented above is depicted in Figure B.10. The paradigmatic structure char-
+acterized by simple kinematics, correctly predicts the hierarchical form of the
+energy (17) and allows comparing the kinematics of the continuum with some-
+thing more easily controllable, accompanying the reader in an ideal transition
+2In finite-element analysis mesh refinement is a powerful stratagem that allows improving
+precision and accuracy of the approximated solutions for the problem at hand, in parts of the
+domain where it is required. Refinement procedures inevitably cause non-uniform discretiza-
+tion of the domain and, in the case of peridynamics, can cause inaccuracy of the solution
+[69].
+25
+
+from discrete to continuum for making evident how the above-mentioned non-
+standard terms have to naturally appear in the peridynamic plate model.
+We hereby limit the paradigmatic structure to stretching and bending kinemat-
+ics which prove to be sufficient for a qualitative interpretation. In particular,
+the structure is loaded by imposing horizontal displacements at the outer nodes
+(nodes 1 and 2 in Figure B.10). In this condition, the total energy can easily
+be obtained analytically. By denoting
+Next- to- nearest neighbour
+Figure B.10: Simplest structure representing a bond-based PD body.
+The nodes are the
+material particles of the body, while the bonds are represented by the truss connecting each
+couple of nodes. The structure has been made symmetric, both in loading conditions that in
+elements stiffness and coordinates. The gray region (denoted by V1) represents the volume
+ascribable to node 1. The horizon δ is taken to be equal to the diagonals of the structure.
+ε = u2 + u1
+4 l14
+,
+χ = 2(u1 − u2)
+l12 l14
+,
+the average stretch and curvature respectively, the total energy amounts to the
+following expression:
+Ediscrete = E α l4+n
+14
+�
+H2 ε2 4
+�
+1 + 2ψ2 + ψn�
+1 + ψ2 + ψn
++ H4 χ2
+�
+,
+(B.3)
+recalling that l12 =H is the length of truss connecting points 1 and 2 of Fig.
+B.10, ψ = cot θ, E is the Young’s Modulus of the beams (assumed constant) and
+α = c V 2
+1 /2. Equation (B.3) represents the equivalent energy of a peridynamic
+continuum (the horizon is included implicitly since l13 = δ = l14/ sin θ). Clearly,
+membrane energy scales with the square of the thickness H, while bending energy
+scales with H4 differently from the classical results of local elasticity [70, 71].
+According to the reduced energy obtained in Section 3.4, the total energy of a
+plate resembling the shape and loads of the paradigmatic case is recovered by
+choosing L= 2l5, δ = l5/ cos θ (hence Ψ = (cos θ)/2):
+EPD = c sec5 θ (29 + 20 cos θ) l6
+5
+�
+H2 ε2
+108
+29 + 20 cos 2θ + H4 χ2
+�
+,
+(B.4)
+which corresponds, at least qualitatively, to (B.3) for n = 2.
+26
+
+Sensitivity analysis of the paradigmatic structure to the horizon. The total en-
+ergy of the paradigmatic case has been derived for a fixed value of the horizon,
+i.e. δ = l5/ cos θ, whereas the reduced form of eq. (17) is defined for any value
+of such parameter. In order to extend the above interpretation analysis using
+the paradigmatic tool to the general case, a series of numerical analyses using
+ANSYS APDL has been carried out on several discrete approximations of a peri-
+dynamic bond-based continuum each characterized by different horizon sizes.
+In Figure B.11 the scaling of three different peridynamic discrete bodies with a
+horizon ranging from a minimal value (dotted green line) to one with a wider
+interaction (dot-dashed dark line) is depicted. On the left, the scaling of the
+membrane energy is found to be quadratic regardless of the horizon size, whereas
+the bending energy (Figure B.11, on the right) scales with the fourth power of
+the thickness. A geometric motivation is available for the scaling of the mem-
+brane energy: the increase in the number of bonds available when the thickness
+is doubled, for example, is (roughly) proportional to the square of the number
+of nodes: nbonds ∼ nnodes(nnodes − 1)/2.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+H
+ε/εPD,max
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+H
+ε/εPD,max
+Figure B.11: On the left: scaling, as a function of the total height H, of the energy associated
+with a uniform stretching deformation (purely membrane) for a standard Pratt truss (in
+green), a hyperstatic truss (in red) and two different patterns of a PD discrete beam; on
+the right: scaling, as a function of the total height H, of the energy associated to a uniform
+bending deformation, for a standard Pratt truss (in green), a hyperstatic beam (in red) and
+two different patterns of a PD discrete beam.
+Appendix C. Coupling in the local theory of plates
+Eq. (17) is derived from an assumption on the kinematics that is typical of
+the first-order shear deformation theory (FSDT), i.e. Reissner-Mindlin theory,
+for local plates [72, 73]. To further address the previous result about the coupling
+in eq. (17), we compare the energies for various local plate model, in accordance
+with the FSDT hypothesis, against the peridynamic result. Under the condition
+of FSDT, small displacements (linear elastic response) and inextensibility in the
+thickness direction, the specific (per unit area) elastic energy amounts to:
+ωred = H
+2
+�
+(λ + 2µ)A′2
+1 + µ(B1 + A′2
+3 )
+�
++ H3
+24 (λ + 2µ)B′2
+1 ,
+(C.1)
+27
+
+where the A1, A3, B1 and B3 are to be intended as functions of x1, and ωred is
+now the reduced energy density of a local plate.
+If one relaxes the inextensibility constraint, meaning the vertical component
+of the displacement follows the linear approximation in the through-thickness
+direction, the elastic energy becomes:
+ωred =H
+�
+(λ + 2µ)(A′
+1 + B3)2 + µ(A′
+3 + B1)2 + (λ − 2µ)A′
+1B3
+�
+/2+
+(C.2)
+H3 �
+(λ + 2µ)B′2
+1 + µB′2
+3
+�
+/24 .
+Lastly, adding a geometric nonlinearity, i.e. small displacements but large/finite
+rotations hypothesis, to the model leads to:
+ωred =H {A′4
+3
+(λ + 2µ)
+4
++ A′2
+1(λ + 2µ) + A′2
+3 (A′
+1(λ + 2µ) + µ + λB3) +
+(C.3)
+2λB3A′
+1 + 2µB1A′
+3 + B2
+3(λ + 2µ) + µB2
+1 }/2+
+H3 �
+40A′
+3(λ + 2µ)B′
+3B′
+1 + B′2
+3
+��
+3A′2
+3 + 2A′
+1
+�
+(λ + 2µ) + 2µ + 2λB3
+�
++
+80(λ + 2µ)B′2
+1
+�
+/48
+H5 �
+(λ + 2µ)B′4
+3
+�
+/640 .
+From the Euler-Lagrange equations that the previous type of elastic energies
+can generate, one can see how only the last nonlinear model accounts for the
+coupling of stretching and bending. The reason for this lies in the nonlinear
+relation between the deformations and the displacement field.
+The intrinsic
+microstructure of a peridynamic continuum naturally accounts for a similar
+effect, since the finite distances between particles makes it so that rotational
+effect can be induced by the intricate connections even under axial loading and
+net null moment.
+References
+[1] T. Inoue, F. Yin, Y. Kimura, K. Tsuzaki, S. Ochiai, Delamination effect on
+impact properties of ultrafine-grained low-carbon steel processed by warm
+caliber rolling, Metall. Mater. Trans. A 41 (2010) 341–355.
+[2] B. Naganarayana, S. Atluri, Strength reduction and delamination growth in
+thin and thick composite plates under compressive loading, Computational
+Mechanics 16 (1995) 170–189.
+[3] D. Dugdale, Yielding of steel sheets containing slits, Journal of the Me-
+chanics and Physics of Solids 8 (1960) 100–104.
+[4] G. Barenblatt, The mathematical theory of equilibrium cracks in brittle
+fracture, Advances in Applied Mechanics 7 (1962) 55–129.
+28
+
+[5] A. Hillerborg, M. Modeer, P.-E. Petersson, Analysis of crack formation
+and crack growth in concrete by means of fracture mechanics and finite
+elements, Cement and Concrete Research 6 (1976) 773–781.
+[6] A. Turon, C. Da’vila, P. Camanho, J. Costa, An engineering solution for
+mesh size effects in the simulation of delamination using cohesive zone
+models, Engineering Fracture Mechanics 74 (2007) 1665–1682.
+[7] M. Elices, G. Guinea, J. Gomez, J. Planas, The cohesive zone model: ad-
+vantages, limitations and challages, Eng. Fract. Mech. 69 (2002) 137–163.
+[8] C. Fan, P.-Y. B. Jar, J. R. Cheng, Cohesive zone with continuum damage
+properties for simulation of delamination in fibre composites and failure of
+adhesive joints, Engineering Fracture Mechanics 75 (2008) 3866–3880.
+[9] L. Zhao, J. Zhi, J. Zhang, Z. Liu, N. Hu, Xfem simulation of delamination
+in composite laminates, Composites Part A: Applied Science and Manufac-
+turing 80 (2016) 61–71.
+[10] S. Yazdani, W. Rust, P. Wriggers, An xfem approach for modelling delam-
+ination in composite laminates, Composite structures 135 (2016) 353–364.
+[11] P. A. V. Kumar, A. Dean, J. Reinoso, M. Paggi, A multi phase-field-
+cohesive zone model for laminated composites: Application to delamination
+migration, Composite structures 276 (2021) 114471.
+[12] P. Roy, S. Deepu, A. Pathrikar, D. Roy, J. Reddy, Phase field based peridy-
+namics damage model for delamination of composite structures, Composite
+structures 180 (2017) 972–993.
+[13] I. Giorgio, N. Rizzi, U. Andreaus, D. Steigmann, A two-dimensional contin-
+uum model of pantographic sheets moving in a 3-d space and accounting for
+the offset and relative rotations of the fibers, Mathematics and mechanics
+of complex systems 7 (4) (2019) 311–325.
+[14] B. Li, D. Mill´an, A. Torres-S´anchez, B. Roman, M. Arroyo, A variational
+model of fracture for tearing brittle thin sheets, Journal of the Mechanics
+and Physics of Solids 119 (2018) 334–348.
+[15] W. Lai, J. Gao, Y. Li, M. Arroyo, Y. Shen, Phase field modeling of brittle
+fracture in an euler–bernoulli beam accounting for transverse part-through
+cracks, Computer Methods in Applied Mechanics and Engineering 361
+(2020) 112787.
+[16] M. Pigazzini, D. Kamensky, D. van Iersel, M. Alaydin, J. Remmers,
+Y. Bazilevs, Gradient-enhanced damage modeling in kirchhoff–love shells:
+application to isogeometric analysis of composite laminates, Computer
+Methods in Applied Mechanics and Engineering 346 (2019) 152–179.
+29
+
+[17] S. Chowdhury, J. Reddy, Geometrically exact micropolar timoshenko beam
+and its application in modelling sandwich beams made of architected lattice
+core, Composite Structures 226 (2019) 111228.
+[18] S. Conti, P. Dondl, J. Orlik, Variational modeling of paperboard delami-
+nation under bending, Mathematics in Engineering 5 (2) (2023) 1–28.
+[19] S. Silling, Reformulation of elasticity theory for discontinuities and long-
+range forces, Journal of the Mechanics and Physics of Solids 48 (1) (2000)
+175–209.
+[20] I. Kunin, Theory of elastic media with a microstructure: Nonlocal theory
+of elasticity, Moscow, Izdatel’stvo Nauka, 1975.
+[21] N. Prakash, B. Deng, R. Stewart, C. Smith, J. Harris, Investigation of mi-
+croscale fracture mechanisms in glass–ceramics using peridynamics simula-
+tions, Journal of the American Ceramic Society 105 (6) (2022) 4304–4320.
+[22] M. R. Karim, K. Kadau, S. Narasimhachary, F. Radaelli, C. Amann,
+K. Dayal, S. Silling, T. Germann, Crack nucleation at forging flaws stud-
+ied by non-local peridynamics simulations, Mathematics and Mechanics of
+Solids 27 (2022) 1129–1149.
+[23] M. Rezaul Karim, K. Kadau, S. Narasimhachary, F. Radaelli, C. Amann,
+K. Dayal, S. Silling, T. C. Germann, Crack nucleation from non-metallic
+inclusions in aluminum alloys described by peridynamics simulations, In-
+ternational Journal of Fatigue 153 (2021) 106475.
+[24] J. Chua, V. Agrawal, T. Breitzman, G. Gazonas, K. Dayal, Phase-field
+modeling and peridynamics for defect dynamics, and an augmented phase-
+field model with viscous stresses, Journal of the Mechanics and Physics of
+Solids 159 (2022) 104716.
+[25] R. Lipton, R. Lehoucq, P. Jha, Complex fracture nucleation and evolu-
+tion with nonlocal elastodynamics, Journal of Peridynamics and Nonlocal
+Modeling 1 (2) (2019) 122–130.
+[26] P. Diehl, R. Lipton, T. Wick, M. Tyagi, A comparative review of peridy-
+namics and phase-field models for engineering fracture mechanics, Compu-
+tational Mechanics (2022).
+[27] L. Jooeun, L. Wenyang, H. Jung-Wuk, Impact fracture analysis enhanced
+by contact of peridynamic and finite element formulations, International
+Journal of Impact Engineering 87 (2016) 108–119.
+[28] N. Liu, D. Liu, W. Zhou, Peridynamic modelling of impact damage in
+three-point bending beam with offset notch, Appl. Math. Mech.-Engl. Ed.
+38 (2017) 99–110.
+30
+
+[29] J. Xu, A. Askari, O. Weckner, S. Silling, Peridynamic analysis of im-
+pact damage in composite laminates, Journal of Aerospace Engineering
+21 (2008).
+[30] G. Zhang, G. Gazonas, F. Bobaru, Supershear damage propagation and
+sub-rayleigh crack growth from edge-on impact: A peridynamic analysis,
+International Journal of Impact Engineering 113 (2018) 73–87.
+[31] F. Bobaru, Y. D. Ha, Adaptive refinement and multiscale modeling in
+2d peridynamics, Journal for Multiscale Computational Engineering 9 (6)
+(2011) 635–659.
+[32] Q. Le, W. Chan, J. Schwartz, A two-dimensional ordinary state based
+peridynamic model for linearly elastic solids, Int. J. Numer. Meth. Eng. 98
+(2014) 547–561.
+[33] G. Sarego, Q. Le, F. Bobaru, M. Zaccariotto, U. Galvanetto, Linearized
+state-based peridynamics for 2d problems, Int. J. Numer. Meth. Eng.
+108 (10) (2016) 1174–1197.
+[34] S. Silling, F. Bobaru, Peridynamic modeling of membranes and fibers, In-
+ternational Journal of Non-Linear Mechanics 40 (2005) 395–409.
+[35] M. Taylor, D. Steigmann, A two-dimensional pridynamic model for thin
+plates, Mathematics and Mechanics of Solids 20 (8) (2013) 998–1010.
+[36] K. Naumenko, V. Eremeyev, A non-linear direct peridynamics plate theory,
+Composite Structures 279 (2022) 114728.
+[37] J. O’Grady, J. Foster, Peridynamic plates and flat shells: A non-ordinary,
+state-based model, International Journal of Solids and Structures 51 (2014)
+4572–4579.
+[38] J. O’Grady, J. Foster, Peridynamic beams: A non-ordinary, state-based
+model, International Journal of Solids and Structures 51 (18) (2014) 3177–
+3183.
+[39] J. O’Grady, J. Foster, Peridynamic beams and plates: A non-ordinary
+state-based model, ASME International Mechanical Engineering Congress
+and Exposition Volume 1: Advances in Aerospace Technology (11 2014).
+[40] M. Behzadinasab, M. Alaydin, N. Trask, Y. Bazilevs, A general-purpose,
+inelastic, rotation-free kirchhoff–love shell formulation for peridynamics,
+Computer Methods in Applied Mechanics and Engineering 389 (2022)
+114422.
+[41] S. Chowdhury, P. Roy, D. Roy, J. Reddy, A peridynamic theory for linear
+elastic shells, International Journal of Solids and Structures 84 (2016) 110–
+132.
+31
+
+[42] J. Reddy, A. Srinivasa, A. Arbind, P. Khodabakhshi, On gradient elasticity
+and discrete peridynamics with applications to beams and plates, in: Ad-
+vanced Materials Research, Vol. 745, Trans Tech Publ, 2013, pp. 145–154.
+[43] Z. Yang, E. Oterkus, S. Oterkus, Peridynamic formulation for higher-order
+plate theory, Journal of peridynamics and nonlocal modeling 3 (2021) 185–
+210.
+[44] U. Yolum, E. Gok, D. Coker, M. Guler, Peridynamic modelling of delami-
+nation in dcb specimen, Procedia Structural Integrity 13 (2018) 2126–2131.
+[45] Y. Hu, N. D. Carvalho, E. Madenci, Peridynamic modeling of delamination
+growth in composites laminates, Comp. Struct. (2015).
+[46] X.-W. Jiang, S. Guo, H. Li, H. Wang, Peridynamic modeling of mode-i
+delamination growth in double contilever composites beam test: a two-
+dimensional modeling using revised energy-based failure criteria, Appl. Sci.
+9 (2019) 656.
+[47] R. Choksi, G. D. Piero, I. Fonseca, D. Owen, Structured deformations as
+energy minimizers in models of fracture and hysteresis, Mathematics and
+Mechanics of Solids 4 (3) (1999) 321–356.
+[48] M. Gobbino, Finite difference approximation of the mumford–shah func-
+tional, Commun. Pure Appl. Math. 51 (2) (1998) 197–228.
+[49] K. Dayal, K. Bhattacharya, Kinetics of phase transformations in the peri-
+dynamic formulation of continuum mechanics, Journal of the Mechanics
+and Physics of Solids 54 (9) (2006) 1811–1842.
+[50] Q. Du, K. Zhou, Mathematical analysis for the peridynamic nonlocal con-
+tinuum theory, Mathematical Modeling and Numerical Analysis 45 (2011)
+217–234.
+[51] K. Zhou, D. Qiang, Mathematical and numerical analysis of linear peridy-
+namic models with nonlocal boundary conditions, Siam J. Numer. Anal.
+48 (5) (2010) 1759–1780.
+[52] F. Erdogan, M. Ozturk, On the singularities in fracture and contact me-
+chanics, Journal of applied mechanics 75 (2008) 051111–1–12.
+[53] F. Bobaru, M. Yang, S. Silling, L. Alves, E. Askari, J. Xu, Convergence,
+adaptive refinement, and scaling in 1d peridynamics, Int. J. Numer. Meth.
+Eng. 77 (2009) 852–877.
+[54] J. Foster, S. Silling, W. Chen, An energy based failure criterion for use
+with peridynamic states, International Journal for Multiscale Computa-
+tional Engineering 9 (2011) 675–687.
+32
+
+[55] H. Zhang, P. Qiao, A two-dimensional ordinary state-based peridynamic
+model for elastic and fracture analysis, Engineering Fracture Mechanics
+232 (2020) 107040.
+[56] E. Madenci, S. Oterkus, Ordinary state-based peridynamics for plastic de-
+formation according to von mises yield criteria with isotropic hardening, J.
+Mech. Phys. Solids 86 (2016) 192–219.
+[57] A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans.
+Royal Soc. London A 221 (1921) 163–198.
+[58] R. Lipton, P. Jha, Nonlocal elastodynamics and fracture, Nonlinear Differ.
+Equ. Appl. 28 (2) (2021) 23.
+[59] P. Jha, R. Lipton, Numerical convergence of nonlinear nonlocal contin-
+uum models to local elastodynamics, International Journal for Numerical
+Methods in Engineering 114 (13) (2018) 1389–1410.
+[60] S. Silling, R. Lehoucq, Peridynamic theory of solid mechanics, Advances in
+Applied Mechanics 44 (2010) 73–168.
+[61] T. Breitzman, K. Dayal, Bond-level deformation gradients and energy av-
+eraging in peridynamics, Journal of the Mechanics and Physics of Solids
+110 (2018) 192–204.
+[62] T. Mengesha, Q. Du, On the variational limit of a class of nonlocal func-
+tionals related to peridynamics, Nonlinearity 28 (2015) 3999–4035.
+[63] J. Bellido, C. Mora-Corral, Existence for nonlocal variational problems in
+peridynamics, Siam J. Numer. Anal. 46 (1) (2014) 890–916.
+[64] B. Aksoylu, T. Mengesha, Results on nonlocal boundary value problems,
+Numerical Functional Analysis and Optimization 31 (12) (2010) 1301–1317.
+[65] D. Foss, P. Radu, C. Wright, Existence and regularity of minimizers for
+nonlocal energy functionals, Differential and Integral Equations 31 (11-12)
+(2018) 807–832.
+[66] J. Bellido, C. Mora-Corral, P. Pedregal, Hyperelasticity as a gamma-limit
+of peridynamics when the horizon goes to zero, Calculus of Variations and
+Partial Differential Equations 54 (2015) 1643–1670.
+[67] J. Bellido, J. Cueto, C. Mora-Corral, Bond-based peridynamics does not
+converge to hyperelasticity as the horizon tend to zero, Journal of Elasticity
+141 (09 2020).
+[68] S. Silling, R. Lehoucq, Convergence of peridynamics to classical elasticity
+theory, Journal of Elasticity 93 (2008) 13.
+33
+
+[69] H. Chen, A comparison study on peridynamic models using irregular non-
+uniform spatial discretization, Computer Methods in Applied Mechanics
+and Engineering 345 (2019) 539–554.
+[70] D. Steigmann, A well-posed finite strain model for thin elastic sheets with
+bending stiffness, Mathematics and Mechanics of Solids 18 (1) (2012) 103–
+112.
+[71] D. Steigmann, Asymptotic Estimate of the Potential Energy of a Plastically
+Deformed Thin Shell, Springer, 2020, Ch. 22, pp. 409–420.
+[72] J. Reddy, Mechanics of laminated composite plates and shells: Theory and
+Analysis, CRC press, 2004.
+[73] J. Reddy, A. Srinivasa, Non-linear theories of beams and plates accounting
+for moderate rotations and material length scales, International Journal of
+Non-Linear Mechanics 66 (2014) 43–53.
+34
+
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+page_content='Distal and non-symmetrical crack nucleation in  delamination of plates via dimensionally-reduced  peridynamics  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Cavuotoa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Cutoloa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Dayalb,c, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Deserid,e,b,f,g,∗, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Fraldia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='∗  aDepartment of Structures for Engineering and Architecture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' University of Naples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Italy  bDepartment of Civil and Environmental Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Carnegie Mellon University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' USA  cCenter for Nonlinear Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Department of Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' CMU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' USA  dDepartment of Civil,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Environmental and Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' University of Trento,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Italy  eDepartment of Civil and Environmental Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Pittsburgh University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' USA  fDepartment of Nanomedicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Houston Methodist Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Houston TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' USA  gDepartment of Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Carnegie Mellon University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Pittsburgh PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' USA  hD´epartment de Physique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Ecole Normale Sup´erieure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' France  Abstract  Exploiting the framework of peridynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' a dimensionally-reduced formula-  tion for plates is developed that allows for the through-thickness nucleation  and growth of fracture surfaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' enabling the treatment of delamination in a  lower-dimensional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Delamination fracture nucleation and propagation  are treated by choosing the kinematics to be composed of an absolutely con-  tinuous part and a zone where jumps in the displacements are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' This  assumption allows the explicit derivation of the dimensionally-reduced elastic  energy, which shows a hierarchy of terms characterising the stored energy in a  the plane element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' An interpretation of the various terms of the reduced energy  is shown by means of the simplest paradigm of bond-based peridynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' A  striking feature of the reduced energy is that, despite the small-displacement  assumption, there is a coupling between the membrane and bending terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Semi-analytical solutions for simplified settings are obtained through a mini-  mization procedure, and a range of nonstandard behaviors such as distal crack  nucleation and curved crack path are captured by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Finally, the con-  vergence of the proposed peridynamic reduced model to a local elastic theory  for vanishing nonlocal lengthscale is determined, giving a local cohesive model  for fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Full article available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='jmps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='105189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Keywords:  crack onset, peridynamics, plates, delamination  ∗Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Email addresses: luca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='deseri@unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='it (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Deseri), fraldi@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='it (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Fraldi)  Preprint submitted to Journal of the Mechanics and Physics of Solids  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='02481v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='app-ph]  6 Jan 2023  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Introduction  Delamination is a mode of failure that is typical of thin plate and shell  structures in which the thickness is much smaller than the other two dimen-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' This mode of failure is characterised by a fracture in which the crack  front propagates within the plane of the structure, resulting in the structure  being broken up into layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Composite laminates, which naturally present a  weak plane at the interface between different materials, are especially vulnera-  ble to delamination, but it can occur in microstructured and homogeneous thin  structures as well [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  This mode of fracture has been studied through a variety of approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Leading approaches include Cohesive Zone Models (CZM) [3–8] and the ex-  tended finite element methods (XFEM) [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' More recently, the phase-field  technique for fracture has been used to model debonding in laminates [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  These models, however, treat the thin structure as a fully three-dimensional  body and treat delamination explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  In a distinct approach to the modeling of thin plates without accounting  for delamination, an established procedure with a long history is to derive two-  dimensional formulations for plates or films based on systematically reducing  the three-dimensional theory using that the thickness is much smaller than  the other dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' This has been studied in a variety of settings, and are  particularly attractive as they can lead to faster computational algorithms with  good convergence properties while still capturing the key physical phenomena  [13–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The aim of this work is to develop an approach that combines the advan-  tages of dimensionally-reduced models while also accounting for delamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  That is, we aim to derive a two-dimensional model of plates that allows for  delamination failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Our approach is based on using peridynamics [19, 20], a  nonlocal theory that models continuum bodies as a collection of infinitesimal  material particles that interact through long-range forces, rather than the typi-  cal contact tractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In contrast to local theories of continuum mechanics that  rely on the definition of strain, consequently constraining the displacement to  have sufficient regularity, peridynamics works directly with the displacement  and does not require regularity a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  This makes it attractive to model  damage, damage-fracture transition [21–26], and dynamic phenomena such as  impact and blasts [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Papers dealing with two-dimensional peridynamic bodies can be divided into  two main categories based on the approach: (1) full 3D numerical simulations  [31–33], and (2) 2D reduced models [34–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' While the former approach has  been extensively used to treat delamination explicitly [44–46], existing reduced  formulation only account for crack propagation with the crack tip oriented nor-  mal to the plane and thus cannot capture phenomena such as delamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  In this work, a reduced formulation of bond-based peridynamics, tailored to  account for through-thickness delamination in thin plates characterized by a  single material and no preexisting weak interface, is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' As a first step,  in Section 3, the displacement field is additively decomposed into its absolutely  2  continuous part and its jump to account for delamination fracture nucleation  and propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' That is, the natural function space for the displacement field  is the space of functions of Special Bounded Variations (SBV), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Further assumptions on both parts of the displacement field lead to a reduced  form of the peridynamics energy in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The reduction procedure gen-  erates a hierarchy of terms characterising the strain energy stored inside the  two-dimensional continuum element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' A striking feature of the reduced energy  is that, despite the small displacement assumption, there is a coupling between  the membrane and bending terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The hierarchy of the resulting functional  allows for a consistent variational approach, enabling the displacement fields to  be obtained by a minimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Semi-analytical solutions for test cases are then obtained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The  tests are performed on a thin cantilever plate, modeled with the proposed re-  duced formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In the first case, such a plate undergoes an imposed upward  vertical displacement of the upper part of the free edge and a downward vertical  displacement of the lower edge in a symmetrical manner - much like a peeling  test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The model shows that variation of the nonlocal interaction lengthscale  δ, also called the horizon, induces different behaviors, namely distal or prox-  imal damage nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In the second case, an asymmetry is introduced by  imposing the vertical upward displacement at various points of the upper edge  of the plate, leading to non-symmetric crack propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In the third case,  Mode-II fracture or sliding delamination is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In order to explore the cou-  pling between bending and membrane terms in the reduced formulation (which  is geometrically linear) in a local setting, in Section 5, we examine the conver-  gence of the proposed model to a local theory when the nonlocal interaction  scale δ tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' By enforcing a condition of bounded and non-vanishing  energy, the scaling of the displacement field with δ is established;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' this, in turn,  determines the scaling of all the terms in the energy, thereby allowing for the  localization of the nonlocal model, leading to a reduced local formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The  reduced local formulation has a cohesive structure, due to some terms of the  energy associated with the jump part of the displacement surviving the limit  operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In Section 2, the constitutive framework of bond-based peridy-  namics is summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Section 3 sets up the theoretical framework for the  reduced formulation, and the reduced elastic energy density of a peridynamic  plate is obtained and interpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The theoretical model is then implemented  in a variational setting to be tested in simple loading conditions in Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Lastly, in Section 5, the behavior of the reduced formulation proposed for  vanishing horizon is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Formulation of the peridynamics model  Bond-based peridynamics models a continuum body B as a collection of  material particles interacting with one another, in pairs, through bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The  3  equation for the static equilibrium of the body [19, 49] can be written as the  integrodifferential equation  �  H  f(x, x′, u, u′) dVx′ + b(x) = 0 ,  (1)  where: f, called the pairwise force field, is the force exerted between material  particle x and x′, it can depend on the displacements u of such particles and  hosts all the constitutive information of the model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' b is the vector of external  body force;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' H represents the so-called family of x and is the set of all the  material points that are within a characteristic distance from it, called horizon,  δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In bond-based peridynamics, the balances of linear and angular momentum  require the pairwise force field, f, to be anti-symmetric with respect to particles  switch [19]  f(x, x′, u, u′) = −f(x′, x, u′, u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (2)  Constitutive relations for the definition of f have been proposed by many au-  thors [19, 50], among which one of the simplest is the standard linear elastic  perfectly brittle relation revisited by Zhou [51],  f = µ c s |ξ|2  σ ξ ,  (3)  where c is called the bond constant (a positive scalar quantity), ξ and η are  the relative position and displacement of the particles respectively, and s is the  stretch of the bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Lastly, σ = σ(ξ) is a function ensuring integrability of  (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Additional conditions on σ(ξ) are necessary to bound the stiffness and  the energy respectively of the PD model to finite positive values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The effect  of σ(ξ) on the PD model, equation (3), is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The function  µ = µ(ξ, η) in equation (3) is a history-dependent scalar-valued function (also  called failure parameter) which enforces bond breakage under tension only:  µ(ξ, η) =  � 1  for s < scr  0  otherwise  ,  (4)  where scr is a critical threshold for the bond elongation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Form (3) admits a  potential, called pairwise potential function:  ω(ξ, η) =  �  f(ξ, η) · dη =  �  ωel = 1  2c s2 |ξ|4  σ  for s < scr  ωcr = 1  2c s2  cr  |ξ|4  σ  otherwise  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (5)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' A microstructural interpretation of peridynamics  The present study proposes a dimensionally reduced formulation of peridy-  namic plates with a particular focus on through-thickness fracture propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Since the model, developed in Section 3, shows unconventional scalings of the en-  ergy terms due to its nonlocal (peridynamic) nature, in the present preliminary  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4  Figure 1: Upper left: normalized stiffness of a bond for different σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' kr = |f|/|η|, δ is the hori-  zon and c3 the bond constant for the case of σ = |ξ|3 which recovers the original formulation  of Silling [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Upper right: normalized bond force for different σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' fr is the modulus of the  force as a function of r (the normalized relative distance) that is applied to bonds under an  imposed uniform stretch ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Below: the bond energy ωr (energy per unit volume squared) for  increasing bond length and imposed uniform stretch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  section a possible micro-structural interpretation of peridynamics, functional to  the mechanical characterization of our model, is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The possibility to pass from the continuum to the discrete level is crucial in  problems involving the transition from elastic to dissipative phenomena such  as damage and fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Nevertheless, the exact equivalence between a given  peridynamic model and a corresponding microstructure is usually not trivial to  find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' One possible interpretation, available from purely energetic arguments,  can be given by means of a discrete structure, see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' To stress such  observations, we build a simple numerical example in which a structure made  of interconnected linear elastic elements leads to a densely packed truss ensem-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Despite one would assume that the asymptotic behavior for an increasing  number of micro-beams of the structure tends to that of a standard local contin-  uum, we demonstrate that –for a prescribed topology– significant discrepancies  in terms of displacements emerges between discrete and homogenised local con-  tinuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Figure 2 depicts the case of a cantilever beam (1 meter long and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4  meters thick), built by assembling trusses in a net-like pattern as displayed in  the inset of such figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' These trusses are connecting each material point of  the body with all the others that satisfy a relative distance requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The  parameters involved, namely axial stiffness of the beams and horizon length, are  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='0  x/L  v0/vmat  Figure 2: (Left) Cantilever beam characterized by a net-like micro-structure and subjected  to an imposed displacement at its free end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' (Right) The normalized deformed shapes (vmat  maximum vertical displacement of structured material) according to peridynamic (PD) and  local theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  calibrated in such a way that the behavior under tension reproduces that of an  ideal homogenized continuous local beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In Figure 2 the deflection of the struc-  tured beam when subjected to a vertical force applied at one end is compared  with local theories (in red Euler-Bernoulli and Timoshenko beam overlapping  one onto the other) and with the predictions of peridynamics (PD, in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Failing of local theories to correctly characterize the bending behavior for the  example introduced above, can be ascribed to the intricate internal structure of  the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' A dimensionally-reduced model for thin plates  In order to develop the analytical calculations necessary for the formulation  of a dimensionally reduced model for plates, certain assumptions are made on  both the kinematics of the plate and on the constitutive relation of the bond-  based peridynamic continua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Kinematics  Since the fracture of a material can be seen as the nucleation and growth  of a discontinuity in its displacement field, one can additively partition the  kinematics into a continuous part, accounting for elastic deformations, and a  jump part, accounting for the displacements due to the delamination, namely:  u(x) = ua(x) + uJ(x) ,  (6)  where the index a indicates the absolutely continuous part while the index J  denotes the jump part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In this sense, it can be said that u is a function in the  space of Special Bounded Variations (SBV) [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  We now restrict ourselves to the study of thin bodies, B, characterized by a  constant thickness H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Given a region of the three-dimensional euclidean space  E3, and a Cartesian reference frame (O, x1, x2, x3), Figure 3, the absolutely  6  continuous part of the displacement is approximated by a polynomial expansion  as follows  ua(x) ≈ A(x1, x2) + B(x1, x2)x3 + · · ·  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (7)  It is important to highlight that the choice of the reduction plane (the expansion  point in the expansion above) can have effects on the hierarchical distribution  of terms in the reduced formulation [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In the sequel the midplane of the plate  is chosen to perform the dimension reduction, so as to align with classical local  elastic reduced formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  O  x3  x1  x2  x  y  H (x)  H (y)  u = ua + uj  h(x1,x2)  j(x1,x2)  Figure 3: Kinematics of a plate of thickness H undergoing through-thickness fracture prop-  agation, here also referred as delamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Each point of the reference configuration lying  on possible delamination surfaces to be determined jumps to a new position specified by the  vector j(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The jump part of the displacement field will be represented in a rather general  way as  uJ(x) = j(x1, x2) · Θ(x3 − h(x1, x2)),  (8)  where the h(x1, x2) can be regarded as the crack surface, defining the surface on  which a displacement discontinuity may arise, while Θ is the Heaviside function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  lastly, j(x1, x2) is the vector function defining the jump itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' It is worth noting  that mixed-mode fracture processes are allowed by the ansatz made above on uJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Due to the kinematic split imposed on equation (6), the relative displacement  field now reads as follows:  η = ηa + ηJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (9)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Damage  It is important to highlight that in nonlocal theories a discontinuity in the  displacement field does not necessarily mean fracture nucleation/propagation, as  particles that are already separated by a finite distance can very well withstand a  jump in their relative displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In PD, what ensures the effective occurrence  of damage is the µ function (4), which represents the failure criterion for the  bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Indeed, the state of interaction can be determined by means of equation  (4), which enforces a critical stretch condition (s < scr) [19, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In many  other cases available in the literature, instead of a critical elongation criterion,  7  an energy-based one is employed [54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Such a criterion relates the breakage  of a bond to the attainment of a threshold in the stored energy, called critical  bond energy ωcr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Both the critical stretch and critical energy are typically  evaluated by means of an energy comparison with the standard local theory of  fracture mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In particular, the PD energy necessary for the growth of a  new surface in the body, defined as the energy required to break all the bonds  which pass through that particular surface (Figure 4), is imposed to be equal to  the critical energy release rate of Griffith theory [57], an operation that ensures  the recovery of the Griffith theory in the limit of small horizon [25, 58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Figure 4: Computation of the total energy necessary to break all the bonds connecting the  material point x with those x′ on the other side of the fracture surface h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  In this fashion, for the case of the critical stretch, one obtains [34, 60]  scr =  �  Gc  β(H, σ) δ ,  (10)  where Gc is the critical energy release rate and β is a scalar function of the  shape of the family H and on σ(|ξ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The transition from damage to fracture is therefore naturally tracked by the  failure mechanism of PD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Lagrangian formulation  Under certain conditions [61], the solution of the equilibrium problem of the  nonlocal PD body coincides with the stationary points of the following functional  [62–64]:  L = −Eel + W,  (11)  where Eel is the elastic energy, and W is the work of the external loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' By  explicitly expressing the various terms, equation (11) becomes  L[u] = −1  2  �  B  �  H  ω(ξ, η) dV′dV +  �  B  b · u dV ,  (12)  8  where ω is the energy density defined in (5), while B and H are the continuum  body and the family of a point, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' We here define the decomposition  of B through a Cartesian product as B = Bα × B3, where  Bα := {(x1, x2), ∀x ∈ B}  and B3 := {x3, ∀x ∈ B} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Accordingly, one can define H = Hα × H3, where  Hα :=  �  (x′  1, x′  2), ∀x′ ∈ B :  �  (x′  1 − x1)2 + (x′  2 − x2)2 ≤ δ  �  ,  H3 :=  �  x′  3, ∀x′ ∈ B :  �  (x′  1 − x1)2 + (x′  2 − x2)2 ≤ δ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Since in the present study x3 is chosen as the out-of-plane coordinate (see Figure  3), its value ranges in between {−H/2, H/2}, H being the plate thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  In view of the previous Cartesian products, one can now write  L[u] =  �  Bα  �  �−1  2  �  Hα  �  H3  �  B3  ω(ξ, η) dx3dx′  3dSα +  �  B3  b · u dx3  �  � dSα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (13)  Performing the integrations through the thickness of (13) allows one to obtain  the reduced form of the total Lagrangian of the plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In particular, the first  addend in parenthesis of equation (13), which is the elastic energy per unit  surface ΛE, becomes:  ΛE = 1  2  �  Hα  �  H3  �  B3  ω(ξ, η) dx3dx′  3dSα =  = 1  2  �  Hα  H  2  �  − H  2  H  2  �  − H  2  ω(ξ, η)dx3dx′  3dSα =  �  Hα  ωreddSα ,  (14)  where ωred is the reduced form of the pairwise potential function ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Similarly, we refer to the result of the through-thickness integration of the work  of the external loads (second addend in parenthesis in equation (13)) as ΛW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  All the functionals involved in (13) are nonlocal, as the unknown function u  is evaluated at different points of the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' An equivalent form of the Euler-  Lagrange equation for nonlocal functionals is now necessary to find the station-  ary points of (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The search for stationary points within the interior of the  domain of the functional (or its minimization) has been investigated in [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' For  the particular case of static and elastic PD nonlocal functional [62–64] one has  that the following implication holds:  min  u L → 2∂ΛE  ∂q − ∂ΛW  ∂q  = 0 ,  (15)  9  where the q is the vector of the unknown functions of the problem, which because  of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' (6) to (8) reads as follows:  q = {h(x1, x2), j(x1, x2), A(x1, x2), B(x1, x2)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  In order to retrieve equation (15), condition (2) must be enforced on the results  of [62–64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Hierarchical form of the reduced pairwise potential function  We here retrieve an explicit form of the reduced pairwise potential function,  ωred =  H  2  �  − H  2  H  2  �  − H  2  ω(ξ, η) dx3dx′  3 ,  (16)  for a bond-based peridynamic body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' For the linear elastic case, the influence  of the function σ appearing in (3) on the overall behavior has been indirectly  investigated in Bobaru at al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' There, the authors have shown how the shape  of the micromodulus function has indeed consequences on the overall behavior  of the material, albeit this does not influence the rate of convergence for a  vanishing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The result is critical to this work, where the consequences  of a localization procedure on the reduced peridynamic model will be explored  (section 5) with the objective of retrieving a local reduced formulation for plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  For the purpose of simplifying the calculations, drawing on the results discussed  above [53], condition σ = 1 is enforced in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Neglecting the failure  parameter (denoted by µ in equation (4)) allows the evaluation of the reduced  form of the energy for the fully elastic case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' when the load has yet to break  any bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' If c is the bond constant, and φ is the ratio between the in-plane  component of the horizon and the thickness, then  ωred  c  = H2 p1(φ, ua) + H4 p2(φ, ua) + H6 p3(φ, ua)  �  ��  �  ωred,a  +ωred,J(H, φ, ua, uJ) ,  (17)  where  p1(φ, ua) =(2φ − φ2)(x′  1 − x1)2 (A1(x′  1) − A1(x1))2 /4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  p2(φ, ua) = { 2φ3(3φ − 4)(A3(x′  1) − A3(x1))2+  (x′  1 − x1)2φ3(3φ − 4)(B1(x′  1)2 + B1(x1)2)+  (x′  1 − x1)2(−2φ + 3φ2 − φ4)(B1(x′  1) − B1(x1))2+  2φ3(3φ − 4)(x′  1 − x1)(A1(x′  1) − A1(x1))(B3(x′  1) + B3(x1))+  2φ3(3φ − 4)(x′  1 − x1)(A3(x′  1) − A3(x1))(B1(x′  1) + B1(x1))  �  /48 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  p3(φ, ua) ={  �  20 − 45φ + 72φ2 − 80φ3�  (B3(x′  1) − B3(x1))2 +  (20 − 45φ − 72φ2 + 80φ3)(B3(x′  1)B3(x1))  �  /1440 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  10  while ωred,J, an implicit function of φ, H and the unknown fields, denotes the  part of the reduced energy associated with the jump field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' As shown in equation  (17) the part of the reduced energy associated with the continuous displacements  is henceforth denoted by ωred,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  In the case of a through-thickness horizon equal to the whole thickness of the  thin element, the physical condition of isotropic interaction is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In such  a case:  p1(1, u) =(x′  1 − x1)2 (A1(x′  1) − A1(x1))2 /4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (18)  p2(1, u) = { − 2(x′  1 − x1)(A3(x′  1) − A3(x1))(B1(x′  1) + B1(x1))+  (19)  − 2(A3(x′  1) − A3(x1))2 − (x′  1 − x1)2(B1(x′  1)2 + B1(x1)2)+  − 2(x′  1 − x1)(A1(x′  1) − A1(x1))(B3(x′  1) + B3(x1))}/48 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  p3(1, u) ={10B3(x′  1)B3(x1) + 7B3(x′  1)2 + 7B3(x1)2}/1440 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (20)  and:  ωred,J  c  = r0(uJ) + H r1(ua, uJ) + H2 r2(uJ)+  (21)  H3 r3(ua, uJ) + H4 r4(ua, uJ) + H5 r5(ua, uJ)  ,  where the ri functions are reported in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  For simplicity, equation (17) has been specialized for the plane strain case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The variables x1 and x′  1 represent respectively the in-plane component of the  position vector for particle x and x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Furthermore, we have used A = {A1, A3},  B = {B1, B3} and j = {0, j3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The latter limits the kinematics to that of a  pure Mode-I fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' It is possible to see how the dimension reduction of the  pairwise potential function generates a hierarchy of terms characterizing the  strain energy stored inside the planar element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  In Appendix  B, following the microstructural interpretation given in section  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1 and through the definition of a paradigmatic case of discrete peridynamics,  a simple tool for the physical interpretation of the various terms in the reduced  energy of our peridynamic continuum is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Thanks to the paradigmatic  case, it is possible to give an immediate physical interpretation to the terms of  (17) that are scaling with the square p1 and the fourth power p2 of the thickness,  that is the former are membrane terms and the latter bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  From an analysis of the expression of p1 (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' (18)), since A1(x1) is the  in-plane component of the displacement field for the points on the reduction  plane, it can be confirmed that the terms scaling with H2 of ωred,a can be  regarded as purely membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In the higher-order term, on the contrary, such  as p2 (equation 19) which multiplies H4, one can assess the presence of purely  bending contributions (for example, those depending solely on B1(x1)2), but also  of mixed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The mixed terms introduce the coupling of membrane behavior  and bending behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' This is a unique feature of the nonlocal formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Indeed, coupling between the membrane and bending behaviors is a feature not  easily recoverable in local theories, as shown in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In Section 5 it is  shown that the coupling is lost when the peridynamic model is localized, namely  the reduced energy is evaluated in the limit of vanishing horizon δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  11  Lastly, the terms scaling with H6 are higher order ones depending only on  B3(x1), which is the nonlocal equivalent of a strain deformation through the  thickness ∂x3(ua · e3), where e3 is the unit vector normal to the plane (x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  We note that in order to recover the kinematics of the Kirchhoff plate theory  B3(x1) must be null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The contribution of the jump part of the displacement field to the reduced en-  ergy, reflected in ωred,J, is more scattered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' We see contributions of the jump field  to both membrane, mixed and bending-related quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Here, again, coupling  occurs between the different fields of the jump part of the displacement and the  continuous part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The highest order term in the thickness (H) is determined  by the order of the truncation in the Taylor expansion of the continuous part  of the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' By retaining only terms up to the first order in x3, the  highest power becomes 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' This particular choice was made in order to check  the convergence of the nonlocal model, which will be done in the last section of  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Model implementation and applications  Under the aforementioned conditions, the solution for the Euler-Lagrange  system of equations (15) of the PD model was achieved by using a Galerkin  approach, resulting in a system of the kind  �  Bα  �  2∂ΛE  ∂q − ∂ΛW  ∂q  �  δq = 0 ,  (22)  which can be solved iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  A Mathematica code has then been developed in order to test the model under  different loading conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  We here present displacement-induced tests for  symmetric and non-symmetric load distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The results have shown that the reduced model is capable of reproducing both  traditional and unconventional mechanical behavior such as distal crack nucle-  ation and loss of symmetry in the crack pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Mechanical and geometric quantities  Value  Young’s Modulus [MPa]  5000  Critical surface energy Gc [J/m2]  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3  Thickness over length (H/L)  1/25  Nonlocal parameters  Value  Horizon(δ)/Length(L)  1/5  Bond constant c [N/mm6]  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='7x106  Critical stretch scr [-]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2x10−4  Table 1: Parameters of the local equivalent material and geometry of the plate (up);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' nonlocal  parameter of the peridynamic bond-based reduced model (down).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  12  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='  u/H  u  (x 10-4)  L  x1  x2  x3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='195 x 10-4H  u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='6  x3  x1  “Distal” crack nucleation  Figure 5: Results of the analysis of a peridynamic cantilever plate subjected to two opposing  vertical displacements at the upper and lower edges using the present formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In the  upper part: on the left, the evolution of the force-displacement response and the schematic  representation of the test carried out, while on the right, the deformed shape corresponding  to an imposed displacement of u/H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='195 × 10−4 amplified by a factor of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In the lower  part: level curves representing the displacements - normalized with respect to the imposed  one - showing the evolution of crack surface h(x), represented by the dashed blue line, as the  imposed displacements at the edges increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Displacement-induced peeling test  The peridynamic reduced formulation proposed above is used to study the  case of displacement-controlled test inducing through-thickness fracture of a  cantilever plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' As shown in Figure 5, the plate is loaded by the application  of a pair of vertical displacements to the upper and lower part of the free edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The initial geometry necessitates neither an a priori crack nor a notch in order  to develop a crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' This is due to the damage being implemented at the con-  stitutive level in the peridynamic theory and to the kinematic assumptions on  the damage-fracture transition taken before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The test has been carried out until a final vertical displacement of around H/600,  a quantity which is sufficient for the development of fracture for the chosen  elastic and critical parameters (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In Figure 5 (above), in blue, the  normalized force vs displacement plot is presented, while in red is the fraction  of bonds that have yet to break near the loaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The latter has been used  13  to investigate the propagation of damage before and during fracture growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In  particular, both damage and fracture surfaces first develop at a distal section  from the plate edge (loci of the applied load) as shown in Figure 5 (below), and  then propagate in both directions, as observed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=', in laminated paper [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  This unusual response is obtained for a significant nonlocal character of the peri-  dynamic continuum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' horizon larger than the thickness of the plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In fact,  by reducing this parameter the interaction becomes more local and a different  response is obtained where the crack nucleates closer to the free edge, ultimately  reaching it in the limit of vanishing horizon which is a typical result of standard  local continuum theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Figure 6 shows the results of a parametric analysis of  δ ringing from a value of twice the thickness H down to approximately zero, the  value at which the fracture is nucleating and propagating from the cross-section  at the free edge (where the load is applied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' When in a peridynamic discrete  δ=H/2  A  A’  A  A’  A  A’  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='  u/H  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='  u/H  δ>H  δ   0  Figure 6: Qualitative behavior of crack nucleation and propagation for a peridynamic reduced  cantilever plate subjected to a displacement-induced test for a decreasing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The plates  are clamped at cross-section AA’ and loaded at the opposite edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  body or continuum the horizon is reduced, the number of total interactions,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' the bonds, of a point is also reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' As a consequence, the behavior of  the structure becomes less cohesive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' this feature is clearly shown in the force-  displacement plots of the various cases depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Surprisingly, the  cohesive trait is not completely lost in the local case as is shown in the next Sec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Finally, the red lines in the force plots are the relative number of broken  14  bonds, thus they represent the total damage in the zone of the load application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Along with the loss in cohesiveness, a reduction in the number of bonds is due  to affect the overall stiffness of the plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' We hence display the result of a com-  parison of the plate behavior for different horizon sizes, given a constant overall  stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' This condition can be obtained by increasing the bond constant c of  the peridynamic model as δ decreases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' having in mind the paradigmatic micro-  structure of a peridynamic discrete body, an increase in c is achieved by thick-  ening each beam that represents a bond, see Figure 7 (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In the  same figure, the comparison shows that the nonlocal micro-structure is capable  of absorbing more energy, displaying thus superior toughness when compared  with the cases of smaller horizons, which are in this sense more brittle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The  relationship between horizon size and total dissipated energy seems to be less  than linear as, from our study, an increase of four times the volume of inter-  action has brought about an increase of total energy dissipated by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='5 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  It is worth mentioning here that contour plots and force displacements plots  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='  u/H  fv /fmax  Figure 7: Comparison of the force-displacement response of nonlocal plates with different  horizon sizes but equal overall stiffness (on the left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' On the right, is the equivalent micro-  structure for the peridynamic body in the different cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' the increase in bond stiffness is  achieved by thickening the cross-section of each beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  all depict early-stage crack propagation phenomenon in the peridynamic plate,  that is the damaging onset, the nucleation of fracturing embryos and the crack  advancement in the very close regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Non-symmetric load distribution inducing a through-thickness crack  Starting from the previous case of a symmetrically loaded plate, we here  explore the effects of an asymmetry in the application of the loads on the crack  surface of a cantilever plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The non-symmetric loading condition is achieved  by pulling the upper edge in multiple points while the lower edge of the plate  is still pulled from a single one, see Figure 8 on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The parameters used  for the simulation are c = 7618N/mm6 (bond constant), δ = H/2 (the horizon),  15  scr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='02036 (critical elongation of a single bond) and the test has been carried  out until a final vertical displacement of approximately H/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  As far as the crack path is concerned, the non-symmetric load distribution in-  duces an unexpected non-symmetric crack trajectory which nucleates and prop-  agates from the external section towards the center of the plate, see Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  This sensitivity of crack path to even slight loss of symmetry in the prescribed  boundary conditions is not typically achieved in thin structures obeying Saint  Venant’s principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='6  Figure 8: Non-symmetrical load distribution leading to a loss of symmetry of the crack path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  On the left, the force-displacement plot (in blue) normalized with respect to the peak force,  and the number of intact bonds is in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The latter is representative of damage evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  On the right, the crack surface (blue dotted line) in the sample loaded by the non-symmetrical  distribution of forces and displacements expressed in terms of level set curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Mode-II fracture propagation  Nonlocal peridynamic plates can show loss of continuity through the thick-  ness due to the action of an external couple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' To show this, a clamped peridy-  namic plate is loaded through the application of two opposing forces, applied at  the upper and lower edge of the free end of the plate, with a growing inclination  (see Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The mechanical and geometrical parameters chosen for the sim-  ulation are the same as the previous case shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Upon reaching  a condition of forces almost horizontal (zero inclination, Figure 9 on the right)  the characteristic distal nucleation shown for the previous case of opposing ver-  tical forces and Mode-I failure is lost and a more “classical” crack growth is  exhibited with nucleation occurring at the free-end section in a Mode-II fash-  ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Nonetheless, in the latter case, the evolution of the crack is not continuous  and, at a later stage, a more distal crack nucleates far from the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  As a last observation it is useful to highlight that from the various examples  presented above it emerges a complex and rich interaction between the applied  loads and the displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' This makes it very hard to substitute a spe-  cific load distribution with a possible static equivalent, such as the resultant,  to be applied to a dimensionally reduced plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The need to follow fractur-  ing/delamination processes makes forces with overall vanishing resultants as  16  relevant as not vanishing ones, the nonzero force and couple resultants being  thus not the sole effective loads to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Indeed, the two opposite forces  of the first example, applied at the same free surface of the plate along the same  vertical direction, give zero global resultant but are however very relevant for  delamination, consistently with the classical peeling tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  a)  b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='6  “Distal” nucleation  Classical nucleation  u  L  x3x3  x2  x1  x3  x1  x3  x1  u  L  x3x3  x2  x1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+page_content='1  x3  x1  Figure 9: A nonlocal peridynamic plate loaded with a couple obtained by two opposing forces  of increasing inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In the upper part of the figure a schematic representation of the loads  and the plate is reported for both the case of a) inclination of the forces of 30° and b) inclination  of the forces close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In the central part of the figure are reported the displacements  (normalized with respect to the maximum displacement imposed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' u/H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='01) induced  in the plate by the forces for the two cases, displaying different crack nucleation and growth  (blue dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Lastly, in the lower part of the Figure, it is depicted the deformed shape  of the nonlocal plate corresponding to an imposed horizontal displacement of u/H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='003  (with an amplification of 40 times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  17  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Convergence to a local elastic model  Convergence of the proposed PD model to local elasticity is assured for  the continuous part of the displacement field only [62, 66–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Nonetheless,  with appropriate scaling of the jump field functions, convergence for vanishing  horizon leads to a bounded form of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  By applying (9) to (5), the first term of (12), which is the elastic energy of a  bond-based PD body, becomes  1  4  c  σ(ξ)µ  �  B  �  H  �  (ξ · ηa)2 + 2(ξ · ηa)(ξ · ηJ) + (ξ · ηJ)2�  dV ′dV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (23)  To ensure convergence for vanishing nonlocality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' δ → 0, the scaling of each  term must be checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Peridynamic parameter evaluation  Isotropic homogeneous linear elastic materials in the bond-based peridy-  namic theory are characterized by one single constant, called the bond constant  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' This is due to the fact that the value of the Poisson’s ratio ν for a bond-based  material is fixed to 1/4 or 1/3 depending on the dimension of the problem, leav-  ing only one parameter tunable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The constant is typically defined by means of  an energetic equivalence with standard local elastic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' It has huge effects  on the value of this constant under what conditions this equivalence is imposed,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' isotropic expansion, pure elongation or even shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The energy obtained  after the convergence to the local model is, in fact, affected by the choice of  the bond constant to the point that certain terms can converge to classical ones  typical of local theories while others may not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' For example, if one were to make  the choice of imposing equivalence of the stretching energy in the peridynamic  model and in the local elastic one, only first-order terms in H of the localized  PD model would converge while quadratic, cubic and higher-order ones would  not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Commonly, for the evaluation of the bond constant through energy equiv-  alence with local continua, the choice of isotropic expansion is made for the  deformation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' This choice is not expected to make all the terms converge to  the classical ones, but it can give a general idea of the possibilities of the model  obtained by the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The energy density for a bond-based PD linear  elastic material under isotropic expansion (η = αξ) is defined as  WP D = 1  2cα2  �  H  |ξ|4−bdV = 1  2cα2 γ(H, b, d) δ4−b+d,  (24)  where γ is a scalar function which depends on the shape of the family H, the  dimension of the problem d, and the parameter b which comes from the choice  of σ(|ξ|) = |ξ|b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Likewise, the energy of an isotropic expanding linear elastic  material in classic local elasticity is defined as  WCL = 1  2α2 I · E[I] = 1  2α2  3E  1 − 2ν ,  (25)  18  where I is the identity tensor, while E is the fourth-order elasticity tensor, E is  Young’s modulus of the local elastic material and ν is Poisson’s ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  By enforcing equivalence between the energies (24) and (25) one recovers  c =  3E  γ(b) δ4−b+d(1 − 2ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (26)  In the case of a spherical horizon, one obtains  c = 15 E  56 δ6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (27)  According to (26), the scaling of the bond constant is then defined as c ∼ δb−4−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Displacement scaling  The continuous part of the energy (the first term of equation (23)) is found  to be scaling as  �  B  �  H  c  |ξ|b (ξ · ηa)2dVdV′ ∼ δ−(4−b+d)−b+2(1+m)+d = δ2(m−1),  (28)  since the scaling of c is defined in (27), and the other terms scale as follow: |ξ| ∼  δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' |η| ∼ δm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' V ′ ∼ δd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' For the integral term to stay bounded and nonvanishing  one requires m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Hence, ηa ∼ δ1, which means that  ηa ≈ ∇xA(x1, x2) ξ + ∇xB(x1, x2)ξ x3 + B(x1, x2) ξ · e3 + · · ·  (29)  defines the scaling of the shape functions, since ξ ∼ δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In particular, no scaling  is required  ∇xA(x1, x2) ∼ δ0 ,  ∇xB(x1, x2) ∼ δ0 ,  B(x1, x2) ∼ δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (30)  In a similar fashion, the second term of (23) must follow the following scaling:  �  B  �  H  c  |ξ|b (ξ · ηJ)(ξ · ηa)dVdV′ ∼ δ−(4−b+d)−b+(2+m+n)+d = δ−2+m+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (31)  In order for the energy to stay bounded the scaling of the jump part of the  displacement field (defined by n) must fulfill the condition of n ≥ 1, since from  (28) m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Accordingly, from the last term of the energy one obtains:  �  B  �  H  c  |ξ|b (ξ · ηJ)2dVdV′ ∼ δ−(4−b+d)−b+2(1+n)+d = δ2(n−1),  (32)  which gives the redundant condition: n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In view of (8) and for vanishing  nonlocality one can approximate the relative jump displacement ηJ as  ηJ ≈ ∇ξu′  J|ξ=0 · ξ,  (33)  19  where u′  J is the displacement of the particle x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' If we call p the scaling of ∇ξu′  J  then by virtue of (32), p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Though, since  ∇x u′  J|ξ=0 = ∇xj Θ(x3 − h(x1, x2)) + j ⊗ (e3 − ∇xh(x1, x2)) φ(x3 − h(x1, x2)),  (34)  where φ is the Dirac delta distribution, one can easily assess that in order for  h(x1, x2) and the energy to be bounded, the following scaling must hold  ∇xj ∼ δ0 ,  j ∼ δ0 ,  ∇xh ∼ δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (35)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The scaling of the failure criterion  Alongside the energy, also the damage criterion (s < scr) scales as δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The scaling of the critical stretch scr is defined by equation (10), so scr ∼ δ−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The scaling of the stretch s, on the other hand, can be obtained by employing  equations (33) and (34)  s =  ξ · ∇ξu′  J|ξ=0 · ξ  |ξ|2  = ∇ξu′  J|ξ=0 : ξ ⊗ ξ  |ξ|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (36)  The second tensor in the double dot product1 is a quantity that scales as δ0  whereas the first tensor harbors a singularity, the Dirac’s Delta function φ,  which for x3 = h(x1, x2) makes the stretch infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Hence, whenever on the  crack surface, the criterion is immediately not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Finally:  s < scr →  �  False  for x3 = h(x1, x2)  True  otherwise  (37)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Localized energy in plane strain  localization of the PD non-local model has been obtained by means of a  limit operation, for vanishing δ, on the PD non-local elastic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The local-  ized energy obtained in this way is composed of a part entirely defined by the  continuous part of the displacement field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' the term (28),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' and a part composed  by mix and purely jump terms  Elocal = Elocal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='a + Elocal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (38)  where for the assumption of continuous displacement field (7) truncated at first  order in x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' and plane strain  Elocal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='a =HE  � 3  56A′  1(x1)2 + 5  84B3(x1)A′  1(x1) +  5  168A′  3(x1)2+  5  84B1(x1)A′  3(x1) +  5  168B1(x1)2 + 3  56B3(x1)2  �  +  H3E  � 1  224B′  1(x1)2 + 5B′  3(x1)2  2016  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (39)  1Given two second-order tensors, A and B, we mean by double dot product the operation  A:BT = Tr(AB) = AijBji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  20  Here A = {A1, A3}, B = {B1, B3} and the primes indicates derivative with  respect to x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Meanwhile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Elocal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='J =HE  � 5  168B1(x1)j′  3(x1) +  5  336j′  3(x1)2 +  5  168j′  3(x1)A′  3(x1)  �  +  (40)  H2E  � 5  672j′  3(x1)B′  3(x1)  �  +  E  � 3  28j3(x1)B3(x1) − 5  84j′  3(x1)B1(x1)h(x1) −  5  168h(x1)j′  3(x1)2+  − 5  84h′(x1)j3(x1)B1(x1) − 5  84j3(x1)j′  3(x1)h′(x1) +  − 5  84j′  3(x1)h(x1)A′  3(x1) − 5  84j3(x1)h′(x1)A′  3(x1)+  5  84j3(x1)A′  1(x1) + 5  84h(x1)j3(x1)B′  1(x1)+  − 5  168j′  3(x1)h(x1)2B′  3(x1) − 5  84j3(x1)h(x1)h′(x1)B′  3(x1)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  where the hypothesis of purely Mode-I crack development j = {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' j3} has been  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Equation (38) basically represents a material with cohesive constitutive law due  to the energy associated with a jump in the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The localized formulation of the nonlocal model introduced in section 3 can  now be obtained by writing the Lagrangian for a local plate where the internal  energy is that of equations (39-40) and then minimizing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Kirchhoff-like plate under Mode-I fracture  The kinematics of a Kirchhoff plate is readily recovered by imposing  A1(x1) → 0,  B3(x1) → 0,  B1(x1) → −A′  3(x1),  (41)  such that  ua(x1, x3) = {−A′  3(x1)x3, A3(x1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  (42)  Mode I delamination is achieved by choosing the jump function and its derivative  in the following way:  j1(x1) → 0,  j′  1(x1) → 0,  (43)  such that  uJ[x1, x3] = {0, j3(x1) Θ(x3 − h(x1))},  (44)  where Θ is the Heaviside function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Under these conditions the terms of localized  energy become  Elocal,a =  1  224EH3A′′  3(x1)2,  (45)  21  and  Elocal,J =E  � 5  84j3(x1)j′  3(x1)h′(x1) +  5  168h(x1)j′  3(x1)2+  (46)  − 5  84j3(x1)h(x1)A′′  3(x1) −  5  336j′  3(x1)2  �  ,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' It is worth mentioning that the limiting local energy for the con-  tinuous part of the displacement is a quantity resembling the classical result  for Kirchhoff plates: Elocal,a  =  H3E  12(1−ν2)A′′  3(x1)2, where for obvious reasons a  Poisson ratio of 1/4 has to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Conclusions  In the present paper, a reduced model for the explicit study of though-  thickness fracture nucleation and propagation in thin structures is put forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The model is obtained by making a hypothesis on the kinematics of the thin  element, which is assumed as the sum of a continuous part and a jump part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' A  particular choice of these fields is made which expresses the dependence on the  out-of-plane variable explicitly, thus making the integration through the thick-  ness feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The resulting reduced model retains information on the loss of  continuity of the material through the functions defining the jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The pro-  posed model has distinguished weak planes/surfaces, yet it leads to the recovery  of both horizontal and deviated crack patterns across the thickness of the plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The dimension reduction procedure generates a hierarchy of terms in the elastic  energy stored inside the plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' A mechanical interpretation of those terms is  possible (especially for the part of the energy associated with the continuous  part of the displacement) and is proposed through the definition of a simple  paradigm of peridynamic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' It is found that the hierarchical form of  the energy shows the coupling of membrane and bending behavior despite the  formulation being expressed in a linear setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The reduced model is then tested in a symmetric displacement-induced delam-  ination test for a cantilever plate and a broad range of qualitative responses  are obtained for a varying horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  In particular, for a horizon larger than  the height of the plate  distal nucleation is observed, whereas in the limit of  vanishing horizon a classic result of linear fracture mechanics is recovered with  the propagation of the fracture starting from the loaded cross-section of the  plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Apart from the crack path development, different horizons have proven  to greatly influence the force-displacement response of the structure, leading to  superior toughness and energy dissipation in the nonlocal model with a greater  horizon and a more brittle behavior in the case of a smaller horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  A non-symmetrical displacement-induced test is also performed and a relevant  sensitivity of the peridynamic plate emerges from the simulation where a curved  crack path characterizes the response at failure of the thin nonlocal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  To further investigate the local limit of the model, localization of the nonlocal  22  reduced formulation is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Firstly, the convergence of the nonlocal en-  ergy to a finite and non-vanishing local equivalent is assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The localized  reduced model shows a cohesive nature, which is expressed by the fact that en-  ergy can be stored by the part of the energy associated with the discontinuous  displacement field when a fracture is propagating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  CRediT authorship contribution statement  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Cavuoto: Developed the theory, performed the calculations and compu-  tations, wrote and edited the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Cutolo: Performed the computa-  tions, wrote and edited the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Dayal: Developed the theory, wrote  and edited the manuscript, supervised the whole work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Fraldi: Developed  the theory, wrote and edited the manuscript, supervised the whole work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Deseri: Developed the theory, wrote and edited the manuscript, supervised  the whole work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Declaration of Competing Interest  The authors declare that they have no known competing financial interests or  personal relationships that could have appeared to influence the work reported  in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Acknowledgements  LD, AC and MF gratefully acknowledge the support of the Italian Min-  istry of Research (MIUR) through the grants PRIN-20177TTP3S and PON  “Stream”-ARS01 01182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' LD also gratefully thanks the support of the Euro-  pean Commission through (i) FET Open “Boheme” grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  863179, and  (ii) LIFE GREEN VULCAN LIFE19 ENV/IT/000213, and (iii) ERC-ADG-  2021-101052956-BEYOND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Kaushik Dayal thanks Army Research Office (MURI  W911NF-19-1-0245), Office of Naval Research (N00014-18-1-2528), and Na-  tional Science Foundation (DMREF 2118945, DMS 2108784) for financial sup-  port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Expanded form of the reduced energy  The hierarchical form of the reduced energy is reported here again for the reader:  ωred = ωred,a + ωred,J  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)  where in the case of φ = 1, specializes to  ωred,a = H2 p1(φ, ua) + H4 p2(φ, ua) + H6 p3(φ, ua) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  ωred,J = r0(uJ) + H r1(ua, uJ) + H2 r2(uJ)+  H3 r3(ua, uJ) + H4 r4(ua, uJ) + H5 r5(ua, uJ)  ,  23  where the pi functions are those specified by eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' (18) to (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' On the other  hand, the coefficients of the reduced energy associated with the jump part of  the displacements are:  r0 = −1  6j3(x′  1)j3(x1)h(x′  1)h(x1)  �  2h(x′  1)2 − 3h(x′  1)h(x1) + 2h(x1)2�  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='r1 = −1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)A3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)3 − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='6j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)2h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)3 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)3A3(x1)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)2(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1 − x)A1(x1) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2j3(x1)A1(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)(x − x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x1)2+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3j3(x1)A3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x1)3 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='6j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)j3(x1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)3 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='6j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)j3(x1)h(x1)3+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3j3(x1)(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1 − x)h(x1)3B1(x1) − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)4B3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4j3(x1)h(x1)4B3(x1) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)A1(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)2(x − x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2j3(x1)(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1 − x)A1(x1)h(x1)2 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)3(x − x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)B1(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='6j3(x1)2 − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3j3(x1)A3(x1)h(x1)3h(x1)3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  r2 = −1  8j3(x′  1)j3(x1)  �  h(x′  1)2 + h(x1)2�  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='r3 = − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='12j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)A3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1) − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='24j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)2h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='12j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)A3(x1)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='12j3(x1)A3(x1)h(x1) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='8j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)A1(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1 − x)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='8j3(x1)(x − x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)A1(x1) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='12j3(x1)(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1 − x)B1(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x1)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='12j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)(x − x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)B1(x1) − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='12j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)2B3(x1)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='24j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)2B3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1) − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='24j3(x1)2h(x1) − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='24j3(x1)h(x1)2B3(x1)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='8j3(x1)A1(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1 − x) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='12j3(x1)A3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x1) − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='12j3(x1)B3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)h(x1)2+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='8j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)(x − x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)A1(x1) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='24j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)j3(x1)h(x1) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='24j3(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)j3(x1)h(x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  r4 = 1  12j3(x′  1)A3(x′  1) − 1  12j3(x′  1)A3(x1) − 1  12j3(x1)A3(x′  1)+  1  24j3(x′  1)(x′  1 − x)B1(x1) + 1  24j3(x1)(x − x′  1)B1(x′  1)+  1  12j3(x1)A3(x1) + 1  24j3(x1)2 + 1  24j3(x′  1)(x′  1 − x)B1(x′  1)+  − 1  96j3(x′  1)j3(x1) + j3(x′  1)2  24  + 1  24j3(x1)(x − x′  1)B1(x1) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  r5 = 1  48j3(x1)B3(x′  1) + 1  48j3(x′  1)B3(x1) +  5  192j3(x′  1)B3(x′  1) +  5  192j3(x1)B3(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  24  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' A micro-structural interpretation for bond-based PD  The structure of bond-based peridynamic constitutive equation (3) lends  itself to an intuitive and simple physical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In fact, one can imag-  ine the body under consideration to be uniformly divided into blocks and to  substitute each block with a node embodying the mass of that specific block2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Then, massless connections can be introduced between nodes to represent their  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' These connections must reflect how pairs of particles exert forces  onto one another in the PD formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  In order to do so, one can look at how the energy is stored between pairs of  interacting volumes (or areas for a two-dimensional problem), Vi and Vj, of the  PD continuum model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Taking into account equation (5), by means of the mean  value theorem for integrals, one has  Ei,j =  �  Vi  �  Vj  c  σ  (ξ · η)2  4  dVidVj = c  σ∗ (ξ∗ · η∗)2 ViVj  4  = c  σ∗  ViVj  4  |ξ∗|2 |η∗|2 cos2 β ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)  where ξ∗ and η∗ are the relative position vector and relative displacement vector  for a pair of points (x∗  i , x∗  j), belonging to the volumes Vi and Vj, for which  the theorem holds, and β is the angle between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  In the limit of very  small volumes Vi and Vj it is legitimate to assume the ξ and η functions to be  constant between the volumes of interest and approximate the (x∗  i , x∗  j) with the  mid-points of each volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The right term of equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1) resembles the energy of a truss tilted by the  angle between ξ and a horizontal axis e1, and subjected to the edge displacement  |η|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In this fashion, the stiffness of the bond representing the interaction between  volumes would be  ki,j = 1  2  c  σ∗ ViVj|ξ∗|2 ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2)  where ξ∗ is now the relative position vector between mid-points of volumes Vi  and Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2) implies that the stiffness is proportional to some power  n of the distance between particles ki,j ∼ ln  i,j (recalling that σ∗ = σ(|ξ∗|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Performing this type of substitution for all the pairs of nodes transforms the  continuum peridynamic body into an intricate reticular beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The simplest  paradigm of structure that can be built by following the PD approximation  presented above is depicted in Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The paradigmatic structure char-  acterized by simple kinematics, correctly predicts the hierarchical form of the  energy (17) and allows comparing the kinematics of the continuum with some-  thing more easily controllable, accompanying the reader in an ideal transition  2In finite-element analysis mesh refinement is a powerful stratagem that allows improving  precision and accuracy of the approximated solutions for the problem at hand, in parts of the  domain where it is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Refinement procedures inevitably cause non-uniform discretiza-  tion of the domain and, in the case of peridynamics, can cause inaccuracy of the solution  [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  25  from discrete to continuum for making evident how the above-mentioned non-  standard terms have to naturally appear in the peridynamic plate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  We hereby limit the paradigmatic structure to stretching and bending kinemat-  ics which prove to be sufficient for a qualitative interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In particular,  the structure is loaded by imposing horizontal displacements at the outer nodes  (nodes 1 and 2 in Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In this condition, the total energy can easily  be obtained analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' By denoting  Next- to- nearest neighbour  Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='10: Simplest structure representing a bond-based PD body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The nodes are the  material particles of the body, while the bonds are represented by the truss connecting each  couple of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The structure has been made symmetric, both in loading conditions that in  elements stiffness and coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The gray region (denoted by V1) represents the volume  ascribable to node 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The horizon δ is taken to be equal to the diagonals of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  ε = u2 + u1  4 l14  ,  χ = 2(u1 − u2)  l12 l14  ,  the average stretch and curvature respectively, the total energy amounts to the  following expression:  Ediscrete = E α l4+n  14  �  H2 ε2 4  �  1 + 2ψ2 + ψn�  1 + ψ2 + ψn  + H4 χ2  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3)  recalling that l12 =H is the length of truss connecting points 1 and 2 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='10, ψ = cot θ, E is the Young’s Modulus of the beams (assumed constant) and  α = c V 2  1 /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3) represents the equivalent energy of a peridynamic  continuum (the horizon is included implicitly since l13 = δ = l14/ sin θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Clearly,  membrane energy scales with the square of the thickness H, while bending energy  scales with H4 differently from the classical results of local elasticity [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  According to the reduced energy obtained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4, the total energy of a  plate resembling the shape and loads of the paradigmatic case is recovered by  choosing L= 2l5, δ = l5/ cos θ (hence Ψ = (cos θ)/2):  EPD = c sec5 θ (29 + 20 cos θ) l6  5  �  H2 ε2  108  29 + 20 cos 2θ + H4 χ2  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4)  which corresponds, at least qualitatively, to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3) for n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  26  Sensitivity analysis of the paradigmatic structure to the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The total en-  ergy of the paradigmatic case has been derived for a fixed value of the horizon,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' δ = l5/ cos θ, whereas the reduced form of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' (17) is defined for any value  of such parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' In order to extend the above interpretation analysis using  the paradigmatic tool to the general case, a series of numerical analyses using  ANSYS APDL has been carried out on several discrete approximations of a peri-  dynamic bond-based continuum each characterized by different horizon sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  In Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='11 the scaling of three different peridynamic discrete bodies with a  horizon ranging from a minimal value (dotted green line) to one with a wider  interaction (dot-dashed dark line) is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' On the left, the scaling of the  membrane energy is found to be quadratic regardless of the horizon size, whereas  the bending energy (Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='11, on the right) scales with the fourth power of  the thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' A geometric motivation is available for the scaling of the mem-  brane energy: the increase in the number of bonds available when the thickness  is doubled, for example, is (roughly) proportional to the square of the number  of nodes: nbonds ∼ nnodes(nnodes − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2  H  ε/εPD,max  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2  H  ε/εPD,max  Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='11: On the left: scaling, as a function of the total height H, of the energy associated  with a uniform stretching deformation (purely membrane) for a standard Pratt truss (in  green), a hyperstatic truss (in red) and two different patterns of a PD discrete beam;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' on  the right: scaling, as a function of the total height H, of the energy associated to a uniform  bending deformation, for a standard Pratt truss (in green), a hyperstatic beam (in red) and  two different patterns of a PD discrete beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Coupling in the local theory of plates  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' (17) is derived from an assumption on the kinematics that is typical of  the first-order shear deformation theory (FSDT), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Reissner-Mindlin theory,  for local plates [72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' To further address the previous result about the coupling  in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' (17), we compare the energies for various local plate model, in accordance  with the FSDT hypothesis, against the peridynamic result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Under the condition  of FSDT, small displacements (linear elastic response) and inextensibility in the  thickness direction, the specific (per unit area) elastic energy amounts to:  ωred = H  2  �  (λ + 2µ)A′2  1 + µ(B1 + A′2  3 )  �  + H3  24 (λ + 2µ)B′2  1 ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='1)  27  where the A1, A3, B1 and B3 are to be intended as functions of x1, and ωred is  now the reduced energy density of a local plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  If one relaxes the inextensibility constraint, meaning the vertical component  of the displacement follows the linear approximation in the through-thickness  direction, the elastic energy becomes:  ωred =H  �  (λ + 2µ)(A′  1 + B3)2 + µ(A′  3 + B1)2 + (λ − 2µ)A′  1B3  �  /2+  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='2)  H3 �  (λ + 2µ)B′2  1 + µB′2  3  �  /24 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Lastly, adding a geometric nonlinearity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' small displacements but large/finite  rotations hypothesis, to the model leads to:  ωred =H {A′4  3  (λ + 2µ)  4  + A′2  1(λ + 2µ) + A′2  3 (A′  1(λ + 2µ) + µ + λB3) +  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='3)  2λB3A′  1 + 2µB1A′  3 + B2  3(λ + 2µ) + µB2  1 }/2+  H3 �  40A′  3(λ + 2µ)B′  3B′  1 + B′2  3  ��  3A′2  3 + 2A′  1  �  (λ + 2µ) + 2µ + 2λB3  �  +  80(λ + 2µ)B′2  1  �  /48  H5 �  (λ + 2µ)B′4  3  �  /640 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  From the Euler-Lagrange equations that the previous type of elastic energies  can generate, one can see how only the last nonlinear model accounts for the  coupling of stretching and bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' The reason for this lies in the nonlinear  relation between the deformations and the displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  The intrinsic  microstructure of a peridynamic continuum naturally accounts for a similar  effect, since the finite distances between particles makes it so that rotational  effect can be induced by the intricate connections even under axial loading and  net null moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  References  [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Inoue, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Yin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Kimura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Tsuzaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Ochiai, Delamination effect on  impact properties of ultrafine-grained low-carbon steel processed by warm  caliber rolling, Metall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' A 41 (2010) 341–355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Naganarayana, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Atluri, Strength reduction and delamination growth in  thin and thick composite plates under compressive loading, Computational  Mechanics 16 (1995) 170–189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Dugdale, Yielding of steel sheets containing slits, Journal of the Me-  chanics and Physics of Solids 8 (1960) 100–104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Barenblatt, The mathematical theory of equilibrium cracks in brittle  fracture, Advances in Applied Mechanics 7 (1962) 55–129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  28  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Hillerborg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Modeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Petersson, Analysis of crack formation  and crack growth in concrete by means of fracture mechanics and finite  elements, Cement and Concrete Research 6 (1976) 773–781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Turon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Da’vila, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Camanho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Costa, An engineering solution for  mesh size effects in the simulation of delamination using cohesive zone  models, Engineering Fracture Mechanics 74 (2007) 1665–1682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Elices, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Guinea, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Gomez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Planas, The cohesive zone model: ad-  vantages, limitations and challages, Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Fract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' 69 (2002) 137–163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Fan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Jar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Cheng, Cohesive zone with continuum damage  properties for simulation of delamination in fibre composites and failure of  adhesive joints, Engineering Fracture Mechanics 75 (2008) 3866–3880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Zhi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Liu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Hu, Xfem simulation of delamination  in composite laminates, Composites Part A: Applied Science and Manufac-  turing 80 (2016) 61–71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Yazdani, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Rust, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Wriggers, An xfem approach for modelling delam-  ination in composite laminates, Composite structures 135 (2016) 353–364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Dean, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Reinoso, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Paggi, A multi phase-field-  cohesive zone model for laminated composites: Application to delamination  migration, Composite structures 276 (2021) 114471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Roy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Deepu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Pathrikar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Roy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Reddy, Phase field based peridy-  namics damage model for delamination of composite structures, Composite  structures 180 (2017) 972–993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [13] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Giorgio, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Rizzi, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Andreaus, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Steigmann, A two-dimensional contin-  uum model of pantographic sheets moving in a 3-d space and accounting for  the offset and relative rotations of the fibers, Mathematics and mechanics  of complex systems 7 (4) (2019) 311–325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [14] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Mill´an, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Torres-S´anchez, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Roman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Arroyo, A variational  model of fracture for tearing brittle thin sheets, Journal of the Mechanics  and Physics of Solids 119 (2018) 334–348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [15] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Lai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Arroyo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Shen, Phase field modeling of brittle  fracture in an euler–bernoulli beam accounting for transverse part-through  cracks, Computer Methods in Applied Mechanics and Engineering 361  (2020) 112787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Pigazzini, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Kamensky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' van Iersel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Alaydin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Remmers,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Bazilevs, Gradient-enhanced damage modeling in kirchhoff–love shells:  application to isogeometric analysis of composite laminates, Computer  Methods in Applied Mechanics and Engineering 346 (2019) 152–179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  29  [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Chowdhury, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Reddy, Geometrically exact micropolar timoshenko beam  and its application in modelling sandwich beams made of architected lattice  core, Composite Structures 226 (2019) 111228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Conti, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Dondl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Orlik, Variational modeling of paperboard delami-  nation under bending, Mathematics in Engineering 5 (2) (2023) 1–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Silling, Reformulation of elasticity theory for discontinuities and long-  range forces, Journal of the Mechanics and Physics of Solids 48 (1) (2000)  175–209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [20] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Kunin, Theory of elastic media with a microstructure: Nonlocal theory  of elasticity, Moscow, Izdatel’stvo Nauka, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [21] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Prakash, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Deng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Stewart, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Harris, Investigation of mi-  croscale fracture mechanisms in glass–ceramics using peridynamics simula-  tions, Journal of the American Ceramic Society 105 (6) (2022) 4304–4320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Karim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Kadau, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Narasimhachary, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Radaelli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Amann,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Dayal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Silling, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Germann, Crack nucleation at forging flaws stud-  ied by non-local peridynamics simulations, Mathematics and Mechanics of  Solids 27 (2022) 1129–1149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Rezaul Karim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Kadau, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Narasimhachary, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Radaelli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Amann,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Dayal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Silling, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Germann, Crack nucleation from non-metallic  inclusions in aluminum alloys described by peridynamics simulations, In-  ternational Journal of Fatigue 153 (2021) 106475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Chua, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Agrawal, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Breitzman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Gazonas, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Dayal, Phase-field  modeling and peridynamics for defect dynamics, and an augmented phase-  field model with viscous stresses, Journal of the Mechanics and Physics of  Solids 159 (2022) 104716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Lipton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Lehoucq, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Jha, Complex fracture nucleation and evolu-  tion with nonlocal elastodynamics, Journal of Peridynamics and Nonlocal  Modeling 1 (2) (2019) 122–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [26] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Diehl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Lipton, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Wick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Tyagi, A comparative review of peridy-  namics and phase-field models for engineering fracture mechanics, Compu-  tational Mechanics (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [27] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Jooeun, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Wenyang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Jung-Wuk, Impact fracture analysis enhanced  by contact of peridynamic and finite element formulations, International  Journal of Impact Engineering 87 (2016) 108–119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [28] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Zhou, Peridynamic modelling of impact damage in  three-point bending beam with offset notch, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='-Engl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  38 (2017) 99–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  30  [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Xu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Askari, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Weckner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Silling, Peridynamic analysis of im-  pact damage in composite laminates, Journal of Aerospace Engineering  21 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [30] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Gazonas, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Bobaru, Supershear damage propagation and  sub-rayleigh crack growth from edge-on impact: A peridynamic analysis,  International Journal of Impact Engineering 113 (2018) 73–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [31] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Bobaru, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Ha, Adaptive refinement and multiscale modeling in  2d peridynamics, Journal for Multiscale Computational Engineering 9 (6)  (2011) 635–659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [32] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Le, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Chan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Schwartz, A two-dimensional ordinary state based  peridynamic model for linearly elastic solids, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' 98  (2014) 547–561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [33] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Sarego, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Le, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Bobaru, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Zaccariotto, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Galvanetto, Linearized  state-based peridynamics for 2d problems, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  108 (10) (2016) 1174–1197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Silling, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Bobaru, Peridynamic modeling of membranes and fibers, In-  ternational Journal of Non-Linear Mechanics 40 (2005) 395–409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Taylor, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Steigmann, A two-dimensional pridynamic model for thin  plates, Mathematics and Mechanics of Solids 20 (8) (2013) 998–1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [36] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Naumenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Eremeyev, A non-linear direct peridynamics plate theory,  Composite Structures 279 (2022) 114728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [37] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' O’Grady, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Foster, Peridynamic plates and flat shells: A non-ordinary,  state-based model, International Journal of Solids and Structures 51 (2014)  4572–4579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' O’Grady, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Foster, Peridynamic beams: A non-ordinary, state-based  model, International Journal of Solids and Structures 51 (18) (2014) 3177–  3183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' O’Grady, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Foster, Peridynamic beams and plates: A non-ordinary  state-based model, ASME International Mechanical Engineering Congress  and Exposition Volume 1: Advances in Aerospace Technology (11 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Behzadinasab, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Alaydin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Trask, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Bazilevs, A general-purpose,  inelastic, rotation-free kirchhoff–love shell formulation for peridynamics,  Computer Methods in Applied Mechanics and Engineering 389 (2022)  114422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Chowdhury, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Roy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Roy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Reddy, A peridynamic theory for linear  elastic shells, International Journal of Solids and Structures 84 (2016) 110–  132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  31  [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Reddy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Srinivasa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Arbind, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Khodabakhshi, On gradient elasticity  and discrete peridynamics with applications to beams and plates, in: Ad-  vanced Materials Research, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' 745, Trans Tech Publ, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' 145–154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [43] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Yang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Oterkus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Oterkus, Peridynamic formulation for higher-order  plate theory, Journal of peridynamics and nonlocal modeling 3 (2021) 185–  210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [44] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Yolum, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Gok, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Coker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Guler, Peridynamic modelling of delami-  nation in dcb specimen, Procedia Structural Integrity 13 (2018) 2126–2131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [45] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Hu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Carvalho, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Madenci, Peridynamic modeling of delamination  growth in composites laminates, Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [46] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Jiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Guo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Wang, Peridynamic modeling of mode-i  delamination growth in double contilever composites beam test: a two-  dimensional modeling using revised energy-based failure criteria, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  9 (2019) 656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [47] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Choksi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Piero, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Fonseca, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Owen, Structured deformations as  energy minimizers in models of fracture and hysteresis, Mathematics and  Mechanics of Solids 4 (3) (1999) 321–356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Gobbino, Finite difference approximation of the mumford–shah func-  tional, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' 51 (2) (1998) 197–228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [49] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Dayal, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Bhattacharya, Kinetics of phase transformations in the peri-  dynamic formulation of continuum mechanics, Journal of the Mechanics  and Physics of Solids 54 (9) (2006) 1811–1842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [50] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Du, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Zhou, Mathematical analysis for the peridynamic nonlocal con-  tinuum theory, Mathematical Modeling and Numerical Analysis 45 (2011)  217–234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [51] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Zhou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Qiang, Mathematical and numerical analysis of linear peridy-  namic models with nonlocal boundary conditions, Siam J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  48 (5) (2010) 1759–1780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [52] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Erdogan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Ozturk, On the singularities in fracture and contact me-  chanics, Journal of applied mechanics 75 (2008) 051111–1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [53] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Bobaru, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Silling, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Alves, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Askari, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Xu, Convergence,  adaptive refinement, and scaling in 1d peridynamics, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' 77 (2009) 852–877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Foster, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Silling, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Chen, An energy based failure criterion for use  with peridynamic states, International Journal for Multiscale Computa-  tional Engineering 9 (2011) 675–687.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  32  [55] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Qiao, A two-dimensional ordinary state-based peridynamic  model for elastic and fracture analysis, Engineering Fracture Mechanics  232 (2020) 107040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [56] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Madenci, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Oterkus, Ordinary state-based peridynamics for plastic de-  formation according to von mises yield criteria with isotropic hardening, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Solids 86 (2016) 192–219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [57] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Griffith, The phenomena of rupture and flow in solids, Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Royal Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' London A 221 (1921) 163–198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [58] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Lipton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Jha, Nonlocal elastodynamics and fracture, Nonlinear Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' 28 (2) (2021) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [59] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Jha, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Lipton, Numerical convergence of nonlinear nonlocal contin-  uum models to local elastodynamics, International Journal for Numerical  Methods in Engineering 114 (13) (2018) 1389–1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [60] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Silling, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Lehoucq, Peridynamic theory of solid mechanics, Advances in  Applied Mechanics 44 (2010) 73–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [61] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Breitzman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Dayal, Bond-level deformation gradients and energy av-  eraging in peridynamics, Journal of the Mechanics and Physics of Solids  110 (2018) 192–204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [62] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Mengesha, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Du, On the variational limit of a class of nonlocal func-  tionals related to peridynamics, Nonlinearity 28 (2015) 3999–4035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [63] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Bellido, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Mora-Corral, Existence for nonlocal variational problems in  peridynamics, Siam J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' 46 (1) (2014) 890–916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [64] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Aksoylu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Mengesha, Results on nonlocal boundary value problems,  Numerical Functional Analysis and Optimization 31 (12) (2010) 1301–1317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [65] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Foss, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Radu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Wright, Existence and regularity of minimizers for  nonlocal energy functionals, Differential and Integral Equations 31 (11-12)  (2018) 807–832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [66] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Bellido, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Mora-Corral, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Pedregal, Hyperelasticity as a gamma-limit  of peridynamics when the horizon goes to zero, Calculus of Variations and  Partial Differential Equations 54 (2015) 1643–1670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [67] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Bellido, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Cueto, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Mora-Corral, Bond-based peridynamics does not  converge to hyperelasticity as the horizon tend to zero, Journal of Elasticity  141 (09 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [68] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Silling, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Lehoucq, Convergence of peridynamics to classical elasticity  theory, Journal of Elasticity 93 (2008) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  33  [69] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Chen, A comparison study on peridynamic models using irregular non-  uniform spatial discretization, Computer Methods in Applied Mechanics  and Engineering 345 (2019) 539–554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [70] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Steigmann, A well-posed finite strain model for thin elastic sheets with  bending stiffness, Mathematics and Mechanics of Solids 18 (1) (2012) 103–  112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [71] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Steigmann, Asymptotic Estimate of the Potential Energy of a Plastically  Deformed Thin Shell, Springer, 2020, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' 22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' 409–420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [72] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Reddy, Mechanics of laminated composite plates and shells: Theory and  Analysis, CRC press, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  [73] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Reddy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content=' Srinivasa, Non-linear theories of beams and plates accounting  for moderate rotations and material length scales, International Journal of  Non-Linear Mechanics 66 (2014) 43–53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
+page_content='  34' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE0T4oBgHgl3EQflAGw/content/2301.02481v1.pdf'}
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+arXiv:2301.12035v1  [eess.SP]  28 Jan 2023
+STATE MACHINE-BASED WAVEFORMS FOR CHANNELS WITH 1-BIT
+QUANTIZATION AND OVERSAMPLING WITH TIME-INSTANCE
+ZERO-CROSSING MODULATION
+Diana M. V. Melo, Lukas T. N. Landau and Rodrigo C. de Lamare
+Centre for Telecommunications Studies,
+Pontifical Catholic University of Rio de Janeiro,
+Rio de Janeiro, Brazil 22453-900
+Email: diana;lukas.landau;delamare@cetuc.puc-rio.br
+ABSTRACT
+Systems with 1-bit quantization and oversampling are
+promising for the Internet of Things (IoT) devices in order
+to reduce the power consumption of the analog-to-digital-
+converters. The novel time-instance zero-crossing (TI ZX)
+modulation is a promising approach for this kind of channels
+but existing studies rely on optimization problems with
+high computational complexity and delay. In this work, we
+propose a practical waveform design based on the estab-
+lished TI ZX modulation for a multiuser multi-input multi-
+output (MIMO) downlink scenario with 1-bit quantization
+and temporal oversampling at the receivers. In this sense,
+the proposed temporal transmit signals are constructed by
+concatenating segments of coefficients which convey the
+information into the time-instances of zero-crossings accord-
+ing to the TI ZX mapping rules. The proposed waveform
+design is compared with other methods from the literature.
+The methods are compared in terms of bit error rate and
+normalized power spectral density. Numerical results show
+that the proposed technique is suitable for multiuser MIMO
+system with 1-bit quantization while tolerating some small
+amount of out-of-band radiation.
+Index Terms— Zero-crossing precoding, oversampling,
+Moore machine, 1-bit quantization.
+I. INTRODUCTION
+Future wireless communication technologies are envi-
+sioned to support a large number of the Internet of Things
+(IoT) devices which require to have low power consumption
+and low complexity. Low resolution analog-to-digital con-
+verters (ADCs) are suitable to meet the IoT requirements
+since the power consumption in the ADCs increase expo-
+nentially with its amplitude resolution [1]. The loss of infor-
+mation caused by the coarse quantization can be partially
+compensated by increasing the sampling rate. Employing
+temporal MRx-fold oversampling, rates of log2(MRx + 1)
+bits per Nyquist interval are achievable in a noise free envi-
+ronment [2]. The authors in [3] study the maximization of the
+achievable rate for systems with 1-bit quantization and over-
+sampling in the presence of noise. Other studies that consider
+systems with 1-bit quantization and oversampling employ
+ASK transmit sequences [4], [5] and 16 QAM modulation
+[6]. Other practical methods are based on the idea presented
+in [2], where the information is conveyed into the zero-
+crossings. An example is the study presented in [7], where
+the waveform is constructed by concatenating sequences
+which convey the information into the zero-crossings. This
+study shows that similar data rates to the one presented
+in [2] can be achieved over noisy channels with relatively
+low out-of-band radiation. Some other practical methods
+which convey the information into the zero-crossings include
+runlength-limited (RLL) sequences [8], [9].
+The benefits of 1-bit quantization and oversampling have
+been studied in [10], [11] for multiple-input multiple-output
+(MIMO) channels in uplink transmission. Moreover, the
+studies [12], [13] investigate sequences for downlink MIMO
+systems with 1-bit quantization and oversampling. In this
+regard, in [12] it is presented the quantization precoding
+method which considers as optimization criterion the maxi-
+mization of the minimum distance to the decision threshold
+(MMDDT) which was proposed in [6]. The quantization
+precoding technique relies on an exhaustive codebook search
+which allows simple Hamming distance detection. Superior
+precoding schemes for MIMO downlink scenarios have been
+investigated in [14], [15], where a novel time-instance zero-
+crossing (TI ZX) modulation is introduced. This novel mod-
+ulation follows the idea of [2] by allocating the information
+into the time-instance of zero-crossings in order to reduce
+the number of zero-crossings of the signal. The study in
+[14] relies on a precoding technique based on the MMDDT
+criterion with spatial zero-forcing (ZF) precoding and TI
+ZX modulation, whereas [15] proposes an optimal temporal-
+spatial precoding technique with TI ZX modulation along
+with an minimum mean square error (MMSE) solution.
+Other studies that consider novel TI ZX modulation schemes
+have been presented in [13], [16], [17] where the computa-
+
+tional complexity is reduced [16]. In [17] the minimization
+of the transmit power under quality of service constraint is
+considered as an objective. The study in [13] investigates
+the spectral efficiency of MIMO systems with sequences
+constructed with the TI ZX modulation and RLL sequences.
+In this work, we propose a TI ZX waveform design
+for multiuser MIMO downlink channels with 1-bit quan-
+tization and oversampling where a defined level of out-of-
+band radiation is tolerated. The proposed waveform design
+considers the novel TI ZX modulation from [14], [15] and
+follows a similar idea as presented in [7]. The proposed
+method conveys the information into the time-instances
+of zero-crossings but instead of considering sequences of
+samples, input bits are mapped into waveform segments
+according to the TI ZX mapping rules [14], [15]. The
+temporal precoding vector is then used in conjunction with
+a simple pulse shaping filter. The optimal set of coeffi-
+cients is computed with an optimization problem which
+is formulated to maximize the minimum distance to the
+decision threshold, constrained with some tolerated out-of-
+band radiation. Finally, the numerical results are evaluated
+considering the bit error rate (BER) and the power spectral
+density (PSD). The proposed waveform design is compared
+with the transceiver waveform design from [7] and the TI ZX
+MMDDT precoding [14]. The transceiver waveform design
+[7] was adapted for MIMO channels. The simulation results
+show that the proposed waveform design is comparable in
+terms of BER performance to the one presented for TI ZX
+MMDDT precoding while having a lower computational
+complexity since the waveform optimization is done once
+and is suitable for any input sequence of bits.
+The rest of the paper is organized as follows: The sys-
+tem model is introduced in Section II. Then, Section III
+describes the novel TI ZX modulation. Section IV explains
+the proposed waveform design optimization including the
+autocorrelation function for TI ZX modulated sequences.
+The simulation results are provided in Section V and finally,
+the conclusions are given in Section VI.
+Notation: In the paper all scalar values, vectors and
+matrices are represented by: a, x and X, respectively.
+II. SYSTEM MODEL
+In this study, a multiuser MIMO downlink scenario with
+Nu single antenna users and Nt transmit antennas at the
+base station (BS), is considered as shown in Fig. 1. Trans-
+mission blocks of N symbols (N Nyquist intervals) are
+considered. The input sequences of symbols xk are mapped
+using the TI ZX mapping and the set of coefficients G
+which yields the temporal precoding vector sgk ∈ CMRxN,
+where MRx/T denotes the sampling rate and T refers to
+the symbol duration. Moreover, the transmit filter gTx(t)
+and receive filter gRx(t) are presented, where the combined
+waveform is given by v(t) = (gTx ∗ gRx) (t). Furthermore
+1-bit quantization is applied at the receivers. The channel
+x 1
+x Nu
+Spatial
+Precoder
+DAC
+s1(t)
+DAC
+gTx(t)
+gTx(t) sNt(t)
+sx1
+sxNt
+Partitioning
+sx
+H
+sg
+1-bit ADC
+Detector
+schk(t) + nk(t)
+�x k
+y k
+z k
+Sampling rate MRx/T
+Q(·)
+gRx(t)
+Mapper
+Fig. 1: Considered multi-user MIMO downlink system model.
+matrix H ∈ CNu×Nt is known at the base station and is
+considered to be frequency-flat fading as typically assumed
+for narrowband IoT systems. Then, with the stacked temporal
+precoding vector sg =
+�
+sT
+g1, sT
+g2, · · · , sT
+gk, · · · , sT
+gNu
+�T
+, the
+received signal z ∈ C3NtotNu can be expressed by stacking
+the received samples of the Nu users as follows:
+z = Q1 ((HP sp ⊗ INtot) (INt ⊗ V ) sg + (INu ⊗ GRx) n)
+= Q1 (Heffsg + GRx,effn) ,
+(1)
+where Q1(·) corresponds the 1-bit quantization operator,
+n ∈ C3NtotNu denotes a vector with zero-mean complex
+Gaussian noise samples with variance σ2
+n with Ntot
+=
+NMRx. The waveform matrix V with size Ntot × Ntot is
+given by
+V =
+
+
+v (0)
+v
+�
+T
+MRx
+�
+· · ·
+v (T N)
+v
+�
+−
+T
+MRx
+�
+v (0)
+· · ·
+v
+�
+T
+�
+N −
+1
+MRx
+��
+...
+...
+...
+...
+v (−T N)
+v
+�
+T
+�
+−N +
+1
+MRx
+��
+· · ·
+v (0)
+
+
+.
+(2)
+The receive filter gRx is represented in discrete time by the
+matrix GRx with size Ntot × 3Ntot and is denoted as
+GRx = aRx
+
+
+�
+gT
+Rx
+�
+0 · · ·
+0
+0
+�
+gT
+Rx
+�
+0 · · · 0
+... ... ...
+0 · · ·
+0
+�
+gT
+Rx
+�
+
+ ,
+(3)
+with gRx = [gRx(−T (N +
+1
+MRx )), gRx(−T (N +
+1
+MRx ) +
+T
+MRx ), . . . , gRx(T (N +
+1
+MRx ))]T and aRx = (T/MRx)1/2. The
+matrix P sp = czfHH �
+HHH�−1 denotes the spatial zero-
+forcing precoder. The matrix P sp is normalized such that
+the spatial precoder does not change the signal power. As in
+[14] the normalization factor czf is given by
+czf =
+�
+Nu/trace
+��
+HHH�−1�� 1
+2 .
+(4)
+
+III. TIME-INSTANCE ZERO-CROSSING MAPPING
+The TI ZX modulation was proposed in the studies [14]
+and [15] for systems with 1-bit quantization and over-
+sampling. The TI ZX modulation conveys the information
+into the time-instances of zero-crossings and also considers
+the absence of zero-crossing during a symbol interval as
+a valid symbol, different to other approaches from the
+literature [2] and [7]. To build the mapped sequence, each
+symbol xi drawn from the set Xin := {b1, b2, · · · , bRin}
+with Rin = MRx + 1, is mapped into a binary codeword
+csi with MRx samples. As mentioned, one of the possible
+symbols corresponds to the pattern that does not contain a
+zero-crossing. The mapping depends on the last sample of
+the previous symbol interval, namely ρ ∈ {1, −1}. Hence,
+the TI ZX mapping provides two possible codewords csi for
+each valid symbol xi which convey the same zero-crossing
+information. Then, for coding and decoding of the first trans-
+mit symbol, a pilot sample ρb ∈ {1, −1} is required. Finally,
+the desired output pattern coutk is obtained by concatenating
+the segments csi such that, coutk = [ρb, cT
+s0, . . . , cT
+sN−1]T with
+total length NMRx + 1.
+IV. WAVEFORM DESIGN OPTIMIZATION
+The proposed waveform design, suitable for systems
+with 1-bit quantization and oversampling, considers the
+novel TI ZX modulation [14], [15], in conjunction with
+the optimization of a set of coefficients. The proposed
+waveform is built by concatenating segment sequences, i.e.,
+subsequences, described by the coefficients which contain
+zero-crossings at the desired time-instances. The proposed
+waveform design relies on the transmit and receive filters
+gTx(t) and gRx(t) which preserve the zero-crossing time-
+instance. Different to prior studies [14], [15], the sequence
+is no longer binary but is defined by the set of coefficients
+G, so that each symbol xi drawn from the set Xin is mapped
+into a codeword gi with MRx different coefficients which
+convey the information into the time-instances of zero-
+crossings. The set of coefficients G is defined in terms of
+G = {G+; G−} where G− = −G+, such that they both
+convey the same zero-crossing information and the sign
+information of the coefficients depends on the last sample
+of the previous interval termed ρ. Considering bit sequences
+as input and the Gray coding for TI ZX modulation shown
+in [14, Table II], ns = 2̺ different states can be defined.
+In this context, the set G =
+�
+gT
+1 ; gT
+2 ; · · · ; gT
+̺
+�
+is presented,
+where gi = [gi,1, gi,2, · · · gi,q] and ρ = sgn (gi,MRx). Then, as
+initially established, the symbol xi is mapped in the segment
+gi. The pilot sample ρb is required for the encoding and
+decoding processes of the first symbol x1. Finally, the input
+sequence of symbols xk is mapped in the sequence sgk with
+length Ntot by concatenating all the segments gi such that,
+sgk = [gT
+0 , . . . , gT
+N−1]T . Nothe that the pilot sample ρb is
+predefined and known at the receivers, hence not included
+in the precoding vector sgk.
+IV-A. Autocorrelation for TI ZX Modulation
+In this section, it is described how to compute the autocor-
+relation function of the TI ZX modulated signal, considering
+the set of coefficients G which conveys the information into
+the time-instances of zero-crossings.
+To obtain the autocorrelation function, the TI ZX modula-
+tion system is converted to a finite-state machine where the
+current output values are determined only by its current state
+which corresponds to an equivalent Moore machine [18]. For
+MRx = 3, one symbol in terms of two bits is mapped in one
+output pattern, so ̺ = 4 and ns = 8 different states are
+presented. While for MRx = 2 sequences of symbols are
+considered in terms of mapping three bits segments in four
+samples, such that ̺ = 8 with ns = 16 different states.
+Table I and Table II provide the equivalent Moore machine
+for MRx = 3 and MRx = 2, respectively. The states with
+positive subscripts represent sequences for ρ = 1 and states
+with negative subscripts represent sequences for ρ = −1.
+Considering a symmetric machine there are m = ̺MRx =
+12 different coefficients for MRx = 3. On the other hand,
+for MRx = 2 sequences of symbols are considered such that
+there are m = 2̺MRx = 32 different coefficients.
+The state transition probability matrix Q of the equivalent
+Moore machine, with dimensions ns ×ns is defined for i.i.d.
+input bits, all valid state transitions have equal probability
+p with p = 1/4 for MRx = 3 and p = 1/8 for MRx = 2.
+Furthermore, the vector π = (1/ns)1 of length ns corre-
+sponds to the stationary distribution of the equivalent Moore
+machine, which implies πT Q = πT . Then, the matrix Γ
+with dimensions ns × MRx for MRx = 3 and ns × 2MRx for
+MRx = 2 is defined which contains the Moore machine’s
+output gi. The block-wise correlation matrix of the TI ZX
+mapping output is given by [19, eq. 3.46]
+Rκ
+g = E{gκ′gT
+κ′+κ} = ΓT ΠQ|κ|Γ.
+(5)
+Then, the average autocorrelation function rg of the TI ZX
+modulation output sequence can be obtained as [19, eq. 3.39]
+rg[kq + l] = 1
+q
+
+
+q−l
+�
+i=1
+�
+Rk
+g
+�
+i,l+i +
+q
+�
+i=q−l+1
+�
+Rk+1
+g
+�
+i,l+i−q
+
+ ,
+(6)
+for k ∈ Z, 0 ≤ l ≤ q − 1.
+Table I: Equivalent Moore machine for TI ZX mapping for MRx = 3
+Current
+state
+next state
+output
+gi
+00
+01
+11
+10
+1+
+1+
+2+
+3+
+4+
+g1,1
+g1,2
+g1,3
+2+
+1−
+2−
+3−
+4−
+g2,1
+g2,2 − g2,3
+3+
+1−
+2−
+3−
+4−
+g3,1 − g3,2 − g3,3
+4+
+1−
+2−
+3−
+4−
+−g4,1 − g4,2 − g4,3
+4+
+1−
+2−
+3−
+4−
+−g4,1 − g4,2 − g4,3
+1−
+1−
+2−
+3−
+4−
+−g1,1 − g1,2 − g1,3
+2−
+1+
+2+
+3+
+4+
+−g2,1 − g2,2
+g2,3
+3−
+1+
+2+
+3+
+4+
+−g3,1
+g3,2
+g3,3
+4−
+1+
+2+
+3+
+4+
+g4,1
+g4,2
+g4,3
+
+Table II: Equivalent Moore machine for TI ZX mapping for MRx = 2
+Current
+state
+next state
+output
+gi
+000
+001
+011
+010
+110
+111
+101
+100
+1+
+1+
+2+
+3+
+4+
+5+
+6+
+7+
+8+
+g1,1
+g1,2
+g1,3
+g1,4
+2+
+1−
+2−
+3−
+4−
+5−
+6−
+7−
+8−
+g2,1
+g2,2
+g2,3 − g2,4
+3+
+1−
+2−
+3−
+4−
+5−
+6−
+7−
+8−
+g3,1
+g3,2 − g3,3 − g3,4
+4+
+1−
+2−
+3−
+4−
+5−
+6−
+7−
+8−
+g4,1 − g4,2 − g4,3 − g4,4
+5+
+1+
+2+
+3+
+4+
+5+
+6+
+7+
+8+
+g5,1 − g5,2 − g5,3
+g5,4
+6+
+1+
+2+
+3+
+4+
+5+
+6+
+7+
+8+
+−g6,1 − g6,2 − g6,3
+g6,4
+7+
+1−
+2−
+3−
+4−
+5−
+6−
+7−
+8−
+−g7,1 − g7,2 − g7,3 − g7,4
+8+
+1+
+2+
+3+
+4+
+5+
+6+
+7+
+8+
+−g8,1 − g8,2
+g8,3
+g8,4
+1−
+1−
+2−
+3−
+4−
+5−
+6−
+7−
+8−
+−g1,1 − g1,2 − g1,3 − g1,4
+2−
+1+
+2+
+3+
+4+
+5+
+6+
+7+
+8+
+−g2,1 − g2,2 − g2,3
+g2,4
+3−
+1+
+2+
+3+
+4+
+5+
+6+
+7+
+8+
+−g3,1 − g3,2
+g3,3
+g3,4
+4−
+1+
+2+
+3+
+4+
+5+
+6+
+7+
+8+
+−g4,1
+g4,2
+g4,3
+g4,4
+5−
+1−
+2−
+3−
+4−
+5−
+6−
+7−
+8−
+−g5,1
+g5,2
+g5,3 − g5,4
+6−
+1−
+2−
+3−
+4−
+5−
+6−
+7−
+8−
+g6,1
+g6,2
+g6,3 − g6,4
+7−
+1+
+2+
+3+
+4+
+5+
+6+
+7+
+8+
+g7,1
+g7,2
+g7,3
+g7,4
+8−
+1−
+2−
+3−
+4−
+5−
+6−
+7−
+8−
+g8,1
+g8,2 − g8,3 − g8,4
+IV-B. Waveform Design
+For a given set G, the autocorrelation function is calculated
+with (6). With this, the PSD is calculated by
+S(f) = Sx(f) |GTx(f)|2 ,
+(7)
+where GTx(f) corresponds to the transfer function of the
+transmit filter gTx and Sx(f) correspond to the PSD of the
+transmit sequence
+Sx(f) = MRx
+T
+∞
+�
+l=−∞
+clej2π
+lT
+MRx f,
+(8)
+where cl denotes the l-th element of the autocorrelation
+function from (6). By defining a critical frequency fc and a
+power containment factor η, the inband power is defined as
+� fc
+−fc
+S(f)df = ηP,
+(9)
+where P =
+� ∞
+−∞ S(f)df. Then, a non convex constrained
+optimization problem which maximizes the minimum dis-
+tance to the decision threshold γ can be formulated as:
+minimizegu
+− γ
+subject to
+gu ≻ γ1
+∥gu∥2 ≤ 1
+η ≥ 0.95.
+(10)
+In contrast to existing methods [12], [14], the optimization
+process is done only once at the BS regardless of the channel
+and input sequence. Therefore, the optimization process can
+be done offline by applying an exhaustive search. When the
+optimal set of coefficients G is obtained, the sequence sgk
+is constructed for each user.
+Finally, a power normalization is considered in order to
+satisfy a total power constraint in terms of E0. Considering
+that the power of the transmit signal sg is given by
+sgAHAsg = ETx,
+(11)
+Table III: Simulation parameters
+Method
+MRx
+Transmit Filter
+Receive filter
+Ib
+Os
+TI ZX MMDDT [14]
+2
+RC α = 0.22
+RRC α = 0.22
+45
+61
+3
+60
+91
+ZX transceiver design [7]
+3
+RC window α = 0.1
+Integrate-and-dump
+180
+270
+TI ZX waveform design
+2
+Integrator T/MRx
+Integrator T/MRx
+45
+60
+3
+60
+90
+where A = INu ⊗ GT
+Tx and GTx denotes a Toeplitz matrix
+of size Ntot × 3Ntot, which is given by
+GTx = aTx
+
+
+�
+gT
+Tx
+�
+0 · · ·
+0
+0
+�
+gT
+Tx
+�
+0 · · · 0
+... ... ...
+0 · · ·
+0
+�
+gT
+Tx
+�
+
+ ,
+(12)
+with aTx = (T/MTx)1/2 and gTx =
+�
+gTx(−T (N + M −1
+Tx )),
+gTx(−T (N + M −1
+Tx ) + T M −1
+Tx ), . . . , gTx(T (N + M −1
+Tx ))
+�T ,
+we scale and update the sequence sg as follows
+sg = sgnorm =
+1
+�
+ETx/E0
+sg.
+(13)
+IV-C. Detection
+The detection process for the proposed waveform, follows
+the same process as for the existing TI ZX waveforms
+which aims for a low complexity receiver [14], [15]. The
+detection process is done in the same way and separately
+for each user stream. From the sequence received in (1)
+the corresponding zk sequences of each user are obtained.
+The sequence zk is segmented into subsequences zbi =
+[ρi−1, zi]T ∈ {+1, −1}MRx+1, where ρi−1 corresponds to
+the last sample of zbi−1 which corresponds to the received
+sequence of the (i − 1) symbol interval. Then the backward
+mapping process is define such that
+⃗
+d : zbi → [ρi−1, cT
+si]
+[14], [15]. In the noise free case it is possible to decode the
+sequence with the backward mapping process
+⃗
+d(·). However,
+in the presence of noise, invalid sequences zbi may arise that
+are not possible to detect via the backward mapping process
+⃗
+d(·). Hence, the Hamming distance metric is required [12]
+which is defined as
+ˆxi =
+⃗
+d(c),
+with c = arg min
+cmap∈MHamming(zbi, cmap),
+(14)
+where Hamming (zbi, cmap) = �MRx+1
+n=1
+1
+2
+��zbi,n − cmap,n
+��
+and cmap = [ρi−1, csi]T , and M denotes all valid forward
+mapping codewords. The detection of the first symbol in
+the sequence, considers the sample ρb which then enables
+the detection process. The real and the imaginary parts are
+detected independently in separate processes.
+V. NUMERICAL RESULTS
+This section presents numerical BER results and normal-
+ized PSD for the proposed TI ZX state machine waveform
+design with power containment factor η = 0.95. Moreover,
+the proposed technique results are compared with other
+methods from the literature, namely TI ZX MMDDT [14]
+and ZX transceiver design [7]. The channel considers Nt = 8
+
+−10
+0
+10
+20
+30
+10−3
+10−1
+SNR [dB]
+BER
+MRx = 2
+MRx = 3
+−10
+0
+10
+20
+30
+10−3
+10−1
+SNR [dB]
+BER
+TI ZX MMDDT [14]
+proposed waveform
+random ZX transceiver [7]
+Golay ZX transceiver [7]
+0
+0.5
+1
+1.5
+−40
+−20
+0
+fT
+normalized PSD [dB]
+MMDDT [14]
+proposed waveform
+proposed waveform (7)
+random ZX transceiver [7]
+Golay ZX transceiver [7]
+Fig. 2: Numerical evaluations. In (a) BER vs SNR for the proposed waveform. In (b) BER vs SNR for MRx = 3 for all the considered methods. In (c)
+PSD for MRx = 3.
+transmit antennas and Nu = 2 single antenna users for all
+the evaluated methods. The SNR is defined as follows
+SNR = E0/(NT )
+N0B
+=
+E0
+NT N02fc
+,
+(15)
+where N0 denotes the noise power spectral density. The
+bandwidth B is define as B = 2fc, where the critical
+frequency is set to fc = 0.65/T . The entries of the channel
+matrix H are i.i.d. with CN(0, 1). For the proposed TI ZX
+state machine waveform design the normalized receive and
+transmit filters are integrators defined as
+gRx(t) = gTx(t) =
+�
+1
+T/MRx rect
+�
+t
+T/MRx
+�
+.
+(16)
+The presented results for the TI ZX MMDDT method from
+[14] considers MRx = 3 and the same data rate as for
+the proposed TI ZX state machine waveform design with
+gTx(t) as an RC filter and gRx(t) as an RRC filter with roll-
+off factors ǫTx = ǫRx = 0.22, where the critical frequency
+corresponds to fc = (1 + ǫTx)/2T . On the other hand, for
+the ZX transceiver design [7], MRx = 3 is considered for
+the random and the Golay mapping methods. The truncation
+interval is set to κ = 3 and the number of bits per
+subinterval n = 2, and at the receiver an integrate-and-
+dump-filter is considered, same as presented in the study
+[7]. Table III summarizes the simulation parameters for the
+proposed TI ZX waveform design and other methods from
+the literature, where Ib corresponds to the number of input
+bits per user and Os represents the number of samples after
+the mapping process. For all the considered methods the
+same normalization is done for the transmit signal.
+The optimal matrix G of positive coefficients is shown
+in Table IV and Table V for MRx = 2 and MRx = 3,
+respectively. The input sequences of symbols x are mapped
+onto the temporal transmit vector sg considering the set of
+coefficients in Table IV and Table V according to MRx. The
+numerical BER results for the proposed TI ZX state machine
+waveform design are presented in Fig. 2 (a) for MRx = 2
+and MRx = 3. As expected the BER for MRx = 2 is lower
+than for MRx = 3. In Fig. 2 (b) the BER is evaluated
+and compared with other methods form the literature for
+MRx = 3. The TI ZX MMDDT [14] and the proposed TI
+ZX state machine waveform design achieves approximately
+the same BER performance while the proposed TI ZX
+state machine waveform design has a lower computational
+complexity. In this context, the complexity order for the
+proposed state machine waveform design is dominated by
+the spatial ZF precoder whose complexity in Big O nation
+is given by O
+�
+N 3
+t
+�
+. This is because the coefficients are
+optimized only once for any transmit sequence of symbols.
+On the other hand, the complexity order for the TI ZX
+MMDDT [14] is given by O
+�
+2Nu(Ntot)3.5 + N 3
+t
+�
+. However,
+note that the proposed TI ZX state machine waveform design
+yields a low level of out-of-band-radiation as seen in Fig. 2
+(c). Additionally, the proposed method is compared with the
+transceiver design from [7]. The transceiver design method
+considers the nonuniform zero-crossing pattern with random
+and Golay mapping and power containment factor η = 0.95.
+It should be noted that the channel coding presented in the
+study [7] is not considered in the simulations.
+Simulation results are presented also in terms of the
+normalized PSD. In Fig. 2 (c) the analytical and numerical
+PSD are compared for the proposed TI ZX state machine
+waveform design with MRx = 3. The analytical PSD is
+calculated with (7) considering the autocorrelation function
+in (6). In Fig. 2 (c), the normalized PSD of the proposed
+waveform design is also compared with the normalized PSD
+of the methods from the literature which is calculated by
+PSDdB = 10log10
+�
+O(−1)
+s
+E{|Fi|2}
+�
+,
+(17)
+where Fi is the discrete Fourier transform of the normalized
+temporal transmit signal per user.
+VI. CONCLUSIONS
+In this study, we have developed a TI ZX state machine
+waveform based on the novel TI ZX modulation for multi-
+user MIMO downlink systems, with 1-bit quantization and
+oversampling. The waveform design considers the optimiza-
+tion of a set of coefficients which conveys the information
+into the time-instances of zero-crossings. The optimization
+
+Table IV: Optimal set G for MRx = 2
+G
+g1
+0.2719,
+0.3751,
+0.3715,
+0.2378
+g2
+0.2081,
+0.2129,
+0.1,
+0.1
+g3
+0.1719,
+0.1,
+0.1,
+0.1440
+g4
+0.1,
+0.1,
+0.1832,
+0.1572
+g5
+0.1,
+0.1,
+0.1,
+0.1
+g6
+0.1,
+0.2030,
+0.1,
+0.1
+g7
+0.1,
+0.2507,
+0.2551,
+0.1655
+g8
+0.1,
+0.1,
+0.1,
+0.1647
+Table V: Optimal set G for MRx = 3
+G
+g1
+0.4566,
+0.4809,
+0.4006
+g2
+0.2631,
+0.1,
+0.1014
+g3
+0.1334,
+0.1,
+0.2312
+g4
+0.1
+0.2875
+0.3692
+is performed considering the power containment bandwidth
+and the maximization of the minimum distance to the deci-
+sion threshold. The simulation results were compared with
+methods from the literature which employ techniques based
+on zero-crossings. The BER performance is favorable for the
+proposed method which achieves a comparable BER result
+as the TI ZX MMDDT [14] method but with significantly
+lower computational complexity.
+ACKNOWLEDGEMENTS
+This work has been supported by FAPERJ, the ELIOT
+ANR-18-CE40-0030 and FAPESP 2018/12579-7 project.
+VII. REFERENCES
+[1] R. H. Walden, “Analog-to-digital converter survey and
+analysis,” IEEE J. Sel. Areas Commun., vol. 17, no. 4,
+pp. 539–550, Apr. 1999.
+[2] S. Shamai (Shitz), “Information rates by oversampling
+the sign of a bandlimited process,” IEEE Trans. Inf.
+Theory, vol. 40, no. 4, pp. 1230–1236, July 1994.
+[3] L. Landau, M. D¨orpinghaus, and G. P. Fettweis, “1-bit
+quantization and oversampling at the receiver: Commu-
+nication over bandlimited channels with noise,” IEEE
+Commun. Lett., vol. 21, no. 5, pp. 1007–1010, 2017.
+[4] L. Landau, M. D¨orpinghaus, and G. P. Fettweis, “1-
+bit quantization and oversampling at the receiver:
+Sequence-based communication,” EURASIP J. Wirel.
+Commun. Netw., April 2018.
+[5] H. Son, H. Han, N. Kim, and H. Park, “An efficient
+detection algorithm for PAM with 1-bit quantization
+and time-oversampling at the receiver,” in IEEE Veh.
+Technol. Conf. (VTC2019-Fall), Honolulu, HI, USA,
+Sep 2019.
+[6] L. Landau, S. Krone, and G. P. Fettweis, “Intersymbol-
+interference design for maximum information rates
+with 1-bit quantization and oversampling at the re-
+ceiver,”
+in Proc. of the Int. ITG Conf. on Systems,
+Communications and Coding, Munich, Germany, Jan.
+2013.
+[7] R. Deng, J. Zhou, and W. Zhang, “Bandlimited commu-
+nication with one-bit quantization and oversampling:
+Transceiver design and performance evaluation,” IEEE
+Trans. Commun., vol. 69, no. 2, pp. 845–862, Feb.
+2021.
+[8] P. Neuhaus, M. D¨orpinghaus, H. Halbauer, S. Wese-
+mann, M. Schl¨uter, F. Gast, and G. Fettweis, “Sub-
+THz wideband system employing 1-bit quantization
+and temporal oversampling,” in Proc. IEEE Int. Conf.
+Commun. (ICC), Dublin, Ireland, Jun. 2020, pp. 1–7.
+[9] P. Neuhaus, M. D¨orpinghaus, H. Halbauer, V. Braun,
+and G. Fettweis, “On the spectral efficiency of over-
+sampled 1-bit quantized systems for wideband LOS
+channels,”
+in Proc. IEEE Int. Symp. on Personal,
+Indoor and Mobile Radio Commun. (PIMRC), London,
+UK, Aug. 2020, pp. 1–6.
+[10] A. B. ¨Uc¸¨unc¨u, E. Bj¨ornson, H. Johansson, A. Yılmaz,
+and E. G. Larsson,
+“Performance analysis of quan-
+tized uplink massive MIMO-OFDM with oversampling
+under adjacent channel interference,”
+IEEE Trans.
+Commun., vol. 68, no. 2, pp. 871–886, November 2020.
+[11] Z. Shao, L. T. N. Landau, and R. C. de Lamare,
+“Dynamic oversampling for 1-bit ADCs in large-scale
+multiple-antenna systems,” IEEE Trans. Commun., vol.
+69, no. 5, pp. 3423–3435, February 2021.
+[12] A. Gokceoglu, E. Bj¨ornson, E. G. Larsson, and
+M. Valkama,
+“Spatio-temporal waveform design for
+multiuser massive MIMO downlink with 1-bit re-
+ceivers,” IEEE J. Sel. Topics Signal Process., vol. 11,
+no. 2, pp. 347–362, March 2017.
+[13] P. Neuhaus, D. M. V. Melo, L. T. N. Landau, R. C.
+de Lamare, and G. P. Fettweis, “Zero-crossing mod-
+ulations for a multi-user MIMO downlink with 1-bit
+temporal oversampling ADCs,”
+in Proc. European
+Sign. Proc. Conf. (EUSIPCO), Dublin, Ireland, August
+2021.
+[14] D. M. V. Melo, L. T. N. Landau, and R. C. de Lamare,
+“Zero-crossing precoding with maximum distance to
+the decision threshold for channels with 1-bit quanti-
+zation and oversampling,”
+in Proc. IEEE Int. Conf.
+Acoust., Speech, Signal Process., Barcelona, Spain,
+May 2020, pp. 5120–5124.
+[15] D. M. V. Melo, L. T. N. Landau, and R. C. de Lamare,
+“Zero-crossing precoding with MMSE criterion for
+channels with 1-bit quantization and oversampling,” in
+Proc. of the Int. ITG Workshop on Smart Antennas,
+Hamburg, Germany, Feb. 2020.
+[16] D. M. V. Melo, L. T. N. Landau, L. N. Ribeiro,
+and M. Haardt,
+“Iterative MMSE space-time zero-
+crossing precoding for channels with 1-bit quantization
+and oversampling,” in 54th Asilomar Conference on
+Signals, Systems, and Computers, Pacific Grove, CA,
+USA, Nov. 2020, pp. 496–500.
+[17] D. M. V. Melo, L. T. N. Landau, L. N. Ribeiro, and
+M. Haardt,
+“Time-instance zero-crossing precoding
+with quality-of-service constraints,” in IEEE Statistical
+Signal Processing Workshop, SSP 2021, Rio de Janeiro,
+
+Brazil, July 2021.
+[18] P. Neuhaus, M. D¨orpinghaus, and G. Fettweis, “Zero-
+crossing modulation for wideband systems employ-
+ing 1-bit quantization and temporal oversampling:
+Transceiver design and performance evaluation,” IEEE
+Open J. Commun. Soc., vol. 2, pp. 1915–1934, 2021.
+[19] K. A. S. Immink,
+Codes for Mass Data Storage
+Systems, Eindhoven, The Netherlands: Shannon Found,
+2004.
+
diff --git a/UdFLT4oBgHgl3EQfRC8l/content/tmp_files/load_file.txt b/UdFLT4oBgHgl3EQfRC8l/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c84f83baf962887360a40d1646dd18e50fb8d466
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@@ -0,0 +1,574 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf,len=573
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='12035v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='SP]  28 Jan 2023  STATE MACHINE-BASED WAVEFORMS FOR CHANNELS WITH 1-BIT  QUANTIZATION AND OVERSAMPLING WITH TIME-INSTANCE  ZERO-CROSSING MODULATION  Diana M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Melo, Lukas T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Landau and Rodrigo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' de Lamare  Centre for Telecommunications Studies,  Pontifical Catholic University of Rio de Janeiro,  Rio de Janeiro, Brazil 22453-900  Email: diana;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='lukas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='landau;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='delamare@cetuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='puc-rio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='br  ABSTRACT  Systems with 1-bit quantization and oversampling are  promising for the Internet of Things (IoT) devices in order  to reduce the power consumption of the analog-to-digital-  converters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The novel time-instance zero-crossing (TI ZX)  modulation is a promising approach for this kind of channels  but existing studies rely on optimization problems with  high computational complexity and delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' In this work, we  propose a practical waveform design based on the estab-  lished TI ZX modulation for a multiuser multi-input multi-  output (MIMO) downlink scenario with 1-bit quantization  and temporal oversampling at the receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' In this sense,  the proposed temporal transmit signals are constructed by  concatenating segments of coefficients which convey the  information into the time-instances of zero-crossings accord-  ing to the TI ZX mapping rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The proposed waveform  design is compared with other methods from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  The methods are compared in terms of bit error rate and  normalized power spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Numerical results show  that the proposed technique is suitable for multiuser MIMO  system with 1-bit quantization while tolerating some small  amount of out-of-band radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Index Terms— Zero-crossing precoding, oversampling,  Moore machine, 1-bit quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' INTRODUCTION  Future wireless communication technologies are envi-  sioned to support a large number of the Internet of Things  (IoT) devices which require to have low power consumption  and low complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Low resolution analog-to-digital con-  verters (ADCs) are suitable to meet the IoT requirements  since the power consumption in the ADCs increase expo-  nentially with its amplitude resolution [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The loss of infor-  mation caused by the coarse quantization can be partially  compensated by increasing the sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Employing  temporal MRx-fold oversampling, rates of log2(MRx + 1)  bits per Nyquist interval are achievable in a noise free envi-  ronment [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The authors in [3] study the maximization of the  achievable rate for systems with 1-bit quantization and over-  sampling in the presence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Other studies that consider  systems with 1-bit quantization and oversampling employ  ASK transmit sequences [4], [5] and 16 QAM modulation  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Other practical methods are based on the idea presented  in [2], where the information is conveyed into the zero-  crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' An example is the study presented in [7], where  the waveform is constructed by concatenating sequences  which convey the information into the zero-crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' This  study shows that similar data rates to the one presented  in [2] can be achieved over noisy channels with relatively  low out-of-band radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Some other practical methods  which convey the information into the zero-crossings include  runlength-limited (RLL) sequences [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  The benefits of 1-bit quantization and oversampling have  been studied in [10], [11] for multiple-input multiple-output  (MIMO) channels in uplink transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Moreover, the  studies [12], [13] investigate sequences for downlink MIMO  systems with 1-bit quantization and oversampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' In this  regard, in [12] it is presented the quantization precoding  method which considers as optimization criterion the maxi-  mization of the minimum distance to the decision threshold  (MMDDT) which was proposed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The quantization  precoding technique relies on an exhaustive codebook search  which allows simple Hamming distance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Superior  precoding schemes for MIMO downlink scenarios have been  investigated in [14], [15], where a novel time-instance zero-  crossing (TI ZX) modulation is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' This novel mod-  ulation follows the idea of [2] by allocating the information  into the time-instance of zero-crossings in order to reduce  the number of zero-crossings of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The study in  [14] relies on a precoding technique based on the MMDDT  criterion with spatial zero-forcing (ZF) precoding and TI  ZX modulation, whereas [15] proposes an optimal temporal-  spatial precoding technique with TI ZX modulation along  with an minimum mean square error (MMSE) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Other studies that consider novel TI ZX modulation schemes  have been presented in [13], [16], [17] where the computa-  tional complexity is reduced [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' In [17] the minimization  of the transmit power under quality of service constraint is  considered as an objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The study in [13] investigates  the spectral efficiency of MIMO systems with sequences  constructed with the TI ZX modulation and RLL sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  In this work, we propose a TI ZX waveform design  for multiuser MIMO downlink channels with 1-bit quan-  tization and oversampling where a defined level of out-of-  band radiation is tolerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The proposed waveform design  considers the novel TI ZX modulation from [14], [15] and  follows a similar idea as presented in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The proposed  method conveys the information into the time-instances  of zero-crossings but instead of considering sequences of  samples, input bits are mapped into waveform segments  according to the TI ZX mapping rules [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The  temporal precoding vector is then used in conjunction with  a simple pulse shaping filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The optimal set of coeffi-  cients is computed with an optimization problem which  is formulated to maximize the minimum distance to the  decision threshold, constrained with some tolerated out-of-  band radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Finally, the numerical results are evaluated  considering the bit error rate (BER) and the power spectral  density (PSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The proposed waveform design is compared  with the transceiver waveform design from [7] and the TI ZX  MMDDT precoding [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The transceiver waveform design  [7] was adapted for MIMO channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The simulation results  show that the proposed waveform design is comparable in  terms of BER performance to the one presented for TI ZX  MMDDT precoding while having a lower computational  complexity since the waveform optimization is done once  and is suitable for any input sequence of bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  The rest of the paper is organized as follows: The sys-  tem model is introduced in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Then, Section III  describes the novel TI ZX modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Section IV explains  the proposed waveform design optimization including the  autocorrelation function for TI ZX modulated sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  The simulation results are provided in Section V and finally,  the conclusions are given in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Notation: In the paper all scalar values, vectors and  matrices are represented by: a, x and X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' SYSTEM MODEL  In this study, a multiuser MIMO downlink scenario with  Nu single antenna users and Nt transmit antennas at the  base station (BS), is considered as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Trans-  mission blocks of N symbols (N Nyquist intervals) are  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The input sequences of symbols xk are mapped  using the TI ZX mapping and the set of coefficients G  which yields the temporal precoding vector sgk ∈ CMRxN,  where MRx/T denotes the sampling rate and T refers to  the symbol duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Moreover, the transmit filter gTx(t)  and receive filter gRx(t) are presented, where the combined  waveform is given by v(t) = (gTx ∗ gRx) (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Furthermore  1-bit quantization is applied at the receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The channel  x 1  x Nu  Spatial  Precoder  DAC  s1(t)  DAC  gTx(t)  gTx(t) sNt(t)  sx1  sxNt  Partitioning  sx  H  sg  1-bit ADC  Detector  schk(t) + nk(t)  �x k  y k  z k  Sampling rate MRx/T  Q(·)  gRx(t)  Mapper  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 1: Considered multi-user MIMO downlink system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  matrix H ∈ CNu×Nt is known at the base station and is  considered to be frequency-flat fading as typically assumed  for narrowband IoT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Then, with the stacked temporal  precoding vector sg =  �  sT  g1, sT  g2, · · · , sT  gk, · · · , sT  gNu  �T  , the  received signal z ∈ C3NtotNu can be expressed by stacking  the received samples of the Nu users as follows:  z = Q1 ((HP sp ⊗ INtot) (INt ⊗ V ) sg + (INu ⊗ GRx) n)  = Q1 (Heffsg + GRx,effn) ,  (1)  where Q1(·) corresponds the 1-bit quantization operator,  n ∈ C3NtotNu denotes a vector with zero-mean complex  Gaussian noise samples with variance σ2  n with Ntot  =  NMRx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The waveform matrix V with size Ntot × Ntot is  given by  V =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  v (0)  v  �  T  MRx  �  · ·  v (T N)  v  �  −  T  MRx  �  v (0)  · ·  v  �  T  �  N −  1  MRx  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  v (−T N)  v  �  T  �  −N +  1  MRx  ��  · ·  v (0)  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  (2)  The receive filter gRx is represented in discrete time by the  matrix GRx with size Ntot × 3Ntot and is denoted as  GRx = aRx  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  �  gT  Rx  �  0 · · ·  0  0  �  gT  Rx  �  0 · · · 0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  0 · · ·  0  �  gT  Rx  �  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb ,  (3)  with gRx = [gRx(−T (N +  1  MRx )), gRx(−T (N +  1  MRx ) +  T  MRx ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' , gRx(T (N +  1  MRx ))]T and aRx = (T/MRx)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The  matrix P sp = czfHH �  HHH�−1 denotes the spatial zero-  forcing precoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The matrix P sp is normalized such that  the spatial precoder does not change the signal power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' As in  [14] the normalization factor czf is given by  czf =  �  Nu/trace  ��  HHH�−1�� 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  (4)  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' TIME-INSTANCE ZERO-CROSSING MAPPING  The TI ZX modulation was proposed in the studies [14]  and [15] for systems with 1-bit quantization and over-  sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The TI ZX modulation conveys the information  into the time-instances of zero-crossings and also considers  the absence of zero-crossing during a symbol interval as  a valid symbol, different to other approaches from the  literature [2] and [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' To build the mapped sequence, each  symbol xi drawn from the set Xin := {b1, b2, · · · , bRin}  with Rin = MRx + 1, is mapped into a binary codeword  csi with MRx samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' As mentioned, one of the possible  symbols corresponds to the pattern that does not contain a  zero-crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The mapping depends on the last sample of  the previous symbol interval, namely ρ ∈ {1, −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Hence,  the TI ZX mapping provides two possible codewords csi for  each valid symbol xi which convey the same zero-crossing  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Then, for coding and decoding of the first trans-  mit symbol, a pilot sample ρb ∈ {1, −1} is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Finally,  the desired output pattern coutk is obtained by concatenating  the segments csi such that, coutk = [ρb, cT  s0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' , cT  sN−1]T with  total length NMRx + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' WAVEFORM DESIGN OPTIMIZATION  The proposed waveform design, suitable for systems  with 1-bit quantization and oversampling, considers the  novel TI ZX modulation [14], [15], in conjunction with  the optimization of a set of coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The proposed  waveform is built by concatenating segment sequences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=',  subsequences, described by the coefficients which contain  zero-crossings at the desired time-instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The proposed  waveform design relies on the transmit and receive filters  gTx(t) and gRx(t) which preserve the zero-crossing time-  instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Different to prior studies [14], [15], the sequence  is no longer binary but is defined by the set of coefficients  G, so that each symbol xi drawn from the set Xin is mapped  into a codeword gi with MRx different coefficients which  convey the information into the time-instances of zero-  crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The set of coefficients G is defined in terms of  G = {G+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' G−} where G− = −G+, such that they both  convey the same zero-crossing information and the sign  information of the coefficients depends on the last sample  of the previous interval termed ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Considering bit sequences  as input and the Gray coding for TI ZX modulation shown  in [14, Table II], ns = 2̺ different states can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  In this context, the set G =  �  gT  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' gT  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' gT  ̺  �  is presented,  where gi = [gi,1, gi,2, · · · gi,q] and ρ = sgn (gi,MRx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Then, as  initially established, the symbol xi is mapped in the segment  gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The pilot sample ρb is required for the encoding and  decoding processes of the first symbol x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Finally, the input  sequence of symbols xk is mapped in the sequence sgk with  length Ntot by concatenating all the segments gi such that,  sgk = [gT  0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' , gT  N−1]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Nothe that the pilot sample ρb is  predefined and known at the receivers, hence not included  in the precoding vector sgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Autocorrelation for TI ZX Modulation  In this section, it is described how to compute the autocor-  relation function of the TI ZX modulated signal, considering  the set of coefficients G which conveys the information into  the time-instances of zero-crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  To obtain the autocorrelation function, the TI ZX modula-  tion system is converted to a finite-state machine where the  current output values are determined only by its current state  which corresponds to an equivalent Moore machine [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' For  MRx = 3, one symbol in terms of two bits is mapped in one  output pattern, so ̺ = 4 and ns = 8 different states are  presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' While for MRx = 2 sequences of symbols are  considered in terms of mapping three bits segments in four  samples, such that ̺ = 8 with ns = 16 different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Table I and Table II provide the equivalent Moore machine  for MRx = 3 and MRx = 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The states with  positive subscripts represent sequences for ρ = 1 and states  with negative subscripts represent sequences for ρ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Considering a symmetric machine there are m = ̺MRx =  12 different coefficients for MRx = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' On the other hand,  for MRx = 2 sequences of symbols are considered such that  there are m = 2̺MRx = 32 different coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  The state transition probability matrix Q of the equivalent  Moore machine, with dimensions ns ×ns is defined for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  input bits, all valid state transitions have equal probability  p with p = 1/4 for MRx = 3 and p = 1/8 for MRx = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Furthermore, the vector π = (1/ns)1 of length ns corre-  sponds to the stationary distribution of the equivalent Moore  machine, which implies πT Q = πT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Then, the matrix Γ  with dimensions ns × MRx for MRx = 3 and ns × 2MRx for  MRx = 2 is defined which contains the Moore machine’s  output gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The block-wise correlation matrix of the TI ZX  mapping output is given by [19, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='46]  Rκ  g = E{gκ′gT  κ′+κ} = ΓT ΠQ|κ|Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  (5)  Then, the average autocorrelation function rg of the TI ZX  modulation output sequence can be obtained as [19, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='39]  rg[kq + l] = 1  q  \uf8eb  \uf8ed  q−l  �  i=1  �  Rk  g  �  i,l+i +  q  �  i=q−l+1  �  Rk+1  g  �  i,l+i−q  \uf8f6  \uf8f8 ,  (6)  for k ∈ Z, 0 ≤ l ≤ q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Table I: Equivalent Moore machine for TI ZX mapping for MRx = 3  Current  state  next state  output  gi  00  01  11  10  1+  1+  2+  3+  4+  g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1  g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2  g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  2+  1−  2−  3−  4−  g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1  g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2 − g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  3+  1−  2−  3−  4−  g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1 − g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2 − g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  4+  1−  2−  3−  4−  −g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1 − g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2 − g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  4+  1−  2−  3−  4−  −g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1 − g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2 − g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  1−  1−  2−  3−  4−  −g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1 − g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2 − g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  2−  1+  2+  3+  4+  −g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1 − g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2  g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  3−  1+  2+  3+  4+  −g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1  g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2  g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  4−  1+  2+  3+  4+  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  Table II: Equivalent Moore machine for TI ZX mapping for MRx = 2  Current  state  next state  output  gi  000  001  011  010  110  111  101  100  1+  1+  2+  3+  4+  5+  6+  7+  8+  g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1  g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2  g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='4  2+  1−  2−  3−  4−  5−  6−  7−  8−  g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1  g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2  g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3 − g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='4  3+  1−  2−  3−  4−  5−  6−  7−  8−  g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1  g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2 − g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3 − g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='4  4+  1−  2−  3−  4−  5−  6−  7−  8−  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1 − g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2 − g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3 − g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='4  5+  1+  2+  3+  4+  5+  6+  7+  8+  g5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1 − g5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2 − g5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  g5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='4  6+  1+  2+  3+  4+  5+  6+  7+  8+  −g6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1 − g6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2 − g6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  g6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='4  7+  1−  2−  3−  4−  5−  6−  7−  8−  −g7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1 − g7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2 − g7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3 − g7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='4  8+  1+  2+  3+  4+  5+  6+  7+  8+  −g8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1 − g8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2  g8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
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+page_content='4  1−  1−  2−  3−  4−  5−  6−  7−  8−  −g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1 − g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2 − g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3 − g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
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+page_content='1 − g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
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+page_content='1 − g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
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+page_content='2  g7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3  g7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='4  8−  1−  2−  3−  4−  5−  6−  7−  8−  g8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1  g8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='2 − g8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='3 − g8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='4  IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Waveform Design  For a given set G, the autocorrelation function is calculated  with (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' With this, the PSD is calculated by  S(f) = Sx(f) |GTx(f)|2 ,  (7)  where GTx(f) corresponds to the transfer function of the  transmit filter gTx and Sx(f) correspond to the PSD of the  transmit sequence  Sx(f) = MRx  T  ∞  �  l=−∞  clej2π  lT  MRx f,  (8)  where cl denotes the l-th element of the autocorrelation  function from (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' By defining a critical frequency fc and a  power containment factor η, the inband power is defined as  � fc  −fc  S(f)df = ηP,  (9)  where P =  � ∞  −∞ S(f)df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Then, a non convex constrained  optimization problem which maximizes the minimum dis-  tance to the decision threshold γ can be formulated as:  minimizegu  − γ  subject to  gu ≻ γ1  ∥gu∥2 ≤ 1  η ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  (10)  In contrast to existing methods [12], [14], the optimization  process is done only once at the BS regardless of the channel  and input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Therefore, the optimization process can  be done offline by applying an exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' When the  optimal set of coefficients G is obtained, the sequence sgk  is constructed for each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Finally, a power normalization is considered in order to  satisfy a total power constraint in terms of E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Considering  that the power of the transmit signal sg is given by  sgAHAsg = ETx,  (11)  Table III: Simulation parameters  Method  MRx  Transmit Filter  Receive filter  Ib  Os  TI ZX MMDDT [14]  2  RC α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='22  RRC α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='22  45  61  3  60  91  ZX transceiver design [7]  3  RC window α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='1  Integrate-and-dump  180  270  TI ZX waveform design  2  Integrator T/MRx  Integrator T/MRx  45  60  3  60  90  where A = INu ⊗ GT  Tx and GTx denotes a Toeplitz matrix  of size Ntot × 3Ntot, which is given by  GTx = aTx  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  �  gT  Tx  �  0 · · ·  0  0  �  gT  Tx  �  0 · · · 0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  0 · · ·  0  �  gT  Tx  �  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb ,  (12)  with aTx = (T/MTx)1/2 and gTx =  �  gTx(−T (N + M −1  Tx )),  gTx(−T (N + M −1  Tx ) + T M −1  Tx ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' , gTx(T (N + M −1  Tx ))  �T ,  we scale and update the sequence sg as follows  sg = sgnorm =  1  �  ETx/E0  sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  (13)  IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Detection  The detection process for the proposed waveform, follows  the same process as for the existing TI ZX waveforms  which aims for a low complexity receiver [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The  detection process is done in the same way and separately  for each user stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' From the sequence received in (1)  the corresponding zk sequences of each user are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  The sequence zk is segmented into subsequences zbi =  [ρi−1, zi]T ∈ {+1, −1}MRx+1, where ρi−1 corresponds to  the last sample of zbi−1 which corresponds to the received  sequence of the (i − 1) symbol interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Then the backward  mapping process is define such that  ⃗  d : zbi → [ρi−1, cT  si]  [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' In the noise free case it is possible to decode the  sequence with the backward mapping process  ⃗  d(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' However,  in the presence of noise, invalid sequences zbi may arise that  are not possible to detect via the backward mapping process  ⃗  d(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Hence, the Hamming distance metric is required [12]  which is defined as  ˆxi =  ⃗  d(c),  with c = arg min  cmap∈MHamming(zbi, cmap),  (14)  where Hamming (zbi, cmap) = �MRx+1  n=1  1  2  ��zbi,n − cmap,n  ��  and cmap = [ρi−1, csi]T , and M denotes all valid forward  mapping codewords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The detection of the first symbol in  the sequence, considers the sample ρb which then enables  the detection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The real and the imaginary parts are  detected independently in separate processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' NUMERICAL RESULTS  This section presents numerical BER results and normal-  ized PSD for the proposed TI ZX state machine waveform  design with power containment factor η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Moreover,  the proposed technique results are compared with other  methods from the literature, namely TI ZX MMDDT [14]  and ZX transceiver design [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The channel considers Nt = 8  −10  0  10  20  30  10−3  10−1  SNR [dB]  BER  MRx = 2  MRx = 3  −10  0  10  20  30  10−3  10−1  SNR [dB]  BER  TI ZX MMDDT [14]  proposed waveform  random ZX transceiver [7]  Golay ZX transceiver [7]  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='5  −40  −20  0  fT  normalized PSD [dB]  MMDDT [14]  proposed waveform  proposed waveform (7)  random ZX transceiver [7]  Golay ZX transceiver [7]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2: Numerical evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' In (a) BER vs SNR for the proposed waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' In (b) BER vs SNR for MRx = 3 for all the considered methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' In (c)  PSD for MRx = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  transmit antennas and Nu = 2 single antenna users for all  the evaluated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The SNR is defined as follows  SNR = E0/(NT )  N0B  =  E0  NT N02fc  ,  (15)  where N0 denotes the noise power spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The  bandwidth B is define as B = 2fc, where the critical  frequency is set to fc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='65/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The entries of the channel  matrix H are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' with CN(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' For the proposed TI ZX  state machine waveform design the normalized receive and  transmit filters are integrators defined as  gRx(t) = gTx(t) =  �  1  T/MRx rect  �  t  T/MRx  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  (16)  The presented results for the TI ZX MMDDT method from  [14] considers MRx = 3 and the same data rate as for  the proposed TI ZX state machine waveform design with  gTx(t) as an RC filter and gRx(t) as an RRC filter with roll-  off factors ǫTx = ǫRx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='22, where the critical frequency  corresponds to fc = (1 + ǫTx)/2T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' On the other hand, for  the ZX transceiver design [7], MRx = 3 is considered for  the random and the Golay mapping methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The truncation  interval is set to κ = 3 and the number of bits per  subinterval n = 2, and at the receiver an integrate-and-  dump-filter is considered, same as presented in the study  [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Table III summarizes the simulation parameters for the  proposed TI ZX waveform design and other methods from  the literature, where Ib corresponds to the number of input  bits per user and Os represents the number of samples after  the mapping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' For all the considered methods the  same normalization is done for the transmit signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  The optimal matrix G of positive coefficients is shown  in Table IV and Table V for MRx = 2 and MRx = 3,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The input sequences of symbols x are mapped  onto the temporal transmit vector sg considering the set of  coefficients in Table IV and Table V according to MRx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The  numerical BER results for the proposed TI ZX state machine  waveform design are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2 (a) for MRx = 2  and MRx = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' As expected the BER for MRx = 2 is lower  than for MRx = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2 (b) the BER is evaluated  and compared with other methods form the literature for  MRx = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The TI ZX MMDDT [14] and the proposed TI  ZX state machine waveform design achieves approximately  the same BER performance while the proposed TI ZX  state machine waveform design has a lower computational  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' In this context, the complexity order for the  proposed state machine waveform design is dominated by  the spatial ZF precoder whose complexity in Big O nation  is given by O  �  N 3  t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' This is because the coefficients are  optimized only once for any transmit sequence of symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  On the other hand, the complexity order for the TI ZX  MMDDT [14] is given by O  �  2Nu(Ntot)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='5 + N 3  t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' However,  note that the proposed TI ZX state machine waveform design  yields a low level of out-of-band-radiation as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2  (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Additionally, the proposed method is compared with the  transceiver design from [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The transceiver design method  considers the nonuniform zero-crossing pattern with random  and Golay mapping and power containment factor η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  It should be noted that the channel coding presented in the  study [7] is not considered in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Simulation results are presented also in terms of the  normalized PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2 (c) the analytical and numerical  PSD are compared for the proposed TI ZX state machine  waveform design with MRx = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The analytical PSD is  calculated with (7) considering the autocorrelation function  in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2 (c), the normalized PSD of the proposed  waveform design is also compared with the normalized PSD  of the methods from the literature which is calculated by  PSDdB = 10log10  �  O(−1)  s  E{|Fi|2}  �  ,  (17)  where Fi is the discrete Fourier transform of the normalized  temporal transmit signal per user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' CONCLUSIONS  In this study, we have developed a TI ZX state machine  waveform based on the novel TI ZX modulation for multi-  user MIMO downlink systems, with 1-bit quantization and  oversampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
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+page_content='3692  is performed considering the power containment bandwidth  and the maximization of the minimum distance to the deci-  sion threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The simulation results were compared with  methods from the literature which employ techniques based  on zero-crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' The BER performance is favorable for the  proposed method which achieves a comparable BER result  as the TI ZX MMDDT [14] method but with significantly  lower computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This work has been supported by FAPERJ, the ELIOT  ANR-18-CE40-0030 and FAPESP 2018/12579-7 project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' REFERENCES  [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Walden, “Analog-to-digital converter survey and  analysis,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Areas Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 4,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 539–550, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Shamai (Shitz), “Information rates by oversampling  the sign of a bandlimited process,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 1230–1236, July 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Landau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' D¨orpinghaus, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Fettweis, “1-bit  quantization and oversampling at the receiver: Commu-  nication over bandlimited channels with noise,” IEEE  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 1007–1010, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Landau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' D¨orpinghaus, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Fettweis, “1-  bit quantization and oversampling at the receiver:  Sequence-based communication,” EURASIP J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Wirel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=', April 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Son, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Han, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Kim, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Park, “An efficient  detection algorithm for PAM with 1-bit quantization  and time-oversampling at the receiver,” in IEEE Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' (VTC2019-Fall), Honolulu, HI, USA,  Sep 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [6] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Landau, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Krone, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Fettweis, “Intersymbol-  interference design for maximum information rates  with 1-bit quantization and oversampling at the re-  ceiver,”  in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' of the Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' ITG Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' on Systems,  Communications and Coding, Munich, Germany, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Deng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Zhou, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Zhang, “Bandlimited commu-  nication with one-bit quantization and oversampling:  Transceiver design and performance evaluation,” IEEE  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 845–862, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Neuhaus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' D¨orpinghaus, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Halbauer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Wese-  mann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Schl¨uter, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Gast, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Fettweis, “Sub-  THz wideband system employing 1-bit quantization  and temporal oversampling,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' (ICC), Dublin, Ireland, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Neuhaus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' D¨orpinghaus, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Halbauer, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Braun,  and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Fettweis, “On the spectral efficiency of over-  sampled 1-bit quantized systems for wideband LOS  channels,”  in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' on Personal,  Indoor and Mobile Radio Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' (PIMRC), London,  UK, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' ¨Uc¸¨unc¨u, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Bj¨ornson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Johansson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Yılmaz,  and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Larsson,  “Performance analysis of quan-  tized uplink massive MIMO-OFDM with oversampling  under adjacent channel interference,”  IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 68, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 871–886, November 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [11] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Shao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Landau, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' de Lamare,  “Dynamic oversampling for 1-bit ADCs in large-scale  multiple-antenna systems,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 3423–3435, February 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Gokceoglu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Bj¨ornson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Larsson, and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Valkama,  “Spatio-temporal waveform design for  multiuser massive MIMO downlink with 1-bit re-  ceivers,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Topics Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 11,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 347–362, March 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Neuhaus, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Melo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Landau, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  de Lamare, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Fettweis, “Zero-crossing mod-  ulations for a multi-user MIMO downlink with 1-bit  temporal oversampling ADCs,”  in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' European  Sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' (EUSIPCO), Dublin, Ireland, August  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Melo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Landau, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' de Lamare,  “Zero-crossing precoding with maximum distance to  the decision threshold for channels with 1-bit quanti-  zation and oversampling,”  in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  Acoust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=', Speech, Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=', Barcelona, Spain,  May 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 5120–5124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Melo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Landau, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' de Lamare,  “Zero-crossing precoding with MMSE criterion for  channels with 1-bit quantization and oversampling,” in  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' of the Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' ITG Workshop on Smart Antennas,  Hamburg, Germany, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Melo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Landau, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Ribeiro,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Haardt,  “Iterative MMSE space-time zero-  crossing precoding for channels with 1-bit quantization  and oversampling,” in 54th Asilomar Conference on  Signals, Systems, and Computers, Pacific Grove, CA,  USA, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 496–500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Melo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Landau, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Ribeiro, and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Haardt,  “Time-instance zero-crossing precoding  with quality-of-service constraints,” in IEEE Statistical  Signal Processing Workshop, SSP 2021, Rio de Janeiro,  Brazil, July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Neuhaus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' D¨orpinghaus, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Fettweis, “Zero-  crossing modulation for wideband systems employ-  ing 1-bit quantization and temporal oversampling:  Transceiver design and performance evaluation,” IEEE  Open J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' 1915–1934, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content='  [19] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
+page_content=' Immink,  Codes for Mass Data Storage  Systems, Eindhoven, The Netherlands: Shannon Found,  2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFLT4oBgHgl3EQfRC8l/content/2301.12035v1.pdf'}
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+Universal degeneration of Riemann surfaces
+and fibered complex surfaces
+Tadashi Ashikaga and Yukio Matsumoto
+Dedicated to Professor Norbert A’Campo on his 80th birthday
+Contents
+§1 Introduction
+§1.1 Precise orbifold structure of M
+orb
+g
+§1.2 The results of the present paper
+§2 Mapping class groups of stable curves
+§2.1 Lifting of stable curves and Fenchel-Nielsen twist at infinity
+§2.2 Obstruction to lifting homeomorphisms
+§2.3 Weyl groups as mapping class groups
+§3 Kuranshi families of stable curves and log structures
+§3.1 Deformations of stable curves and Kuranishi families
+§3.2 Log structure and real blow up of Kuranishi families
+§4 Construction of the universal degenerating family
+§4.1 Weyl marking and controlled deformation spaces
+§4.2 Kuranishi families over controlled deformation spaces
+§4.3 Orbifold fiber space over the Deligne-Mumford compactification
+§5 Automorphisms of stable curves and cyclic equisymmetric strata on M
+orb
+g
+§5.1 Automorphisms of Riemann surfaces and equisymmetric strata
+§5.2 Logarithmic quadratic representation of automorphisms
+§5.3 Little Teichm¨uller space in an orbifold chart of M
+orb
+g
+§5.4 Automorphisms and cyclic branched coverings of stable curves
+§5.5 Equisymmetric strata at the boundary charts of M
+orb
+g
+§5.6 Harris-Mumford coordinates around equisymmmetric strata
+§6 Monodromy and orbifold moduli maps of degenerations of Riemann surfaces
+§6.1 Pseudo-periodic maps and automorphisms of stable curves
+§6.2 Orbifold structures of degenerations of Riemann surfaces
+§6.3 Local orbifold moduli maps and Kodaira-periodicity
+§6.4 Examples of degenerations and their invariants
+§7 Recovery of fibered complex surfaces from the universal degenerating family
+§7.1 Local recovery of degenerations from the universal family
+§7.2 Global orbifold moduli maps for fibered complex surfaces
+§7.3 Global recovery of basic members of fibered complex surfaces
+References
+1
+arXiv:2301.00381v1  [math.AG]  1 Jan 2023
+
+1
+Introduction
+Kodaira [46] constructed elliptic surfaces in a canonical way for given monodromies and
+J-invariants, which he called the basic members of elliptic surfaces. Our aim is to extend
+this result to fibered complex surfaces of genus g ≥ 2.
+In our previous paper [54], we constructed a new orbifold structure M
+orb
+g
+over the
+Deligne–Mumford compactification by using a certain bordification of Teichm¨uller space.
+In this paper, we construct an orbifold fiber space
+π : Y
+orb
+g
+→ M
+orb
+g
+such that any fibered complex surface admitting unstable fibers can be pulled back from π
+by the orbifold moduli map Jorb which we will construct. The fiber space π : Y
+orb
+g
+→ M
+orb
+g
+will be constructed by patching Kuranishi families of stable curves. The map Jorb has a
+nature similar to Kodaira’s J-invariant in the sense that it is delicately influenced by the
+monodromy and has the property of pseudo-periodicity which we will explain below.
+In §1.1, we will fix the notation from Teichm¨uller theory, and will review the structure
+of M
+orb
+g . Then we will state the results of the present paper in §1.2.
+1.1
+Precise orbifold structure of M
+orb
+g
+Throughout the paper, we will fix a closed oriented topological surface Σg of genus g(> 1).
+A Riemann surface S is usually considered to be a marked Riemann surface (S, w), in other
+words, a Riemann surface for which an orientation-preserving homeomophism w : Σg → S
+(called a marking) is fixed. Two marked surfaces (S1, w1) and (S2, w2) are equivalent if
+there exists a holomorphic map f : S1 → S2 such that f ◦ w1 ≃ w2, where ≃ means “is
+isotopic to” (or equivalently, in the case of closed surfaces, “is homotopic to”). The set of
+equivalence classes is nothing but the Teichm¨uller space modeled on Σg, and it is denoted
+by Tg. By Teichm¨uller’s theorem, Tg is a metric space homeomorphic to R6g−6 (see [41],
+Chapter 5).
+Teichm¨uller [70],[69] constructed the space Tg and the universal curve on it. He found
+that Tg is a complex manifold of dimension 3g − 3. Grothendieck [28] had a deep insight
+into Tg from functorial viewpoint. For a review of Teichm¨uller–Grothendieck theory, see
+[4], [3].
+Ahlfors–Bers [5], [6] rediscovered the complex structure on Tg from analytic
+viewpoint.
+The mapping class group of genus g, Γg, is defined to be
+Γg = {ϕ : Σg → Σg | orientation preserving homeomorphisms}/ ≃ .
+The group structure of Γg is given by the usual composition of maps: for [ϕ], [ψ] ∈ Γg, we
+have [ϕ][ψ] = [ϕ ◦ ψ]. The mapping class group Γg acts on the Teichm¨uller space Tg by
+2
+
+the rule: for [ϕ] ∈ Γg and [S, w] ∈ Tg, we define
+ϕ∗([S, w]) = [S, w ◦ ϕ−1].
+This action is properly discontinuous, and preserves the Teichm¨uller metric and the com-
+plex structure (see [41]). The quotient space Mg = Tg/Γg is the moduli space. It is a
+normal complex variety (see [17]). The moduli space Mg parametrizes all the isomor-
+phism classes of closed Riemann surfaces of genus g, but it is not compact. By adding
+frontier points corresponding to “stable curves” (i.e. Riemann surfaces with nodes and
+with finite automorphisms), it can be compactified. This is the Deligne–Mumford com-
+pactification M g (DM-compactification for short) of the moduli space Mg (see [21]). Many
+authors tried to reconstruct M g from the viewpoint of Teichm¨uller theory. Bers [14] be-
+gan this project, and Harvey [32] topologically reconstructed M g by using discrete group
+theory. Hubbard and Koch’s paper [37] contains a bit of history concerning the DM-
+compactification of the moduli space. For recent developments of “compactifications of
+Teichm¨uller space”, see Ohshika [62], Miyachi [56] or Masai [52].
+According to Kra’s overview [48] in this field, an analytic construction of the DM-
+compactification had to await the work of Hubbard and Koch [37] in the twenty-first
+century. But as we explained in [54] (by a few sentences after Theorem 6.5 and by Re-
+mark 6.3), Hubbard–Koch’s orbifold charts contain certain insufficient points as orbifold
+charts of M g. On the other hand, we gave a complete set of orbifold charts to the DM-
+compactification of the moduli space M g ([54, §6]).
+Harvey’s curve complex
+For our construction, Harvey’s curve complex Cg plays an important role. Given a
+surface Σg, Harvey [33] introduced an abstract simplicial complex called the curve complex
+Cg = C(Σg).
+By definition, a vertex of Cg is an isotopy class of an essential (i.e. not null-homotopic)
+simple closed curve on Σg. A simplex σ ∈ Cg is a collection of disjoint, mutually non-
+isotopic essential simple closed curves on Σg:
+σ = ⟨C1, . . . , Ck⟩.
+The number k of simple closed curves contained in σ will be denoted by |σ|. It is known
+that |σ| ≤ 3g − 3. We have dim σ = |σ| − 1.
+Let [S, w] be a point of Tg, and let σ ∈ Cg be a simplex: σ = ⟨C1, C2, . . . , Ck⟩. We
+represent each simple closed curve w(Ci) on the Riemann surface S by a geodesic, and
+contract each of them to a point. Then we have a stable curve. Thus the topological type
+of a stable curve is described by a simplex σ of the curve complex Cg, and the topological
+model of the stable curve is denoted by Σg(σ).
+3
+
+Weyl groups
+Another important object associated with a simplex σ ∈ Cg is the Weyl group W(σ),
+which is defined as follows:
+Let Γ(σ) be a subgroup of the mapping class group Γg generated by the Dehn twists
+about the simple closed curves Ci, i = 1, . . . , k on Σg. Let NΓ(σ) be the normalizer of
+Γ(σ) in Γg. Then we can prove that a mapping class ϕ ∈ Γg belongs to the normalizer
+NΓ(σ) if and only if ϕ permutes the isotopy classes of the curves Ci, i = 1, . . . , k of σ
+(Theorem 4.5 in [54]).
+Now the definition of the Weyl group W(σ) is the following:
+W(σ) := NΓ(σ)/Γ(σ).
+Note that the Weyl group W(σ) in our sense is not necessarily a finite group. In §2, we
+will prove that the Weyl group W(σ) is the mapping class group of a (topological) surface
+with nodes Σg(σ).
+Little Tichm¨uller space T(σ)
+Just as the Teichm¨uller space Tg was constructed by using marked Riemann sur-
+faces (S, w) (w being an orientation-preserving homeomorphism w : Σg → S), the
+little Teichm¨uller space T(σ) is constructed using marked stable curves (S, w) (where
+w : Σg(σ) → S is an orientation-preservig homeomorphism, S being a stable curve). T(σ)
+is a bounded domain of C3g−3−|σ|.
+Controlled deformation space Dε(σ).
+Let M be a (2-dimensional) Margulis constant: two distinct simple closed geodesics
+on any Riemann surface S are disjoint if their lengths are smaller than M. Of course,
+any positive number smaller than M has again the same property. Thus the Margulis
+constant is not unique. Let ε be a positive number smaller than a Margulis constant M.
+We will fix such an ε throughout the present paper.
+Let σ be a simplex of Cg. In §4.1, we will give the definition of the controlled deforma-
+tion space Dε(σ) (Def. 4.3). Dε(σ) is a complex manifold of complex dimension 3g − 3,
+and is homeomorphic to an open (6g − 6)-cell (see Lemma 6.7 in [54]). Dε(σ) contains
+T(σ). The Weyl group W(σ) acts on Dε(σ). This action is properly discontinuous and
+holomorphic (see Lemmas 6.4 and 6.6 of [54]). We can prove that the compactified moduli
+space M g is covered by Dε(σ)/W(σ), where σ runs over the simplexes of Cg, and can be
+empty (see Lemma 6.9 in [54]). Thus we have
+Theorem(Theorem 6.11 [54]) In the Deligne–Mumford compactification M g, the finite
+4
+
+family
+{(Dε(σ), W(σ))}σ∈Cg/Γg∪∅
+forms an atlas of orbifold charts of a complex (3g − 3)-orbifold.
+We will denote the compactified moduli space M g with this atlas of orbifold charts by
+M
+orb
+g .
+1.2
+The results of the present paper
+The main results consist of the following three parts (A) ∼ (C).
+(A) Arbarello-Cornalba [8] reconstructed the universal family of Riemman surfaces
+due to Teichm¨uller [70] and Bers [14] over Teichm¨uller space by patching Kuranishi fam-
+ilies of Riemann surfaces. Inspired by their work, we construct a family of stable curves
+over the controlled deformation spaces by patching Kuranishi families of stable curves
+(Th.4.4), which may be considered as a reconstruction of Hubbard–Koch’s family [37,
+Th.10.1]. By patching these families over all the orbifold charts of M
+orb
+g , we have;
+Main Theorem I (Th.4.8 and Def. 4.9) There exists a (strong) orbifold fibration
+π : Y
+orb
+g
+→ M
+orb
+g
+which is obtained by patching standard Kuranishi families of stable curves.
+We call π the universal degenerating family of Riemann surfaces. The naming comes
+from the universality in the sense that any fibered complex surface is obtained by pulling
+back from π. (We will explain this point in (C).) We are also inspired by Arbarello–
+Cornalba–Griffiths’ description [9, Chap.XV §8] of the bordification of Teichm¨uller space
+using real blow-ups and the method of log geometry by Kato-Nakayama [42] and Usui
+[72]. We also refer to Hinich-Vaintrob [34] for a related work. Philosophically we are
+inspired by the project of Catanese [19] for studying the relation between the Kuranishi
+families and the Teichm¨uller theory for many kinds of algebraic varieties.
+(B) For a mapping class ϕ ∈ Γg, let us denote by [ϕ] the set of conjugate elements of
+ϕ. Harvey [31] described the fixed point locus T [ϕ]
+g
+in Tg for a finite (elliptic, or periodic)
+element ϕ ∈ Γg in terms of the space of the base Riemann surfaces pointed by the branch
+points for the cyclic covering associated with ϕ. T [ϕ]
+g
+is called the equisymmetric strata
+for ϕ. Broughton [16] described the similar set M [ϕ]
+g
+in Mg. We extend these results to
+the boundaries of Tg and Mg, i.e. to the little Teichm¨uller space T(σ) and its image on
+M g.
+5
+
+Main Theorem II (Th.5.17 and Def. 5.15) Let T [ϕ](σ) be the set in T(σ) consisting
+of the marked stable curves with a given numerical data Num(ϕ) of an automorphism.
+Then a connected component of T [ϕ](σ) is a complex manifold which is isomorphic to a
+direct product of pointed Teichm¨uller spaces.
+The image M [ϕ](σ) of T [ϕ](σ) in M g \ Mg is an irreducible subvariety.
+T [ϕ](σ) may be identified with the fixed point locus for a parabolic (pseudo-periodic)
+element in Γg. The direct factor of the structure of T [ϕ]
+g
+appears as the space of pointed
+Riemann surfaces which come from each component of the base nodal Riemann surfaces of
+the cyclic covering associated with ϕ (see the discussion in §5.4 and the precise statement
+of Th.5.17). Our discussion is similar to those of the moduli construction of the branched
+covering of Riemann surfaces (cf. [73], [57] etc.). We are also inspired by Terasoma’s
+argument [71]. Note that the equisymmetric theory on Mg via various group actions was
+recently developed by Takamura [67], Hirakawa-Takamura [35] et al.
+(C) Kodaira [46] constructed the basic member of an elliptic surface (without multiple
+fibers) in a canonical way from the data of the monodromy and the J-invariant. We will
+extend this result to any fibered complex surface of genus g ≥ 2 (cf. Remark 7.9).
+We start from the local viewpoint. Let f : M → ∆ be a degeneration of genus g ≥ 2
+with the degenerate fiber F = f −1(0) over a small disk ∆, and µf be the topological
+monodromy of f.
+Then µf belongs to the conjugacy class ˆP(−)
+g
+(σ) ⊂ ˆΓg of pseudo-
+periodic maps of negative twists for some σ ∈ Cg (cf. [61], [55]). Now as an extended
+notion of local J-invariant, we define the local orbifold moduli map
+{ �J : �∆ → B ⊂ Dϵ(σ), J : ∆ → Mϵ(σ) ⊂ M g; G = ⟨µ⟩ ≃ Z/NZ}
+as follows. Here J is the usual holomorphically extended moduli map for f (cf. [39]),
+�∆ → ∆ is the totally ramified covering of disks with covering degree N which is the
+pseudo-period of the monodromy µf, �f : �
+M → �∆ is the precise stable reduction of f ([10,
+§2]) and �J is the natural map associated with the deformation �f to the Kuranishi space
+B (as a chart of Dϵ(σ)) of the stable curve �F = �f −1(0). Then �J can be considered as
+the lifting map of J by the cyclic group G whose generator µ essentially comes from the
+automorphism of �F. In order to describe the map �J explicitly, we define the following;
+Definition (Defs. 6.8, 6.11) A holomorphic function ϕ(u) is called a pseudo-periodic
+function with multiplicity γ ∈ N and period L ∈ N if the Taylor expansion at the origin
+is given by ϕ(u) = �∞
+i=0 ciuγ+iL (c0 ̸= 0). We call γ/L ∈ Q the analytic screw number of
+ϕ(u).
+Main Theorem III (Th.6.12) Let (z1, · · · , zk, zk+1, · · · z3g−3) be the Harris–Mumford
+coordinates at �J(0) of B ⊂ Dϵ(σ), where zi (1 ≤ i ≤ k) is a smoothing coordinate at a
+6
+
+node of �F, and zj (k + 1 ≤ j ≤ 3g − 3) a variable coordinate of a component of �F (see
+Def. 5.21). Let
+�J : �∆ ∋ u �−→ (z1, · · · , z3g−3) = (ϕ1(u), · · · , ϕ3g−3(u)) ∈ B ⊂ Dϵ(σ)
+be the expression of the map �J by these coordinates.
+(i) If 1 ≤ i ≤ k, then ϕi(u) is a pseudo-periodic function whose multiplicity and
+period are explicitly written by the numerical data of the topological monodormy µf.
+In particular, the analytic screw number of ϕi(u) coincides with the absolute value of
+Nielsen’s screw number of µf for a cut curve in σ.
+(ii) If k + 1 ≤ j ≤ 3g − 3 and ϕj(u) is not identically zero, then ϕj(u) is a pseudo-
+periodic function whose period and fractional part of the analytic screw number are
+explicitly written by the numerical data of µf.
+Note that the fractional part of the analytic screw number of ϕj(u) (k+1 ≤ j ≤ 3g−3)
+is written by a logarithmic quadratic character induced from the total valency of µf by
+the discussion in §5.2. But the integral part of the analytic screw number of ϕj(u) is
+not a monodromy invariant, and it is purely an analytic invariant of the fiber germ (see
+Remark 6.14).
+Pseudo-periodicity has already appeared in Kodaira’s local description [46, §8] of J-
+invariant by the action of the element of SL(2, Z) which is the homological monodromy
+of a degeneration of elliptic curves.
+Now we discuss from the global point of view. Let f : M → B be a global fibered
+complex surface of genus g ≥ 2 with descriminant locus Discf(B) = {Q1, · · · , Qs} ⊂ B.
+We set B(0) = B \ Discf(B), and let
+µf : π1(B(0), b0) −→ Γg,
+(b0 ∈ B(0))
+be the monodromy representation of f. Let
+Borb =
+�
+�B(0), π1(B(0), b0), ϕ�B(0), B(0)� �
+1≤i≤s
+�
+�∆i, Z/NiZ, ϕ�∆i, ∆i
+�
+be the natural orbifold structure over B, where �B(0) is the universal covering of B(0) and
+�∆i → ∆i the covering of disks around Qi whose degree coincides with the pseudo-period
+of the local monodromy at Qi. Then the orbifold moduli map
+Jorb
+f
+: Borb −→ M
+orb
+g
+is well-defined by patching the local orbifold moduli maps and the natural map from �B(0)
+to Tg (Def. 7.7).
+7
+
+Main Theorem IV (Th.7.8, Def.4.10) Any fibered complex surface f : M → B can
+be pulled back in the orbifold sense from the universal degenerating family of Riemann
+surfaces π : Y
+orb
+g
+−→ M
+orb
+g
+via the orbifold muduli map Jorb
+f
+.
+The above theorem may be considered as an accomplishment of the results of Imayoshi
+[39] and [53].
+2
+Mapping class groups of stable curves
+In this section, we will define and study the mapping class group of a stable curve S of
+genus g. We already described it in [54, Cor.4.7] as a certain quotient group of a subgroup
+of the mapping class group of a Riemann surface, and called it the Weyl group of genus
+g. However, some parts of the argument were omitted there. The aim of this section is
+to supply these points.
+In §2.1, we define the lifting map from S to a Riemann surface Σg via real blow-ups with
+some parameters. For the study of the boundary of Teichm¨uller space, the contraction
+maps of simple closed curves on a Riemann surface to nodes are fundamental (cf.[15], [1]
+etc.). However, this contraction loses the information on twisting parameters of Fenchel–
+Nielsen coordinates along the geodesics isotopic to these curves. By this lifting, we recover
+the above lost twisting parameters and make it possible to discuss the Fenchel–Nielsen
+deformation (or twist) at the boundary. This argument is essentially due to [42],[72] and
+[9], and will be discussed in §3.2 in detail.
+In §2.2, we discuss the liftability of a continuous map f : S1 → S2 of stable curves to
+a map between two Riemann surfaces. If f is a holomorphic or real-analytic map, this is
+possible. However, in the case where f is a continuous map, there exists an obstruction to
+the lifting by the existence of a map which behaves as an infinite-angled rotation around
+a node.
+In §2.3, we show that this obstruction is cancelled in the isotopy class of oriented
+homeomorphisms by the Alexander trick [7], and we can discuss the mapping class group
+of a stable curve in the language of the mapping class group of a Riemann surface. Then
+the geometric meaning of the Weyl groups will be clarified in Theorem 2.10.
+2.1
+Lifting of stable curves and Fenchel–Nielsen twist at infinity
+In this subsection, we will define and discuss the lifting map from a stable curve to a
+Riemann surface via real blow-ups with some parameters.
+Let S be a stable curve over C, i.e. a complete algebraic curve with at most nodes
+(ordinary double points) as singularities such that a nonsingular rational component has
+8
+
+at least three nodes of S. Assume S has genus g > 1, and has k nodes P1, · · · , Pk. If
+S is nonsingular (i.e. a compact Riemann surface), then k = 0. Let S = �r
+i=1 Si be
+the irreducible decomposition and let h : �S = �r
+i=1 �Si −→ S be the normalization in the
+function field. Topologically, the component �Si is obtained from a connected component
+Si of the complement of nodes of S by closing the punctures which are the pull back by
+h of nodes to this component, say Pi := �ri
+j=1 Pi,j. If �Si is a projective line, then ri ≥ 3.
+We call the ri-pointed Riemann surface
+(�Si, Pi)
+(1)
+the normalized pointed component of S. Let
+�πi : BlPi(�Si) −→ �Si
+be the real blowing up at Pi,1, · · · , Pi,ri. By using a complex local coordinate (U, z) at
+Pi,j = (z = 0), BlPi(�Si) is defined locally by {(z, θ) ∈ U × S1 | z = |z|e
+√−1θ} and �πi
+is induced from the first projection. Globally �πi is a real analytic isomorphism on the
+complement of the inverse images of Pi,j’s, and each fiber (�πi)−1(Pi,j)
+(1 ≤ j ≤ ri) is
+the circle S1, which is called the exceptional circle of �πi. The inverse image of the node
+Pj under the composition map �h = (�r
+i=1 �πi) ◦ h : �r
+i=1 BlPi(�Si) −→ S consists of two
+exceptional circles, which we write
+(�h)−1(Pj) = C(1)
+j
+�
+C(2)
+j
+(1 ≤ j ≤ k)
+(2)
+where the order of two circles C(1)
+j , C(2)
+j
+is arbitrary. We consider a map
+Φ =
+k
+�
+j=1
+ϕj :
+k
+�
+j=1
+�
+C(1)
+j
+−→ C(2)
+j
+�
+(3)
+where each ϕj : C(1)
+j
+−→ C(2)
+j
+is a homeomorphism of circles. Let S(Φ) be the topo-
+logical surface obtained from Riemann surfaces with boundaries BlP1(�S1), · · · , BlPr(�Sr)
+by pasting the corresponding boundaries (2) which are the exceptional circles via the
+identification map Φ of (3). We obtain the natural continuous map
+π(Φ) : S(Φ) −→ S
+(4)
+such that the fiber of the node (π(Φ))−1(Pj) is a circle, say Cj, which is obtained from
+C(1)
+j
+and C(2)
+j
+via the identification ϕj, and the restriction of π(Φ) to the complement of
+the inverse images of nodes S(Φ) \ �
+1≤j≤k Cj −→ S \ �
+1≤j≤k Pj is a homeomorphism.
+Figure I is a simple example in the case where g = k = r = 2.
+9
+
+Definition 2.1. We call the map π(Φ) of (4) the lifting of S, and S(Φ) the lifted Riemann
+surface of S with twist Φ. We also call {C1, · · · , Ck} the exceptional circles for π(Φ).
+Example 2.2. (a typical example of lifting)
+We identify S1 with R/2πZ via the map
+R/2πZ ∈ x �−→ exp(
+√
+−1x) ∈ S1.
+(5)
+Let ϕθ : S1 −→ S1 be the rotation map of angle θ defined by x �→ x + θ. By the definition
+of the real blowing up, we have a canonical identification C(1)
+j
+= C(2)
+j
+= S1. Therefore, by
+a k-ple of angles Θ = (θ1, · · · , θk) ∈ (S1)k, the map
+Φ(Θ) =
+k
+�
+j=1
+ϕθj :
+k
+�
+j=1
+�
+C(1)
+j
+−→ C(2)
+j
+�
+(6)
+is defined. By using Φ(Θ) as the identification map, the lifting π(Φ(Θ)) : S(Φ(Θ)) −→ S
+is constructed. We call it the lifting of S with the rotation angles Θ.
+We choose the element o = (0, · · · , 0) ∈ (S1)k.
+By resetting S(Φ(o)) = Σg and
+π(Φ(o)) = πS, we write the lifting of S with the rotation angle o as
+πS : Σg −→ S,
+(7)
+and call it the canonical lifting of S. Since S is a stable curve, the exceptional circles
+{C1, · · · , Ck} on Σg for πS are not isotopic to each other. Therefore, their isotopy classes
+determine a (k −1)-simplex of Harvey’s curve complex Cg (see §3.1, and [33]) of Σg, which
+we write
+σπS = ⟨C1, · · · , Ck⟩ ∈ Cg,
+(8)
+10
+
+(2)
+S(Φ)
+2
+past with twist
+(1)
+(“)=
+contraction
+realblowup at
+of Ci, C2
+P11,P12, P21, P22
+P12
+22
+P11
+normalization
+S1
+S2
+31
+S
+(Figure 1)The composition of real modifications of a stable curveand call it the exceptional simplex for πS.
+The lifted Riemann surface S(Φ) of arbitrary twist Φ is reconstructed from Σg by
+changing the pasting parameters along exceptional cycles by using (3).
+We naturally
+obtain a homeomorphism
+τ�Φ : Σg −→ S(Φ)
+(9)
+which satisfies πS = π(Φ) ◦ τ�Φ. If we admit a hyperbolic metric on Σg such that the
+exceptional circles C1, · · · , Ck are geodesics with respect to this metric, then the map (9)
+is traditionally called the Fenchel–Nielsen deformation or the Fenchel–Nielsen twist along
+Cj’s (cf. [77, p.24], [41, Chap.8]). We also call τ�Φ the Fenchel–Nielsen twist at infinity,
+since S itself sits on the boundary of moduli space as we will discuss afterwards.
+Note that τ�Φ is not uniquely determined from Φ as a homeomorphism. Since Φ of
+(3) is a pasting map along circles in the construction of S(Φ), the integral rotation of
+each circle is ignored. However, if an identification of S(Φ) with Σg is somehow fixed,
+the homeomorphism τ�Φ may be considered as an element in the mapping class group Γg
+of Σg as follows. By identifying S(Φ) with Σg, the identification map ϕj in (3) naturally
+defines a self-homeomorphism ϕj : A(Cj) −→ A(Cj) of an annular neighborhood A(Cj)
+of Cj. In the mapping class group of the annulus A(Cj), ϕj is isotopic to a rotation map
+ϕθj with some real-valued angle θj ∈ R = (−∞, +∞). However , if the θj’s are integral
+multiples of 2π, the map τ�Φ : Σg −→ Σg = S(Φ) is isotopic to a composition of integral
+Dehn twists along the exceptional circles.
+Let Γ(σπS) be the subgroup of Γg generated by Dehn twists along the exceptional
+circles, and let [τ�Φ] be the isotopy class of τ�Φ. By the above argument, if the θj’s are
+integral multiples of 2π, we have
+[τ�Φ] ∈ Γ(σπS).
+(10)
+2.2
+Obstruction to lifting homeomorphisms
+Here we discuss the liftability of a homeomorphism of stable curves to a homeomorphism
+of lifted Riemann surfaces with twists.
+We start by sketching it locally. Let f : ∆ −→ ∆ be a map between the unit disks with
+f(0) = 0 such that f is a homeomorphism onto its image. Let Lθ = {z ∈ ∆ | arg z = θ}
+be a radial segment in ∆ with the fixed angle θ ∈ R. If the limit
+f|Lθ(0) :=
+lim
+z∈Lθ,|z|�→0
+f(z)
+|f(z)|
+(11)
+exists for each θ ∈ R, then f is said to be finite-angled. Otherwise f is said to be infinite-
+angled. The image of Lθ under an infinite-angled homeomorphism goes around the origin
+infinitely many times. We first give one of this type of examples, and then state our
+claims.
+11
+
+Example 2.3. Let [r, θ] be the polar coordinate of ∆; z = re
+√−1θ. We define f[r, θ] =
+[r, θ +2π/r] for [r, θ] ̸= [0, 0], and f[0, 0] = [0, 0]. Then f is an infinite-angled homeomor-
+phism of ∆.
+Lemma 2.4. Let f : ∆ −→ ∆ be a finite-angled homeomorphism onto its image between
+the unit disks with f(0) = 0, and let π0 : Bl0(∆) −→ ∆ be the real blowing up at 0 ∈ ∆.
+Then there exists uniquely a homeomorphism �f : Bl0(∆) −→ Bl0(∆) onto its image which
+is a lift of f, i.e. π0 ◦ �f = f ◦ π0 holds.
+Proof
+We rename the sourse disk by ∆1 = ∆ and the target disk by ∆2 = ∆,
+and let zi be the complex coordinate of ∆i (i = 1, 2). By definition, we have Bl0(∆i) =
+{(zi, θi) ∈ ∆i × S1 | zi = |zi|e
+√−1θi} where the coordinate θi of S1 is written via the
+identification (5). By using (11), we define the map �f : Bl0(∆1) −→ Bl0(∆2) by
+(z1, θ1) �−→ (z2, θ2) = (f(z1), arg(f(z1))
+if z1 ̸= 0,
+(0, θ1) �−→ (z2, θ2) = (0, f|Lθ1(0)).
+Then �f satisfies the desired property.
+Definition 2.5. A homeomorphism f : S1 −→ S2 of stable curves is said to be finite-
+angled if the restrictions of f to small disk neighborhoods of both banks (local components)
+of each node of S1 are finite-angled.
+Proposition 2.6. Let f : S1 −→ S2 be a finite-angled homeomorphism of stable curves.
+We choose a lifting π(Φ1) : S1(Φ1) −→ S1 with an arbitrary twist Φ1.
+Then there exist a unique lifting π(Φ2) : S2(Φ2) −→ S2 with some twist Φ2 and a
+unique homeomorphism �f : S1(Φ1) −→ S2(Φ2) which satisfy the following commutative
+diagram
+S1(Φ1)
+�f
+�
+π(Φ1)
+�
+S2(Φ2)
+π(Φ2)
+�
+S1
+f
+� S2.
+Proof
+Let P1 be a node of S1, and P2 = f(P1) be the image of P1, which is a node of
+S2. Let ∆(j)
+i
+(i = 1, 2, j = 1, 2) be small disk neighborhoods in both banks of Pi such that
+the restriction f|∆(j)
+1
+: ∆(j)
+1
+−→ ∆(j)
+2
+is a homeomorphism onto its image, for j = 1, 2. Let
+π(j)
+Pi : BlPi(∆(j)
+i ) −→ ∆(j)
+i
+be the real blowing up at the origin Pi with exceptional circle
+C(j)
+i
+= (π(j)
+Pi )−1(Pi). Since f|∆(j)
+1
+is finite-angled by assumption, it follows from Lemma
+2.4 that there exists a unique homeomorphism �f|∆(j)
+1
+: BlP1(∆(j)
+1 ) −→ BlP2(∆(j)
+2 ) which
+satisfies
+π(j)
+P2 ◦ �f|∆(j)
+1
+= f|∆(j)
+1 ◦ π(j)
+P1 .
+(12)
+12
+
+Let ϕ1 : C(1)
+1
+−→ C(2)
+1
+be the pasting homeomorphism associated with Φ1. Then we define
+the homeomorphism ϕ2 : C(1)
+2
+−→ C(2)
+2
+by
+ϕ2 = �f|∆(2)
+1 ◦ ϕ1 ◦ ( �f|∆(1)
+1 )−1|C(1)
+2 .
+(13)
+By pasting C(1)
+2
+and C(2)
+2
+via ϕ2, the desired S2(Φ2) and �f are locally well-defined by (12)
+and (13). We do the same construction on disk neighborhoods of all the nodes of S1 and
+S2 and paste the resultants trivially to their complements in S1 and S2. Then we globally
+obtain the desired S2(Φ2) and �f. The uniqueness is clear from the construction.
+2.3
+Weyl groups as mapping class groups
+Here we characterize the mapping class group of a stable curve as the Weyl group.
+First we discuss the orientation of a stable curve S. Since the local equation of a node
+of S is xy = 0 in C2, the link of the singularity is a positive Hopf link. Then S has a
+natural orientation ∇S so that its normalized components have orientations as Riemann
+surfaces which are compatible with the positivity of Hopf links at nodes.
+On the other hand, we consider the canonical lifting πS : Σg −→ S described in (7).
+We compare the natural orientaion ∇Σg with ∇S. Then:
+Lemma 2.7. The orientations ∇Σg and ∇S are compatible via the map πS.
+Proof
+Since Σg and S are realized as fibers in an oriented manifold by Theorem 3.1
+(in the next section), the assertion is clear.
+Let Si (i = 1, 2) be oriented stable curves with orientations ∇Si. A homeomorphism
+f : S1 −→ S2 is said to be oriented if the pushed down f∗(∇S1) defines the orientation on
+S2 which is equivalent to ∇S2. Two oriented homeomorphisms fi : S1 −→ S2 (i = 0, 1) are
+said to be isotopic to each other if there exists an isotopy {ft : S1 −→ S2}0≤t≤1 such that
+each ft is an oriented homeomorphism. The isotopy class of an oriented homeomorphism
+f : S1 −→ S2 is said to be the mapping class of f, and is denoted by [f]. For a fixed stable
+curve S, the set of mapping classes becomes a group by the composition of maps, which
+is called the mapping class group of S, and is denoted by Σ(S). We have the following:
+Proposition 2.8. Any oriented homeomorphism f : S1 −→ S2 of stable curves is isotopic
+to a finite-angled oriented homeomorphism f ′ : S1 −→ S2.
+Proof
+Suppose that the restricted homeomorphism to a disk neighborhood of a
+bank f|∆(1)
+P : ∆(1)
+P −→ f(∆(1)
+P ) ⊂ ∆(1)
+f(P) of some node P of S1 is infinite-angled. We choose
+another map h : ∆(1)
+P −→ ∆(1)
+f(P) such that
+13
+
+(i)
+h(∆(1)
+P ) = f|∆(1)
+P (∆(1)
+P ) as sets and the restricted maps to the boundary coincide:
+h|∂∆(1)
+P ≡ f|∂∆(1)
+P ,
+(ii) h is a finite-angled homeomorphism onto its image so that h and f|∆(1)
+P
+define the
+same orientation.
+Note that we have infinitely many choices of h. Since the composite homeomorphisms
+h−1 ◦ f|∆(1)
+P
+: ∆(1)
+P
+−→ ∆(1)
+P
+coincide with each other at the boundary ∂∆(1)
+P , they are
+isotopic to the identity map id∆(1)
+P
+on the whole disk ∆(1)
+P
+by the Alexander trick ([7]).
+That is to say an h is isotopic to f|∆(1)
+P
+without moving the points of the boundary. We
+define an oriented homeomorphism f ′ : S1 −→ S2 by
+f ′(x) =
+�
+�
+�
+f(x)
+if x ∈ S \ ∆(1)
+P
+h(x)
+if x ∈ ∆(1)
+P .
+Then f ′ is isotopic to f. If f ′ is infinite-angled at a certain node, then we repeat the same
+process as above. After a finite number of steps, we reach a new f ′ which satisfies the
+desired property.
+Corollary 2.9. Any mapping class [f] of a stable curve S has a lifting to a mapping class
+[ �f] of a Riemann surface Σg such that (πS)∗([ �f]) = [f] holds.
+Proof
+By identifying the Riemann surface S(Φ) with Σg = S(Φ(o)) as in §2.1, the
+assertion follows from Propositions 2.8 and 2.6.
+The lifted class [ �f] is not uniquely determined by [f]. This ambiguity comes from the
+ambiguity of the choice of the map h near a node P as in the proof of Proposition 2.8.
+By the same argument as (10), this ambiguity is cancelled modulo integral Dehn twists
+along the exceptional circles and the lifting [ �f] is uniquely determined as an element of
+Γg/Γ(σπS). On the other hand, since f permutes the nodes P1, · · · , Pk of S, [ �f] permutes
+the ambient isotopy classes [C1], · · · , [Ck] of exceptional circles.
+By [54, Th.4.5], the
+subgroup of Γg consisting the elements which permute [C1], · · · , [Ck] is nothing but the
+normalizer NΓ(σπS) of Γ(σπS). We set
+W(σπS) = NΓ(σπS)/Γ(σπS)
+and call it the Weyl group of the exceptional simplex σπS. Note that our W(σπS) is not
+necessarily a finite group.
+Theorem 2.10. ([54, Cor.4.7])
+The mapping class group Γ(S) of a stable curve S is
+isomorphic to the Weyl group
+Γ(S) ∼= W(σπS).
+14
+
+3
+Kuranshi families of stable curves and log struc-
+tures
+Here we review well-known facts about the Kuranishi families of stable curves which will
+be used afterwards. We mainly refer to Arbarello–Cornalba–Griffiths [9].
+In §3.1, we first review the formal properties of the standard Kuranishi families of
+stable curves, and secondly the cohomological properties of them. In particular, we discuss
+the parameter spaces of the smoothing (plumbing) deformations by deforming the nodes
+to annuli, and those of variable deformations by deforming the complex structures of
+irreducible components and shifting the positions of nodes.
+In §3.2, we first briefly review the notion of log geometry following Kato–Nakayama
+[42] and Usui [72], and then review in Theorem 3.1 that the boundary of the log lifting
+of the Kuranishi family of stable curves parametrises the possible Fenchel-Nielsen twist
+at infinity which was discussed in §2.1. The method of [9] does not use the terminology
+of log geometry and shows this fact direcly by real blowing ups.
+3.1
+Deformations of stable curves and Kuranishi families
+In this subsection, we review the standard results of the Kuranishi families of stable
+curves.
+We recall Harvey’s curve complex Cg of a Riemann surface Σg (see [33]). By definition,
+Cg is a (3g−3−1)-dimensional simplicial complex whose (k−1)-simplex σ = ⟨C1, · · · , Ck⟩
+consists of mutually disjoint and non-homotopic isotopy classes of simple closed curves
+Cj (1 ≤ j ≤ k) on Σg. An (ℓ − 1)-simplex ρ = ⟨Ci1, · · · , Ciℓ⟩ is a face of σ, denoted by
+ρ < σ, if {Ci1, · · · , Ciℓ} is a subset of {C1, · · · , Ck} as isotopy classes. We denote by
+contσ : Σg −→ Σg(σ)
+(14)
+the contraction map of σ, i.e. the continuous map which contracts each Cj to a point and
+is homeomorphic on the complement of the Cj’s. Here Σg(σ) is a nodal Riemann surface
+with k nodes. The map (14) is topologically identified with the map (7).
+Let S be a stable curve whose topological type coincides with Σg(σ). By the Kuranishi
+family of S, we mean the fibration
+ψ : X −→ B
+(15)
+which has the following properties (cf. [9, Chap.XI, §4, §6]):
+(i) B (resp. X) is a (3g − 3)-dimensional (resp. (3g − 2)-dimensional) complex manifold
+with S = ψ−1(b0) (b0 ∈ B), b0 being a fixed point,
+15
+
+(ii) ψ has the universal property with respect to local deformations. In other words, for
+any deformation ϕ : V −→ Z with ϕ−1(e0) = S, the restricted family ϕW : VW −→ W
+to any sufficiently small connected neighborhood W (⊂ Z) of e0 has unique holomorphic
+maps f : W −→ B and �f : VW −→ X with f(e0) = b0 such that ψ ◦ �f = f ◦ ϕW,
+(iii) For any b ∈ B, the Kodaira-Spencer map
+TB,b −→ Ext1
+OXb(Ω1
+Xb, OXb)
+is an isomorphism, where TB,b is the tangent space of B at b and Ω1
+Xb is the sheaf of K¨ahler
+differentials of Xb = ψ−1(b),
+(iv) The discriminant locus D of ψ is a normal crossing divisor on B such that the fibers of
+ψ over the irreducible component Di1···iℓ of ℓ-codimensional open strata of D are the stable
+curves whose topological types are Σg(ρ) corresponding to the face ρ = ⟨Ci1, · · · , Ciℓ⟩ < σ.
+Moreover, if ψ satisfies the additional condition (v), ψ is said to be a standard Kuran-
+ishi family of S:
+(v) The action of an analytic automorphism of S extends to a compatible action on X
+and B (the totality of which will be denoted by Aut(X/B)). Conversely, any isomorphism
+between fibers of ψ is induced by an analytic automorphism of S (the totality of which
+will be denoted by G). That is to say, if there exists an isomorphism of fiberes hb1b2 :
+ψ−1(b1) → ψ−1(b2) (for two points b1, b2 ∈ B), then there exists some h0 ∈ G and
+its extension h0 ∈ Aut(X/B) (h0|ψ−1(b0) = h0) such that the restriction map h0|ψ−1(b1)
+coincides with hb1b2.
+(vi) A Kuranishi family and a standard Kuranishi family of S exist up to isomorphisms
+of families and up to shrinking the base B near b0 ([9]).
+Now we summarize well-known facts about the characterization of smoothing (or
+plumbing) and non-smoothing deformations of S for ψ (cf.[9, Chap.XI], [29, §1]). Let
+S = �r
+i=1 Si be a stable curve, and let ( ˆSi, Pi) and Pj (1 ≤ j ≤ k) be as in §2.1. Since the
+tangent space at b0 of B is isomorphic to Ext1
+OS(Ω1
+S, OS), we can choose a local coordinate
+neighborhood Bloc at b0 of B as an open neighborhood of the origin of this vector space ;
+Bloc ⊂ Ext1
+OS(Ω1
+S, OS) ∼= C3g−3.
+(16)
+From the spectral sequence Hq(S, Extp(Ω1
+S, OS)) =⇒ Extp+q(Ω1
+S, OS), we have the exact
+sequence
+0 −→ H1(S, HomOS(Ω1
+S, OS)) −→ Ext1
+OS(Ω1
+S, OS) −→ H0(S, Ext1(Ω1
+S, OS)) −→ 0. (17)
+By using the Grothendieck–Serre duality, the dual vector spaces of each space in (17) are
+H1(S, HomOS(Ω1
+S, OS))∗ ∼=
+r
+�
+i=1
+H1( ˆSi, T ˆSi(−Pi))∗ ∼=
+r
+�
+i=1
+H0( ˆSi, 2K ˆSi + Pi),
+(18)
+16
+
+Ext1
+OS(Ω1
+S, OS)∗ ∼= H0(S, Ω1
+S ⊗ ωS),
+H0(S, Ext1(Ω1
+S, OS))∗ ∼=
+k
+�
+j=1
+τPj,
+where ωS is the dualizing sheaf of S, T ˆSi and K ˆSi are the tangent sheaf and the canonical
+sheaf of ˆSi respectively, and τPj is the torsion sheaf supported on Pj which is described as
+follows (cf. [29, p.33]): If S is locally defined by xy = 0 near the node Pj, the differentials
+ω1 = dx⊗2/x, ω2 = dy⊗2/y have the relation yω1 = xω2. Therefore
+yω1 = ydx⊗2
+x
+= xdy⊗2
+y
+(19)
+generates a one-dimensional submodule over C, which is nothing but the generator of τPj.
+Hence the dual exact sequence of (17) is
+0 −→
+k
+�
+j=1
+τPj −→ H0(S, Ω1
+S ⊗ ωS) −→
+r
+�
+i=1
+H0( ˆSi, 2K ˆSi + Pi) −→ 0
+(20)
+Then the deformation-theoretic meaning of (17) and (20) is the following;
+(I) The space H1( ˆSi, T ˆSi(−Pi)) or H0( ˆSi, 2K ˆSi + Pi) parametrizes the variable deforma-
+tions for varying the complex structures of ˆSi without smoothing the attaching nodes,
+(II) The torsion sheaf τPj parametrizes the smoothing deformations of the nodes Pj to
+annuli. In particular, τPj is generated by the plumbing at Pj (Note that here the meaning
+of “plumbing” is different from that in differential topology.
+For the meaning in the
+present context, see [47, §2, §3], [9, pp.184–186]).
+The above facts (I) and (II) also follow from the general theory of deformations of
+varieties of normal crossing (cf.[26]).
+3.2
+Log structure and real blowing up of Kuranishi families
+In this subsection, we apply a part of the standard arguments of log geometry[?] to the
+Kuranishi family ψ : X → B of S. For the teminology, see [42, §1], [72, §1], [43, Chap.2]
+etc.
+Since the discriminant locus D is a normal crossing divisor on B, the sheaf of finitely
+generated and saturated monoid M = {f ∈ OB | f is invertible outside D} embedded in
+OB defines the log structure as follows. We set
+Blog =
+�
+(b, h) | b ∈ B, h ∈ Hom(M gp
+B,b, S1), h(f) = f(x)
+|f(x)|, ∀f ∈ O∗
+B,b
+�
+,
+where M is embedded in the abelian group M gp = {a/b | a, b ∈ M} as a sheaf on B, and
+O∗
+B,b is the stalk at b of the non-vanishing holomorphic functions. The first projection
+(b, h) �→ b induces a map
+τB : Blog −→ B.
+(21)
+17
+
+Then Blog has the structure of a real analytic manifold with corners such that the map
+(21) may be considered as the real blowing up of B along D. Namely, let (U, z1, · · · , z3g−3)
+be a system of local coordinates of B around b0 so that zi = 0 (1 ≤ i ≤ k) defines locally
+an irreducible component Di of D. A component Di1,··· ,iℓ of the ℓ-codimensional open
+strata of D on U is defined by zi1 = · · · zil = 0, zj ̸= 0 (j ∈ {1, · · · , 3g − 3} \ {i1, · · · , iℓ}).
+Then Blog is defined near a point of Di1,··· ,iℓ ∩ U as
+{(z1, · · · , z3g−3, θ1, · · · , θℓ) ∈ U × (S1)ℓ ; zij = |zij|e
+√−1θj, 1 ≤ j ≤ ℓ}
+by identifying S1 = R/2πZ. The exceptional set E of Blog has the stratification so that
+the component Ei1,··· ,iℓ = τ −1
+B (Di1,··· ,iℓ) is locally written by
+{zi1 = · · · zil = 0, (zj1, · · · , zj3g−3−ℓ, θ1, · · · , θℓ) ∈ (C∗)3g−3−ℓ × (S1)ℓ}
+(22)
+where j1, · · · , j3g−3−ℓ ∈ {1, · · · , 3g − 3} \ {i1, · · · , iℓ}.
+In particular, the restriction
+Ei1,··· ,iℓ −→ Di1,··· ,iℓ of τB is an (S1)ℓ-bundle.
+On the other hand, since ψ−1(D) is a normal crossing divisor on X, we have the
+construction τX : Xlog −→ X similar to (21). Then from [42, Lemma 1.3], we have the
+log lifting ψlog : Xlog → Blog of ψ with which the diagram
+Xlog
+τX
+�
+ψlog
+�
+X
+ψ
+�
+Blog
+τB
+� B
+(23)
+is commutative. The geometric meaning of the log lifting is clarified by [72] in a more
+general setting. In the case of the Kuranishi family of a stable curve, it is a real analytic
+family of Riemann surfaces including the possible Fenchel–Nielsen twists at infinity ([9,
+p.149–156]) :
+Theorem 3.1. ([9], [72], [42]) Let ψ : X −→ B be a Kuranishi family of a stable curve
+S. Then ψlog : Xlog → Blog is a real analytic family of Riemann surfaces such that
+(i)
+The restriction (τB ◦ ψlog)−1(B \ D) −→ τ −1
+B (B \ D) of ψlog is isomorphic to the
+restriction ψ−1(B \ D) −→ B \ D of ψ.
+(ii) Let Q be a point of the fiber τ −1
+B (P) of (S1)ℓ-bundle Ei1,··· ,iℓ −→ Di1,··· ,iℓ (P ∈ Di1,··· ,iℓ)
+such that Θ = (θ1, · · · , θℓ) is the fiber coordinates of Q given in (22). Then the fiber
+(ψlog)−1(Q) is isomorphic to the Riemann surface which is the lifting of the stable curve
+ψ−1(P) by rotation angle Θ. That is to say, in the notation (6), we have
+(ψlog)−1(Q) ∼= ψ−1(P)(Φ(Θ)).
+(24)
+Remark 3.2. The lifting map of a real analytic map between real analytic manifolds
+via real blowing ups is also constructed by Hubbard–Papadopol–Veselov [38, §5] and is
+discussed for other purpose.
+18
+
+4
+Construction of the universal degenerating family
+Starting from Hubbard-Koch’s result [37], we constructed in [54] a new orbifold structure
+on the Deligne-Mumford compactification , which we will denote here by M
+orb
+g . In this
+section, using this structure, we will construct an orbifold fiber space πorb : Y
+orb
+g
+−→ M
+orb
+g
+which is a family of stable curves with some universal properties.
+In §4.1, we review the orbifold structure of M
+orb
+g
+with some comments. The “biggest
+chart” of M
+orb
+g
+is the moduli space Mg which is the quotient of Teichmuller space by
+the mapping class group (Mg; Tg, Γg). The “boundary charts” are indexed by σ ∈ Cg
+(Harvey’s curve complex), each of which is an open set Mϵ(σ) containing the locus V (σ)
+of stable curves with topological type Σg(σ).
+Mϵ(σ) is the quotient of the controlled
+deformation space Dϵ(σ) by the Weyl group W(σ). Here Dϵ(σ) is the space consisting of
+σ-marked stable curves whose generalized Fenchel–Nielsen coordinates are “controlled”
+so that W(σ) acts on Dϵ(σ) properly discontinuously.
+In §4.2, we show that Dϵ(σ) is expressed by patching the bases of the Kuranishi
+families of σ-marked stable curves. From this, we construct a family of stable curves
+πσ : Xσ −→ Dϵ(σ) by patching the Kuranishi families of stable curves. This family is a
+refinement of Hubbard–Koch’s family [37, Th.10.1], and also is an extension of Arbarello–
+Cornalba’s theorem [8] which says that the universal curve Cg −→ Tg over Teichm¨uller
+space is expressed by patching the Kuranishi families of Riemann surfaces. As the main
+tool of the proof of our theorem, we use the log lifting of the Kuranishi families discussed
+in §3.2.
+In §4.3, by patching the families πσ naturally, we construct the desired orbifold fiber
+space πorb. We call it the universal degenerating family of Riemann surfaces for fibered
+complex surfaces, because any fibered complex surface admitting unstable fibers can be
+pulled back from πorb. This point will be discussed in §7.
+4.1
+Weyl marking and controlled deformation spaces
+In this subsection, we introduce the notions of the Weyl marking and controlled defor-
+mation space from [54, Def.6.2], and review related results proved in [54] by adding some
+comments.
+We fix a simplex σ = ⟨C1, · · · , Ck⟩ of Cg, and let S be a stable curve whose topological
+type coincides with Σg(σ) in (14). By a pre-marking of S, we mean the isotopy class [w]
+of an oriented homeomorphism
+w : Σg(σ) −→ S.
+(25)
+Two such pre-marked stable curves (S1, [w1]) and (S2, [w2]) are said to be equivalent
+(and denoted by (S1, [w1]) ∼ (S2, [w2]) ) if there exists an analytic isomorphism f : S1 −→
+19
+
+S2 such that f ◦ w1 is isotopic to w2. We denote the equivalence class by
+[S, w] = (S, [w])/ ∼
+and call it a stable curve with Weyl marking (or σ-marked stable curve). This notion of
+the marking is different from the ones given in [37, p.270] and [9, p.490], since the action
+of Γ(σ) is neglected in our case. Then the rigidity, i.e. the automorphism-freeness of a
+stable curve with Weyl marking, is spelled out as follows:
+Lemma 4.1. Let w : Σg(σ) → S be a Weyl marking, and f : S → S be an analytic
+automorphism such that [S, f ◦ w] = [S, w]. Then f is the identity map.
+Proof
+If f permutes a proper subset of nodes of S non-trivially, then a lifting of f to
+Γg would permute a non-trivial subset of the isotopy classes of C1, · · · , Ck, and f◦w cannot
+be isotopic to w. Therefore, f stabilizes each irreducible component of S, and stabilizes
+each normalized component of it as a pointed Riemann surface.
+Hence the assertion
+follows from the usual rigidity of the Teichm¨uller marking (cf. [36, Prop. 6.8.1]).
+The following is a criterion of the equivalence of the marking:
+Lemma 4.2. Two oriented homeomorphisms wi : Σg(σ) −→ Si (i = 1, 2) give the same
+σ-Weyl marking, namely [S1, w1] = [S2, w2], if and only if there exist an oriented homeo-
+morphism �f : S1(Φ1) −→ S2(Φ2) which is a lift of an analytic isomorphism f : S1 −→ S2,
+oriented homeomorphisms �wi : Σg −→ Si(Φi) (i = 1, 2) and an action α of the Dehn twists
+Σg −→ Σg (α ∈ Γ(σ)) such that the following diagram is homotopically commutative
+Σg
+�w1 �
+α
+�
+S1(Φ1)
+�f
+�
+π(Φ1) � S1
+f
+�
+Σg
+�w2 � S2(Φ2)
+π(Φ2) � S2.
+Proof
+The assertion is clear from (10) and Corollary 2.9.
+We denote by T(σ) = �
+w:σ-Weyl marking[S, w] the set of σ-marked stable curves. For a
+face ρ < σ, we denote by T(ρ) = �
+w:ρ-Weyl marking[S, w] the set of ρ-marked stable curves
+similarly. If ρ = ∅, then T(∅) = Tg = �
+w:∅-Weyl marking[S, w] is the Teichm¨uller space, since
+a ∅-Weyl marking [S, w] is nothing but a Riemann surface S with a usual Teichm¨uller
+marking w.
+We have the following real structure on Tg
+�
+ρ<σ T(ρ) as a subspace of the augmented
+Teichm¨uller space �Tg (cf. [77, Chap.4]). We fix a maximal simplex
+˜σ = ⟨C1, · · · , Ck, Ck+1, · · · , C3g−3⟩
+20
+
+containing σ = ⟨C1, · · · , Ck⟩, and let (ℓ1, · · · , ℓ3g−3, τ1, · · · , τ3g−3) be the associated Fenchel-
+Nielsen coordinates. Wolpert [76] proves that these coordinates, except for the first k twist
+coordinates τ1, · · · , τk, extend to Tg ∪ T(σ) continuously as
+((ℓj, τj), ℓi) : Tg
+�
+ρ<σ
+T(ρ) −→
+3g−3
+�
+j=k+1
+(R>0 × R) ×
+k
+�
+j=1
+(R≥0).
+(26)
+Moreover, if we consider a point p = [S, w] ∈ T(ρ) for a face ρ = ⟨Ci1, · · · , Cis⟩, then
+the nodes of S correspond to ℓij(p) = 0 for 1 ≤ j ≤ s.
+Let M be a universal constant such that two distinct simple closed geodesics on any
+Riemann surface of genus g are disjoint if their lengths are smaller than M ([44], [1, §3.3]),
+which is sometimes called a 2-dimensional Margulis constant. Take a number 0 < ϵ ≤ M
+and fix it throughout the discussion. By using the extended Fenchel–Nielsen coordinates
+(26), we define the subspace Uϵ(σ) of Tg
+�
+ρ<σ T(ρ) by
+Uϵ(σ) = { 0 ≤ ℓi < ϵ (1 ≤ i ≤ k), max{ℓ1. · · · , ℓk} < ℓj (k + 1 ≤ j ≤ 3g − 3) } .
+(27)
+Since T(σ) is defined by ℓj = 0 (1 ≤ j ≤ k), we have a natural inclusion T(σ) ⊂ Uϵ(σ)
+and Γ(σ) naturally acts on Uϵ(σ).
+Definition 4.3. ([54, Def.6.2])
+The quotient space
+Dϵ(σ) = Uϵ(σ)/Γ(σ)
+is called the controlled deformation space with respect to the simplex σ.
+These spaces are considered as refinements of Bers’ deformation spaces ([15]), and
+have the following properties:
+(I) (topological properties) Dϵ(σ) is a Hausdorff topological space and the Weyl group
+W(σ) acts on Dϵ(σ) as the change of the marking
+[S, w] −→ [S, w ◦ ϕ−1]
+for ϕ ∈ W(σ).
+(28)
+This action is properly discontinuous ([54, Lemmata 6.1 ∼ 6.4]).
+(II) (analytic properties) Dϵ(σ) has a complex structure on which W(σ) acts holomorphi-
+cally, and the quotient space Mϵ(σ) = Dϵ(σ)/W(σ) has an analytic open embedding into
+M g ([54, Lemmata 6.5 ∼ 6.8]).
+The property (I) depends on the real analytic augumented Teichm¨uller theory (cf.[1])
+and some Weil–Petersson geometry (cf. [76], [78]). Note that the delicate condition (27)
+for the geodesic lengths ℓi comes from the determination of the region on which W(σ) acts
+properly discontinuously (see [54, Remark 6.3]). The property (II) is essentially comes
+from Hubbard–Koch’s theorem [37, Th.10.1]. In the next subsection, we add a stronger
+analytic property.
+21
+
+4.2
+Kuranishi families over controlled deformation spaces
+In this subsection, we first intoduce the complex structure on Dϵ(σ) by a method which is
+independent of Hubbard-Koch’s theorem [37, Th.10.1], and construct an analytic family
+over Dϵ(σ) by patching Kuranishi families of stable curves. The basic idea of the method
+here is due to Arbarello–Cornalba [8, §§3,4] and Arbarello–Cornalba–Griffiths [9, §8], but
+our proof of the following theorem will be rather direct using the topological properties
+(I) of Dϵ(σ) in §4.1.
+Theorem 4.4. There exist a (3g − 2)-dimensional complex manifold Xϵ(σ), a complex
+structure Dϵ(σ) = �
+i∈I Bi and a holomorphic map
+πσ : Xϵ(σ) −→ Dϵ(σ)
+such that each restricted family πσ|Xi : Xi = (πσ)−1(Bi) −→ Bi coincides with a standard
+Kuranishi family of a stable curve (πσ)−1(bi) for some bi ∈ Bi.
+The Weyl group W(σ) acts on πσ holomorphically and properly discontinuously so that
+Xϵ(σ)
+Dϵ(σ)
+Yϵ(σ) ∼= Xϵ(σ)/W(σ)
+Mϵ(σ) ∼= Dϵ(σ)/W(σ).
+πσ
+πσ
+ϕσ
+ψσ
+is a commutative diagram, where ϕσ and ψσ are the quotient holomorphic maps to the
+normal analytic spaces Yϵ(σ) and Mϵ(σ), and πσ is the induced holomorphic map.
+Moreover Mϵ(σ) has an holomorphic open embedding into M g.
+We need the following:
+Lemma 4.5. Let p = [S, w] ∈ Dϵ(σ)∩T(σ) be a point, and let ψ : X −→ B be a standard
+Kuranishi family of ψ−1(b0) = S with a sufficiently small base B. Then there exists a
+natural topological open embedding
+ιB : B −→ Dϵ(σ).
+Proof
+Step 1
+We set S = ψ−1(b0). Let �w : Σg −→ S(Φ0) be the lift of w with a
+certain twist Φ0. By Theorem 3.1, there exists a point e0 ∈ τ −1
+B (b0) such that S(Φ0) is
+canonically identified with the fiber
+(ψlog)−1(e0) = S(Φ0).
+(29)
+We consider the stable curve XP = ψ−1(P) for any P ∈ B, and choose a point eP ∈
+τ −1
+B (P). Then we have
+(ψlog)−1(eP) = XP(ΦP)
+(30)
+22
+
+with a certain twist ΦP. By the connectivity of Blog, we can choose a smooth path Le0eP
+connecting the points e0 and eP on Blog. The family ψlog : Xlog −→ Blog is a differentiable
+family of Riemann surfaces over a manifold with corners. Usui’s theorem [72] which is
+an extension of Ehresmann’s theorem ([24]) to this situation states that there exists a
+diffeomorphism
+ϕe0eP : (ψlog)−1(e0) −→ (ψlog)−1(eP)
+(31)
+by connecting diffeomorphisms of fibers along Le0eP . From (29), (30) and (31), we have a
+diffeomorphism
+�wP := ϕe0eP ◦ �w : Σg −→ XP(ΦP).
+(32)
+Note that �wP is uniquely determined modulo the action of Γ(σ), independently of the
+choices of Φ0, eP and Le0eP . Therefore, from Lemma 4.2 and (32), we have a ρ-Weyl
+marking
+wP : Σg(ρ) −→ XP
+(33)
+where the face ρ (possibly, ρ = ∅, σ) is determined by the stratum of B which contains
+P. Since the extended Fenchel–Nielsen coordinates (26) moves continuously ([76]), the
+length condition (27) is satisfied for [XP, wP]. Hence the continuous map
+ιB : B ∋ P �−→ [XP, wP] ∈ Dϵ(σ)
+(34)
+is well-defined.
+Step 2
+We will prove that the map ιB is injective. We assume ιB(P1) = ιB(P2)
+for some P1, P2 ∈ B. By definition, the stable curves XPi = ψ−1(Pi) (i = 1, 2) have
+ρ-Weyl markings wi : Σg(ρ) −→ XPi for some ρ such that there exists an isomorphism
+gP1P2 : XP1 −→ XP2 satisfying w2 ≃ gP1P2 ◦ w1 (homotopic).
+It follows from the property (v) of a standard Kuranishi family, stated in §3.1, that
+there exist an automorphism gb0 : Xb0 −→ Xb0 (Xb0 = S) and a relative automorphism of
+the family
+X
+gX �
+ψ
+�
+X
+ψ
+�
+B
+gB
+� B
+(35)
+such that gb0 and gP1P2 are induced by the restriction to the fibers of the automorphism
+gX. The stable curves XP1 and XP2 have the same topological type Σg(ρ). Let B(ρ) be
+the stratum in B such that the fibers of ψ over B(ρ) are of type Σg(ρ).
+Let L1 be a real smooth path on B connecting the points P1 and b0 such that L∗
+1 =
+L1 \ {b0} is contained in B(ρ). Then the real proper transform �L1 of L1 on Blog via the
+map τB : Blog → B is defined as follows.
+23
+
+Assume that dim ρ = k0 − 1 and dim σ = k − 1 (k0 ≤ k). Let (U; z1, · · · , z3g−3)
+be local coordinates at b0 of B such that B(ρ) ∩ U = {z1 = · · · = zk0 = 0, zk0+1 ̸=
+0, · · · , z3g−3 ̸= 0} and b0 = {(z1, · · · , zk, zk+1, · · · , z3g−3) = (0, · · · , 0, ck+1, · · · , c3g−3)} for
+some ck1+1, · · · , c3g−3 ∈ C∗. Assume that L1 ∩ U is defined by
+zi = 0,
+zj = (1 − t)rj(t)exp(
+√
+−1αj(t)),
+zℓ = cℓ
+for 1 ≤ i ≤ k0, k0 + 1 ≤ j ≤ k, k + 1 ≤ ℓ ≤ 3g − 3, where rj(t)’s are non-vanishing
+smooth functions and αj(t)’s are smooth functions with respect to a variable t ∈ [δ, 1]
+(0 ≤ ∃δ < 1). On the other hand, as stated in §3.2, τ −1
+B (U) is defined by
+{(z1, · · · , z3g−3, θ1, · · · , θk) ∈ U × (S1)k ; zj = |zj|e
+√−1θj, 1 ≤ j ≤ k}
+by identifying S1 = R/2πZ. We define the smooth section (s1)|loc : L1 ∩ U → τ −1
+B (U) by
+θj = 0 (1 ≤ j ≤ k0),
+θj = αj(t) (k0 + 1 ≤ j ≤ k)
+using the fiber coordinates of U × (S1)k → U. By the similar arguments for other charts
+on B and patching them, we obtain the smooth section s1 : L1 → Blog. Then we set
+�L1 = s1(L1). The restriction map (τB)|�L1 : �L1 → L1 is a diffeomorphism by definition.
+Now we set
+L2 = gB(L1).
+(36)
+Then L2 is a real smooth path on B connecting P2 and b0 such that L∗
+2 = L2 \ {b0}
+is contained in B(ρ). Let ϕi : [0, 1] → Li (i = 1, 2) be the parametrization such that
+ϕi(0) = Pi, ϕi(1) = b0 and ϕ2(t) = gB◦ϕ1(t) (t ∈ [0, 1]). By the restriction of gX : X → X
+in (35) to the fibers, we have the family of isomorphisms
+{gt = (gX)|Xϕ1(t) : Xϕ1(t) −→ Xϕ2(t)}0≤t≤1
+(37)
+such that g0 = gP1P2 and g1 = gb0.
+Let �L2 = s2(L2) be the real proper transform of L2 via τB given by the smooth
+section s2 : L2 → Blog. Let �ϕi = si ◦ ϕi : [0, 1]
+ϕi
+−→ Li
+si
+−→ �Li be the parametrization.
+By the definition of the real proper transform, or the argument in Proposition 2.6 and
+Theorem 3.1, there exists the homeomorphism �gt : Xlog
+�ϕ1(t) = (ψlog)−1(�ϕ1(t)) −→ Xlog
+�ϕ2(t) =
+(ψlog)−1(�ϕ2(t)) with the commutative diagram
+Xlog
+�ϕ1(t)
+�gt
+�
+τX
+�
+Xlog
+�ϕ2(t)
+τX
+�
+Xϕ1(t)
+gt
+� Xϕ2(t)
+(38)
+where we use the same symbol τX as the restriction to the fibers of τX : Xlog → X.
+24
+
+First we consider the case t = 0 in (38). By the same reasoning as above, the ρ-
+marking wi : Σg(ρ) → XPi and the homotopy relation w2 ≃ g0 ◦ w1 are lifted to the
+homeomorphism �wi : Σg → Xlog
+�ϕi(0) and the homotopy relation �w2 ≃ �g0 ◦ �w1.
+Secondly we consider (38) for any t ∈ [0, 1]. By applying [24] to the family ψlog, we
+have the diffeomorphism f�ϕi(0)�ϕi(t) : Xlog
+�ϕi(0) → Xlog
+�ϕi(t) along the path �Li connecting �ϕi(0)
+and �ϕi(t). Let �wi(t) = f�ϕi(0)�ϕi(t) ◦ �wi : Σg → Xlog
+�ϕi(t) be the composite homeomorphism. By
+the commutativity of (38), we obtain the family of the homotopy relations
+{ �w2(t) ≃ �gt ◦ �w1(t)}0≤t≤1.
+(39)
+Lastly we consider the case t = 1. The homeomorphism �wi(1) : Σg → Xlog
+�ϕi(1) descends
+to the homeomorphism wi(1) : Σg(σ) → Xb0 via the contraction maps contσ : Σg → Σg(σ)
+and (τX)|Xlog
+�
+ϕi(1) : Xlog
+�ϕi(1) → Xb0. Then the relation (39) for t = 1 descends to
+w2(1) ≃ gb0 ◦ w1(1)
+(homotopic).
+(40)
+From Lemma 4.1 and (40), the automorphism gb0 is the identity map of S. Hence P1 = P2,
+i.e. the injectivity of the map ιB is proved.
+Step 3
+By shrinking B′ ⊂ B preserving the properties (i) ∼(v) in §3.1, we may
+assume that the map ιB′ : B′ −→ Dϵ(σ) for the closure B′ in B is injective. Since B′
+is compact and Dϵ(σ) is Hausdorff, ιB′ is homeomorphic onto its image. Therefore the
+assertion of Lemma 4.5 follows for B′.
+Proof of Theorem 4.4 :
+Step 1
+Let p = [S, w] ∈ Dϵ(σ) be a point, and let ψ : X −→ B be a standard
+Kuranishi family of S. By the same argument as in Lemma 4.5, there exist an open
+neighborhood Up of p in Dϵ(σ) and a homeomorphism ιB : B −→ Up. Let
+Dϵ(σ) =
+�
+p∈Dϵ(σ)
+Up
+(41)
+be the open covering by Up’s of these types.
+By identifying Up with B, the complex
+structures {B} induce a complex structure on Dϵ(σ).
+In fact, the coordinate transformations are biholomorphic as follows. Let Uq = B′
+be another base of a standard Kuranishi family of S′ for q = [S′, w′] ∈ Dϵ(σ) such that
+Up ∩ Uq ̸= ∅. Since the Kodaira-Spencer map at any point r ∈ Up ∩ Uq = B ∩ B′ is an
+isomorphism by the property (iii) of the Kuranishi family, stated in §3.1, B ∩ B′ is a base
+of a Kuranishi family of the stable curve ψ−1(r) (cf. [9, Chap.XI, Cor.(4.6)]). By the
+uniqueness of the Kuranishi family modulo isomorphism, the coordinate transformations
+are nothing but these isomorphisms. Note that Dϵ(σ) is a Hausdorff space. Therefore,
+(41) defines the desired complex structure on Dϵ(σ).
+25
+
+These standard Kuranishi families {ψp : Vp = X(p) −→ Up = B(p)}p∈Dϵ(σ) are patched
+together, and define a complex manifold Xϵ(σ) and a holomorphic map
+πσ = ∪ψp : Xϵ(σ) =
+�
+Vp −→
+�
+Up = Dϵ(σ).
+These are also well-patched by the uniqueness of Kuranishi families modulo isomorphisms.
+Therefore, we have constructed the desired family.
+Since the action of ϕ ∈ W(σ) on Dϵ(σ) is given by (28), ϕ sends the open neighborhood
+Up = B(p) of p = [S, w] which is the base of the standard Kuranishi family of S to an
+open neighborhood of Uϕ(p) = B(ϕ(p)) of ϕ(p) = [S, w ◦ ϕ−1]. This is nothing but the
+isomorphism of the bases of Kuranishi families, and is holomorphic. Hence ϕ acts on
+Dϵ(σ) holomorphically.
+Step 2
+It suffices to prove that W(σ) acts on Xϵ(σ) properly discontinuously, i.e.
+(i) let x′, x′′ ∈ Xϵ(σ) be points such that g(x′) ̸= x′′ for an element g ∈ W(σ). Then there
+exist open neighborhoods U ′ and U ′′ of x′ and x′′ respectively such that g(U ′) ∩ U ′′ = ∅,
+(ii) the isotropy sub-group Gx (⊂ W(σ)) of a point x ∈ Xϵ(σ) is finite,
+(iii) there exists a Gx-invariant neighborhood U of x such that, if g(U) ∩ U ̸= ∅ for some
+g ∈ W(σ), then g ∈ Gx.
+Since W(σ) acts on Dϵ(σ) properly discontinuously by [54, Lemma 6.4], the assertion
+(i) is clear in the case where x′ and x′′ belong to distinct fibers of πσ. Assume that x′ and
+x′′ belong to the same fiber of πσ, say x′ = [[S], w′] and x′′ = [[S], w′′] with an isomorphism
+class [S] ∈ M g and distinct markings. Then, after forgetting the markings, W(σ) acts on
+S as the analytic automorphism group of S which is finite. Therefore the assertion (i)
+holds.
+The assertion (ii) is clear, since Gx is a subgroup of the isotropy sub-group of πσ(x)
+by the action of W(σ) on Dϵ(σ), which is finite by [54, Lemma 6.4].
+By the first half of this theorem, there exists an open neighborhood V of πσ(x) in Dϵ(σ)
+such that π−1
+σ (V ) −→ V is a standard Kuranishi family of the stable curve π−1
+σ (πσ(x)).
+Therefore the assertion (iii) follows from the property (v) stated in §3.1. Hence W(σ)
+acts on Xϵ(σ) properly discontinuously, and the quotient space Xϵ(σ)/W(σ) is a normal
+analytic space by Cartan’s theorem ([18]).
+The holomorphic embedding of Dϵ(σ)/W(σ) into M g follows from the above holomor-
+phy and the property (II) stated in §4.1.
+4.3
+Orbifold fiber space over the Deligne-Mumford compactifi-
+cation
+In this subsection, as a globalization of Theorem 4.4, we construct an orbifold fiber space
+(or orbifold fibration for short) πg : X
+orb
+g
+−→ M
+orb
+g
+so that the Bers fiber space ([14]) over
+26
+
+Teichm¨uller space is contained as an open chart of πg.
+By a complex orbifold M, we mean here that M is a normal complex space such that M
+is covered by (an atlas of) orbifold charts {(�Ui, Gi, ϕi, Ui)}i∈I which satisfy the following
+conditions:
+(i) Each chart consists of a complex manifold �Ui, a (not necessary finite) group Gi acting
+on �Ui holomorphically and properly discontinuously (admitting non-effective action), an
+open set Ui of M and a folding map ϕi : �Ui −→ Ui which induces a natural analytic
+isomorphism �Ui/Gi −→ Ui.
+(ii) (compatibility condition) For x ∈ �Ui and y ∈ �Uj such that ϕi(x) = ϕj(y) ∈ Ui ∩ Uj,
+there exists a biholomorphic map ϕx,y : �U ′
+x −→ �U ′
+y from an open neighborhood of x in �Ui
+to an open neighborhood of y in �Uj such that ϕx,y(x) = y and ϕj ◦ ϕx,y = ϕi.
+For simplicity, we sometimes write this orbifold structure by M = {(Ui; �Ui, Gi)}i∈I or
+M = �
+i∈I �Ui/Gi if there is no fear of confusion.
+Let M ′ be another complex orbifold with its orbifold charts {(�Vk, Hk, φk, Vk)}k∈K. By
+an orbifold map h : M ′ → M, we mean that h is a holomorphic map of normal complex
+spaces which satisfies the following conditions:
+(i) For any points p ∈ M ′ and h(p) ∈ M, let (�Vk, Hk, φk, Vk) be a small orbifold chart
+of M ′ with p ∈ Vk and (�Ui, Gi, ϕi, Ui) be that of M with h(p) ∈ Ui. Then there exists a
+holomorphic map �hki : �Vk → �Ui such that ϕi ◦ �hki = h|Vk ◦ φk.
+(ii) (compatibility condition) Assume that x ∈ �Vk, y ∈ �Vℓ,�hki(x) ∈ �Ui,�hℓj(y) ∈ �Uj such
+that φk(x) = φℓ(y) and ϕi(�hki(x)) = ϕj(�hℓj(y)). Then there exist open neighborhoods
+�Vx, �Vy, �U�hki(x), �U�hℓj(y) of x, y,�hki(x),�hℓj(y) in �Vk, �Vℓ, �Ui, �Uj respectively and biholomorphic
+maps φx,y : �Vx → �Vy and ϕh(x),h(y) : �U�hki(x) → �U�hℓj(y) such that ϕh(x),h(y) ◦ �hki|�Vx =
+�hℓ,j|�Vy ◦ φx,y.
+Moreover, we define the following:
+Definition 4.6. Let f : X −→ M be an orbifold map of complex orbifolds of relative
+dimension ≥ 1 (i.e. dimCX ≥ dimCM +1). We say that f has a structure of an orbifold
+fibration if the following conditions hold.
+(i) Let {(�Ui, Gi, ϕi, Ui)}i∈I be the orbifold charts of M. For each i ∈ I, there exists a nor-
+mal complex space �
+Wi and a holomorphic map �fi : �
+Wi −→ �Ui, and Gi acts holomorphically
+and relatively on �fi such that the diagram
+�
+Wi
+�Ui
+f −1(Ui) ∼= �
+Wi/Gi
+Ui ∼= �Ui/Gi.
+�fi
+f
+ϕi
+ψi
+is commutaive, where ψi is the projection map by Gi.
+27
+
+(ii)
+Each �
+Wi is an open complex sub-orbifold of X in the following sense: �
+Wi has the
+orbifold charts {(�
+Wi,j, Hi,j, ϕi,j, �
+Wi,j)}j∈J(i) (the set J(i) of suffixes depends on i), and
+there exist a group Gi,j containing Hi,j as a normal subgroup and an exact sequence of
+groups
+1 −→ Hi,j −→ Gi,j −→ Gi −→ 1,
+such that the orbifold charts of X are given by {(�
+Wi,j, Gi,j, ψi◦ϕi,j, ψi◦ϕi,j(�
+Wi,j))}i∈I,j∈J(i).
+The typical example of an orbifold fibration will be given in §6.2. Note that the set
+{(�
+Wi, Gi, ψi, f −1(Ui))}i∈I
+(42)
+is not an atlas of orbifold charts of X in general, because �
+Wi may have singularities.
+Nevertheless it is important in our discussion in §6.2. See Definition 4.7 (i).
+Definition 4.7. (i)
+We call (42) the pseudo-orbifold charts of X with respect to the
+orbifold fibration f : X → M.
+(ii) If �
+Wi is nonsingular for each i, i.e. if the set (42) gives an atlas of orbifold charts of
+X, we call f : X → M a strong orbifold fibration.
+Now the theorem ([54, Th.6.11]) says that the Deligne-Mumford compactification M g
+has the orbifold charts
+{(Dϵ(σ), W(σ), ϕσ, Mϵ(σ))}σ∈Cg/Γg,
+(43)
+where σ moves in the curve complex modulo the action of the mapping class group Γg.
+From now on, since this orbifold structure is different from the usual one (cf. [9, Chap.XII])
+of M g, we use the symbol M
+orb
+g
+in order to specify it. Over this base structure, we have
+the following strong orbifold fiber space.
+Theorem 4.8. There exists a strong orbifold fibration
+π : Y
+orb
+g
+−→ M
+orb
+g
+(44)
+such that the orbifold charts of M
+orb
+g
+are given by (43), and those of Y
+orb
+g
+are
+{(Xϵ(σ), W(σ), ψσ, Yϵ(σ))}σ∈Cg/Γg
+(45)
+given in Theorem 4.4.
+Proof
+It suffices to prove the compatibility condition (ii) of Def.4.7. Let [S] ∈ M g
+be an isomorphism class, and let pi = [S, wi], wi : Σg(σ) −→ S (i = 1, 2) be two marked
+stable curves such that the point pi belongs to Dϵ(ρi) for ρi ≤ σ. By Theorem 4.4, there
+28
+
+exists an open neighborhood Ui ⊂ Dϵ(ρi) of pi such that the restricted family πi : Xi −→
+Ui of πσ : Xϵ(ρi) −→ Dϵ(ρi) over Ui coincides with the standard Kuranishi family of
+S. By the uniqueness of the standard Kuranishi family modulo isomorphism, there exist
+biholomorphic maps h : X1 −→ X2 and h : U1 −→ U2 which satisfy h ◦ π1 = π2 ◦ h after
+a suitable shrinking of Ui. Therefore, the desired compatibility condition is satisfied.
+The other required properties for the strong orbifold fibration are given in Theorem
+4.4. In particular, the normal analytic structure of Y
+orb
+g
+is given by patching those of
+Yϵ(σ)’s obtained by Cartan’s theorem via the above local biholomorphic maps.
+Definition 4.9. We call π : Y
+orb
+g
+−→ M
+orb
+g
+the universal degenerating family of Riemann
+surfaces of genus g.
+We believe that the family π is universal in the sense that every orbifold fibration
+with a Riemann surface of genus g as a general fiber can be pulled back from this univer-
+sal orbifold fibration. Our present achievement is, however, rather modest, and we have
+proved the universality of π only for fibered complex surfaces, namely, for orbifold fibra-
+tions whose base spaces are of dimension 1. See §7. For a precise definition of “orbifold
+pull-back”, see the following.
+Definition 4.10. Let f : X → M and f ′ : X′ → M ′ be orbifold fibrations and h :
+M ′ → M be an orbifold map. We say that f ′ is the orbifold pull back from f via h if the
+following conditions are satisfied:
+(i) Let {(�Ui, Gi, ϕi, Ui)}i∈I and {(�Vj, Hj, φj, Vj)}j∈J be the orbifold charts of M and M ′
+respectively. For any j ∈ J, there exist some i ∈ I depending on j and a holomorphic
+map hji = h|Vj : Vj → Ui. Let �hji : �Vj → �Ui be the lifting of hji and ϕ(j)
+i
+= ϕi|�U(j)
+i
+: �U (j)
+i
+=
+�hji(�Vj) −→ U (j)
+i
+= hji(Vj) be the restriction map to the image, i.e.
+�Vj
+�Vj/Hj ∼= Vj
+�U (j)
+i
+⊂ �Ui
+U (j)
+i
+⊂ Ui ∼= �Ui/Gi
+φj
+ϕ(j)
+i
+hji
+�hji
+is a commutative diagram. Then there exists an injective group homomorphism
+Hj �→ Gi
+(46)
+such that the subgroup Hj of Gi acts on �U (j)
+i
+holomorphically.
+(ii)
+There exists a lifted orbifold map k : X′ → X of h, i.e. f ◦ k = h ◦ f ′, which
+has the following property: Let {(�
+Wi, Gi, ψi, f −1(Ui))}i∈I and {( �Tj, Hj, ωj, (f ′)−1(Vj))}j∈J
+be the pseudo-orbifold charts of X and X′ respectively. Let �kji : �Tj → �
+Wi be the lifting
+of kji = k|(f′)−1(Vj) : (f ′)−1(Vj) −→ f −1(Ui), and ψ(j)
+i
+= ψi|�
+W (j)
+i
+: �
+W (j)
+i
+= �kji( �Tj) =
+29
+
+ψ−1
+i (�U (j)
+i ) −→ �U (j)
+i
+be the restriction map. Then the group Hj relatively acts on ψ(j)
+i
+such
+that the diagram
+�Tj
+�Vj
+�
+W (j)
+i
+⊂ �
+Wi
+�U (j)
+i
+⊂ �Ui
+ωj
+ψ(j)
+i
+kji
+�kji
+expresses the fiber product compatible with the action of Hj, i.e.
+�Tj is isomorphic to
+�
+W (j)
+i
+�U(j)
+i
+�Vj such that α ◦ �kji = kji ◦ α for any α ∈ Hj.
+5
+Automorphisms of stable curves and cyclic equi-
+symmetric strata on M
+orb
+g
+We study an automorphism ϕ of a stable curve S and the associated cyclic branched
+covering from several points of view. In particular, we define the numerical data Num(ϕ)
+which consist of two kinds of data, i.e. the action on the dual graph of S and the system of
+branch data of cyclic branched coverings which every irreducible component of S naturally
+has. We study the locus T [ϕ]
+σ
+on the boundary charts of M
+orb
+g
+consisting of marked stable
+curves with the automorphisms of this type of numerical data, and describe its structure
+in terms of pointed Teichm¨uller spaces of lower genera (Theorem 5.18). This result is an
+extension of Harvey–Broughton’s theorem about the equisymmetric strata on Tg and Mg,
+to the boundaries for cyclic groups.
+In §5.1, we review some known results about automorphisms of Riemann surfaces.
+First, we review the notion of total valency essentially due to Nielsen [59] and Harvey
+[30], whic provides precise information on the branch data of the cyclic covering associated
+with the automorphism. Secondly, we review Harvey–Broughton’s equisymmetric strata
+on Tg and Mg ([31], [16]) in the case of cyclic groups.
+In §5.2, as a modified discussion of Eichler’s trace formula, we propose a method
+to determine the characters of the representation of automorphisms of pointed Riemann
+surfaces into the space of logarithmic quadratic differential forms.
+In the terminology of augmented Teichm¨uller theory, the true boundary T(σ) on the
+boundary chart of M
+orb
+g
+should be called the little Teichm¨uller space. In §5.3, we interpret
+this notion by the cohomological terminologies of Kuranishi spaces of stable curves.
+The aim of §5.4 is to define Num(ϕ) naturally. The point is to analyze the cyclic
+branched covering πϕ : S → W associated with ϕ. Although the base W is a nodal
+Riemann surface, the dual graph of W is not a simple quotient graph of the dual graph of
+S. In fact, we extend the notion of graphs to those admitting open edges, and then define
+30
+
+the notion of compact quotient graphs as the desired one. By combining these data of
+the graphs and the system of the total valencies which every component of S naturally
+has, we have the definition of Num(ϕ).
+In §5.5, we prove the structure theorem of the equisymmetric strata T [ϕ]
+σ
+on T(σ). The
+space T [ϕ]
+σ
+is described locally by the Kuranishi space of each normalized component of
+W, and globally by the pointed Teichm¨uller spaces. Our basic method is, in the case of
+Riemann surfaces, closely related to the one in [73] or [57, §§4,5]. In the case of stable
+curves, the pioneering work of Terasoma [71] already shows the connectivity of the moduli
+space M
+[ϕ]
+g
+in our teminology by using the level structures.
+In §5.6, we define a special system of local coordinates around a point of T [ϕ]
+σ . These
+coordinates consist of the eigenvectors for the action of ϕ on the standard chart of the
+Kuranishi space of S. Harris–Mumford [29, §1] intrinsically used this type of coordinates
+systematically for a certain local analysis of M g.
+5.1
+Automorphisms of Riemann surfaces and equisymmetric strata
+In this subsection, we review some results about the automorphisms of Riemann surfaces,
+the equisymmetric stratification on Teichm¨uller space Tg and the moduli space Mg ([31],
+[16]).
+We consider a Fuchsian group F, i.e. a discrete subgroup of Aut(H) ∼= PSL(2, R)
+where H is the upper half plane, such that H/F is compact. As an abstract group, F is
+generated by a1, b1, · · · , ag, bg, x1, · · · , xs with the relations
+x1 · · · xs
+g�
+i=1
+[ai, bi] = 1,
+xλi
+i = 1 (1 ≤ i ≤ s).
+The ordered set (g; λ1, · · · , λs) is called the signature of F. We assume that there exists
+a Fuchsian group K with signature (g, −) (where − means that {x1, · · · } is empty) and
+exact sequence
+1 → K
+i∗−→ F
+j∗
+−→ Gϕ → 1
+(47)
+such that Gϕ = ⟨ϕ⟩ ∼= Z/nZ is the cyclic group of order n generated by ϕ and i∗(K) is
+a normal subgroup of F. In this case, Gϕ is geometrically characterized as follows. The
+Riemann surface S ∼= H/K of genus g has an analytic automorphism ϕ : S −→ S of order
+n such that the associated branched covering πϕ : S −→ W = S/Gϕ has the following
+properties: The genus of W coincides with g. Let P1, · · · .Ps ∈ W be the branch points for
+πϕ. Then λi (1 ≤ i ≤ s) coincides with the ramification index at the point �Pi ∈ π−1
+ϕ (Pi).
+The map ϕn/λi fixes �Pi and rotates the disc neighborhood of �Pi by the angle 2πδi/λi. Let
+1 ≤ σi ≤ λi − 1 be the natural number with σiδi ≡ 1 (mod λi). The triple (mi, λi, σi)
+31
+
+is called the valency of ϕ at �Pi ([59], [55, Def. 1.5]), and sometimes is written by σi/λi.
+Then:
+(a1) (Hurwitz formula)
+2(g − 1)/n = 2(g − 1) + �s
+i=1(1 − 1/λi),
+(a2) (Nielsen [59, (4.6)])
+�s
+i=1 σi/λi is an integer,
+(a3) (Wiman [74])
+n ≤ 4g + 2,
+(a4) (Harvey [30, Th.4])
+We set M = lcm(λ1, · · · , λs). Then
+(a4-1)
+lcm(λ1, · · · , �λi, · · · , λs) = M for all i, where �λi denotes the omission of λi.
+(a4-2) M divides n, and if g = 0, then M = n.
+(a4-3) s ̸= 1, and, if g = 0, then s ≥ 3.
+(a4-4) If 2|M, the number of λ1, · · · , λs which are divisible by the maximal power of
+2 dividing M is even.
+We symbolically write these data by
+TV(ϕ) =
+�
+g, g, n : σ1
+λ1
++ · · · + σs
+λs
+�
+, TVc(ϕ) =
+�
+g, g, n :
+� δ1
+λ1
+, · · · , δs
+λs
+��
+(48)
+and call TV(ϕ) (resp.TVc(ϕ)) the total valency (resp. the total co-valency) of ϕ. Note
+that the total (co-)valency is determined by j∗ of (47) from [31, Th.7, Lem.6].
+Remark 5.1. (i) The valency was introduced by Nielsen [59] as the unique essential
+conjugacy invariant of periodic maps (i.e. finite automorphisms) in Γg. A periodic map
+is realized as an analytic automorphism of a certain complex structure on Σg by a standard
+augument or as a corollary of Kerchoff [45]. Therefore, the total valency may be considered
+as an invariant of both periodic maps in Γg and anlytic automorphisms.
+(ii) Even if g = 0, 1, a finite automorphism ϕ satisfies the conditions (a1) ∼ (a3) and we
+use the same terminology (48) in these cases.
+(iii) For the classification of TV(ϕ) in the case where g = 2, 3, see e.g. [11, p.199].
+As discussed in [31], the choice of the inclusion map i∗ in (47) determines the Te-
+ichm¨uller marking, and the choice of the surjective map j∗ determines the generators of
+Gϕ, which determine the data (a2) in turn.
+The conditions (a1) ∼ (a4) are not only necessary conditions but also sufficient condi-
+tions for the existence of automorphisms. This point is widely discussed from the moduli
+theoretic viewpoint by Harvey and Broughton as follows. We fix the numerical data (48)
+for g ≥ 2, and consider the locus in Tg (or Mg) of Riemann surfaces which have this type
+of automorphisms. Note that, in the following results (I) and (II), the case where g = 2
+and ϕ is the hyperelliptic involution is excluded as an exceptional case:
+(I) ([31, Th.2 and (6) in p.392], [16, Prop.2.5]) Let ϕ : S → S be a periodic map in
+Γg. Let T ϕ
+g be the subset of Tg consisting of the points p = [S, f] ∈ Tg which are fixed
+32
+
+by the action of ϕ. We define two periodic maps ϕ and ψ to be equivalent if their total
+valencies are the same: TV(ϕ) = TV(ψ). As we remarked in Remark 5.1 (i), Nielsen [59]
+proved that this is equivalent to saying that ϕ and ψ are conjugate. Let [ϕ] denote the
+equivalence class to which ϕ belongs, and we define T [ϕ]
+g
+as
+T [ϕ]
+g
+=
+�
+ψ∈[ϕ]
+T ψ
+g .
+Recall that πϕ : S → W = S/Gϕ is the branched covering associated with the analytic
+action ϕ : S → S of order n, and g is the genus of W. P1, · · · , Ps ∈ W are the branch
+points for πϕ. Then T ϕ
+g is real analytically isomorphic to the Teichm¨uller space Tg,s of
+s-pointed Riemann surfaces of genus g. Since T ϕ
+g is a connected component (T [ϕ]
+g )(0) of
+T [ϕ]
+g , we have
+T ϕ
+g ∼= (T [ϕ]
+g )(0) ∼= Tg,s
+(real analytically).
+Each connected component of T [ϕ]
+g
+is T ψ
+g for some ψ ∈ [ϕ], and we can say the same thing
+for T ψ
+g . Thus the space T [ϕ]
+g
+is a countable union of (3g − 3 + s)-dimensional complex
+manifolds so that each of them is real analytically isomorphic to (T [ϕ]
+g )(0).
+(II) ([16, Th.2.2])
+The locus M [ϕ]
+g
+of the isomorphism classes of Riemann surfaces which
+have automorphisms whose total valencies coincide with TV(ϕ) is a closed irreducible
+algebraic subvariety of Mg.
+The loci T [ϕ]
+g
+and M [ϕ]
+g
+are called the equisymmetric strata for [ϕ] of Tg and Mg respec-
+tively. Note that, in the excluded case where g = 2 and ϕ is the hyperelliptic involution,
+we have T [ϕ]
+2
+= T2 and M [ϕ]
+2
+= M2.
+Remark 5.2. For the study of the equisymmetric strata on Tg, Kuribayashi’s method
+([50]) using invariant quadratic differentials is important. Recently, Takamura ([67]) and
+Hirakawa and Takamura ([35]) developed this type of argument and gave a method for the
+detailed analysis of the stratification for various group actions on Riemann surfaces.
+5.2
+Logarithmic quadratic representation of automorphisms
+Here we discuss a representation of an automorphism of a pointed Riemann surface to
+the logarithmic quadratic differentials for later use.
+Let ϕ : S −→ S be an automorphism of a Riemann surface of genus g ≥ 0 with the
+given total (co-)valency (48), and πϕ : S −→ W = S/Gϕ be the associated cyclic branched
+covering (see also Remark 5.1 (ii)). We consider the vector space
+V = H0(S, 2KS +
+s′
+�
+i=1
+π−1
+ϕ (Pi) +
+t
+�
+i=1
+π−1
+ϕ (Ps+i))
+(49)
+33
+
+where {P1, · · · , Ps′} is a subset (it may be empty) of the set of branch points {P1, · · · , Ps′,
+Ps′+1, · · · , Ps} (s′ ≤ s) for πϕ, and {Ps+1, · · · , Ps+t} is a subset of the set of un-branched
+points for πϕ (which is intended to be the set of possible poles, and may be empty). We
+assume
+2g − 2 + s + t > 0.
+(50)
+An element v ∈ V is considered as a logarithmic quadratic differenticial on S whose poles
+are at most in �s′
+i=1 π−1
+ϕ (Pi) + �t
+i=1 π−1
+ϕ (Ps+i). Since ϕ acts on each fiber of πϕ : S → W
+as a permutation of points, ϕ naturally acts on the differential forms v by
+ϕ(v) = v ◦ ϕ−1 ∈ V
+(51)
+(similarly to [25, p.269]), and ϕ induces a linear automorphism of V .
+We call it the
+representation of ϕ to the logarithmic quadratic forms V . Fundamental facts of the rep-
+resentation to holomorphic differential forms discussed in [25, §V2] are easily extended
+to this type of representation to logarithmic forms. For example, its eigenvalues are n-th
+roots of unity.
+The proof of the following Proposition is a slight modification of I. Guerrero’s argument
+written in [25, p.274–277]. For a rational number x, we write [x] the maximal integer not
+exceeding x and {x} = x − [x] its fractional part. We also use the symbol e(x) = e2πix.
+Proposition 5.3. Let ϕ : S −→ S be an automorphism of order n of a Riemann surface
+S with the total (co-)valency (48). Then, under the assumption (50), the dimension of
+the eigenspace of eigenvalue e(α/n) (0 ≤ α ≤ n − 1) for the the action ϕ on V defined by
+(51) is given as follows.
+h0
+�
+S, 2KS +
+s′
+�
+i=1
+π−1
+ϕ (Pi) +
+t
+�
+i=1
+π−1
+ϕ (Ps+i)
+�
+e(α/n)
+= 3g − 3 + 2s + t −
+s′
+�
+i=1
+��−ασi − 1
+λi
+�
++ 1
+λi
+�
+−
+s
+�
+i=s′+1
+��−ασi − 2
+λi
+�
++ 2
+λi
+�
+.
+Proof
+If 1 ≤ i ≤ s′ or s + 1 ≤ i ≤ s + t, then we set ri = 1. If s′ + 1 ≤ i ≤ s,
+then we set ri = 0. We denote the eigenspace of eigenvalue e(α/n) by Vα = H0(S, 2KS +
+�s+t
+i=1 riπ−1
+ϕ (Pi))e(α/n) for simplicity. Moreover, we set
+λi = 1, δi = σi = 0,
+for s + 1 ≤ i ≤ s + t.
+(52)
+For 1 ≤ i ≤ s + t, we denote the fiber by
+π−1
+ϕ (Pi) =
+�
+1≤j≤n/λi
+P (i)
+j .
+34
+
+Assuming Vα ̸= ∅, we fix an element v0 ∈ Vα. Then for any v ∈ Vα, the element v/v0
+is ϕ-invariant, and it is a meromorphic function on W = S/Gϕ. Therefore, there exits a
+divisor Dα on W such that
+Vα ∼= H0(W, Dα).
+(53)
+We determine Dα explicitly. By using a local coordinate z around P (i)
+j , the Laurant
+expansion of v is written as v = �
+k≥0 Akzk−ridz2 for Ak ∈ C. Since the map ϕ−n/λi is
+written here by z �→ e(−δi/λi)z, we have
+(ϕn/λi)∗v =
+�
+k≥0
+Ake
+�−δi(k + 2 − ri)
+λi
+�
+zk−ridz2
+which coincides with �
+k≥0 e(α/λi)Akzk−ridz2 by definition. Therefore, Ak = 0 for −δi(k+
+2 − ri) ̸≡ α (mod λi), i.e. k ̸≡ −ασi − 2 + ri (mod λi). It follows that
+v =
+�
+k∈ZK
+Akzk−ridz2,
+where ZK = {k = λia − ασi − 2 + ri ≥ 0 | ∃a ∈ Z } .
+Let bi be the vanishing order of v0 at P (i)
+j , which is automatically independent of j.
+Then there exists some �bi ∈ Z such that
+bi = λi�bi − ασi − 2 + ri ≥ 0.
+(54)
+Let Q1, · · · , Qu be the images by πϕ of zeros of v0 except for P1, · · · , Ps+t. Put π−1
+ϕ (Qi) =
+�
+1≤j≤n Q(i)
+j . Let βi be the vanishing order of v0 at Q(i)
+j . For a meromorphic function �h on
+W, the element (�h◦πϕ)·v0 belongs to Vα if and only if �h satisfies ordPi�h ≥ −bi/λi, ordQi�h ≥
+−βi and is holomorphic outside Pi’s and Qi’s. Since −ασi − 2 + ri < 0, the integral
+divisor which satisfies these conditions should be Dα = �s+t
+i=1(�bi+[(−ασi − 2 + ri)/λi])Pi+
+�u
+i=1 βiQi. In particular,
+deg Dα =
+s+t
+�
+i=1
+�
+�bi +
+�−ασi − 2 + ri
+λi
+��
++
+u
+�
+i=1
+βi.
+(55)
+We rewrite the expression (55) of degDα so that it is independent of the choice of v0.
+First we clearly have deg v0 = �s+t
+i=1(n/λi)bi + n �u
+i=1 βi. On the other hand, since v0
+belongs to Vα, we have degv0 = deg
+�
+2KS + �s+t
+i=1 riπ−1
+ϕ (Pi)
+�
+= 4(g − 1) + �s+t
+i=1(n/λi)ri.
+Therefore it follows from (54) that
+degv0
+n
+=
+s+t
+�
+i=1
+�
+�bi + −ασi − 2 + ri
+λi
+�
++
+t
+�
+i=1
+βi = 4(g − 1)
+n
++
+s+t
+�
+i=1
+ri
+λi
+.
+(56)
+The Riemann–Hurwitz formula for the covering πϕ says that
+2g − 2
+n
+= 2g − 2 +
+s
+�
+i=1
+�
+1 − 1
+λi
+�
+.
+(57)
+35
+
+Therefore, from (52), (55), (56) and (57), we obtain
+deg Dα =
+s+t
+�
+i=1
+�
+�bi + −ασi − 2 + ri
+λi
+�
++
+t
+�
+i=1
+βi −
+s+t
+�
+i=1
+�−ασi − 2 + ri
+λi
+�
+= 4g − 4 + 2
+s
+�
+i=1
+�
+1 − 1
+λi
+�
++
+s+t
+�
+i=1
+� ri
+λi
+−
+�−ασi − 2 + ri
+λi
+��
+= 4g − 4 + 2s + t −
+s
+�
+i=1
+��−ασi − 2 + ri
+λi
+�
++ 2 − ri
+λi
+�
+.
+(58)
+By (50) and (58), we have deg(KW ⊗ D−1
+α ) ≤ 2 − 2g − s − t < 0. Hence
+H0(W, KW ⊗ D−1
+α ) = 0.
+(59)
+From the Riemann–Roch formula, the Serre duality and (53), (58), (59), we have
+dimVα = dimH0(W, Dα) = degDα−g+1 = 3g−3+2s+t−
+s
+�
+i=1
+��−ασi − 2 + ri
+λi
+�
++ 2 − ri
+λi
+�
+.
+This equality coincides with the desired one.
+We consider (S, P) as a k′-pointed Riemann surface, where P = �s′
+i=1 π−1
+ϕ (Pi) +
+�t
+i=1 π−1
+ϕ (Ps+i) in (49) and k′ = ♯(P) (cardinality), and write the set of the eigenvalues
+for the action of ϕ on the (3g − 3 + k′)-dimensional vector space V = H0(S, 2KS + P) by
+�
+e
+�θ1
+n
+�
+, e
+�θ2
+n
+�
+, · · · , e
+�θ3g−3+k′
+n
+��
+(0 ≤ θ1 ≤ θ2 ≤ · · · ≤ θ3g−3+k′ ≤ n − 1). (60)
+Here each eigenvalue is counted as many times as the dimension of its eigenspace. Then:
+Definition 5.4. By using (60), we define the ordered set of the log-quadratic characters
+for the automorphism ϕ of the pointed Riemann surface (S, P) as
+ChϕH0(S, 2KS + P) =
+�θ1
+n , θ2
+n , · · · , θ3g−3+k′
+n
+�
+.
+(61)
+We also define the ordered set of the eigenbasis {v1. · · · , v3g−3+k′} of V as the basis con-
+sisting of eigenvectors of each of ChϕH0(S, 2KS + P), i.e.
+ϕ∗(vi) = e
+�θi
+n
+�
+vi
+(1 ≤ i ≤ 3g − 3 + k′).
+(62)
+Remark 5.5. Since the dimension of the eigenspace for some eigenvalue might be strictly
+greater than 1, the eigenbasis of V is not unique in general.
+36
+
+Example 5.6. Let ϕ : (S, P) → (S, P) be the automorphism of order 7 of the one-pointed
+genus 3 Riemann surface with the total valency (6/7 + 6/7 + 2/7, ¯g = 0) where 6/7 is
+attached to P. Since s = 3, s′ = 1 and t = 0, it follows from Prop. 5.3 that
+h0 (C, 2KC + P)e(ν/7) = 3−
+��−6ν − 1
+7
+�
++ 1
+7
+�
+−
+��−2ν − 2
+7
+�
++ 2
+7
+�
+−
+��−6ν − 2
+7
+�
++ 2
+7
+�
+= 0, 1, 2, 1, 1, 1, 1
+(ν = 0, 1, 2, 3, 4, 5, 6, respectively).
+Hence ChϕH0(S, 2KS + P) = {1/7, 2/7, 2/7, 3/7, 4/7, 5/7, 6/7}.
+For other examples, see Example 6.15 or [29, §1].
+5.3
+The little Teichm¨uller space in an orbifold chart of M
+orb
+g
+In this subsection, we interpret the little Teichm¨uller spaces in the augumented Te-
+ichm¨uller theory (cf.[37, §7]) by the language of Kuranishi families, using the facts in
+§3.1.
+We follow the notation of §4.1. The maximal codimensional strata T(σ) ⊂ Dϵ(σ) \ Tg
+is written by the extended Fenchel–Nielsen coordinates (26) as
+ℓ1 = · · · = ℓk = 0, ℓj > 0 (k + 1 ≤ j ≤ 3g − 3).
+The space T(σ) is called the little Teichm¨uller space for σ, which is isomorphic to the
+product of lower-dimensional Teichm¨uller spaces of pointed Riemann surfaces complex
+analytically (cf.[37, p.289]). Here we first review the real analytic structure of T(σ), and
+then explain its complex analytic structure from the viewpoint of §3.1.
+Let Σg(σ) = �r
+i=1 Ri be the irreducible decomposition of the source stable curve and
+( ˆRi, ˆPi) (1 ≤ i ≤ r) be the pointed Riemann surfaces obtained from the normalization of
+Σg(σ) as in §2.1. Let gi be the genus of ˆRi and ki = ♯(ˆPi) the number of points. From
+the count of the dimensions of the vector spaces in the exact sequence (20), we have
+r
+�
+i=1
+(3gi − 3 + ki) = 3g − 3 − k.
+Now each member of the curve system ˜σ \ σ = ⟨Ck+1, · · · , C3g−3⟩ may be considered as
+a simple closed curve on a unique member of these pointed Riemann surfaces via the
+normalization map. Let
+{k + 1, · · · , 3g − 3} =
+r�
+i=1
+Ii,
+♯(Ii) = 3gi − 3 + ki
+be the decomposition of suffixes so that {Cij}ij∈Ii is a system of maximal simple closed
+curves on ( ˆRi, ˆPi) which induces a pants decomposition of this surface. Let [S, w] be
+37
+
+a σ-marked stable curve with the marking w : Σg(σ) −→ S and let S = �r
+i=1 Si be
+the irreducible decomposition, ( ˆSi, Pi) (1 ≤ i ≤ r) being the pointed Riemann surfaces
+obtained by the normalization. Then the restriction of w to each component is lifted via
+the normalization to a Teichm¨uller marking
+wi : ( ˆRi, ˆPi) −→ ( ˆSi, Pi).
+(63)
+Then the Fenchel–Nielsen coordinates (lij(p), τij(p)) ∈ (R>0)3gi−3+ki × R3gi−3+ki are de-
+fined for the point p = [( ˆSi, Pi), wi] of the pointed Teichm¨uller space Tgi,ki.
+They
+are the hyperbolic length lij(p) of the unique geodesic in the isotopy class of wi(Cij)
+(ij ∈ Ii) and the twisting parameter τij(p) along Cij. Then the real analytic isomorphism
+T(σ) ∼= �
+1≤i≤r Tgi,ki is induced via these Fenchel–Nielsen coordinates:
+{ℓi([S, w]), τi([S, w])}k+1≤i≤3g−3 �−→
+�
+1≤i≤r
+{(lij([( ˆSi, Pi), wi]), τij([( ˆSi, Pi), wi])}ij∈Ii.
+Over this real structure of T(σ), the complex structure is described as follows. By
+shrinking the base spaces of standard Kuranishi families, it follows from Theorem 4.4
+and (16) that there exists complex coordinate patching Dϵ(σ) = ∪α∈IBα, where each
+Bα := BSα is the base space of a standard Kuranishi family for a σ-marked stable curve
+[Sα, wα] which is embedded in Ext1OSα(Ω1
+Sα, OSα).
+Then T(σ) ∩ Bα is isomorphic to
+H1(Sα, HomOSα(Ω1
+Sα, OSα)) ∩ Bα by (17),(18) and the property (I) in §3.1.
+Let Sα = �r
+i=1 Sα,i be the irreducible decomposition, and ( ˆSα,i, Pα,i) be the pointed
+Riemann surface of Sα,i via the normalization. Then the direct factor Tgi,ki of T(σ) comes
+from the direct factor H1( ˆSα,i, T ˆSα,i(−Pα,i)) of H1(Sα, HomOSα(Ω1
+Sα, OSα)) in (18), i.e.
+Tgi,ki ∩ Bα is isomorphic to H1( ˆSα,i, T ˆSα,i(−Pα,i)) ∩ Bα ∼= H0( ˆSα,i, 2K ˆSα,i + Pα,i)∗ ∩ Bα by
+the property (I) in §3.1. These spaces are globally well-patched by the arguments in [9,
+Chap.XV,§2]. Thus:
+Lemma 5.7. In the above notation, the complex analytic structures of the little Te-
+ichm¨uller space T(σ) is described by the coordinate patchings
+T(σ) =
+�
+α∈I
+�
+H1(Sα, HomOSα(Ω1
+Sα, OSα)) ∩ Bα
+�
+,
+T(σ) ∼=
+�
+1≤i≤r
+Tgi,ki (analytically),
+Tgi,ki =
+�
+α∈I
+�
+H1( ˆSα,i, T ˆSα,i(−Pα,i)) ∩ Bα
+�
+=
+�
+α∈I
+�
+H0( ˆSα,i, 2K ˆSα,i + Pα,i)∗ ∩ Bα
+�
+.
+5.4
+Automorphisms and cyclic branched coverings of stable curves
+In this subsection, we study automorphisms of stable curves and the associated cyclic
+branched coverings from stable curves to nodal Riemann surfaces.
+38
+
+We start from the discussion of graphs and their automorphisms.
+A graph G =
+{vi,⃗ej}1≤i≤r,1≤j≤k is a 1-dimensional finite oriented “open” cell complex embedded in
+Euclidian 3-space E3 in the following sense. A 0-cell vi is a vertex. A 1-cell ⃗ej is an
+oriented edge with one of the following two types. The first type is an ordinary edge
+⃗ej = ⃗ej(vh1(j), vh2(j)) which connects a vertex vh1(j) to a vertex vh2(j) in this direction.
+(Note that vh1(j) = vh2(j) may occur. This case corresponds to a self-intersection of an
+irreducible component.) The second type is an open edge which emanates from a vertex
+vh1(j) such that the end point is a point in E3 which is not contained in the set of vertices,
+and we symbolically write it as ⃗ej = ⃗ej(vh1(j), ∗). The absolute edge |⃗ej| is the usual edge
+obtained by neglecting the orientation of ⃗ej. If all the oriented edges of G are ordinary,
+we call G a compact graph.
+We define the contraction map of G
+cont : G −→ Gc
+as the identity map on the complement of open edges such that any open edge ⃗ej(vh1(j), ∗)
+is contracted to the vertex vh1(j). Then Gc is a compact graph.
+An automorphism ¯ϕ : G → G of a compact graph G means that ¯ϕ is a homeomorphism
+in the Euclidian topology which preserves the set of vertices and the set of absolute edges
+such that ¯ϕn is the identity map for some n. The least natural number n which enjoys
+the above property is called the order of ¯ϕ.
+For a vertex vi (resp. an edge ⃗ej), there exists a minimal natural number m(vi)
+(resp. m(⃗ej)) which satisfies ¯ϕm(vi)(vi) = vi (resp. ¯ϕm(⃗ej)(⃗ej) = ⃗ej).
+If m(⃗ej) is even
+and ¯ϕm(⃗ej)/2 stabilizes |⃗ej| by reversing its orientation, i.e. ¯ϕm(⃗ej)/2(⃗ej) = −⃗ej, then ⃗ej is
+said to be an amphidrome edge for ¯ϕ. Otherwise, ⃗ej is said to be a non-amphidrome edge.
+Let V, NE, AE be the set of vertices, non-amphidrome edges and amphidrome edges of
+G for ¯ϕ, respectively. Let
+V =
+�
+1≤i≤r
+�
+0≤ℓ≤m(vi)−1
+¯ϕℓ(vi),
+NE =
+�
+1≤j≤k1
+�
+0≤ℓ≤m(⃗ej)−1
+¯ϕℓ(⃗ej),
+AE =
+�
+k1+1≤j≤k1+k2
+�
+0≤ℓ≤m(⃗ej)/2−1
+¯ϕℓ(⃗ej)
+be the orbit decompositions for ¯ϕ. Here {v1, · · · , vr}, {⃗e1, · · · ,⃗ek1} and {⃗ek1+1, · · · ,⃗ek1+k2}
+are assumed to belong to mutually distinct orbits in V, NE and AE respectively. We
+have r = �
+1≤i≤r m(vi), k = �
+1≤j≤k1 m(⃗ej) + �
+k1+1≤j≤k1+k2 m(⃗ej)/2.
+Let G ¯ϕ ∼= Z/nZ be the cyclic group generated by ¯ϕ.
+Definition 5.8. (i) The quotient map and the quotient graph of G by G ¯ϕ
+π ¯ϕ : G −→ W := G/G ¯ϕ
+39
+
+are defined by the following two conditions;
+(ia) The vertices and the edges are given by W = {v♯
+i, (⃗ej)♯}1≤i≤r,1≤j≤k1+k2 so that π ¯ϕ( ¯ϕℓ(vi))
+= v♯
+i and π ¯ϕ( ¯ϕℓ(⃗ej)) = (⃗ej)♯ for any i, j, ℓ.
+(ib) Suppose ⃗ej = ⃗ej(vh1(j), vh2(j)). If 1 ≤ j ≤ k1, then (⃗ej)♯ is an ordinary edge given by
+(⃗ej)♯(π ¯ϕ(vh1(j)), π ¯ϕ(vh2(j))). If k1 + 1 ≤ j ≤ k1 + k2, then (⃗ej)♯ is an open edge given by
+(⃗ej)♯(π ¯ϕ(vh1(j)), ∗).
+(64)
+(ii) The compact quotient map and the compact quotient graph of G by G ¯ϕ are defined by
+πc
+¯ϕ = cont ◦ π ¯ϕ : G −→ (W)c = (G/G ¯ϕ)c.
+With respect to (64), we may write (⃗ej)♯(π′
+¯ϕ(vh2(j)), ∗) because ⃗ej is amphidrome in
+this case and vh1(j) and vh2(j) are on the same orbit.
+Example 5.9. We consider the compact graph G = {vi,⃗ej(v1, v2)}1≤i,j≤2. Let ¯ϕ, ¯ψ : G →
+G be automorphisms of order 2 defined as follows. The first ¯ϕ interchanges v1 and v2, and
+stablizes |⃗ej| by reversing their orientations. By using the usual symbols of permutations,
+¯ϕ is written by (v1, v2), (⃗e1, −⃗e1), (⃗e2, −⃗e2). The second ¯ψ interchanges v1 and v2, and also
+⃗e1 and ⃗e2, i.e. ¯ψ is written by (v1, v2), (⃗e1, −⃗e2).
+Then ⃗e1 and ⃗e2 are amphidrome (resp. non-amphidrome) for ¯ϕ (resp. for ¯ψ), and
+we have G/G ¯ϕ = {v♯
+1, (⃗e1)♯(v♯
+1, ∗), (⃗e2)♯(v♯
+1, ∗)}, (G/G ¯ϕ)c = {v♯
+1} (empty edge), G/G ¯ψ =
+(G/G ¯ψ)c = {v♯
+1, (⃗e1)♯(v♯
+1, v♯
+1)} as shown in Figure II.
+v1
+v2
+⃗e1
+⃗e2
+G
+v♯
+1
+(⃗e1)♯
+(⃗e2)♯
+G/G ¯ϕ
+(G/G ¯ϕ)c
+v♯
+1
+G/G ¯ψ = (G/G ¯ψ)c
+v♯
+1
+(⃗e1)♯
+(Figure II) The quotient and the compact quotient graphs in Example 5.9
+Now let S be a stable curve of genus g ≥ 2 with its irreducible decomposition
+S = �r
+i=1 Si, and let P = {P1, · · · , Pk} be its set of nodes. By this configuration, a com-
+pact graph rg(S) = {vSi,⃗ePj}1≤i≤r,1≤j≤k is defined as follows: An irreducible component
+Si corresponds to a vertex vSi. If a node Pj is an intersection point of the irreducible com-
+ponents Sh1(j) and Sh2(j), then it is expressed by an ordinary edge ⃗ePj = ⃗ePj(vSh1(j), vSh2(j))
+(the orientation is arbitrary). We call rg(S) the reduced dual graph of S.
+Let ϕ : S → S be an analytic automorphism of order N, and G = ⟨ϕ⟩ ∼= Z/NZ the
+subgroup of Aut(S) generated by ϕ. Then ϕ clearly induces the automorphism ϕrg(S) :
+rg(S) → rg(S). We translate the terminologies defined for (rg(S), ϕrg(S)) into (S, ϕ). That
+40
+
+is to say, m(Si) and m(Pj) are the minimal natural numbers which satisfy ϕm(Si)(Si) = Si
+and ϕm(Pj)(Pj) = Pj, and Pj is an amphidrome node (resp. a non-amphidrome node) for
+ϕ if ⃗ePj is an amphidrome edge (resp. a non-amphidrome edge) for ϕrg(S). The orbit-
+irreducible decomposition of S for ϕ is defined by
+S =
+r
+�
+i=1
+m(Si)−1
+�
+j=0
+ϕj(Si),
+(65)
+where each Si for 1 ≤ i ≤ r has mutually distinct orbits and r = �r
+i=1 m(Si).
+Let h : �S = �r
+i=1
+�m(Si)−1
+j=1
+�
+ϕj(Si) −→ S be the normalization map, and ϕi,j :
+�
+ϕj(Si) −→ �
+ϕj(Si) the lifting of ϕm(Si)|ϕj(Si) : ϕj(Si) −→ ϕj(Si).
+Since �
+ϕj(Si) ∼= �Si
+and ϕi,j is congruent to ϕi,0 for 0 ≤ j ≤ m(Si) − 1, we may consider
+ϕi := ϕi,0 : �Si −→ �Si
+(1 ≤ i ≤ r)
+(66)
+as the representatives of the ϕi,j’s. Let ni be the order of ϕi. We have an ni-fold cyclic
+branched covering of Riemann surfaces
+πϕi : �Si −→ �
+Wi ∼= �Si/Gϕi,
+Gϕi = ⟨ϕi⟩ ∼= Z/niZ.
+(67)
+Note that ϕi also defines an automorphism of the pointed Riemann surface
+ϕi : (�Si, Pi) −→ (�Si, Pi).
+(68)
+We divide the set Pi and the set Qi = πϕi(Pi) on �
+Wi into
+Pi = PN
+i
+�
+PA
+i ,
+Qi = QN
+i
+�
+QA
+i
+(69)
+where PN
+i
+(resp. PA
+i ) consists of the points P such that the nodes h(P) in S are non-
+amphidrome (resp. amphidrome) for ϕ, and QN
+i = πϕi(PN
+i ), QA
+i = πϕi(PA
+i ).
+The dual graph dg(S) = {(vSi, g(�Si)),⃗ePj}1≤i≤r,1≤j≤k is the weighted graph obtained
+from the reduced dual graph rg(S) by attaching the weight g(�Si) to each vertex vSi, which
+is the genus of �Si. (If g(�Si) = 0, it is sometimes omitted.) An automorphism ϕ : S → S
+also induces an automorphism ϕdg(S) : dg(S) → dg(S), since ϕ preserves g(�Si).
+The following two lemmata guarantee the existence of a natural quotient of S by G
+as a nodal Riemann surface.
+Lemma 5.10. There exists a finite holomorphic map
+πϕ : S −→ W
+(70)
+to a nodal Riemann surface W which has the following properties:
+41
+
+(i) W has an irreducible decomposition �r
+i=1 Wi such that Wi = πϕ(ϕj(Si)) for any j.
+(ii) The normalizations of S and W naturally induce cyclic branched coverings
+πϕi,j : �
+ϕj(Si) −→ �
+Wi
+(1 ≤ i ≤ r, 0 ≤ j ≤ m(Si) − 1)
+such that πϕi,j is isomorphic to πϕi in (67) for any j.
+(iii) The reduced dual graph rg(W) coincides with the compact quotient graph (rg(S)/Gϕrg(S))c
+so that the dual graph is written as dg(W) = {(vWi, g(�
+Wi)),⃗eQj}1≤i≤r,1≤j≤k1. In partic-
+ular, a non-amphidrome node in S is sent by πϕ to a node of W, while an amphidrome
+node is sent to a non-singular point.
+Proof
+By using {(�
+Wi, QN
+i )}1≤i≤r in (67), (69) as the building blocks for patchings,
+one can construct the desired nodal surface W so that QN
+i are patched as the nodes of W.
+This patching process is constructed by identifying the two local components of xy = 0
+at the origin of C2 with the disk neighborhoods of �
+Wi’s at the suitable points in QN
+i .
+Lemma 5.11. W (in Lemma 5.10) is isomorphic to the quotient analytic space S/G.
+Proof
+We consider the local ring OS,P at the node P of S, and let �
+OS,P ∼= C[[x, y]]/(xy)
+be its completion by the maximal ideal of OS,P. If P is non-amphidrome whose covalencies
+at both banks are δ(1)/λ(1) and δ(2)/λ(2), then the action is written by
+ϕm(P) : (x, y) �−→ (e(δ(1)/λ(1))x, (e(δ(2)/λ(2))y).
+(71)
+We may assume λ(1) ≥ λ(2). Then the invariant subring of �
+OS,P for ϕm(P) is given by
+( �
+OS,P)ϕm(P ) ∼= C[[z, w]]/(zw),
+where z = xλ(1), w = yλ(2) and zw = xλ(1)−λ(2)(xy)λ(2) = 0. If P is amphidrome for ϕm(P)
+whose covalency is δ/λ, then
+ϕm(P)/2 : (x, y) �−→ (e(δ/2λ)y, (e(δ/2λ)x).
+(72)
+Hence
+( �
+OS,P)ϕm(P ) ∼= C[[z]],
+where z = xm(P) = ym(P). If P is a nonsingular point of S, the similar invariant subring
+is obviously regular.
+It follows that S/G exists as a complex curve with at most nodes. The natural mor-
+phism S → S/G has the same properties as those in Lemma 5.10.
+Therefore, W is
+isomorphic to S/G.
+Definition 5.12. We write (70) as πϕ : S −→ W = S/G and call it the cyclic branched
+covering associated with ϕ.
+42
+
+Example 5.13. Let S be a stable curve of genus 3 with two irreducible components and
+two nodes; S = �2
+i=1 Si, S1 ∩ S2 = {P1, P2}. Assume that each Si is isomorphic to an
+elliptic curve with the period √−1; Si ≃ C/(Z + √−1Z). We consider the following two
+automorphisms ϕ, ψ : S → S of order 8 so that the automorphisms ϕrg(S) and ψrg(S) are
+given in Example 5.9. First, ϕ fixes P1 and P2 and interchanges S1 and S2 such that the
+total valencies of ϕ2|Si (i = 1, 2) are 3/4 + 3/4 + 1/2 (where 3/4 are attached to P1 and
+P2). Secondly, ψ interchanges P1 and P2 and interchanges S1 and S2 such that the total
+valencies of ψ2|Si are 3/4 + 3/4 + 1/2.
+Then P1 and P2 are amphidrome nodes (resp. non-amphidrome nodes) for ϕ (resp.
+ψ). Let πϕ : S → Wϕ, πψ : S → Wψ be the cyclic branched covering associated with ϕ and
+ψ. Then Wϕ is isomorphic to P1, while Wψ is a stable curve of genus 1 with one node.
+Their reduced dual graphs are given in Example 5.9.
+Remark 5.14. It may be natural to re-write πϕ : S −→ W as πϕ : S −→ W = �r
+i=1 miWi
+where W is a non-reduced scheme such that W
+red = W. This point will be also discussed
+in §6.1 (see also [55, Chap.3]).
+Let Bi ⊂ �
+Wi be the set of branch points of πϕi : �Si → �
+Wi in (67). By comparing with
+(69), we define the cardinalities of the sets by
+♯(Bi) = si, ♯(Bi∩Qi) = s′
+i, ♯(Bi∩QN
+i ) = (s′
+i)(1), ♯(Bi∩QA
+i ) = (s′
+i)(2), s′
+i = (s′
+i)(1)+(s′
+i)(2),
+♯(Qi \ Bi) = ti, ♯(QN
+i \ Bi) = t(1)
+i , ♯(QA
+i \ Bi) = t(2)
+i ,
+ti = t(1)
+i
++ t(2)
+i .
+(73)
+We add the valency 1 for each non-branch point in Qi \ Bi to the total valency (48) for
+ϕi, and define the dressed total valency for ϕi by
+DTV(ϕi) =
+�
+�gi, gi, ni : σ(i)
+1
+λ(i)
+1
++ · · · +
+σ(i)
+(s′
+i)(1)
+λ(i)
+(s′
+i)(1)
++
+��σ(i)
+(s′
+i)(1)+1
+λ(i)
+(s′
+i)(1)+1
+��
++ · · · +
+��σ(i)
+s′
+i
+λ(i)
+s′
+i
+��
++
+σ(i)
+(s′
+i)+1
+λ(i)
+s′
+i+1
++ · · · σ(i)
+si
+λ(i)
+si
++ 1 + · · · + 1
+�
+��
+�
+t(1)
+i
++ ((1)) + · · · + ((1))
+�
+��
+�
+t(2)
+i
+�
+�
+�
+�
+(74)
+where the valencies are ordered for Bi∩QN
+i , Bi∩QA
+i , Bi\Qi, QN
+i \Bi and QA
+i \Bi. These
+symbols (bold faced valency for a non-amphidrome branch point, soft double bracket
+valency for an amphidrome branch point, etc.) are borrowed from [11, §2]. Then:
+Definition 5.15. We define the numerical data of an automorphism ϕ : S −→ S of a
+stable curve (of order N) by
+Num(ϕ) =
+�
+N, ϕdg(S),
+r�
+i=1
+DTV(ϕi)
+�
+.
+(75)
+43
+
+Remark 5.16. These numerical data are essentially the same as the numerical data of
+pseudo-periodic maps of negative twist (see §6.1). From this viewpoint, Num(ϕ) for g = 3
+are classified in [11, Table 2, Lemma 3.4, Prop. 3.8].
+5.5
+Equisymmetric strata at the boundary charts of M
+orb
+g
+We prove a structure theorem for equisymmetric strata consisting of marked stable curves
+which have the same numerical invariants of automorphisms.
+Let w : Σg(σ) −→ S be a σ-marking such that S has an automorphism ϕ with the
+preassigned numerical data Num(ϕ) as in (75). From the orbit-irreducible decomposition
+S = �r
+i=1
+�mi−1
+j=0 ϕj(Si) (mi = m(Si)) as in (65), the little Teichm¨uller space T(σ) which
+contains the point [S, w] is isomorphic to
+T(σ) ∼=
+r�
+i=1
+(Tgi,ki × · · · × Tgi,ki)
+�
+��
+�
+mi
+,
+(76)
+where gi = g(�Si), ki = ♯(Pi). See Lemma 5.7.
+In §5.1, we defined, with the explanation (I), the subspaces T ϕ
+g and T [ϕ]
+g
+for a periodic
+map ϕ : S → S. There S was a Riemann suface. We consider here a similar definition
+for a stable curve S.
+As in the previous subsection §5.4, let ϕ : S → S be an analytic automorphism of
+order N of a stable curve S. Let T ϕ
+σ be the set of points p = [S, w] in T(σ) which is fixed
+by the action of ϕ∗ : T(σ) → T(σ). (Through the σ-marking w : Σg(σ) → S, the analytic
+automorphism ϕ is considered to be an element w−1 ◦ ϕ ◦ w of the Weyl group W(σ).
+The action ϕ∗ is the action as an element of W(σ). See (28).) As in the case of Riemann
+surfaces, we define the equivalence class [ϕ] to be the set of those analytic automorphisms
+ψ : S → S which have the same numerical data (75) as ϕ : Num(ψ) = Num(ϕ). Then
+T [ϕ]
+σ
+is defined as follows:
+T [ϕ]
+σ
+=
+�
+ψ∈[ϕ]
+T ψ
+σ .
+The subspace M
+[ϕ]
+g
+of M g is defined to be the quotient space of T [ϕ]
+σ
+by the Weyl group:
+M
+[ϕ]
+g
+= T [ϕ]
+σ /W(σ).
+Remark 5.17. As we will prove in §6.1, an analytic automorphism ϕ of a stable curve
+S is lifted to a pseudo-periodic map of negative twist ˜ϕ of a Riemann surface such that
+Num(ϕ) may be identified with the invariants (a), (c) of ˜ϕ in Th. 6.1. Hence Num(ϕ)
+determines the conjugacy class of the analytic automorphism ϕ of the stable curve S by
+the results of [61], [55].
+44
+
+Theorem 5.18. (i) There exist analytic embeddings of pointed Teichm¨uller spaces
+ψi,j : Tgi,si+ti −→ Tgi,ki (1 ≤ i ≤ r, 0 ≤ j ≤ mi − 1),
+where ti is given in (73), and also an analytic embedding
+Φ :
+r�
+i=1
+Tgi,si+ti �→
+r�
+i=1
+(Tgi,ki × · · · × Tgi,ki)
+�
+��
+�
+mi
+∼= T(σ),
+(77)
+Φ : (x1, · · · , xr) �−→ (ψ1,0(x1), · · · , ψ1,mi−1(x1)
+�
+��
+�
+m1
+, · · · , ψr,0(xr), · · · , ψr,mr−1(xr)
+�
+��
+�
+mr
+),
+(78)
+such that the connected component (T [ϕ]
+σ )(0) of T [ϕ]
+σ
+containing the point [S, w] is analyt-
+ically isomorphic to the image Φ(�r
+i=1 Tgi,si+ti). In particular, (T [ϕ]
+σ )(0) is a �r
+i=1(3gi −
+3 + si + ti)-dimensional complex submanifold of T(σ). The space T [ϕ]
+σ
+itself is a countable
+union of submanifolds of these types of connected components.
+(ii) M
+[ϕ]
+g
+is a locally closed irreducible subvariety of M g which is isomorphic to �r
+i=1 Mgi,si+ti.
+Proof
+Step 1
+Let S be a stable curve with the irreducible decomposition S =
+�r
+i=1 Si such that S has an automorphism ϕ with the preassigned numerical data (75).
+Let ψ : X → B be a standard Kuransihi family of S = ψ−1(b0) (b0 ∈ B). We may assume
+that B is a small open ball around the origin (= b0) of the vector space Ext1
+OS(Ω1
+S, OS),
+which we write B = Ext1
+OS(Ω1
+S, OS) ∩ B if we want to emphasize this ambient space.
+From (18) and (I) of §3.1, we restrict B to the subspace which parematrizes the variable
+deformation
+H1(S, HomOS(Ω1
+S, OS)) ∩ B ∼=
+r
+�
+i=1
+Vi, where Vi := H0( ˆSi, 2K ˆSi + Pi)∗ ∩ B.
+(79)
+Now ϕ relatively acts on ψ, and let Bϕ = Ext1
+OS(Ω1
+S, OS)ϕ ∩ B be the invariant subspace.
+Since the action of ϕ on S preserves the set of nodes, the action of ϕ on B preserves the
+subspace �r
+i=1 Vi. Set mi = m(Si). Since the iterations of the action of ϕ on the direct
+factor Vi isomorphically map Vi → ϕ(Vi) → ϕ2(Vi) → · · · and stabilize ϕmi(Vi) = Vi, the
+ϕ-invariant subspace (�r
+i=1 Vi)ϕ is isomorphic to the direct sum of ϕmi-invariant subspaces
+�
+r
+�
+i=1
+Vi
+�ϕ
+∼=
+r
+�
+i=1
+mi−1
+�
+j=0
+�
+ϕj(Vi)
+�ϕi , where ϕi := ϕmi.
+(80)
+The direct factor V ϕi
+i
+of (80) is described as follows. We consider the covering πϕi : �Si −→
+�
+Wi in (67), and set Bi = {Q1, · · · , Qsi}, Qi \ Bi = {Qsi+1, · · · , Qsi+ti} from (73). By
+45
+
+appling Proposition 5.3 to the 0-eigenspace, the dimension of V ϕi
+i
+is equal to 3gi−3+si+ti.
+More precisely, the discussion in the proof of Proposition 5.3 says that
+V ϕi
+i
+∼= H0(�
+Wi, 2K�
+Wi +
+si+ti
+�
+j=1
+Qj)∗ ∩ B,
+(81)
+where B is a small open ball around the origin (= b0) of H0(�
+Wi, 2K�
+Wi + �si+ti
+j=1 Qj)∗.
+Step 2
+We show the existence of a natural family of pointed Riemann surfaces over
+V ϕi
+i . The connected component of the normalization of the restricted family ψ−1(Vi) → Vi
+naturally induces a family of pointed Riemann surfaces
+ψi : (S, P) −→ Vi,
+(82)
+which should be a standard Kuranishi family of the pointed Riemann surface ψ−1
+i (b0) =
+(�Si, Pi) by H1(�Si, T�Si(−Pi)) ∼= H0(�Si, 2K�Si + Pi)∗ (cf. [9, p.177]). Therefore, the natural
+action of the stabilizer Gϕi = Stab(Si) = ⟨ϕi⟩ on �Si is extended relatively to the family
+ψi. We consider the relative quotient map
+ψi : W = S/Gϕi −→ Vi = Vi/Gϕi ∼= V ϕi
+i .
+(83)
+The central fiber ψ
+−1
+i (b0) of (83) is isomorphic to �Si/Gϕi = �
+Wi. For any b ∈ Vi, the fiber
+ψ
+−1
+i (b) is described as follows. Since (82) is a standard Kuranishi family, its fundamental
+property (cf. [9, Chap.XI (6.13)]) says that the fiber ψ−1
+i (b) (b ∈ Vi) admits a subgroup
+Gϕi,b ⊂ Aut(ψ−1
+i (b)) which is isomorphic to Gϕi. Then
+ψ
+−1
+i (b) ∼= ψ−1
+i (b)/Gϕi,b := �
+Wi,b.
+In other words, the family (83) is a deformation of �
+Wi = ψ
+−1
+i (b0) so that each fiber of ψi
+is a quotient surface of each fiber of ψi in (82) by essentially the same Galois group Gϕi.
+Moreover, each point Qj ∈ {Q1, · · · .Qsi, Qsi+1. · · · , Qsi+ti} on �
+Wi is extended as a section
+Qj : Vi → W of ψi.
+In fact, we set πS : S → W = S/Gϕi and πS,b0 = πS|(πS)−1(b0) : �Si → �
+Wi = �Si/Gϕi.
+Then Qj satisfies one of the following two conditions:
+(i) There exists a point Pi,j in the support of Pi such that Qj = πS,b0(Pi,j).
+(ii) Qj is a branch point of the covering πS,b0.
+In case (i), since ψi is a Kuranishi family of (�Si, Pi), there exists a section s : Vi → S
+such that s(Vi) passes through Pi,j. Then the image πS(s(Vi)) defines the desired section
+Qj. In case (ii), since Gϕi and Gϕi,b have the same signature (λ(i)
+1 , · · · , λ(i)
+si ) by [31], the
+components of the discriminant locus of ψi : W → Vi are sections. Then one of them is
+the desired Qj.
+46
+
+Thus the family ψi is a deformation of pointed Riemann surfaces (�
+Wi; Q1, · · · , Qsi+ti).
+More strongly, it follows from (81) that ψi is nothing but a standard Kuranishi family of
+(�
+Wi; Q1, · · · , Qsi+ti).
+Step 3
+We fix a point [S, w] ∈ T(σ) ⊂ Dϵ(σ). We may assume that the source space
+Σg(σ) of the Weyl marking w : Σg(σ) → S is the topological model of a stable curve
+which has an automorphism ϕ with the numerical data (74), (75).
+We consider the Teichm¨uller marking wi : ( ˆRi, ˆPi) −→ ( ˆSi, Pi) in (63). Then there
+exists a branched covering ˆπi : ˆRi → Σgi over a Riemann surface Σgi of genus gi such that
+the covering transformation group of ˆπi coincides with Gϕi in (67). The cardinality of
+the set of points consisting of the branch points for ˆπi and ˆπi(ˆPi) is si + ti, and we write
+this set by { �Q1, · · · , �Qsi+ti}. By the natural descent of wi, we have a pointed oriented
+homeomorphism
+ˆwi : (Σgi; �Q1, · · · , �Qsi+ti) −→ (�
+Wi; Q1, · · · , Qsi+ti).
+(84)
+We may consider (84) as a Teichm¨uller marking of the fiber ψ
+−1
+i (b0) of the family (83).
+Since (83) is a standard Kuranishi family, it follows from the discussion of Arbarello–
+Cornalba–Griffiths [9, Chap.XV, §2] that this marking is extended to the whole family
+(83).
+That is to say, (83) induces a family of Teichm¨uller-marked pointed Riemann
+surfaces.
+Step 4
+By applying the method of the expression of pointed Teichm¨uller spaces via
+the patching of the base spaces of the standard Kuranishi families of pointed Riemann
+surfaces due to [9, Chap.XV, §2] and also the fundamental property of the action on
+Kuranishi families, we globalize the discussion in Steps 1 ∼ 3.
+Let T ϕ
+σ (p) be the connected component of T ϕ
+σ containing the point p = [S, w], where
+the marking w is the one defined in Step 3. This marking induces the marking of the
+ϕj-image of the normalized componet ( ˆSi, Pi) of S as ϕj ◦wi : ( ˆRi, ˆPi) → (ϕj( ˆSi), ϕj(Pi)),
+where wi is also given in Step 3. Then the point
+pi,j = [(ϕj( ˆSi), ϕj(Pi)), ϕj ◦ wi],
+(1 ≤ i ≤ r, 0 ≤ j ≤ mi − 1)
+is contained in the ϕi(= ϕmi)-invariant locus T ϕi
+gi,ki of Tgi,ki. For a fixed i, the connected
+components T ϕi
+gi,ki(pi,j) of T ϕi
+gi,ki containing pi,j (0 ≤ j ≤ mi − 1) are clearly isomorphic
+to each other, and we set (T ϕi
+gi,ki)(0) := T ϕi
+gi,ki(pi,0) for convenience. Then, from (80) and
+Lemma 5.7, we have the composition of analytic embeddings
+r�
+i=1
+(T ϕi
+gi,ki)(0) −→
+r�
+i=1
+T ϕi
+gi,ki(pi,0) × · · · × T ϕi
+gi,ki(pi,mi−1) −→
+r�
+i=1
+Tgi,ki × · · · × Tgi,ki
+�
+��
+�
+mi
+.
+(85)
+47
+
+From (83) and the discussions in Step 3 and in §5.1, we have an isomorphism
+(T ϕi
+gi,ki)(0) ∼= Tgi,si+ti.
+(86)
+Then the first assertion of (i) follows from (85) and (86). Since each of the set of connected
+components of the T ϕi
+gi,ki’s is countable by the same argument as in [31], the set of connected
+components of T ϕ
+σ is also countable. Hence the assertion (i) holds.
+Since the connected components of T ϕ
+σ are isomorphic to each other so that these
+isomorphisms are given by the change of markings, they are mapped by the forgetting map
+T(σ) → M g of the markings onto the same image, which is isomorphic to �r
+i=1 Mgi,si+ti
+from the assertion (i). This is irreducible since each M gi,si+ti is irreducible. By the same
+argument as in [16, p.106] using the factorization of the forgetful map to the composition
+of the infinite unramified covering and the finite Galois covering, it is a closed subvariety
+on M g. Hence the assertion (ii) holds.
+Corollary 5.19. ([71])
+M
+ϕ
+g is connected.
+Remark 5.20. The relation between the equisymmetric strata T ϕ
+σ at the boundary and
+their limits on Mg or Tg seems to be interesting. See for instance [22], etc.
+5.6
+Harris–Mumford coordinates around equisymmmetric strata
+In this subsection, we define a special system of local coordinates on the controlled de-
+formation space Dϵ(σ) around an arbitrarily chosen point p of the equisymmetric strata
+according to the method of [29, §1].
+We fix a point p = [S, w] ∈ T ϕ
+σ ⊂ T(σ) ⊂ Dϵ(σ). Since an open neighborhood of p in
+Dϵ(σ) is the base B of a standard Kuranishi family of S, it follows form (16), (17) and
+(18) that the dual bases of H0(S, Ω1
+S ⊗ ωS) express a system of local coordinates at p in
+Dϵ(σ). From (20), it can be decomposed into the parameters for the variable defomations
+(I) and the smoothing deformations (II) as in §3.1. By considering the action of ϕ on
+H0(S, Ω1
+S ⊗ ωS), we choose the basis as follows:
+(I) Let S = �r
+i=1
+�mi−1
+j=0 ϕj(Si) be the orbit-irreducible decomposition in (65), and
+�S = �¯r
+i=1
+�mi−1
+j=0
+�
+ϕj(Si) be the natural decomposition of the normalization. As the dual
+to (79), we consider the parameter space �¯r
+i=1
+�mi−1
+j=0 ϕj(V ∗
+i ) of the variable deformations,
+where V ∗
+i = H0(�Si, 2K�Si + Pi). Now we choose the eigenbasis in Definition 5.4 of the
+vector space V ∗
+i with respect to the ϕmi-action
+{vi,1, · · · , vi,qi} ∈ V ∗
+i ,
+(ϕmi)∗(vi,j) = e
+�θi,j
+ni
+�
+vi,j,
+(1 ≤ j ≤ qi)
+(87)
+where qi = dim V ∗
+i and ni is the order of the action of ϕmi on �Si.
+48
+
+With respect to the vector space ϕj(V ∗
+i ) for 1 ≤ j ≤ mi − 1, we choose the basis
+{ϕj(vi,1), · · · , ϕj(vi,qi)}. Then we have (ϕmi)∗(ϕj(vi,j)) = e(θi,j/ni)ϕj(vi,j).
+(II) Let �¯k
+i=1
+�m(Pi)−1
+j=0
+ϕj(Pi) be the set of nodes of S such that StabG(Pi) = ⟨ϕm(Pi)⟩
+and k = �¯k
+i=1 m(Pi). Let vPi be the generator of the torsion sheaf τPi given in (19). Then
+the basis of the parameter space of the smoothing deformations is given by
+{ϕj(vPi)}1≤i≤¯k,0≤j≤m(Pi)−1 ∈
+¯k
+�
+i=1
+m(Pi)−1
+�
+j=0
+τ ϕj(Pi).
+(88)
+For the eigenvalues, we have the following. Here we write P = ϕj(Pi), vP = ϕj(vPi),
+(δ(1)/λ(1))(P) = δ(1)/λ(1) and so on for simplicity.
+Lemma 5.21. If P is a non-amphidrome node such that the covalencies of both sides are
+δ(1)/λ(1) and δ(2)/λ(2), then (ϕm(P))∗vP = e
+�
+−δ(1)/λ(1) − δ(2)/λ(2)�
+vP.
+If P is an amphidrome node with covalency δ/λ, then (ϕm(P)/2)∗vP = e (−δ/λ) vP.
+Proof
+Assume P is non-amphidrome. Note that vP is written as ydx⊗2/x = xdy⊗2/y
+modulo xy = 0 by (19). From (71), we have
+(ϕm(P))∗vP =
+e
+�
+− δ(2)
+λ(2)
+�
+y · e
+�
+− 2δ(1)
+λ(1)
+�
+dx⊗2
+e
+�
+− δ(1)
+λ(1)
+�
+x
+= e
+�
+− δ(1)
+λ(1) − δ(2)
+λ(2)
+�
+vP.
+Siminarly in the case where P is amphidrome, we have the desired result from (72).
+Based on this discussion, we define the local coordinates at p of Dϵ(σ) by noticing that
+k = �¯k
+i=1 m(Pi) and 3g − 3 − k = �¯r
+i=1 miqi.
+Definition 5.22. The system of Harris–Mumford coordinates (z1, · · · , z3g−3) of Dϵ(σ) at
+p = [S, w] ∈ T �µ
+σ is defined by the following:
+(i) p = {(z1, · · · , z3g−3) = (0, · · · , 0)}.
+(ii) We rewrite it by using the lexicographic order with respect to i, j, α, β, γ as
+(z1, · · · , z3g−3) = (z(0)
+1 , · · · , z(j)
+i , · · · , z(m(Pk)−1)
+¯k
+, z(0)
+1,1, · · · , z(γ)
+α,β, · · · , z(m¯r−1)
+¯r,q¯r
+)
+(89)
+for 1 ≤ i ≤ ¯k, 0 ≤ j ≤ m(Pi) − 1, 1 ≤ α ≤ ¯r, 1 ≤ β ≤ qα, 0 ≤ γ ≤ mα − 1. Then the
+coordinate z(j)
+i
+is the dual vector of ϕj(vPi) where vPi is given in (88). The coordinate z(γ)
+α,β
+is the dual vector of ϕγ(vα,β) which is given in (87).
+For the coordinates (89) on B ⊂ Dϵ(σ) near p, the action of �µ on B is written as
+follows. The proof of Lemma 5.23 is obvious from Lemma 5.21 and (87).
+Lemma 5.23. The action of �µ on B near p is written via the coordinates (89) as
+�µ : z(j)
+i
+�−→ z(j+1)
+i
+= e
+� 1
+N
+� δ(1)
+λ(1)(Pi) + δ(2)
+λ(2)(Pi)
+��
+z(j)
+i , z(γ)
+α,β �−→ z(γ+1)
+α,β
+= e
+�
+−θα,β
+N
+�
+z(γ)
+α,β
+by identifying z(m(Pi))
+i
+= z(0)
+i
+and z(mα)
+α,β
+= z(0)
+α,β.
+49
+
+6
+Monodromy and orbifold moduli maps of degener-
+ations of Riemann surfaces
+We discuss fundamental properties of the monodromy and the moduli maps of degenera-
+tion of Riemann surfaces of genus g ≥ 2 from several points of view.
+In §6.1, we study pseudo-periodic maps of negative twist µ, whose totality is denoted
+by P(−)
+g
+. Our interest in P(−)
+g
+comes from the fact that the topological monodromy of
+a degeneration of a Riemann surface belongs to this class. In this subsection, we show
+that µ is obtained from the lifting via the real blow-up of an analytic automorphism µan
+of a stable curve. It follows that the fundamental invariants of µ ([61], [55]) essentially
+come from those of µan except for the screw numbers ([55, Def. 2.4]), which express
+the fractional Dehn twists along the exceptional circles. We also review the notion of
+generalized quotient space Σg/µ consisting of cores and non-core components constructed
+in [55, Chap. 3]. This gives important information about the central fiber of a normally
+minimal degeneration whose topological monodormy coincides with µ.
+In §6.2, we define the orbifold model f : S → ∆ of a degeneration by contracting
+the non-core components of the central fiber which is topologically identified with the
+generalized quotient ([55]). Here S is a normal complex space with at most quotient
+singularities such that the orbifold structure of f is explicitly induced from the precise
+stable reduction �f : S → �∆ given in [10, §2]. Note that this type of orbifold model
+historically originates from Imayoshi [39] from the viewpoint of Teichm¨uller theory.
+The discussion in §6.3 is the main part of this section. We define the notion of orbifold
+moduli map Jf : ∆ → M
+orb
+g
+for an orbifold model f of a degeneration, and show that
+Jf has the Kodaira-periodicity property in the following sense. For an elliptic fibration,
+Kodaira [46] defined the functional invariant J as the map from the base to the upper
+half-plane H (=Teichm˝uller space of g = 1) by means of the elliptic modular function.
+Since the monodormy in SL(2, Z) (=the mapping class group of g = 1) acts on J around
+a degenerate fiber, the local expression of J has a certain “periodicity”. This property is
+extended to g ≥ 2 as the orbifold moduli maps Jf and the action of the Weyl groups.
+In §6.4, we give two examples of degenerations of Riemann surfaces and their orbifold
+structures together with their orbifold moduli maps.
+6.1
+Pseudo-periodic maps and automorphisms of stable curves
+We compare a pseudo-periodic map of negative twist with an automorphism of a stable
+curve via the lifting given in §2.1. For the basic terminologies, see [55], [10, §1], [40, §5].
+The isotopy class of an oriented homeomorphism µ : Σg → Σg is called a pseudo-
+periodic map of negative twist with respect to a simplex σ = ⟨C1, · · · , Ck⟩ (of Harvey’s
+50
+
+curve complex, [33]) if
+(i) µ preserves σ, and the restriction µ|B to the complement B = Σg \ �
+1≤i≤k Ci is a
+periodic map, i.e. a certain power (µ|B)N is isotopic to the identity map idB.
+(ii) Let m(Ci) (1 ≤ i ≤ k) be the minimal natural number such that µm(Ci)(−→
+Ci) = −→
+Ci as
+oriented curves. Then µm(Ci) acts on an annular neighborhood Ai of Ci as a right-handed
+fractional Dehn twist.
+Decomposing B = �r
+i=1 Bi into connected components, we call Bi a body component
+([55, Def.4.8]), see also Fig. 4.1 of [55]. We also call the minimal natural number N with
+the property stated in (i) the pseudo-period of µ. The map µ with the property (i) is
+called a pseudo-periodic map ([55, Def.1.1]), or in Bers’ terminology, a map of elliptic or
+parabolic type (see [40, §3]). We denote the subset of Γg consisting of pseudo-periodic
+maps of negative twist with respect to σ by
+P−
+g (σ) ⊂ Γg.
+(90)
+The set of congujacy classes of (90) is denoted by �
+P−
+g (σ) ⊂ �
+Γg, which is characterized as
+follows:
+Theorem 6.1. ([61], [55]) An element µ ∈ �
+P−
+g (σ) is uniquely determined by
+(a) the Nielsen valencies at the multiple points and at the boundary curves of {Bi},
+(b) the screw numbers {s(Cj)} of {Aj} where Aj is a small annular neighborhood of Cj
+such that |s(Cj)| (s(Cj) ≤ 0) is the fractional Dehn twist of Aj,
+(c) the action of µ on the extended partition graph Γ(µ), i.e. the one-dimensional oriented
+graph whose vertices correspond to {Bi} and whose edges correspond naturally to {Cj}.
+The invariants (a) and (b) are due to [61], and (c) is due to [55]. We call them the nu-
+merical data (a), (b), (c) of µ. The pseudo-periodic map and the analytic automorphism
+of a stable curve are related as follows.
+Proposition 6.2. Let µ : Σg → Σg be an element of P−
+g (σ). Then there exists an analytic
+automorphism µσ : Σg(σ) → Σg(σ) with respect to some complex structure on the stable
+curve Σg(σ) such that the following diagram is isotopically commutative;
+Σg
+Σg(σ)
+Σg
+Σg(σ).
+contσ
+contσ
+µσ
+µ
+Conversely, any analytic automorphism µσ of a stable curve Σg(σ) is lifted to an element
+µ of P−
+g (σ).
+51
+
+Proof (Step 1) If σ = ∅, then µ is a periodic map, and it is known that µ is isotopic
+to an analytic automorphism. This fact is a corollary of Kerchhoff’s theorem [45], or this
+isotopy is explicitly constructed, see [60] or [55, Chap.2].
+If σ ̸= ∅, µ is isotopic on B = Σg\�
+1≤i≤k Ai to a direct sum of analytic automorphisms
+of Riemann surfaces with real boundaries, and the restrictions to � Ai could shrink to
+point maps (in Σg(σ)), just as in Prop. 2.8.
+Since Σg(σ) is constructed from Σg by
+shrinking � Ai to nodes via isotopy, the analytic automorphism on B descends to an
+analytic automorphism of Σg(σ).
+(Step 2)
+Conversely, let µσ : Σg(σ) → Σg(σ) be an analytic automorphism with
+respect to some complex structure.
+Assume that Pi = cont(Ci) is a non-amphidrome node with co-valencies δ(1)/λ(1)
+and δ(2)/λ(2) at the disk neighborhoods U (1) and U (2) of the local components of both
+sides, where Pi = U (1) ∩ U (2).
+By definition, µm(Pi)
+σ
+preserves U (j) and rotates it by
+the angle 2πδ(j)/λ(j) clockwise, within the view from the insides of both components
+(j = 1, 2). Let π(j)
+P : �U (j) → U (j) be the real blowing up at Pi with the exceptional circle
+C(j) = (π(j)
+P )−1(P), and A a small annulus with boundary ∂A = ∂A(1) � ∂A(2).
+We construct a part of a Riemann surface (Σg)loc = �U (1) ∪ A ∪ �U (2) from the building
+blocks �U (1), �U (2) and A by pasting C(j) to ∂A(j) naturally, and define a homeomorphism
+µloc : (Σg)loc → (Σg)loc which is a local lift of µm(Pi)
+σ
+as follows. The restriction µloc|�U(j) :
+�U (j) → �U (j) is defined to be the lifting of µm(Pi)
+σ
+via π(j)
+P , and µloc|A : A → A is defined
+to be the fractional Dehn twist, i.e. the linear twist ([55, §2.3]) with screw number [55,
+Def. 2.4]
+s(Ci) = − δ(1)
+i
+λ(1)
+i
+− δ(2)
+i
+λ(2)
+i
+− K(Ci)
+(K(Ci) ∈ Z, K(Ci) ≥ −1).
+(91)
+Since −δ(2)/λ(2) ≡ δ(1)/λ(1) + s(Ci) (mod Z), the map µloc is well-defined and expresses a
+right-handed fractional Dehn twist with s(Ci) ≤ 0.
+When Pi is an amphidrome node with co-valency δ/λ of the local components of both
+sides, then by the same notations as above we define the homeomorphism µloc : (Σg)loc →
+(Σg)loc which is the local lift of µm(Pi)/2
+σ
+as follows. The restrictions µloc|�U(1) : �U (1) → �U (2)
+and µloc|�U(2) : �U (2) → �U (1) are defined to be the liftings of µm(Pi)/2
+σ
+via π(1)
+P
+and π(2)
+P ,
+and µloc|A : A → A is defined to be the special twist ([55, §2.4]) which interchanges the
+boundary components ∂A(1) and ∂A(2) with screw number s(Ci)/2, where
+s(Ci) = −2δi
+λi
+− 2K(Ci)
+(K(Ci) ∈ Z, K(Ci) ≥ 0).
+(92)
+The restrictions of µσ to parts of Σg(σ), other than the disk neighborhoods of the
+nodes, have trivial liftings. Following the action of µσ on the dual graph of Σg(σ), we
+52
+
+patch these parts of Riemann surfaces and the homeomorphisms. Then we obtain Σg and
+the desired homeomorphism µ.
+Note that (91) and (92) are the screw numbers in the data (b) of Theorem 6.1. By
+the descent from µ to µσ, the data (a) and (c) are preserved, while the data (b) vanish by
+this descent, i.e. the screw numbers are not the data of the analytic automorphism of the
+stable curve. (The screw numbers express essentially the fractional Dehn twist coefficients
+along the exceptional circles, cf. Liu [51].)
+With respect to this complex structure on Σg(σ), there exists a quotient holomorphic
+map πµσ : Σg(σ) −→ Σg(σ)/Gµσ (Def. 5.12). This quotient map is topologically lifted to
+the generalized quotient map [55, Def.3.4]
+πµ : Σg(σ) −→ W(µ) = Σg/⟨µ⟩.
+(93)
+Here W(µ) is a non-reduced nodal Riemann surface in general (i.e. a multiplicity is at-
+tached to each component) and πµ is a pinched covering ([55], Def.3.2), i.e.
+a finite
+unramified topological covering except over nodes. Each connected component of the
+inverse image of a node is homeomorphic to a circle. The space W(µ) is decomposed into
+the parts ([55], p.95, Fig.4.1);
+W(µ) =
+�
+i
+Corei +
+�
+j
+Tailj +
+�
+j
+Archj +
+�
+j
+Quasitailj.
+(94)
+The orbit µα(Bi) (0 ≤ α ≤ m(Bi) − 1) of Bi is mapped to Corei by πµ except for small
+disk neighborhoods of multiple points. The parts Tailj, Archj, Quasitailj are chains or
+trees of P1’s which are the images under πµ of the neighborhoods of multiple points,
+non-amphidrome annuli and amphidrome annuli, respectively, and these multiplicities
+are explicitly determined from the data (a) and (b).
+Let cont(nc) : W(µ) → W(µ)♯ be the contraction of the non-cores (i.e. all the Tails,
+Archs and Quasitails). Then W(µ)♯ can be identified with Σg(σ)/Gµσ (see Remark 5.14),
+in other words, the following homotopically commutative diagram is the lifting of πµσ to
+πµ;
+Σg
+Σg(σ)
+W(µ)
+W(µ)♯ ∼= Σg/Gµσ.
+contσ
+cont(nc)
+πµσ
+πµ
+(Diagram I) The lifting of the analytic quotient to the generalized quotient
+6.2
+Orbifold structures of degenerations of Riemann surfaces
+In this subsection, we propose the notion of orbifold model of a degeneration of Riemann
+surfaces.
+53
+
+Let M be a 2-dimensional normal complex space and f : M → ∆ = {t ∈ C | |t| ≤ ϵ0}
+a proper surjective holomorphic map to a disc with a sufficiently small radius such that
+any fiber f −1(t) over t ∈ ∆∗ = ∆ \ {0} is a Riemann surface of genus g. We call f a
+degeneration of genus g. We always assume g ≥ 2 unless otherwise mentioned.
+First we assume that M is nonsingular. Let F = f −1(0) = �r0
+i=1 miFi be the irre-
+ducible decomposition of the central fiber. If the reduced scheme F red = �r0
+i=1 Fi has
+at most nodal singularities such that any (−1)-spherical component of F red has at least
+three intersection points with other components, then f is called the normally minimal
+model which is uniquely determined in the local birational equivalence class of f.
+Two degenerations of genus g are topologically (resp. analytically) equivalent if, with
+their normally minimal models f : M → ∆ and f ′ : M ′ → ∆′, there exist oriented
+homeomorphisms (resp. analytic isomorphisms) hM : M → M ′ and h∆ : ∆ → ∆′ such
+that h∆ ◦ f = f ′ ◦ hM. The equivalence class is called the topological type (resp. the
+analytic type) of the degeneration.
+Now we fix a point t0 ∈ ∆∗, and let w : Σg → f −1(t0) be a Teichm¨uller marking. We
+choose a smooth loop Lt0t0 in ∆∗ which starts from t0, goes around 0 ∈ ∆ once counter-
+clockwise and comes back to t0. Let µ(Lt0t0) : f −1(t0) → f −1(t0) be the diffeomorphism
+along Lt0t0. Then w−1 ◦ µ(Lt0t0) ◦ w : Σg → Σg is an oriented homeomorphism. For
+another point t′
+0 ∈ ∆∗, we define the Teichm¨uller marking of f −1(t′
+0) coming from w, by
+w′ = µ(Lt0t′
+0) ◦ w : Σg −→ f −1(t′
+0) where µ(Lt0t′
+0) : f −1(t0) → f −1(t′
+0) is the diffeomor-
+phism along a path Lt0t′
+0 in ∆∗ connecting t0 and t′
+0. Then the map w′−1 ◦ µ(Lt′
+0t′
+0) ◦ w′ is
+conjugate to w−1 ◦ µ(Lt0t0) ◦ w. In other words, the conjugacy class of
+µf(w) = [w−1 ◦ µ(Lt0t0) ◦ w]
+(95)
+is well-defined, independently of the choices of t0 and Lt0t′
+0.
+The proof goes as follows: We take a “straight” path L′
+t0t′
+0 in ∆∗ connecting t0 and
+t′
+0, and we suppose that the loop L′
+t0t′
+0Lt′
+0t′
+0L′
+t′
+0t0 is isotopic to Lt0t0. We will denote the
+diffeomorphisms µ(Lt0t′
+0) and µ(L′
+t0t′
+0) by a and b : f −1(t0) → f −1(t′
+0), respectively. Then
+by the asumption on L′
+t0t′
+0, we have b−1µ(Lt′
+0t′
+0)b = µ(Lt0t0), and
+[w′−1 ◦ µ(Lt′
+0t′
+0) ◦ w′]
+= [w−1a−1 ◦ µ(Lt′
+0t′
+0) ◦ aw]
+= [w−1a−1bb−1 ◦ µ(Lt′
+0t′
+0) ◦ bb−1aw]
+= [w−1a−1b ◦ (b−1µ(Lt′
+0t′
+0)b) ◦ b−1aw]
+= [w−1a−1b ◦ µ(Lt0t0) ◦ b−1aw]
+= [w−1(a−1b)ww−1 ◦ µ(Lt0t0) ◦ ww−1(b−1a)w]
+= [c−1w−1 ◦ µ(Lt0t0) ◦ wc]
+(96)
+54
+
+where we put c = w−1(b−1a)w : Σg → Σg. The equation (96) shows that the cojugacy
+classes [w′−1 ◦ µ(Lt′
+0t′
+0) ◦ w′] and [w−1 ◦ µ(Lt0t0) ◦ w] coincide. We call this conjugacy class
+µf(w) the topological monodromy of f with respect to the marking w . Note that µf(w) is
+determined by the fibering structure of f −1(∆∗) → ∆∗.
+For the same complex structure of f −1(t0), we choose another Teichm¨uller marking
+�w : Σg → f −1(t0) and consider the topological monodormy µf( �w) with respect to �w. Then
+µf(w) and µf( �w) are obviously conjugate to each other in Γg, i.e.
+µf( �w) = [w−1 ◦ �w]−1µf(w)[w−1 ◦ �w] ∈ Γg.
+Therefore, the conjugacy class �
+µf(w) of µf(w) is uniquely determined by f, independently
+of the choice of the marking w. We write
+µf = �
+µf(w) ∈ �Γg (conjugacy classes of Γg),
+(97)
+and call µf the topological monodromy of f. Then:
+Theorem 6.3. (i) ([20], [2], [39], [65], [23]) The topological monodromy belongs to the
+conjugacy classes of pseudo-periodic maps of negative twist.
+(ii) ([55, Th.7.2]) The topological structure of a degeneration of genus g is uniquely de-
+termined by its topological monodromy. The set of topological monodromies and the set of
+conjugacy classes of pseudo-periodic maps of negative twist correspond bijectively.
+The relation between the normally minimal model and the generalized quotient (93)
+of the topological monodormy µf is the following:
+Theorem 6.4. ([55, Chap.9]) The central fiber F = f −1(0) of the normally minimal
+model f : M → ∆ of a degeneration of genus g and the generalized quotient space W(µf)
+of µf topologically coincide with each other as non-reduced nodal Riemann surfaces. In
+paticular, F is decomposed into cores and non-cores as in (94).
+Now we change the model in the birational equivalence class of f. Since any proper
+subset of irreducible components of the central fiber F of the normally minimal model f
+has negative intersection form, there exists the analytic contraction map τ : M → M ♯ of
+the non-cores of F by Grauert’s theorem [27]. Let f ♯ : M ♯ → ∆ be the natural map. The
+restriction of τ to F coincides with the contraction map
+τ|F = cont(nc) : F −→ F ♯ = (f ♯)−1(0)
+given in Diagram I of §6.1. The total space M ♯ is a normal complex space with cyclic and
+dihedral quotient singularities whose supports are on the contraction points of the non-
+cores, and the types of singularities are explicitly given by the data of µf ([10, Lem. 2.1])
+55
+
+such that the non-cores in F are the exceptional set of the minimal resolution of these
+singularities. Since M ♯ has at most quotient singularities, we may consider it as a complex
+orbifold in the sense of Satake [64]. We call f ♯ the orbifold model of the degeneration f.
+The explicit orbifold structure of f ♯ is described as follows ([10, §§2,3]). Let π : �∆ → ∆
+be the covering map of disks defined by u �→ t = uN where N is the pseudo-period of µf,
+and let �
+M be the normalization of the fiber product of M ♯ and �∆ over ∆. Let �f : �
+M → �∆
+be the natural map. Then the covering transformation group G := ⟨�µ⟩ ≃ Z/NZ acting
+on �f −1(�∆∗) → �∆∗ = �∆ \ {0} is extended holomorphically over �f by the normality of �
+M
+such that the following commutative diagram holds:
+∆ ≃ �∆/G
+M ♯ ≃ �
+M/G
+M
+f
+�
+M
+�∆
+�f
+π
+�π
+τ
+f ♯
+(Diagram II) The process of the precise stable reduction
+Since the topological monodromy µ �f ≃ µN
+f is trivial on the body B (as defined before
+Th.6.1), the non-cores of the generalized quotient W(µ �f) consists of archs ([55, Def.4.7])
+of multiplicity 1, i.e. W(µ �f) is a semi-stable curve of genus g. Then the central fiber
+�F = �f −1(0) is a stable curve with the topological type Σg(σ) such that �
+M has rational
+double points of type A at the nodes of �F. From the viewpoint of the semi-stable curve
+W(µ �f), �F is the image of the contraction of the archs on it. The restriction to the fiber
+�π| �F : �F → F ♯ of the map �π in Diagram II essentially coincides with the quotient map
+πµσ in Diagram I. We call �f the precise stable reduction of f ([10]). This stable reduction
+is minimal in the sense that the degree N of the base change is minimal among all the
+stable reductions of f, because the generalized quotient for µn
+f with n < N cannot be a
+semi-stable curve of genus g by the algorithms given in [55, Chap.3].
+The action of the generator �µ of G on �f near �F is an extension of the automorphism
+�µ| �F : �F → �F of the stable curve �F to that of the family �f, and is locally described as
+follows:
+(i) Assume that P is a nonsingular point of �F which belongs to an irreducible compo-
+nent �Fi with StabG( �Fi) = ⟨�µm( �Fi)⟩. The automorphism �µm( �Fi)| �Fi : �Fi → �Fi of order n( �Fi)
+and the associated n( �Fi)-fold cyclic covering �π| �Fi : �Fi → �π( �Fi) ⊂ F ♯ are given in (66) and
+(67). Let (x, u) be the local coordinates at P = {(x, u) = (0, 0)} of �
+M such that x is the
+fiber coordinate and u is the lift of the base coordinate �∆ = {u ∈ C | |u| ≤ ϵ1/N}. Then
+the map �µm( �Fi) locally acts as
+�µm( �Fi) : (x, u) �−→
+�
+�µm( �Fi)| �Fi(x), e
+�
+m( �Fi)
+N
+�
+u
+�
+.
+(98)
+56
+
+Moreover if P is the ramification point of �π| �Fi with co-valency δ/λ, then StabG(P) =
+⟨�µm( �Fi)n( �Fi)/λ⟩ and the map �µm( �Fi)n( �Fi)/λ acts near P as
+�µm( �Fi)n( �Fi)/λ : (x, u) �−→
+�
+e
+� δ
+λ
+�
+x, e
+�
+m( �Fi)n( �Fi)
+λN
+�
+u
+�
+.
+(99)
+(ii) Assume P is a non-amphidrome node of �F with StabG(P) = ⟨�µm(P)⟩. By (91),
+the screw number at the curve CP = Contσ
+−1(P) via the map Contσ : Σg → Σg(σ) ≃ �F
+is given by s(CP) = −δ(1)/λ(1) − δ(2)/λ(2) − K. The monodromy map µ �f ≃ µN
+f behaves
+on an anular neighborhood of C as the right-handed integral Dehn twist of times
+n(P) := N|s(CP)|
+m(P)
+= ℓ(δ(1)λ(2) + δ(2)λ(1) + Kλ(1)λ(2))
+gcd(λ(1), λ(2))
+,
+(100)
+where ℓ = N/
+�
+lcm(λ(1), λ(2))m(P)
+�
+∈ Z. The restriction of this map to the tubular neigh-
+borhood of P gives the Milnor fibration of the singularity P whose monodormy map is
+the one described above. This means that �
+M is defined locally near P by the equation
+xy = un(P) where x and y are local parameters of the components of both sides of P.
+Here n(P) coincides with the Milnor number of the singularity P of �
+M. The map �µm(P)
+is given locally near P from (71) by
+�µm(P) : (x, y, u) �−→
+�
+e
+� δ(1)
+λ(1)
+�
+x, e
+� δ(2)
+λ(2)
+�
+y, e
+�m(P)
+N
+�
+u
+�
+.
+(101)
+(iii) Assume P is an amphidrome node with StabG(P) = ⟨�µm(P)/2⟩. The screw number
+is given by s(CP) = −2(δ/λ) − 2K from (92). Then �
+M is defined locally near P by the
+equation xy = un(P) such that the Milnor number n(P) is given by
+n(P) = N|s(CP)|
+m(P)
+= 2ℓ
+�
+δ + Kλ
+�
+,
+(102)
+where ℓ = N/(λm(P)) ∈ Z. The map �µm(P)/2 is locally given from (72) by
+�µm(P)/2 : (x, y, u) �−→
+�
+e
+� δ
+2λ
+�
+y, e
+� δ
+2λ
+�
+x, e
+�m(P)
+2N
+�
+u
+�
+.
+(103)
+Depending on the above arguments, we define the following;
+Definition 6.5. The marked orbifold structure of a degeneration f is defined by;
+(i) the set consisting of the orbifold model, the precise stable reduction and the group
+{ �f : �
+M → �∆, f ♯ : M ♯ → ∆, G = Z/NZ}
+(104)
+satisfying the Diagram II so that the action of G on �f is locally induced from (98) ∼
+(103), and
+57
+
+(ii) the lifting of the Weyl marking of the stable curve �F = �f −1(0) given by
+�w = cont(nc) ◦ πµ �
+f : Σg −→ W(µ �f) −→ �F.
+(105)
+We sometimes write this marked orbifold structure by f orb = { �f, f ♯, G, �w} for simplicity.
+Remark 6.6. (i) Since �
+M has singularities, ( �f, G) does not define directly the complex
+orbifold structure on f ♯. However this point is easily recovered by Takamura’s method
+[66]: the actions (101) and (103) near the nodes are lifted to the local linear actions of
+type A of C2 explicitly. Hence the orbifold charts of M ♯ in the neighborhoods of the points
+of the contraction of the non-cores are defined as the open neighborhoods at the origin
+of C2 and the above lifted actions (see also [10, §3.2]). For other orbifold charts of M ♯,
+we could use the open sets of �
+M and the restricted actions of G on them. In conclusion,
+f ♯ : M ♯ −→ ∆ has the structure of an orbifold fibration (see Definition 4.6).
+(ii) The marking �w (105) trivially decends to the Weyl marking
+w : Σg(σ) −→ �F.
+(106)
+If one compares the marking �w with w, �w is determined through the lifting Σg → Σg(σ)
+which includes the data of the screw numbers. This is the reason why we use �w instead of
+w. Note that this type of marking for a stable curve is used by Hubbard–Koch [37, Def.2.1]
+for the construction of the augmented Teichm¨uller space (see also [9, p.490]).
+6.3
+Local orbifold moduli maps and Kodaira-periodicity
+In this subsection, we define the local orbifold moduli map of a degenetaion and show that
+it has the Kodaira-periodicity.
+For a given degeneration f : M → ∆ of genus g with topological monodormy µf ∈
+P(−)
+g
+(σ), the restricted holomorphic family f −1(∆∗) → ∆∗ induces the moduli map ∆∗ →
+Mg. By the classical stable reduction theorem and the valuative criterion algebraically,
+or by Imayoshi [39, Th.2,Th.4] analytically, this map has a holomorphic extension to the
+Deligne–Mumford compactification
+Jf : ∆ −→ M g,
+(107)
+which is called the canonically extended moduli map ([58]). This map is determined by
+f −1(∆∗) → ∆∗, and is independent of the choice of the local birational model of f.
+Now we consider the marked orbifold structure f orb = { �f, f ♯, G, �w} of f in Definition
+6.5 and the orbifold M
+orb
+g
+= {(Dϵ(σ), W(σ), ϕσ, Mϵ(σ))}σ∈Cg/Γg in (43). The map Jf is
+lifted to the map of these spaces as follow. By the descent of the marking �w to w by (106),
+we consider the σ-Weyl marked stable curve [ �F, w] ( �F = �f −1(0)). Since the generator
+58
+
+�µ of G induces an analytic automorphism of �F, the point p = [ �F, w] is contained in the
+equisymmetric strata T �µ
+σ in Theorem 5.18:
+p = [ �F, w] ∈ T �µ
+σ ⊂ T(σ) ⊂ Dϵ(σ).
+(108)
+We consider the universal family π : Y
+orb
+g
+−→ M
+orb
+g
+of (44), (45). By Theorem 4.4, there
+exists an open neighborhood B of p in Dϵ(σ) such that the restricted family πX = πσ|X :
+X = π−1
+σ (B) ⊂ Xϵ(σ) −→ B ⊂ Dϵ(σ) is a standard Kuranishi family of �F with marking
+w. Since �f : �
+M → �∆ is a local deformation of �F, it follows from the universality of the
+Kuranishi family that there uniquely exists a holomorphic map
+�J �f : �∆ −→ B
+(109)
+such that �f is the pull back of πX by �J �f. Then we have the commutative diagram
+�∆
+∆
+�J �f(�∆) ⊂
+B
+⊂ Dϵ(σ)
+Jf(∆) ⊂ ϕσ(B) ⊂ Mϵ(σ) ⊂ M g
+π
+ϕσ,�∆
+ϕσ
+�J �f
+Jf
+(Diagram III) The local orbifold moduli map
+where ϕσ,�∆ is the restriction to �J �f(�∆) of the orbifold structure map ϕσ : Dϵ(σ) −→
+Mϵ(σ) = Dϵ(σ)/W(σ). More precisely, the space B in Diagram III is really the Kuranishi
+space with Weyl marking (B, w) and the image of the restriction of the forgetting map
+(B, w) → B → ϕσ(B) is nothing but the quotient of B by G; ϕσ(B) = B/G. By the
+construction of �f in Diagram II in §6.2, the group G also acts on the analytic subspace
+�J �f(�∆) of B such that Jf(∆) = �J �f(�∆)/G. In this sense, �J �f is the lifting of Jf. Considering
+these points, we define the following:
+Definition 6.7. For a degeneration f : M → ∆ of Riemann surfaces with topological
+monodormy µf ∈ P(−)
+g
+(σ) and orbifold structure f orb = { �f, f ♯, G, �w}, we define the local
+orbifold moduli map by
+{ �J �f : �∆ → Dϵ(σ), Jf : ∆ → Mϵ(σ), G}
+(110)
+which satisfies Diagram III. For simplicity, we sometimes write (110) merely by �J �f : �∆ →
+Dϵ(σ).
+In order to describe �J �f : �∆ → Dϵ(σ) explicitly, we will define a special class of
+holomorphic functions.
+59
+
+Definition 6.8. A holomorphic function ϕ(t) of a variable t at the origin in C is called
+a pseudo-periodic function with multiplicity γ ∈ N and period L ∈ N if there exists a
+holomorphic function �ϕ(t) = �∞
+i=0 citi (ci ∈ C) with
+ϕ(t) = tγ �ϕ(tL) = c0tγ + c1tγ+L + c2tγ+2L + · · ·
+(c0 ̸= 0).
+(111)
+Remark 6.9. The notion of pseudo-periodic function in this framework is inspired by
+Kodaira [46, §8]. As we already explained, the functional invariant of a degeneration of
+elliptic curves defined in [46, §7] is the orbifold moduli map in our terminology. The func-
+tional invariant belongs to the class of pseudo-periodc functions in the present terminology
+around each degenerate fiber germ in [46, §8].
+Pseudo-periodic functions are characterized as follows.
+Lemma 6.10. Let ϕ(t) be a holomorphic function at the origin with ϕ(0) = 0 and ϕ(t) ̸≡
+0. Then the following conditions (i) and (ii) are equivalent;
+(i) ϕ(t) is a pseudo-periodic function with multiplicity γ and period L,
+(ii) ϕ(t) admits an action C → C given by t �→ e(1/L)t with the charactor e(γ/L) such
+that the derivations of ϕ(t) at the origin satisfy the following;
+ϕ
+�
+e
+� 1
+L
+�
+t
+�
+= e
+� γ
+L
+�
+ϕ(t),
+ϕ(γ)(0) ̸= 0, ϕ(j)(0) = 0 for ∀j < γ.
+(112)
+Proof
+We assume (i). From the expression (111) of ϕ(t), we have
+ϕ
+�
+e
+� 1
+L
+�
+t
+�
+= c0e
+� γ
+L
+�
+tγ +
+�
+i≥1
+cie
+�γ + iL
+L
+�
+tγ+iL = e
+� γ
+L
+�
+ϕ(t).
+Since c0 ̸= 0, the conditions of the derivatives in (ii) are also satisfied.
+Conversely we assume (ii). We decompose ϕ(t) into
+ϕ(t) = ϕ1(t) + ϕ2(t),
+where ϕ1(t) =
+�
+i≡γ mod L
+citi,
+ϕ2(t) =
+�
+j̸≡γ mod L
+cjtj.
+Then we have
+ϕ
+�
+e
+� 1
+L
+�
+t
+�
+= e
+� γ
+L
+�
+ϕ1(t) +
+�
+j̸≡γ mod L
+cje
+� j
+L
+�
+tj.
+Thus the condition (ii) says that
+cje
+� j
+L
+�
+= cje
+� γ
+L
+�
+, for ∀j ̸≡ γ (mod L).
+This means that cj = 0, ∀j ̸≡ γ (mod L), i.e. ϕ2(t) ≡ 0. Hence we have ϕ(t) = ϕ1(t). By
+the conditions of the derivatives in (ii), the leading term of ϕ1(t) should be the non-zero
+constant multiple of tγ. Thus we obtain (i).
+60
+
+Definition 6.11. Let ϕ(t) be a pseudo-periodic function with multiplicity γ and the period
+L.
+Let γ′ and K be integers which satify γ = γ′ + KL, K ≥ 0,
+0 ≤ γ′ ≤ L − 1,
+γ′ ≡ γ (mod L). We define the analytic screw number of ϕ(t) by
+s(ϕ) = γ
+L = γ′
+L + K,
+(113)
+and call K and γ′/L the integral term and the fractional term of s(ϕ) respectively.
+The geometric meaning of Def. 6.11 will be clarified by Th. 6.12 (iv) and Rem. 6.14.
+Now we express the map �J �f : �∆ → Dϵ(σ) in (110) around p = [ �F, w] ∈ T �µ
+σ explicitly.
+From the discussions in §5.6, the orbit-irreducible decomposition �F = �¯r
+i=1
+�m( �Fi)−1
+j=0
+�µj( �Fi)
+and the decomposition �¯k
+i=1
+�m(Pi)−1
+j=0
+�µj(Pi) of nodes of �F induce the Harris–Mumford
+coordinates (z(0)
+1 , · · · , z(j)
+i , · · · , z(γ)
+α,β, · · · , z(m( �F¯r)−1)
+¯r,q¯r
+) of Dϵ(σ) around p as in (89). Then the
+map �J �f is expressed by 3g−3 holomorphic functions ϕ(j)
+i (u) (1 ≤ i ≤ ¯k, 0 ≤ j ≤ m(Pi)−1),
+ψ(γ)
+α,β(u) (1 ≤ α ≤ ¯r, 1 ≤ β ≤ qα, 0 ≤ γ ≤ m( �Fα) − 1) with ϕ(j)
+i (0) = ψ(γ)
+α,β(0) = 0 sending
+to each of these coordinates as
+�J �f :
+z(j)
+i
+= ϕ(j)
+i (u),
+z(γ)
+α,β = ψ(γ)
+α,β(u)
+(u ∈ �∆).
+(114)
+The following theorem is an extension of Kodaira’s result from the viewpoint of Remark
+6.9.
+Theorem 6.12. Let ( �J �f : �∆ → Dϵ(σ), G) be the chart of the orbifold moduli map (110)
+of the marked orbifold structure (104),(105) of a degeneration f : M → ∆ of genus g ≥ 2.
+Let ϕ(j)
+i (u), ψ(γ)
+α,β(u) be the holomorphic functions in (114) expressing �J �f. Then
+(i) ϕ(0)
+i (u) is a pseudo-periodic function with period N/m(Pi) and with multiplicity
+n(Pi). Here n(Pi) is the Milnor number described in (100) (resp. (102)) in the case
+where Pi is a non-amphidrome node (resp. an amphidrome node).
+(ii) If ψ(0)
+α,β(u) ̸≡ 0, then ψ(0)
+α,β(u) is a pseudo-periodic function with period N/m( �Fα)
+and multiplicity
+γα,β :=
+�
+n( �Fα) − θα,β
+n( �Fα)
++ Kα,β
+�
+N
+m( �Fα)
+(115)
+where Kα,β is a non-negative integer and θα,β/n( �Fα) is given in (87).
+(iii) We have ϕ(j)
+i (u) = ϕ(0)
+i (u) for any j, and ψ(γ)
+α,β(u) = ψ(0)
+α,β(u) for any γ.
+(iv) The analytic screw numbers of the pseudo-periodic functions ϕ(j)
+i (u) and ψ(γ)
+α,β(u)
+are given by
+s(ϕ(j)
+i ) = |s(CPi)|,
+s(ψ(γ)
+α,β) = n( �Fα) − θα,β
+n( �Fα)
++ Kα,β
+(116)
+for any j and γ, where |s(CPi)| is the absolute value of the screw number of the cut curve
+CPi given in (91) and (100), or (92) and (102).
+61
+
+Proof
+We prove (i). First we assume that Pi is a non-amphidrome node of �F. We
+set L = N/m(Pi). Since z(0)
+i
+is the dual vector of the generator vPi of the torsion sheaf
+τPi, it follows from Lemma 5.21 and (100) that �µm(Pi) acts on z(0)
+i
+by
+�µm(Pi) : z(0)
+i
+�−→ e
+� δ(1)
+λ(1) + δ(2)
+λ(2)
+�
+z(0)
+i
+= e
+�n(Pi)
+L
+�
+z(0)
+i .
+(117)
+Since �µ(�∆) is naturally the G-invariant subspace in H0( �F, Ω1
+�F ⊗ ω �F)∗ ⊂ B ⊂ Dϵ(σ)
+and �µm(Pi) acts on �∆ by u �→ e(1/L)u, the function ϕ(0)
+i (u) satisfies ϕ(0)
+i (e(1/L)u) =
+e
+�
+n(Pi)/L
+�
+ϕ(0)
+i (u). Therefore by Lemma 6.10, ϕ(0)
+i (u) is a pseudo-periodic function with
+period L, and its multiplicity is congruent to n(Pi) modulo L.
+On the other hand, as we explained in §3.1 for the comment (II) of (20), the Kuranishi
+family of �F has a local structure {xy = t|(x, y, t) ∈ C3, |x|, |y|, |t| < 1} at a node ([9,
+pp.184–186]), i.e. it is a plumbing variety (cf.[47]). Since �
+M has singularity xy = un(Pi)
+at Pi and �f : �
+M → �∆ should be pulled back from the Kuranishi family by �J �f, the map
+�J �f is locally given by
+�J �f : u �→ t = un(Pi)ψ(u)
+where ψ(u) is a holomorphic function with ψ(0) ̸= 0.
+Therefore the multiplicity of ϕ(0)
+i (u) exactly coincides with n(Pi).
+Hence we obtain the assertion (i) for non-amphidrome case. The case where Pi is an
+amphidrome node, the discussion is similar and is omitted.
+We prove (ii). We consider a component �Fα and set L′ = N/m( �Fα). Since z(0)
+α,β is the
+dual vector of vα,β in (87), the map �µm( �Fα) acts by
+�µm( �Fα) : z(0)
+α,β �−→ e
+�
+− θα,β
+n( �Fα)
+�
+z(0)
+α,β = e
+�
+1
+L′
+�
+−θα,βL′
+n( �Fα)
+��
+z(0)
+α,β.
+Therefore we similarly obtain the assertion (ii) by Lemma 6.10.
+Since the iterations of the map �µ permute cyclically and isomorphically each neigh-
+borhood of the orbit of a node and also each neighborhood of the orbit of a component
+of �F, the assertion (iii) follows.
+The assertion (iv) is clear from (i), (ii), (iii) and Definition 6.11.
+Remark 6.13. In the property (ii) of Th.6.12, the trivial function ψ(0)
+α,β(u) ≡ 0 may
+occure. In particular, in the case where ¯k = 0 and ψ(0)
+1,β(u) ≡ 0 for all 1 ≤ β ≤ 3g − 3, �J �f
+is the constant map �J �f(�∆) = p, i.e. �
+M ≃ �F × �∆ and �f is the projection �F × �∆ −→ �∆
+such that f is obtained from the resolution of ( �F × �∆)/G.
+Remark 6.14. The fractional term (n( �Fα) − θα,β)/n( �Fα) of the analytic screw number
+s(ψ(γ)
+α,β) in (116) is a topological invariant, because it is determined by the total valency
+62
+
+using Prop. 5.3. On the other hand, the integral term Kα,β of s(ψ(γ)
+α,β) is not a topological
+invariant, and is determined purely by the analytic structure of f. By comparison with
+the usual screw numbers (91) and (92), the number Kα,β seems to be an “analytic analog
+of the number of integral Dehn twists”. See also Kuno’s paper [49, §4.3].
+In a certain situation, Kα,β is related to the modular invariant ([63],[68]) of the fiber
+germ (f, F) from the viewpoint of [13]. This point will be discussed in a forthcoming
+paper.
+6.4
+Examples of degenerations and their invariants
+Here we give two examples of degenerations of Riemann surfaces and show their orbifold
+structures and the properties of their orbifold moduli maps.
+Example 6.15. Let µ : Σ8 −→ Σ8 be the element of P(−)
+8 (σ) with σ = ⟨C1, · · · , C5⟩ and
+the pseudo-period N = 84 described in Figure III. The total valencies of B = �4
+i=1 Bi =
+Σ8 \ �5
+j=1 Cj are (g = 3, ¯g = 0, n = 7; 2/7 + 6/7 + 6/7) on B1, (g = 1, ¯g = 0, n =
+4; 1/4 + 1/4 + 1/2) on B2, (g = 1, ¯g = 0, n = 3; 2/3 + 2/3 + 2/3) on B3 and B4. Here
+the valencies written by boldface are attached to the boundary curves and those by roman
+are to the multiple points assigned by the cross symbols inside the body.
+The action on the graph has order 2 with µ(B3) = B4, µ(C4) = C5 such that C1, C4,
+C5 are non-amphidrome and C2, C3 are amphidrome. The screw numbers are s(C1) =
+−6/7 − 1/4 − K1 (K1 ≥ −1), s(C2) = −2(2/3 + K2) (K2 ≥ 0), s(C3) = −2(2/3 + K3)
+(K3 ≥ 0), s(C4) = s(C5) = −1/2 − 2/3 − K4 (K4 ≥ −1).
+Let f : M −→ ∆ be the normally minimal model of a degeneration with topological
+monodormy µf = µ. The central fiber F = f −1(0), i.e. the generalized quotient W(µf)
+is given as in Figure IV by the algorithm in [55]. Here the circles mean P1’s and the
+numbers in the circles mean their multiplicities. (The case of Ki = −1 for some i is
+B1
+C1
+2
+7 + 6
+7 + 6
+7
+− 6
+7 − 1
+4 − K1
+B2
+C4
+C5
+1
+4 + 1
+4 + 1
+2
+− 1
+2 − 2
+3 − K4
+− 1
+2 − 2
+3 − K4
+B3
+B4
+C3
+C2
+µ
+−2( 2
+3 + K3)
+2
+3 + 2
+3 + 2
+3
+2
+3 + 2
+3 + 2
+3
+−2( 2
+3 + K2)
+(Figure III) The data of the topological monodromy of Ex. 6.15
+63
+
+7
+2
+1
+6
+5
+4
+3
+2
+1
+6 5 4 3 2 1
+K1
+1 1 4 2 2
+K4
+2 4
+1
+6
+4 2
+4 2
+K2
+2
+2
+K3
+2
+2
+1
+1
+1
+1
+(Figure IV) The central fiber F of the normally minimal model of Ex. 6.15
+omitted in this figure.) F has three core components of multiplicities 7, 4 and 6.
+By the contraction M −→ M ♯ of the non-core of F, we obtain the orbifold model
+f ♯ : M ♯ −→ ∆. The fiber F ♯ = (f ♯)−1(0) is described in Figure V. Here M ♯ has five
+isolated cyclic quotient singularities and two dihedral quotient singularities whose supports
+are indicated by the cross symbols in this figure.
+Let �∆ −→ ∆ be the cover u �→ t = u84, and �
+M be the normalization of M ♯ ×∆ �∆. We
+obtain the precise stable reduction �f : �
+M −→ �∆, and the orbifold structure (f ♯, �f, G =
+⟨�µ⟩ ≃ Z/84Z). The stable fiber �F = �f −1(0) is as in Figure V. The topological monodromy
+µ �f ≃ µ84
+f
+is trivial on B and acts as ni = 84|s(Ci)|-right Dehn twist at an annular
+neighborhood of Cj, i.e.
+n1 = 84K1 + 93, n2 = 84K2 + 56, n3 = 84K3 + 56, n4 = n5 = 42K4 + 49.
+(118)
+The marking is defined by the composition w : Σ8 → Σ8(σ) ≃ W(µ �f) → �F. Let �F =
+�2
+i=1 �Fi + �1
+j=0 �µj( �F3) be the orbit-irreducible decomposition such that w(Bi) = �Fi (1 ≤
+i ≤ 3), w(B4) = �µ( �F3). Let P = �3
+i=1 Pi + �1
+j=0 �µ(P4) be the decomposion of the set of
+nodes on �F such that w(Ci) = Pi (1 ≤ i ≤ 4) and w(C5) = �µ(P4). �
+M has the singularity
+of type xy = uni for (118) at each node. The action of G to �
+M is not effective. In fact,
+the order of StabG( �F1) = G is 86 while the order of the automorphism �µ| �F1 of �F1 is 7.
+64
+
+Fi
+4
+42K2+48
+A84Ks+55
+F
+F3
+84 : 1
+4
+6
+X
+F#C M#~M/G
+A84K1+92
+A42K2+48A84K4+55
+u(F3)
+(Figure V)The central fibers of the precise stable reduction of Ex.6.15We set V1 = H0( �F1, 2K �F1 + P1), V2 = H0( �F2, 2K �F2 + P1 + P4 + �µ(P4)), V3 =
+H0( �F3, 2K �F3 +P2 +P3 +P4) and �µ(V3) = H0(�µ( �F3), 2K �F3 +P2 +P3 + �µ(P4)). By Example
+5.6 and the similar argument using Prop. 5.3, the log-quadratic characters are
+Ch�µV1 =
+�1
+7, 2
+7, 2
+7, 3
+7, 4
+7, 5
+7, 6
+7
+�
+, Ch�µV2 =
+�1
+4, 1
+2, 3
+4
+�
+, Ch�µ2�µj(V3) =
+�1
+3, 1
+3, 2
+3
+�
+,
+(119)
+for j = 0, 1. The little Teichm¨uller space T(σ) is locally an open set of �2
+i=1 Vi
+�1
+j=0 �µj(V3)
+and dim T(σ) = 16. The equisymmetric strata T �µ
+σ consists of a unique point p = [ �F, w],
+since the 0-eigen space is 0-dimensional by (119) and T �µ
+σ is connected by Cor. 5.19. Let
+(z1, z2, z3, z(0)
+4 , z(1)
+4 , z1,1, · · · , z1,7, z2,1, z2,2, z2,3, z(0)
+3,1, z(0)
+3,2, z(0)
+3,3, z(1)
+3,1, z(1)
+3,2, z(1)
+3,3)
+(120)
+be the system of Harris–Mumford coordinates at p on Dϵ(σ) which are ordered as the dual
+vectors of the ordered torsion sheaf at the nodes and the eigenvectors corresponding to the
+characters in (119). From (119), the multiplicities given in (115) are
+γ1,j = 84K1,j + 12γ′
+1,j, γ′
+1,j = 6, 5, 5, 4, 3, 2, 1 (j = 1, 2, 3, 4, 5, 6, 7), K1,j ∈ Z≥0,
+(121)
+γ2,j = 84K1,j + 21γ′
+2,j, γ′
+2,j = 3, 2, 1 (j = 1, 2, 3), K2,j ∈ Z≥0,
+(122)
+γ3,j = 42K3,j + 14γ′
+3,j, γ′
+3,j = 2, 2, 1 (j = 1, 2, 3), K3,j ∈ Z≥0,
+(123)
+where the notations mean that γ′
+1,1 = 6, γ′
+1,2 = 5, · · · , γ′
+3,3 = 1. From Th. 6.12, (120),
+(118) and (121)∼(123), the the orbifold moduli map �J �f : �∆ → Dϵ(σ) (u ∈ ∆) has the
+Kodaira-periodicity given by
+zi =
+∞
+�
+k=0
+ci,kuni+84k
+(i = 1, 2, 3, ci,0 ̸= 0, ci,k ∈ C),
+z(j)
+4
+=
+∞
+�
+k=0
+c4,kun4+42k
+(j = 0, 1, c(k)
+4,0 ̸= 0, c4,k ∈ C),
+zi,j =
+∞
+�
+k=0
+ci,j,kuγi,j+84k
+(i = 1, 2, j = 1, 2, 3, ci,j,k ∈ C),
+z(j)
+3,i =
+∞
+�
+k=0
+c3,i,kuγ3,i+42k
+(i = 1, 2, 3, j = 0, 1, c3,i,k ∈ C).
+Example 6.16. Let µ : Σ5 −→ Σ5 be the element of P(−)
+5 (σ) with σ = ⟨C1, · · · , C5⟩
+and N = 15 as in Figure VI. The total valencies of B = �5
+i=0 Bi = Σ5 \ �5
+j=1 Cj are
+(g = 0, ¯g = 0, n = 5, 2/5+3/5+1) on B0 and (g = 1, ¯g = 0, n = 3, 2/3+2/3+2/3) on Bi
+(1 ≤ i ≤ 5). The action on the graph has order 5 with permutations (B1, B3, B5, B2, B4),
+(C1, C3, C5, C2, C4), i.e. the 2/5-turn in the terminology of [55, p.148] such that Cj is
+non-amphidrome with the screw number s(Cj) = −2/3 − K (K ≥ 0).
+65
+
+Let f : S −→ ∆ be the degeneration with topological monodormy µf = µ. The fiber
+F = f −1(0) and the process to obtain its precise stable reduction �f : �S −→ �∆ is shown
+in Figure VII. Here �F = �f −1(0) = �5
+i=0 �F (i) is the irreducible decomposition such that
+�F (i) = �µi−1( �F (1)) (1 ≤ i ≤ 5) for the induced automorphism �µ : �F −→ �F.
+Then
+g( �F (0)) = 0, g( �F (i)) = 1 and �S has A3K+1-singularity at the node Pi (1 ≤ i ≤ 5).
+15
+10
+5
+10
+5
+10 5
+5
+K
+5
+3 1
+2 1
+contr.
+15
+5
+15 : 1
+P1
+P2
+P5
+P3
+P4
+�F (1)
+�F (2)
+�F (5)
+�F (3)
+�F (4)
+�F (0)
+A3K+1(∀Pi)
+2
+5-turn
+F ⊂ S
+F ♯ ⊂ S♯
+�F ⊂ �S
+(Figure VII) The central fibers of the precise stable reduction of Ex. 6.16
+We set V0 = H0( �F0, 2K �F0 + �5
+i=1 Pi), V1 = H0( �F1, 2K �F1 + P1). A calculation by using
+Prop. 5.3 shows that the log-quadratic characters are
+Ch�µV0 =
+�1
+5, 4
+5
+�
+, Ch�µV1 =
+�2
+3
+�
+.
+(124)
+Then T(σ) is locally an open set of V0
+�4
+j=0 �µj(V1) ≃ C7, and T �µ
+σ consists of a unique
+point p = [ �F, w] by (124). From (124), the multiplicities given in (115) are
+γ0,1 = 15K0,1 + 12, γ0,2 = 15K0,2 + 3, γ1,1 = 3K1,1 + 1 (K0,1, K0,2, K1,1 ∈ Z≥0).
+Let (z(0)
+1 , z(1)
+1 , z(2)
+1 , z(3)
+1 , z(4)
+1 , z0,1, z0,2, z(0)
+1,1, z(1)
+1,1, z(2)
+1,1, z(3)
+1,1, z(4)
+1,1) be the system of Harris–Mumford
+coordinates at p on Dϵ(σ). Then the orbifold moduli map �J �f : �∆ → Dϵ(σ) (u ∈ ∆) has
+the Kodaira-periodicity as
+z(j)
+1
+=
+∞
+�
+k=0
+c1,ku2+3K+3k (0 ≤ j ≤ 4, c1,0 ̸= 0, c1,k ∈ C),
+66
+
+(Bo
+("A)- 
+2-turn
+Bo
+网B1
+/5
+2
+(Bi,i≠0)
+3
+3
+B:
+(FigureVI)The data of topological monodromy ofEx.6.16z0,j =
+∞
+�
+k=0
+c0,j,kuγ0,j+15k (j = 1, 2, c0,j,k ∈ C), z(j)
+1,1 =
+∞
+�
+k=0
+c1,1,kuγ1,1+3k (0 ≤ j ≤ 4, c1,1,k ∈ C).
+7
+Recovery of fibered complex surfaces from the uni-
+versal degenerating family
+The goal of this section is to show that any fibered complex surface can be pulled back
+from the universal degenerating family of Riemann surfaces π : Y
+orb
+g
+−→ M
+orb
+g
+given in
+§4. By globalizing the notions discussed in §6, we define the global monodromy µ and the
+global orbifold moduli map Jorb for a fibered complex surface f : M → B of genus g ≥ 2.
+Conversely, starting from the invariants (µ, Jorb), we construct a fibered complex surface
+f realizing these invariants, by pulling back the universal family π via Jorb.
+Note that Kodaira [46] constructed an elliptic surface with given (µ, Jorb) (the homo-
+logical invariant and the functional invariant in his terminology), and called it the basic
+member of the elliptic surface ([46, p.603]). Our Theorem 7.8 may be considered as the
+construction of the basic members of the fibered complex surfaces of genus g ≥ 2.
+In §7.1, our discussion is from the local point of view. For a given pseudo-periodic map
+µloc ∈ P−
+g (σ), we define the set P∗(�∆, ∆; Dϵ(σ), Mϵ(σ); T µσ
+σ ) of the dual pseudo-periodic
+maps of µloc, which consists of the orbifold maps Jorb
+loc from the orbifold disk to the chart
+Dϵ(σ) of M
+orb
+g
+at the equisymmetric strata T µσ
+σ
+which have the Kodaira-periodicity given in
+Theorem 6.12. Concisely speaking, for a given (µloc, Jorb
+loc ), we construct the degeneration
+f : M → ∆ which realizes these invariants, by pulling back in the orbifold theoretic
+sense via Jorb
+loc the local structure of the universal family π.
+Moreover, we show that
+P∗(�∆, ∆; Dϵ(σ), Mϵ(σ); T µσ
+σ ) is the classifying space of analytic structures over the fixed
+topological structure of f.
+In §7.2, we define the global monodormy and the global orbifold moduli map of a
+fibered complex surface f : M → B. Over the restricted holomorphic family f (0) : M(0) →
+B(0) outsides the discriminant locus of f, these notions are the usual ones, namely, the
+monodromy representation to the mapping class group µ : π1(B(0), b0) → Γg and the
+pair (J(0))orb = (�J(0), J(0)) consisting of the moduli map J(0) : B(0) → Mg and its lift
+�J(0) : �B(0) → Tg from the universal cover of B(0) to the Teichm˝uller space. We prove
+that (µ, (J(0))orb) and (µloc, Jorb
+loc )’s around the critical set given in §7.1 are well-patched
+globally.
+In §7.3, we achieve our main purpose by proving Theorem 7.8.
+67
+
+7.1
+Local recovery of degenerations from the universal family
+We discuss the recovery of a degeneration f : M → ∆ from the local structure of π.
+We fix an element µ ∈ P−
+g (σ) with pseudo-period N (see (90)). We set G = Z/NZ,
+and let (�∆, G, π�∆, ∆) be the orbifold disk defined by π�∆ : �∆ ∋ u �→ t = uN ∈ ∆. Let
+µσ : Σg(σ) → Σg(σ) be the analytic automorphism which is a descent of µ (see Prop. 6.2).
+We consider the equisymmetric strata T µσ
+σ
+⊂ Dϵ(σ). We fix a point p = [S, w] ∈ T µσ
+σ , and
+let (· · · , z(j)
+i , · · · , z(γ)
+α,β, · · · ) be the system of Harris–Mumford coordinates at p on Dϵ(σ).
+Definition 7.1. An orbifold map
+( �J, J, G) : (�∆, G, π�∆, ∆) −→ (Dϵ(σ), W(σ), ϕσ, Mϵ(σ))
+to the chart (43) of M
+orb
+g
+is said to be a dual pseudo-periodic map of µ at p if the following
+conditions are satisfied: A holomorphic map �J : �∆ → Dϵ(σ) with �J(0) = p is expressed
+by (3g − 3) pseudo-periodic functions z(j)
+i
+= ϕ(j)
+i (u), z(γ)
+α,β = ψ(γ)
+α,β(u) such that
+(d-i) ϕ(0)
+i (u) has period N/m(Pi) and multiplicity n(Pi) as in (100) or (102),
+(d-ii) ψ(0)
+α,β(u) has period N/m( �Fα) and multiplicity γα,β as in (115), or ψ(0)
+α,β(u) ≡ 0,
+(d-iii) ϕ(j)
+i (u) = ϕ(0)
+i (u) for j, 0 ≤ j ≤ m(Pi) − 1, and ψ(γ)
+α,β(u) = ψ(0)
+α,β(u) for γ, 0 ≤ γ ≤
+m( �Fα) − 1.
+The holomorphic map J = ϕσ ◦ �J ◦ (π�∆)−1 : ∆ −→ Mϵ(σ) is well-defined.
+The motivation of this definition comes from Th. 6.12.
+All the invariants in (d-i)
+∼ (d-iii) except for Kα,β are numerically determined from the data (a),(b),(c) of µ in
+Th. 6.1. But the Kα,β’s can be any non-negative integers provided that γα,β > 0 (see
+(115)). Since there are infinitely many choices of such Kα,β’s (cf. Th.6.12 (ii)), there are
+infinitely many choices of ( �J, J, G) for a given µ and p. Let P∗(�∆, ∆; Dϵ(σ), Mϵ(σ); p) be
+the set of the dual pseudo-periodic maps ( �J, J, G) of µ at p. By varying p over T µσ
+σ , we
+have the following definition:
+Definition 7.2. The set of dual pseudo-periodic maps of µ for T µσ
+σ
+is defined by
+P∗(�∆, ∆; Dϵ(σ), Mϵ(σ); T µσ
+σ ) =
+�
+p∈T µσ
+σ
+P∗(�∆, ∆; Dϵ(σ), Mϵ(σ); p).
+Theorem 7.3. (i) For any µ ∈ P−
+g (σ) and ( �J, J, G) ∈ P∗(�∆, ∆; Dϵ(σ), Mϵ(σ); T µσ
+σ ), there
+uniquely exists a degeneration f : M → ∆ of Riemann surfaces of genus g such that the
+marked topological monodormy of f coincides with µ, and the chart map of the orbifold
+moduli map of f coincides with ( �J, J, G).
+(ii) For elements ( �J(i), J(i), G) ∈ P∗(�∆, ∆; Dϵ(σ), Mϵ(σ); T µσ
+σ ) (i = 1, 2), let f (i) :
+M (i) → ∆ be the degenerations given in (i) . Then f (1) is analytically equivalent to f (2) if
+and only if ( �J(1), J(1), G) coincides with ( �J(2), J(2), G).
+68
+
+Proof
+We prove (i). We consider an element ( �J, J, G) ∈ P∗(�∆, ∆; Dϵ(σ), Mϵ(σ); p)
+for p = [S, w] ∈ T µ
+σ , Harris-Mumford coordinates (· · · , z(j)
+i , · · · , z(γ)
+α,β, · · · ) of p on Dϵ(σ),
+and the expression of �J via the pseudo-periodic functions ϕ(j)
+i (u), and ψ(γ)
+α,β(u).
+Let πσ : Xϵ(σ) → Dϵ(σ) be the family in Th. 4.4, and let �f : �
+M → �∆ be the pulled
+back family (in the sense of Def. 4.10) from πσ by the map �J : �∆ → Dϵ(σ), i.e. �f is the
+second projection of the fiber product �
+M = π−1
+σ ( �J(�∆)) × �
+J(�∆) �∆ −→ �∆. Then the central
+fiber �f −1(0) is isomorphic to S by construction.
+Here we prove the following claim:
+Claim �
+M is normal with A-type singularities and any fiber �f −1(u) for u ̸= 0 is non-
+singular.
+In fact, since z(j)
+i ’s are the smoothing coordinates at the nodes and the ϕ(j)
+i (u)’s are not
+identically zero, all the nodes of S vanish via the deformation �f of S, i.e. the fiber �f −1(u)
+for u ̸= 0 is non-singular. Next, the total space of the restricted family π−1
+σ ( �J(�∆)) −→
+�J(�∆) has non-normal singularities along the central fiber which is the non-isolated cusp
+of the complex space curve locally given in C3g−2 by the image of the map
+�J : �∆ ∈ u �−→ (· · · , z(j)
+i , · · · , z(γ)
+α,β, · · · , x) = (· · · , un(Pi), · · · , uγα,β, · · · , x)
+(125)
+modulo units under the conditions n(Pi) ≥ 2 and γα,β ≥ 2 for any i, α, β, where x is a local
+fiber coordinate of πσ at a nonsingular point of π−1
+σ (0). Fortunately, these codimension-
+one singularities are resolved along the open locus of the central fiber automatically by
+the pull back by �J, for u may be considered via (125) as the local uniformization pa-
+rameter of this singularity. On the other hand, the germ (�Yg, P (j)
+i
+) at the node P (j)
+i
+in
+�Yg coincides with the germ at the origin given by xy = z(j)
+i
+in the ambient coordinates
+(· · · , z(j)
+i , · · · , z(γ)
+α,β, · · · , x, y) of C3g−1 (cf. [9, Chap.XI, §3]). Then the germ (�
+M, P (j)
+i
+) is
+the A-type singularity xy = un(Pi). The above claim is proved.
+Now by the definitions (d-i) ∼ (d-iii), Lemma 6.10 and (88), the complex curve �J(�∆) ⊂
+B ⊂ Dϵ(σ) is invariant under the action of G = ⟨µ⟩ on B compatible with respect to the
+action on �∆, i.e.
+�J
+�
+e
+� 1
+N
+�
+u
+�
+= �J (µ(u)) .
+Therefore G acts relatively on the family �f, which is a natural extension of the automor-
+phism of the central fiber S. The explicit descriptions of this action on �f are essentially
+the same as (98) ∼ (103) given for Diagram II. Let f ♯ : M ♯ = �
+M/G −→ ∆ = �∆/G be the
+quotient family. By the composition of f ♯ and the minimal resolution M → M ♯ of the
+singularities on M ♯, we obtain the normally minimal model f : M → ∆ of a degeneration.
+69
+
+The marked topological monodormy of f coincides with the preassigned µ ∈ P−
+g (σ),
+since it is determined by the action of G as in [10, Th.3.1.1]. The orbifold structure of f is
+nothing but the above {f ♯, �f, G}. Let Jf : ∆ → Mσ be the canonically extended moduli
+map of f. Then �J is clearly the lift of Jf by G, i.e. �J is the chart map of the orbifold
+moduli map of f. Thus we obtain the desired unique degeneration f.
+We prove (ii). Assume that f (i) are analytically equivalent to each other for i = 1, 2.
+Then Jf(i) : ∆ −→ Mσ ⊂ M g coincides with each other (cf. [58]). Therefore there exists
+an element g ∈ G such that �J �f(2) = g ◦ �J �f(1). If g ̸= id, then µf(1) ̸= µf(2), contradicting the
+assumption. Hence g = id and ( �J(1), J(1), G) = ( �J(2), J(2), G). The converse is obvious.
+Corollary 7.4. Let f : M → ∆ be a degeneration of genus g with a marking w : Σg →
+f −1(t0) (t0 ∈ ∂∆), and µf ∈ P−
+g (σ) be the marked topological monodromy of (f, w).
+Let ( �J �f, Jf, G) : (�∆, G, π�∆, ∆) −→ (Dϵ(σ), W(σ), ϕσ, Mϵ(σ)) be the orbifold moduli
+map of (f, w), and πσ : Xϵ(σ) → Dϵ(σ) be the family in Theorem 4.4. Then the orbifold
+model of f is isomorphic to the orbifold pull back (in Def.4.10) of πσ via ( �J �f, Jf, G).
+Corollary 7.5. Let AS(µ) be the set of (complex) analytic structures of degenerations
+f : M → ∆ of genus g under a fixed marking w whose topological monodromies �µf
+coincide with a fixed µ ∈ P−
+g (σ).
+Then AS(µ) is in a bijective correspondence with
+P∗(�∆, ∆; Dϵ(σ), Mϵ(σ); T µσ
+σ ).
+7.2
+Global orbifold moduli maps for fibered complex surfaces
+Here we define the notion of orbifold moduli map for a fibered complex surface.
+Let f : M → B be a proper surjective holomorphic map from a 2-dimensional com-
+plex manifold M to a compact Riemann surface B of genus h ≥ 0. Let Discf(B) =
+{Q1, · · · , Qs} ⊂ B be the discriminant locus, and set B(0) = B \ Discf(B). We call f a
+fibered complex surface of genus g ≥ 2 if any fiber of f over B(0) is a Riemann surface of
+genus g. Each degenerate fiber Fi = f −1(Qi) is assumed to be normally minimal in the
+sense of §6.2. Let f (0) : M(0) → B(0) be the restricted holomorphic family of f over B(0).
+We fix a point b0 ∈ B(0), and consider the fiber f −1(b0) := Σg as the base Riemmann
+surface of the marking.
+The fundamental group π1(B(0), b0) is generated by the loops β1, · · · , β2h, α1, · · · , αs
+on B(0) starting and ending at b0 with the unique relation
+β1β2β−1
+1 β−1
+2 β3β4 · · · β−1
+2h−1β−1
+2h α1 · · · αs = 1.
+Here each αi is a loop which goes around Qi once counterclockwise and β1, · · · , β2h are
+canonical generators of π1(B, b0). Since f (0) is a differentiable fiber bundle, we have the
+70
+
+monodromy representation to the mapping class group
+µf : π1(B(0), b0) −→ Γg.
+(126)
+Let ∆i = {ti ∈ C | |ti| < ϵ0} be a local disk coordinate at Qi on B, and fi = f|Mi :
+Mi = f −1(∆i) −→ ∆i be the degeneration over ∆i. Then we may identify µf(αi) as
+the marked topological monodromy µfi of fi defined in §6.2. Therefore, there exists an
+element σ(i) ∈ Cg such that µf(αi) = µfi ∈ P−
+g (σ(i)). Let Ni denote the pseudo-period of
+µfi.
+Let ϕ�B(0) : �B(0) −→ B(0) be the universal covering of B(0). Then π1(B(0), b0) acts on
+�B(0), and we have B(0) ∼= �B(0)/π1(B(0), b0). On the other hand, the cyclic group Z/NiZ
+acts on �∆i = {ui ∈ C | |ui| < ϵ1/Ni} by ui �→ e(1/Ni)ui, and the projection ϕ�∆i : �∆i → ∆i
+is defined by ui �→ ti = uNi. We define the complex orbifold structure over B by
+Borb =
+�
+�B(0), π1(B(0), b0), ϕ�B(0), B(0)� �
+1≤i≤s
+�
+�∆i, Z/NiZ, ϕ�∆i, ∆i
+�
+.
+(127)
+Note that, although π1(B(0), b0) is an infinite group and Z/NiZ is a finite group, the
+orbifold structure Borb is well-defined in the sense of §4.3.
+Let �f (0) : �
+M(0) = M(0) ×B(0) �B(0) −→ �B(0) be the pull back of f (0) over �B(0). Let
+M → M♯ be the contraction map of all the non-cores of Fi (1 ≤ i ≤ s), and f ♯ : M♯ → B
+be the natural holomorphic map. Let {f ♯
+i : M ♯
+i → ∆i, �fi : �
+Mi → �∆i, Gi = Z/NiZ} be the
+orbifold structure of fi in Definition 6.5 and Remark 6.6. We define the complex orbifold
+structure over M♯ by
+(M♯)orb =
+�
+�
+M(0), π1(B(0), b0), ϕ�
+M(0), M(0)� �
+1≤i≤s
+�
+�
+Mi, Z/NiZ, ϕ�
+Mi, M ♯
+i
+�
+(128)
+where ϕ�
+M(0) and ϕ�
+Mi are natural projections.
+Then we have the orbifold fibration of
+Definition 4.6 by
+(f ♯)orb : (M♯)orb −→ Borb.
+(129)
+Now let
+J(0) : B(0) −→ Mg
+(130)
+be the moduli map.
+The lifting of J(0) over �B(0) is defined as follows.
+For a point
+x ∈ B(0), let γ(b0, x) be an arc on B(0) starting from b0 and ending at x, and [γ(b0, x)] be
+its homopoty class. Then �B(0) consists of all the pairs (x, [γ(b0, x)]). Let
+wγ(b0,x) : Σg = f −1(b0) −→ f −1(x)
+be the oriented homeomorphism of fibers of f (0) along γ(b0, x). Since the isotopy class of
+wγ(b0,x) defines a Teichm¨uller marking of f −1(x), the map
+�J(0) : �B(0) −→ Tg
+(131)
+71
+
+is defined by �J(0) ((x, [γ(b0, x)])) = [f −1(x), wγ(b0,x)]. Then �J(0) is a holomorphic map which
+satisfies ϕTg ◦ �J(0) = J(0), i.e. it is a lifting of J(0) in (130).
+On the other hand, it follows from (107) that J(0) is holomorphically extended to
+J : B −→ M g.
+(132)
+By (110), the restricted map Jfi = J|∆i : ∆i −→ M g is lifted to
+�J �fi : �∆i −→ Dϵ(σ(i)).
+(133)
+We assure the compatibility condition for the patching of (131) and (133) as an orbifold
+map. If J is a constant map, i.e. J(B) = b0, then the maps (131) and (133) are clearly
+well-patched. Therefore we assume that J(B) is an analytic curve on M g . Let x ∈ B(0) ∩
+(∆i\Qi) be a point. Let �x ∈ �B(0) and x♯ ∈ �∆i be the points with ϕ�B(0)(�x) = ϕ�∆i(x♯) = x.
+Let Vx be a sufficiently small open neighborhood of x of B(0) ∩(∆i\Qi). Let �V�x (resp. V ♯
+x♯)
+be the open neighborhood of �x of �B(0) (resp. of x♯ of �∆i) with ϕ�B(0)(�V�x) = ϕ�∆i(V ♯
+x♯) = Vx.
+We set �y = �J(0)(�x) ∈ Tg = Dϵ(∅) and y♯ = �J �fi(x♯) ∈ Dϵ(σ(i)).
+Lemma 7.6. There exist open neighborhoods �U�y of �y in Tg, and U ♯
+y♯ of y♯ in Dϵ(σ(i))
+respectively such that
+(i) �J(0)(�V�x) (resp. �J �fi(V ♯
+x♯)) is an analytic curve in �U�y (resp. U ♯
+y♯),
+(ii) there exists a biholomorphic map ψ : �U�y −→ U ♯
+y♯ so that the restriction map of ψ to
+�J(0)(�V�x) induces an isomorphism onto �J �fi(V ♯
+x♯).
+Proof
+By Arbarello–Cornalba’s theorem [8], we may assume that �U�y is the base of
+a sufficiently small Kuranishi family of the Riemann surface f −1(x). On the other hand,
+Dϵ(σ(i)) is a union of the bases of standard Kuranishi families of stable curves by Theorem
+4.4. The Kodaira–Spencer map is an isomorphism at any point, particularly at y♯, of the
+base of the standard Kuranishi family by the property (iii) in §3.1. Therefore we also may
+assume that U ♯
+y♯ is the base of a sufficiently small Kuranishi family of f −1(x).
+Hence we may assume that there exists a biholomorphic map ψ′ : �U�y −→ U ♯
+y♯ such
+that ψ′ is induced from the extension g0 to the Kuranishi family of an automorphism
+g0 ∈ Aut(f −1(x)) by the property (v) in §3.1.
+Since ϕTg : Tg → Mg and ϕDϵ(σ(i)) : Dϵ(σ(i)) → Mϵ(σ(i)) are the forgetting maps of the
+Teichm¨uller marking and the Weyl marking respectively, we have ϕTg(�U�y) = ϕDϵ(σ(i))(U ♯
+y♯)
+as an open set (in the classical topology) of Mg containing J(0)(x).
+Now we consider the analytic curves �J(0)(�V�x) and �J �fi(V ♯
+x♯) on the above spaces �U�y and
+U ♯
+y♯ respectively. It is clear that ϕTg(�J(0)(�V�x)) = ϕDϵ(σ(i))( �J �fi(V ♯
+x♯)) = J(0)(Vx). Hence there
+exists an element g0 ∈ Aut(f −1(x)) such that the restriction of the biholomorphic map
+g0 ◦ ψ′ : �U�y −→ U ♯
+y♯ to �J(0)(�V�x) sends isomorphically onto �J �fi(V ♯
+x♯).
+72
+
+We extend the canonically extended moduli map J : B −→ M g in (132) to the orbifold
+map as follows. Lemma 7.6 assures its well-definedness.
+Definition 7.7. The maps �J(0) : �B(0) −→ Tg in (131) and �J �fi : �∆i −→ Dϵ(σ(i)) in (133)
+(1 ≤ i ≤ s) are well-patched and define the orbifold map
+Jorb
+f
+: Borb −→ M
+orb
+g .
+We call Jorb
+f
+the (global) orbifold moduli map of a fibered complex surface f : M → B.
+7.3
+Global recovery of basic members of fibered complex sur-
+faces
+We construct all the fibered complex surfaces for possible monodromy representations and
+orbifold moduli maps by pulling back from our universal degenerating family.
+We let D(B) = {Q1, · · · , Qs} be a finite set of a compact Riemann surface B, and
+set B(0) = B \ D(B). For a fixed point b0 ∈ B(0), we consider a representation of the
+fundamental group to the mapping class group of genus g ≥ 2
+µ : π1(B(0), b0) −→ Γg.
+(134)
+We assume that µ satisfies the following condition. Let αi be a loop which goes around
+Qi once counterclockwise for 1 ≤ i ≤ s. Then there exists an element σ(i) ∈ Cg such that
+µ(αi) ∈ P−
+g (σ(i)). We call such a µ a pseudo-periodic representaion on B.
+Let Ni be the pseudo-period of µ(αi) and Borb be the orbifold structure over B as in
+(127). Let J : B → M g be a holomorphic map, and
+Jorb : Borb −→ M
+orb
+g
+(135)
+be an orbifold map over J. We assume that Jorb satisfies the following conditions:
+(i) the restricted chart map on �B(0) maps �B(0) to Tg, i.e. �J : �B(0) −→ Tg = Dϵ(∅),
+(ii) the restricted chart map on each �∆i belongs (in Def. 7.1) to
+�J|�∆i ∈ P∗(�∆i, ∆i; Dϵ(σ(i)), Mϵ(σ(i)); T
+µσ(i)
+σ(i) ).
+We call such Jorb a dual pseudo-periodic map associated with µ.
+Theorem 7.8. (i)
+For any (µ, Jorb) ∈ F(B), there uniquely exists (modulo analytc
+equivalence) a fibered complex surface of genus g ≥ 2
+f : M −→ B
+(136)
+73
+
+such that the monodromy representation of f coincides with µ, and the orbifold moduli
+map of f coincides with Jorb.
+(ii)
+The orbifold fibration (f ♯)orb : (M♯)orb −→ Borb associated with (136) (see (129))
+coincides with the orbifold pull-back (Def. 4.10) from the universal degenerating family
+(Def. 4.9) π : Y
+orb
+g
+−→ M
+orb
+g .
+Proof
+The restricted family of π to Dϵ(∅) = Tg is nothing but the universal family
+([14]) of Teichm¨uller-marked Riemann surfaces Yg → Tg. Since �J(�B(0)) is contained in Tg
+by assumption, the family over �B(0) obtained by the pull-back via �J from this universal
+family is a holomorphic family �f (0) : �
+M(0) → �B(0) of Teichm¨uller-marked Riemann sur-
+faces. Since B(0) is the quotient space of �B(0) by π1(B(0), b0), the forgetting map of the
+markings induces a holomorphic family f (0) : M(0) → B(0) of Riemann surfaces.
+On the other hand, it follows from Theorem 7.3 that the pull-back via �J|�∆i of π induces
+a family of stable curves �fi : �
+Mi → �∆i and its quotient family f ♯
+i : M ♯
+i → ∆i.
+By the same argument as that of the well-definedness of Definition 7.7, the maps
+(�f (0), f (0)) and ( �fi, f ♯
+i ) (1 ≤ i ≤ s) are well-patched as orbifold maps and define an orbifold
+fibration (f ♯)orb : (M♯)orb → Borb. Hence by the argument in §6.2, we have a fibered
+complex surface f : M → B.
+It is clear from the construction that the monodromy representation of f coincides
+with µ, and the orbifold moduli map of f coincides with Jorb. The uniqueness of f modulo
+analytic equivalence is also clear since it has the fixed orbifold moduli map Jorb.
+Remark 7.9. The construction of the fibered complex surface (136) of genus g ≥ 2 is
+analogous to that of the basic members of elliptic surfaces (basic elliptic surfaces, for
+short) due to Kodaira [46, §8]; nevertheless they are different in the following points.
+(i)
+Basic elliptic surfaces are assumed to have no multiple fibers (i.e fibers of types
+mIb, m ≥ 2 given in [46, p.565]), while our fibered surfaces (136) are admitted to have
+any singular fibers including multiple fibers.
+(ii) Let F(J , G) be the set of elliptic surfaces without multiple fibers which have J-
+invariant J and homological invariant G (see [46, Def. 8.1]). Then the basic elliptic
+surface in F(J , G) is characterized as the unique member which admits a global section
+([46, Th. 10.2]), and other members in F(J , G) are obtained from the basic elliptic surface
+by twisting defined in [46, Def. 9.2] (see [46, Th. 10.1]).
+On the other hand, a fibered complex surface of genus g ≥ 2 is uniquely determined by
+the data (µ, Jorb) by Th.7.8.
+This difference essentially comes from the fact that an elliptic curve has translation
+automorphisms, while a Reimann surface of genus g ≥ 2 has no such infinite automor-
+phisms.
+74
+
+Acknowledgments. We thank Prof. Sampei Usui for his advice and discussions on log
+geometry, Prof. Toshiyuki Akita for his advice on representations of automorphisms of
+Riemann surfaces, Prof. Kunio Obitsu for his advice on augmented Teichm¨uller spaces,
+Prof. Kazuhiro Konno for discussions on fibered complex surfaces. Finally we thank
+Prof. Athanase Papadopoulos for his interest in our results and for many comments which
+improved our presentation very much. The second named author is partially supported
+by JSPS Grant KAKENHI 17H01091.
+References
+[1] W. Abikoff, The real analytic theory of Teichm¨uller space, Lecture Notes in Math. 820
+(1980), Springer.
+[2] N. A’Campo, Sur la monodromie des singularit´es isol´ees d’hypersurfaces complexes, Invent.
+math. 20 (1973), 147–169.
+[3] N. A’Campo, L. Ji, A. Papadopoulos, On Grothendieck’s construction of Teichm¨uller space,
+in Handbook of Teichm¨uller theory, Vol. VI (2016), 35–69.
+[4] A. A’Campo-Neuen, N. A’Campo, L. Ji, A. Papadopoulos, A commentary on Teichm¨uller’s
+paper “Ver¨anderliche Riemannsche Fl¨achen”, in Handbook of Teichm¨uller theory, Vol. IV
+(2014), 805–814.
+[5] L. V. Ahlfors, Some remarks on Teichm¨uller’s space of Riemann surfaces, Ann. of Math.
+(2) 74 (1961), 171–191.
+[6] L. V. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math.
+(2) 72 (1960), 385–404.
+[7] J. W. Alexander, On the deformation of a n-cell, Proc. Nath. Acad. Sci. 9 (1923), 406–407.
+[8] E. Arbarello, M. Cornalba, Teichm¨uller space via Kuranishi families, Ann. Scuola
+Norm. Pisa Cl. Sci. (5) Vol. VIII (2009), 89–116.
+[9] E. Arbarello, M. Cornalba, P.A. Griffiths, Geometry of Algebraic Curves, Vol. II, Springer,
+2011.
+[10] T. Ashikaga, Local signature defect of fibered complex surfaces via monodromy and stable
+reduction, Comment. Math. Helv. 85 (2010), 417-461.
+[11] T. Ashikaga and M. Ishizaka, Classification of degenerations of curves of genus 3 via
+Matsumoto-Montesinos’ theorem, Tohoku Math. J. 54 (2002), 195–226.
+[12] T. Ashikaga and K. Konno, Global and local properties of pencils of algebraic curves, in
+Algebraic Geometry 2000 Azumino, ed. by S. Usui et al., Adv. St. in Pure Math. 36 (2002),
+1–49.
+[13] T. Ashikaga and K. Yoshikawa, A divisor on the moduli space of curves associated to the
+signature of fibered surfaces, Adv. St. in Pure Math. 56 (2009), 1–34.
+[14] L. Bers, Fiber space over Teichm¨uller spaces, Acta. Math. 130 (1973), 89–126.
+[15] L. Bers, Space of degenerating Riemann surfaces with nodes, in Discontinuous Groups and
+Riemann surfaces, Ann. Math. Studies 79, Princeton U. P. (1974), 43–55.
+75
+
+[16] S. A. Broughton, The equisymmetric stratification of the moduli space and the Krull di-
+mension of mapping class group, Topology and its Applications 37 (1990), 101–113.
+[17] H. Cartan, Quotient d’un espace analytique par un group d’automorphismes, Algebraic
+geometry and topology, A Symposium in honor of S. Lefschetz, Edited by R. H. Fox, D. C.
+Spencer, and A. W. Tucker, Princeton Math. Series, 12. Princeton Univ. Press, Princeton,
+N.J., 1957, 90–102.
+[18] H. Cartan, Quotients of complex analytic spaces, in Contributions to Function Theory,
+Tata Institute, Bombay (1960), 1–15.
+[19] F. Catanese, A superficial working guide to deformation and moduli, in Handbook of Moduli,
+vol. I (2013), 161–215.
+[20] C.H. Clemens, Picard-Lefschetz theorem for families of nonsingular algebraic varieties ac-
+quiring ordinary singularities, Trans. Amer. Math. Soc. 136 (1969), 93–108.
+[21] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Publ.
+Math. I.H.E.S. 36 (1969), 75–110.
+[22] Dias Raquel and Gonz´ales-Aguilera Victor, Limit points of the branch locus of Mg,
+Adv. Geom. 19 (2019), 505–526.
+[23] C. Earle and P. Sipe, Families of Riemann surfaces over the punctured disk, Pacific J. Math.
+150 (1991), 79–86.
+[24] C. Ehresmann, Sur les sepaces fib´es diff´erentiables, C. R. Acad. Sci. Paris 224 (1947),
+1611-1612.
+[25] H. M. Farkas and I. Kra, Riemann surfaces, 1979, Springer.
+[26] R. Friedman, Global smoothings of varieties with normal crossings, Ann. of Math.
+118
+(1983), 75–114.
+[27] H. Grauert, ¨Uber Modifikationen und exzeptionelle analytische Mengen, Math. Ann, 146
+(1962), 331–368.
+[28] A. Grothendieck, Techniques de construction en g´eom´etrie analytique. I∼X. S´eminaire
+Henri Cartan, tome 13, (1960-1961).
+[29] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent.
+Math. 67 (1982), 23–88.
+[30] W. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart.
+J. Math. Oxford Ser.(2) 17 (1966), 86–97.
+[31] W. J. Harvey, On branch loci in Teichm¨uller space, Trans. Amer. Math. Soc. 153 (1971),
+387–399.
+[32] W. H. Harvey, Chabauty spaces of discrete groups, Discontinuous groups and Riemann
+surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Princeton Univ. Press,
+Princeton, N.J., 1974, 239–246. Ann. of Math. Studies, no. 79, MR 0364629.
+[33] W. J. Harvey, Geometric structure of surface mapping class groups, in Homological gourp
+theory. Proceedings of a Symposium on Homological and Combinatorial Techniques in
+Group Theory held at Durham, September, 1977. Ed. by C. T. C. Wall. London Math.
+Soc. Lecture Note Series, 36. Cambridge University Press, 1979, 255–269.
+76
+
+[34] V. Hinich and A. Vaintrob, Augmented Teichm˝uller spaces and orbifolds, Sel. Math. New
+Ser. 16 (2010), 533–629.
+[35] R. Hirakawa and S. Takamura, Atlas for Moduli Geometry of Riemann Surfaces, Kyoto
+University, preprint, 2022.
+[36] J. H. Hubbard, Teichm¨uller theory and applications to geometry, topology, and dynamics,
+Vol. I, Matrix Editions, Ithaca, NY, 2006.
+[37] J. H. Hubbard and S. Koch, An analytic construction of the Deligne-Mumford compactifi-
+cation of the moduli space of curves, J. Diff. Geom. 98 (2014), 261–313.
+[38] J. H. Hubbard, P. Papadopol and V. Veselov, A compactification of H´eron mapping in C2
+as dynamical systems, Acta Math. 184 (2000), 203–270.
+[39] Y. Imayoshi, Holomorphic families of Riemann surfaces and Teichm¨uller spaces, in Riemann
+surfaces and related topics, Ann. of Math. Studies 97 (1981), 277–300.
+[40] Y. Imayoshi, A construction of holomorphic families of Riemann surfaces over punctured
+disk with given monodromy, in Handbook of Teichm¨uller theory, Vol. II (2009), 93–130.
+[41] Y. Imayoshi and Taniguchi, An introduction to Teichm¨uller spaces, Springer, 1992.
+[42] K. Kato and C. Nakayama, Log Betti cohomology, log ´etale cohomology, and log de Rham
+cohomology of log schemes over C, Kodai Math.J. 22 (1999), 161–189.
+[43] K. Kato and S. Usui, Classifying spaces of degeneration polarized Hodge structures, Ann.
+of Math. Studies 169, Princeton Univ. Press, 2009.
+[44] L. Keen, Collars on Riemann surfaces, In Discontinuous groups and Riemann surfaces,
+Princeton Univ. Press (1974), 263–268.
+[45] S. P. Kerchhoff, The Nielsen realization problem, Ann. of Math. 117 (1983), 235–265.
+[46] K. Kodaira, On compact analytic surfaces II, Ann. of Math. 77 (1963), 563–626.
+[47] I. Kra, Horocyclic Coordinates for Riemann Surfaces and Moduli Spaces. I: Teichm¨uller
+and Riemann Spaces of Kleinian Groups, J. Amer. Math. Soc. 3 (1990), 499–578.
+[48] I. Kra, Book review of : S. Donaldson, Riemann surfaces. Bull. Amer. Math. Soc. (N.S.) 49
+(2012), no.3, 455–463. MR 2952711 Zbl 1321.00036.
+[49] Y. Kuno, Meyer functions and the signature of fibered 4-manifolds, in Handbook of Te-
+ichm¨uller theory, Vol. V (2016), 75–96.
+[50] A. Kuribayashi, On analytic families of compact Riemann surfaces with non-trivial auto-
+morphisms, Nagoya J. Math. 28 (1966), 119–165.
+[51] X. L. Liu, Fractional Dehn twists and modular invariants, Science China Math. 64 (2021),
+1735 –1744.
+[52] H. Masai, Compactification and distance on Teichm¨uller space via renormalized volume,
+arXiv:2108.06059v3 [math.GT] 27Sep 2022.
+[53] Y. Matsumoto, On the universal degenerating family of Riemann surfaces, in Singularities
+in Geometry and Topology–Strasbourg 2009, Euro. Math. Soc. (2012), 71–102.
+[54] Y. Matsumoto, The Deligne-Mumford compactification and crystallographic groups, in
+Handbook of Teichm¨uller theory, Vol. VII (2020), 21–62.
+77
+
+[55] Y. Matsumoto and J. M. Montesinos-Amilibia, Pseudo-periodic maps and degeneration of
+Riemann surfaces, Lecture Note in Math. No. 2030 (2011), Springer.
+[56] H. Miyachi, Lipschitz algebras and compactifications of Teichm¨uller space, in Handbook of
+Teichm¨uller theory, Vol. IV (2014), 375–413.
+[57] S. Mizuta and M. Namba, Greenberg’s theorem and equivalence problem on compact Rie-
+mann surfaces, Osaka J. Math. 43 (2006), 137–178.
+[58] Y. Namikawa, Studies on degeneraions, in Classification of Algebraic Varieties and Compact
+Complex Manifolds, Lecture Note in Math. vol. 412 (1974), Springer.
+[59] J.
+Nielsen,
+Die
+Structur
+periodischer
+Transformationen
+von
+Fl¨achen,
+Mat.-
+Fys. Medd. Danske Vid. Selsk. 15 (1937), 77 pp. English translation:
+in Collected
+Papers 2, Birkh¨auser, 1986.
+[60] J. Nielsen, Abbildungsklassen endlicher Ordnung, Acta Math., 75 (1942), 23–115. English
+translation: in Collected Papers 2, Birkh¨auser, 1986.
+[61] J. Nielsen, Surface transformation classes of algebraically finite type, Mat.-Fys. Medd.
+Danske Vid. Selsk. 21 (1944), 3–89 : in Collected Papers 2, Birkh¨auser, 1986.
+[62] K. Ohshika, Compactifications of Teichm¨uller spaces, in Handbook of Teichm¨uller theory,
+Vol. IV (2014), 235–254.
+[63] M. Reid, Problems on pencils of small genus, Preprint (1990) (see his home page).
+[64] I. Satake, The Gauss-Bonnet Theorem for V-manifolds, J. Math. Soc. Japan, 9 (1957),
+464–492.
+[65] H. Shiga and H. Tanigawa, On the Maskit coordinates of Teichm¨uller spaces and modular
+transformations, Kodai Math. J. 12 (1989), 437–443.
+[66] S. Takamura, Towards the classification of atoms of degenerations II, (Cyclic quotient con-
+struction of degenerations of complex curves), Res. Inst. Math. Sci. Kyoto Univ. Preprint
+series No. 1344 (2001).
+[67] S. Takamura, Linearization of quotient families, J. Math. Sci. Univ. Tokyo 26 (2019), 361–
+389.
+[68] S. L. Tan, Chern numbers of a singular fiber, modular invariants and isotrivial families of
+curves, Acta Math. Vietnamica 35 (2010), 159 –172.
+[69] O. Teichm¨uller, Extremale quasikonforme Abbildungen und quadratische Differentiale.
+Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 22 (1940), 1–197. English translation by
+G. Th´eret, Extremal quasiconformal mappings and quadratic differentials. In Handbook of
+Theichm¨uller theory (A. Papadopoulos, ed.) Vol. V, EMS Publishing House, Z¨urich, 2016,
+321–483.
+[70] O. Teichm¨uller, Ver¨anderliche Riemannsche Fl¨achen. Deutsche Math. 7, 344–359 (1944).
+English translation by A. A’Campo Neuen, Variable Riemann surfaces. In Handbook of
+Teichm¨uller theory (A. Papadopoulos, ed.) Vol. IV, EMS Publishing House, Z¨urich, 2014,
+787–803.
+[71] T. Terasoma, Degeneration of curves and analytic deformations, arXiv:math/9801118v1
+[math.AG] 26 Jan.1998.
+78
+
+[72] S. Usui, Recovery of vanishing cycles by log geometry, Tohoku Math. J. 53 (2001), 1–36.
+[73] H. V¨olklein, Moduli spaces for covers of the Riemann sphere, Israel J. Math. 85 (1994),
+407–430.
+[74] A. Wiman, ¨Uber die hyperelliptischen Curven und diejenigen vom Geschlechte p=3,
+welche eindeutige Transformationen in sich zulassen, Bihang Kongl. Svenska Vetenskaps-
+Akademiens Handligar (1985), no. 21 (1), 1–23.
+[75] S. Wolpert, Geometry of Weil-Petersson completion of Teichm¨uller space, In Surveys in
+Differential Geometry VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck, pp.
+357–393. Intl. Press, Cambridge, MA. 2003.
+[76] S. Wolpert, Behaviour of geodesic-length function on Teichm¨uller space, J. Differential
+Geom. 79 (2008), 277–334.
+[77] S. Wolpert, Families of Riemann surfaces and Weil-Petersson geometry, CBMS Regional
+Conference Series in Math. 113, Amer. Math. Soc. 2010.
+[78] S. Yamada, On the geometry of Weil-Petersson completion of Teichm¨uller spaces, Math.
+Res. Lett. 11 (2004), 327–344.
+Tadashi Ashikaga, Faculty of Engineering, Tohoku-Gakuin University, Tagajo,
+Miyagi 985-8537, Japan; e-mail: ashikaga@mail.tohoku-gakuin.ac.jp
+Yukio Matsumoto, Department of Mathematics, Gakushuin University, Mejiro,
+Toshima-ku, Tokyo 171-8588, Japan; e-mail: yukiomat@math.gakushuin.ac.jp
+79
+
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+page_content='6 Harris-Mumford coordinates around equisymmmetric strata  §6 Monodromy and orbifold moduli maps of degenerations of Riemann surfaces  §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1 Pseudo-periodic maps and automorphisms of stable curves  §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2 Orbifold structures of degenerations of Riemann surfaces  §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3 Local orbifold moduli maps and Kodaira-periodicity  §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4 Examples of degenerations and their invariants  §7 Recovery of fibered complex surfaces from the universal degenerating family  §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1 Local recovery of degenerations from the universal family  §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2 Global orbifold moduli maps for fibered complex surfaces  §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3 Global recovery of basic members of fibered complex surfaces  References  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='00381v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='AG]  1 Jan 2023  1  Introduction  Kodaira [46] constructed elliptic surfaces in a canonical way for given monodromies and  J-invariants, which he called the basic members of elliptic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Our aim is to extend  this result to fibered complex surfaces of genus g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In our previous paper [54], we constructed a new orbifold structure M  orb  g  over the  Deligne–Mumford compactification by using a certain bordification of Teichm¨uller space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In this paper, we construct an orbifold fiber space  π : Y  orb  g  → M  orb  g  such that any fibered complex surface admitting unstable fibers can be pulled back from π  by the orbifold moduli map Jorb which we will construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The fiber space π : Y  orb  g  → M  orb  g  will be constructed by patching Kuranishi families of stable curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The map Jorb has a  nature similar to Kodaira’s J-invariant in the sense that it is delicately influenced by the  monodromy and has the property of pseudo-periodicity which we will explain below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, we will fix the notation from Teichm¨uller theory, and will review the structure  of M  orb  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then we will state the results of the present paper in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1  Precise orbifold structure of M  orb  g  Throughout the paper, we will fix a closed oriented topological surface Σg of genus g(> 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  A Riemann surface S is usually considered to be a marked Riemann surface (S, w), in other  words, a Riemann surface for which an orientation-preserving homeomophism w : Σg → S  (called a marking) is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Two marked surfaces (S1, w1) and (S2, w2) are equivalent if  there exists a holomorphic map f : S1 → S2 such that f ◦ w1 ≃ w2, where ≃ means “is  isotopic to” (or equivalently, in the case of closed surfaces, “is homotopic to”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The set of  equivalence classes is nothing but the Teichm¨uller space modeled on Σg, and it is denoted  by Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By Teichm¨uller’s theorem, Tg is a metric space homeomorphic to R6g−6 (see [41],  Chapter 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Teichm¨uller [70],[69] constructed the space Tg and the universal curve on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' He found  that Tg is a complex manifold of dimension 3g − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Grothendieck [28] had a deep insight  into Tg from functorial viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For a review of Teichm¨uller–Grothendieck theory, see  [4], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Ahlfors–Bers [5], [6] rediscovered the complex structure on Tg from analytic  viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The mapping class group of genus g, Γg, is defined to be  Γg = {ϕ : Σg → Σg | orientation preserving homeomorphisms}/ ≃ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The group structure of Γg is given by the usual composition of maps: for [ϕ], [ψ] ∈ Γg, we  have [ϕ][ψ] = [ϕ ◦ ψ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The mapping class group Γg acts on the Teichm¨uller space Tg by  2  the rule: for [ϕ] ∈ Γg and [S, w] ∈ Tg, we define  ϕ∗([S, w]) = [S, w ◦ ϕ−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  This action is properly discontinuous, and preserves the Teichm¨uller metric and the com-  plex structure (see [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The quotient space Mg = Tg/Γg is the moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' It is a  normal complex variety (see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The moduli space Mg parametrizes all the isomor-  phism classes of closed Riemann surfaces of genus g, but it is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By adding  frontier points corresponding to “stable curves” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Riemann surfaces with nodes and  with finite automorphisms), it can be compactified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This is the Deligne–Mumford com-  pactification M g (DM-compactification for short) of the moduli space Mg (see [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Many  authors tried to reconstruct M g from the viewpoint of Teichm¨uller theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Bers [14] be-  gan this project, and Harvey [32] topologically reconstructed M g by using discrete group  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hubbard and Koch’s paper [37] contains a bit of history concerning the DM-  compactification of the moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For recent developments of “compactifications of  Teichm¨uller space”, see Ohshika [62], Miyachi [56] or Masai [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  According to Kra’s overview [48] in this field, an analytic construction of the DM-  compactification had to await the work of Hubbard and Koch [37] in the twenty-first  century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' But as we explained in [54] (by a few sentences after Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5 and by Re-  mark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3), Hubbard–Koch’s orbifold charts contain certain insufficient points as orbifold  charts of M g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' On the other hand, we gave a complete set of orbifold charts to the DM-  compactification of the moduli space M g ([54, §6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Harvey’s curve complex  For our construction, Harvey’s curve complex Cg plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Given a  surface Σg, Harvey [33] introduced an abstract simplicial complex called the curve complex  Cg = C(Σg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  By definition, a vertex of Cg is an isotopy class of an essential (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' not null-homotopic)  simple closed curve on Σg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A simplex σ ∈ Cg is a collection of disjoint, mutually non-  isotopic essential simple closed curves on Σg:  σ = ⟨C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' , Ck⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The number k of simple closed curves contained in σ will be denoted by |σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' It is known  that |σ| ≤ 3g − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We have dim σ = |σ| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let [S, w] be a point of Tg, and let σ ∈ Cg be a simplex: σ = ⟨C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' , Ck⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We  represent each simple closed curve w(Ci) on the Riemann surface S by a geodesic, and  contract each of them to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then we have a stable curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Thus the topological type  of a stable curve is described by a simplex σ of the curve complex Cg, and the topological  model of the stable curve is denoted by Σg(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  3  Weyl groups  Another important object associated with a simplex σ ∈ Cg is the Weyl group W(σ),  which is defined as follows:  Let Γ(σ) be a subgroup of the mapping class group Γg generated by the Dehn twists  about the simple closed curves Ci, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' , k on Σg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let NΓ(σ) be the normalizer of  Γ(σ) in Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then we can prove that a mapping class ϕ ∈ Γg belongs to the normalizer  NΓ(σ) if and only if ϕ permutes the isotopy classes of the curves Ci, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' , k of σ  (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5 in [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Now the definition of the Weyl group W(σ) is the following:  W(σ) := NΓ(σ)/Γ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Note that the Weyl group W(σ) in our sense is not necessarily a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In §2, we  will prove that the Weyl group W(σ) is the mapping class group of a (topological) surface  with nodes Σg(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Little Tichm¨uller space T(σ)  Just as the Teichm¨uller space Tg was constructed by using marked Riemann sur-  faces (S, w) (w being an orientation-preserving homeomorphism w : Σg → S), the  little Teichm¨uller space T(σ) is constructed using marked stable curves (S, w) (where  w : Σg(σ) → S is an orientation-preservig homeomorphism, S being a stable curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' T(σ)  is a bounded domain of C3g−3−|σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Controlled deformation space Dε(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let M be a (2-dimensional) Margulis constant: two distinct simple closed geodesics  on any Riemann surface S are disjoint if their lengths are smaller than M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Of course,  any positive number smaller than M has again the same property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Thus the Margulis  constant is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let ε be a positive number smaller than a Margulis constant M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We will fix such an ε throughout the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let σ be a simplex of Cg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, we will give the definition of the controlled deforma-  tion space Dε(σ) (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Dε(σ) is a complex manifold of complex dimension 3g − 3,  and is homeomorphic to an open (6g − 6)-cell (see Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7 in [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Dε(σ) contains  T(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The Weyl group W(σ) acts on Dε(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This action is properly discontinuous and  holomorphic (see Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6 of [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We can prove that the compactified moduli  space M g is covered by Dε(σ)/W(σ), where σ runs over the simplexes of Cg, and can be  empty (see Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9 in [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Thus we have  Theorem(Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='11 [54]) In the Deligne–Mumford compactification M g, the finite  4  family  {(Dε(σ), W(σ))}σ∈Cg/Γg∪∅  forms an atlas of orbifold charts of a complex (3g − 3)-orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We will denote the compactified moduli space M g with this atlas of orbifold charts by  M  orb  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2  The results of the present paper  The main results consist of the following three parts (A) ∼ (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (A) Arbarello-Cornalba [8] reconstructed the universal family of Riemman surfaces  due to Teichm¨uller [70] and Bers [14] over Teichm¨uller space by patching Kuranishi fam-  ilies of Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Inspired by their work, we construct a family of stable curves  over the controlled deformation spaces by patching Kuranishi families of stable curves  (Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4), which may be considered as a reconstruction of Hubbard–Koch’s family [37,  Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By patching these families over all the orbifold charts of M  orb  g , we have;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Main Theorem I (Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8 and Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9) There exists a (strong) orbifold fibration  π : Y  orb  g  → M  orb  g  which is obtained by patching standard Kuranishi families of stable curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We call π the universal degenerating family of Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The naming comes  from the universality in the sense that any fibered complex surface is obtained by pulling  back from π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (We will explain this point in (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=') We are also inspired by Arbarello–  Cornalba–Griffiths’ description [9, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='XV §8] of the bordification of Teichm¨uller space  using real blow-ups and the method of log geometry by Kato-Nakayama [42] and Usui  [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We also refer to Hinich-Vaintrob [34] for a related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Philosophically we are  inspired by the project of Catanese [19] for studying the relation between the Kuranishi  families and the Teichm¨uller theory for many kinds of algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (B) For a mapping class ϕ ∈ Γg, let us denote by [ϕ] the set of conjugate elements of  ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Harvey [31] described the fixed point locus T [ϕ]  g  in Tg for a finite (elliptic, or periodic)  element ϕ ∈ Γg in terms of the space of the base Riemann surfaces pointed by the branch  points for the cyclic covering associated with ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' T [ϕ]  g  is called the equisymmetric strata  for ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Broughton [16] described the similar set M [ϕ]  g  in Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We extend these results to  the boundaries of Tg and Mg, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' to the little Teichm¨uller space T(σ) and its image on  M g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  5  Main Theorem II (Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='17 and Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='15) Let T [ϕ](σ) be the set in T(σ) consisting  of the marked stable curves with a given numerical data Num(ϕ) of an automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then a connected component of T [ϕ](σ) is a complex manifold which is isomorphic to a  direct product of pointed Teichm¨uller spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The image M [ϕ](σ) of T [ϕ](σ) in M g \\ Mg is an irreducible subvariety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  T [ϕ](σ) may be identified with the fixed point locus for a parabolic (pseudo-periodic)  element in Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The direct factor of the structure of T [ϕ]  g  appears as the space of pointed  Riemann surfaces which come from each component of the base nodal Riemann surfaces of  the cyclic covering associated with ϕ (see the discussion in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4 and the precise statement  of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Our discussion is similar to those of the moduli construction of the branched  covering of Riemann surfaces (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [73], [57] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We are also inspired by Terasoma’s  argument [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Note that the equisymmetric theory on Mg via various group actions was  recently developed by Takamura [67], Hirakawa-Takamura [35] et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (C) Kodaira [46] constructed the basic member of an elliptic surface (without multiple  fibers) in a canonical way from the data of the monodromy and the J-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We will  extend this result to any fibered complex surface of genus g ≥ 2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We start from the local viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let f : M → ∆ be a degeneration of genus g ≥ 2  with the degenerate fiber F = f −1(0) over a small disk ∆, and µf be the topological  monodromy of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then µf belongs to the conjugacy class ˆP(−)  g  (σ) ⊂ ˆΓg of pseudo-  periodic maps of negative twists for some σ ∈ Cg (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [61], [55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Now as an extended  notion of local J-invariant, we define the local orbifold moduli map  { �J : �∆ → B ⊂ Dϵ(σ), J : ∆ → Mϵ(σ) ⊂ M g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' G = ⟨µ⟩ ≃ Z/NZ}  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Here J is the usual holomorphically extended moduli map for f (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [39]),  �∆ → ∆ is the totally ramified covering of disks with covering degree N which is the  pseudo-period of the monodromy µf, �f : �  M → �∆ is the precise stable reduction of f ([10,  §2]) and �J is the natural map associated with the deformation �f to the Kuranishi space  B (as a chart of Dϵ(σ)) of the stable curve �F = �f −1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then �J can be considered as  the lifting map of J by the cyclic group G whose generator µ essentially comes from the  automorphism of �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In order to describe the map �J explicitly, we define the following;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Definition (Defs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='11) A holomorphic function ϕ(u) is called a pseudo-periodic  function with multiplicity γ ∈ N and period L ∈ N if the Taylor expansion at the origin  is given by ϕ(u) = �∞  i=0 ciuγ+iL (c0 ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call γ/L ∈ Q the analytic screw number of  ϕ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Main Theorem III (Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='12) Let (z1, · · · , zk, zk+1, · · · z3g−3) be the Harris–Mumford  coordinates at �J(0) of B ⊂ Dϵ(σ), where zi (1 ≤ i ≤ k) is a smoothing coordinate at a  6  node of �F, and zj (k + 1 ≤ j ≤ 3g − 3) a variable coordinate of a component of �F (see  Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  �J : �∆ ∋ u �−→ (z1, · · · , z3g−3) = (ϕ1(u), · · · , ϕ3g−3(u)) ∈ B ⊂ Dϵ(σ)  be the expression of the map �J by these coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (i) If 1 ≤ i ≤ k, then ϕi(u) is a pseudo-periodic function whose multiplicity and  period are explicitly written by the numerical data of the topological monodormy µf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In particular, the analytic screw number of ϕi(u) coincides with the absolute value of  Nielsen’s screw number of µf for a cut curve in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) If k + 1 ≤ j ≤ 3g − 3 and ϕj(u) is not identically zero, then ϕj(u) is a pseudo-  periodic function whose period and fractional part of the analytic screw number are  explicitly written by the numerical data of µf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Note that the fractional part of the analytic screw number of ϕj(u) (k+1 ≤ j ≤ 3g−3)  is written by a logarithmic quadratic character induced from the total valency of µf by  the discussion in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' But the integral part of the analytic screw number of ϕj(u) is  not a monodromy invariant, and it is purely an analytic invariant of the fiber germ (see  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Pseudo-periodicity has already appeared in Kodaira’s local description [46, §8] of J-  invariant by the action of the element of SL(2, Z) which is the homological monodromy  of a degeneration of elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Now we discuss from the global point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let f : M → B be a global fibered  complex surface of genus g ≥ 2 with descriminant locus Discf(B) = {Q1, · · · , Qs} ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We set B(0) = B \\ Discf(B), and let  µf : π1(B(0), b0) −→ Γg,  (b0 ∈ B(0))  be the monodromy representation of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  Borb =  �  �B(0), π1(B(0), b0), ϕ�B(0), B(0)� �  1≤i≤s  �  �∆i, Z/NiZ, ϕ�∆i, ∆i  �  be the natural orbifold structure over B, where �B(0) is the universal covering of B(0) and  �∆i → ∆i the covering of disks around Qi whose degree coincides with the pseudo-period  of the local monodromy at Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the orbifold moduli map  Jorb  f  : Borb −→ M  orb  g  is well-defined by patching the local orbifold moduli maps and the natural map from �B(0)  to Tg (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  7  Main Theorem IV (Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10) Any fibered complex surface f : M → B can  be pulled back in the orbifold sense from the universal degenerating family of Riemann  surfaces π : Y  orb  g  −→ M  orb  g  via the orbifold muduli map Jorb  f  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The above theorem may be considered as an accomplishment of the results of Imayoshi  [39] and [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  2  Mapping class groups of stable curves  In this section, we will define and study the mapping class group of a stable curve S of  genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We already described it in [54, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7] as a certain quotient group of a subgroup  of the mapping class group of a Riemann surface, and called it the Weyl group of genus  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' However, some parts of the argument were omitted there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The aim of this section is  to supply these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, we define the lifting map from S to a Riemann surface Σg via real blow-ups with  some parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For the study of the boundary of Teichm¨uller space, the contraction  maps of simple closed curves on a Riemann surface to nodes are fundamental (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [15], [1]  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' However, this contraction loses the information on twisting parameters of Fenchel–  Nielsen coordinates along the geodesics isotopic to these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By this lifting, we recover  the above lost twisting parameters and make it possible to discuss the Fenchel–Nielsen  deformation (or twist) at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This argument is essentially due to [42],[72] and  [9], and will be discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2 in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2, we discuss the liftability of a continuous map f : S1 → S2 of stable curves to  a map between two Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If f is a holomorphic or real-analytic map, this is  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' However, in the case where f is a continuous map, there exists an obstruction to  the lifting by the existence of a map which behaves as an infinite-angled rotation around  a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3, we show that this obstruction is cancelled in the isotopy class of oriented  homeomorphisms by the Alexander trick [7], and we can discuss the mapping class group  of a stable curve in the language of the mapping class group of a Riemann surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then  the geometric meaning of the Weyl groups will be clarified in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1  Lifting of stable curves and Fenchel–Nielsen twist at infinity  In this subsection, we will define and discuss the lifting map from a stable curve to a  Riemann surface via real blow-ups with some parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let S be a stable curve over C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' a complete algebraic curve with at most nodes  (ordinary double points) as singularities such that a nonsingular rational component has  8  at least three nodes of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Assume S has genus g > 1, and has k nodes P1, · · · , Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If  S is nonsingular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' a compact Riemann surface), then k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let S = �r  i=1 Si be  the irreducible decomposition and let h : �S = �r  i=1 �Si −→ S be the normalization in the  function field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Topologically, the component �Si is obtained from a connected component  Si of the complement of nodes of S by closing the punctures which are the pull back by  h of nodes to this component, say Pi := �ri  j=1 Pi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If �Si is a projective line, then ri ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We call the ri-pointed Riemann surface  (�Si, Pi)  (1)  the normalized pointed component of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  �πi : BlPi(�Si) −→ �Si  be the real blowing up at Pi,1, · · · , Pi,ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By using a complex local coordinate (U, z) at  Pi,j = (z = 0), BlPi(�Si) is defined locally by {(z, θ) ∈ U × S1 | z = |z|e  √−1θ} and �πi  is induced from the first projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Globally �πi is a real analytic isomorphism on the  complement of the inverse images of Pi,j’s, and each fiber (�πi)−1(Pi,j)  (1 ≤ j ≤ ri) is  the circle S1, which is called the exceptional circle of �πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The inverse image of the node  Pj under the composition map �h = (�r  i=1 �πi) ◦ h : �r  i=1 BlPi(�Si) −→ S consists of two  exceptional circles, which we write  (�h)−1(Pj) = C(1)  j  �  C(2)  j  (1 ≤ j ≤ k)  (2)  where the order of two circles C(1)  j , C(2)  j  is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We consider a map  Φ =  k  �  j=1  ϕj :  k  �  j=1  �  C(1)  j  −→ C(2)  j  �  (3)  where each ϕj : C(1)  j  −→ C(2)  j  is a homeomorphism of circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let S(Φ) be the topo-  logical surface obtained from Riemann surfaces with boundaries BlP1(�S1), · · · , BlPr(�Sr)  by pasting the corresponding boundaries (2) which are the exceptional circles via the  identification map Φ of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We obtain the natural continuous map  π(Φ) : S(Φ) −→ S  (4)  such that the fiber of the node (π(Φ))−1(Pj) is a circle, say Cj, which is obtained from  C(1)  j  and C(2)  j  via the identification ϕj, and the restriction of π(Φ) to the complement of  the inverse images of nodes S(Φ) \\ �  1≤j≤k Cj −→ S \\ �  1≤j≤k Pj is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Figure I is a simple example in the case where g = k = r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  9  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call the map π(Φ) of (4) the lifting of S, and S(Φ) the lifted Riemann  surface of S with twist Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We also call {C1, · · · , Ck} the exceptional circles for π(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (a typical example of lifting)  We identify S1 with R/2πZ via the map  R/2πZ ∈ x �−→ exp(  √  −1x) ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (5)  Let ϕθ : S1 −→ S1 be the rotation map of angle θ defined by x �→ x + θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the definition  of the real blowing up, we have a canonical identification C(1)  j  = C(2)  j  = S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore, by  a k-ple of angles Θ = (θ1, · · · , θk) ∈ (S1)k, the map  Φ(Θ) =  k  �  j=1  ϕθj :  k  �  j=1  �  C(1)  j  −→ C(2)  j  �  (6)  is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By using Φ(Θ) as the identification map, the lifting π(Φ(Θ)) : S(Φ(Θ)) −→ S  is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call it the lifting of S with the rotation angles Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We choose the element o = (0, · · · , 0) ∈ (S1)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  By resetting S(Φ(o)) = Σg and  π(Φ(o)) = πS, we write the lifting of S with the rotation angle o as  πS : Σg −→ S,  (7)  and call it the canonical lifting of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since S is a stable curve, the exceptional circles  {C1, · · · , Ck} on Σg for πS are not isotopic to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore, their isotopy classes  determine a (k −1)-simplex of Harvey’s curve complex Cg (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, and [33]) of Σg, which  we write  σπS = ⟨C1, · · · , Ck⟩ ∈ Cg,  (8)  10  (2)  S(Φ)  2  past with twist  (1)  (“)=  contraction  realblowup at  of Ci, C2  P11,P12, P21, P22  P12  22  P11  normalization  S1  S2  31  S  (Figure 1)The composition of real modifications of a stable curveand call it the exceptional simplex for πS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The lifted Riemann surface S(Φ) of arbitrary twist Φ is reconstructed from Σg by  changing the pasting parameters along exceptional cycles by using (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We naturally  obtain a homeomorphism  τ�Φ : Σg −→ S(Φ)  (9)  which satisfies πS = π(Φ) ◦ τ�Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If we admit a hyperbolic metric on Σg such that the  exceptional circles C1, · · · , Ck are geodesics with respect to this metric, then the map (9)  is traditionally called the Fenchel–Nielsen deformation or the Fenchel–Nielsen twist along  Cj’s (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [77, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='24], [41, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We also call τ�Φ the Fenchel–Nielsen twist at infinity,  since S itself sits on the boundary of moduli space as we will discuss afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Note that τ�Φ is not uniquely determined from Φ as a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since Φ of  (3) is a pasting map along circles in the construction of S(Φ), the integral rotation of  each circle is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' However, if an identification of S(Φ) with Σg is somehow fixed,  the homeomorphism τ�Φ may be considered as an element in the mapping class group Γg  of Σg as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By identifying S(Φ) with Σg, the identification map ϕj in (3) naturally  defines a self-homeomorphism ϕj : A(Cj) −→ A(Cj) of an annular neighborhood A(Cj)  of Cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In the mapping class group of the annulus A(Cj), ϕj is isotopic to a rotation map  ϕθj with some real-valued angle θj ∈ R = (−∞, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' However , if the θj’s are integral  multiples of 2π, the map τ�Φ : Σg −→ Σg = S(Φ) is isotopic to a composition of integral  Dehn twists along the exceptional circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let Γ(σπS) be the subgroup of Γg generated by Dehn twists along the exceptional  circles, and let [τ�Φ] be the isotopy class of τ�Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the above argument, if the θj’s are  integral multiples of 2π, we have  [τ�Φ] ∈ Γ(σπS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (10)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2  Obstruction to lifting homeomorphisms  Here we discuss the liftability of a homeomorphism of stable curves to a homeomorphism  of lifted Riemann surfaces with twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We start by sketching it locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let f : ∆ −→ ∆ be a map between the unit disks with  f(0) = 0 such that f is a homeomorphism onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let Lθ = {z ∈ ∆ | arg z = θ}  be a radial segment in ∆ with the fixed angle θ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If the limit  f|Lθ(0) :=  lim  z∈Lθ,|z|�→0  f(z)  |f(z)|  (11)  exists for each θ ∈ R, then f is said to be finite-angled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Otherwise f is said to be infinite-  angled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The image of Lθ under an infinite-angled homeomorphism goes around the origin  infinitely many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We first give one of this type of examples, and then state our  claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  11  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let [r, θ] be the polar coordinate of ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' z = re  √−1θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We define f[r, θ] =  [r, θ +2π/r] for [r, θ] ̸= [0, 0], and f[0, 0] = [0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then f is an infinite-angled homeomor-  phism of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let f : ∆ −→ ∆ be a finite-angled homeomorphism onto its image between  the unit disks with f(0) = 0, and let π0 : Bl0(∆) −→ ∆ be the real blowing up at 0 ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then there exists uniquely a homeomorphism �f : Bl0(∆) −→ Bl0(∆) onto its image which  is a lift of f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' π0 ◦ �f = f ◦ π0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  We rename the sourse disk by ∆1 = ∆ and the target disk by ∆2 = ∆,  and let zi be the complex coordinate of ∆i (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By definition, we have Bl0(∆i) =  {(zi, θi) ∈ ∆i × S1 | zi = |zi|e  √−1θi} where the coordinate θi of S1 is written via the  identification (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By using (11), we define the map �f : Bl0(∆1) −→ Bl0(∆2) by  (z1, θ1) �−→ (z2, θ2) = (f(z1), arg(f(z1))  if z1 ̸= 0,  (0, θ1) �−→ (z2, θ2) = (0, f|Lθ1(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then �f satisfies the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A homeomorphism f : S1 −→ S2 of stable curves is said to be finite-  angled if the restrictions of f to small disk neighborhoods of both banks (local components)  of each node of S1 are finite-angled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let f : S1 −→ S2 be a finite-angled homeomorphism of stable curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We choose a lifting π(Φ1) : S1(Φ1) −→ S1 with an arbitrary twist Φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then there exist a unique lifting π(Φ2) : S2(Φ2) −→ S2 with some twist Φ2 and a  unique homeomorphism �f : S1(Φ1) −→ S2(Φ2) which satisfy the following commutative  diagram  S1(Φ1)  �f  �  π(Φ1)  �  S2(Φ2)  π(Φ2)  �  S1  f  � S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  Let P1 be a node of S1, and P2 = f(P1) be the image of P1, which is a node of  S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let ∆(j)  i  (i = 1, 2, j = 1, 2) be small disk neighborhoods in both banks of Pi such that  the restriction f|∆(j)  1  : ∆(j)  1  −→ ∆(j)  2  is a homeomorphism onto its image, for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  π(j)  Pi : BlPi(∆(j)  i ) −→ ∆(j)  i  be the real blowing up at the origin Pi with exceptional circle  C(j)  i  = (π(j)  Pi )−1(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since f|∆(j)  1  is finite-angled by assumption, it follows from Lemma  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4 that there exists a unique homeomorphism �f|∆(j)  1  : BlP1(∆(j)  1 ) −→ BlP2(∆(j)  2 ) which  satisfies  π(j)  P2 ◦ �f|∆(j)  1  = f|∆(j)  1 ◦ π(j)  P1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (12)  12  Let ϕ1 : C(1)  1  −→ C(2)  1  be the pasting homeomorphism associated with Φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then we define  the homeomorphism ϕ2 : C(1)  2  −→ C(2)  2  by  ϕ2 = �f|∆(2)  1 ◦ ϕ1 ◦ ( �f|∆(1)  1 )−1|C(1)  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (13)  By pasting C(1)  2  and C(2)  2  via ϕ2, the desired S2(Φ2) and �f are locally well-defined by (12)  and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We do the same construction on disk neighborhoods of all the nodes of S1 and  S2 and paste the resultants trivially to their complements in S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then we globally  obtain the desired S2(Φ2) and �f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The uniqueness is clear from the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3  Weyl groups as mapping class groups  Here we characterize the mapping class group of a stable curve as the Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  First we discuss the orientation of a stable curve S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since the local equation of a node  of S is xy = 0 in C2, the link of the singularity is a positive Hopf link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then S has a  natural orientation ∇S so that its normalized components have orientations as Riemann  surfaces which are compatible with the positivity of Hopf links at nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  On the other hand, we consider the canonical lifting πS : Σg −→ S described in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We compare the natural orientaion ∇Σg with ∇S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The orientations ∇Σg and ∇S are compatible via the map πS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  Since Σg and S are realized as fibers in an oriented manifold by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1  (in the next section), the assertion is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let Si (i = 1, 2) be oriented stable curves with orientations ∇Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A homeomorphism  f : S1 −→ S2 is said to be oriented if the pushed down f∗(∇S1) defines the orientation on  S2 which is equivalent to ∇S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Two oriented homeomorphisms fi : S1 −→ S2 (i = 0, 1) are  said to be isotopic to each other if there exists an isotopy {ft : S1 −→ S2}0≤t≤1 such that  each ft is an oriented homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The isotopy class of an oriented homeomorphism  f : S1 −→ S2 is said to be the mapping class of f, and is denoted by [f].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For a fixed stable  curve S, the set of mapping classes becomes a group by the composition of maps, which  is called the mapping class group of S, and is denoted by Σ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We have the following:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Any oriented homeomorphism f : S1 −→ S2 of stable curves is isotopic  to a finite-angled oriented homeomorphism f ′ : S1 −→ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  Suppose that the restricted homeomorphism to a disk neighborhood of a  bank f|∆(1)  P : ∆(1)  P −→ f(∆(1)  P ) ⊂ ∆(1)  f(P) of some node P of S1 is infinite-angled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We choose  another map h : ∆(1)  P −→ ∆(1)  f(P) such that  13  (i)  h(∆(1)  P ) = f|∆(1)  P (∆(1)  P ) as sets and the restricted maps to the boundary coincide:  h|∂∆(1)  P ≡ f|∂∆(1)  P ,  (ii) h is a finite-angled homeomorphism onto its image so that h and f|∆(1)  P  define the  same orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Note that we have infinitely many choices of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since the composite homeomorphisms  h−1 ◦ f|∆(1)  P  : ∆(1)  P  −→ ∆(1)  P  coincide with each other at the boundary ∂∆(1)  P , they are  isotopic to the identity map id∆(1)  P  on the whole disk ∆(1)  P  by the Alexander trick ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  That is to say an h is isotopic to f|∆(1)  P  without moving the points of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We  define an oriented homeomorphism f ′ : S1 −→ S2 by  f ′(x) =  �  �  �  f(x)  if x ∈ S \\ ∆(1)  P  h(x)  if x ∈ ∆(1)  P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then f ′ is isotopic to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If f ′ is infinite-angled at a certain node, then we repeat the same  process as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' After a finite number of steps, we reach a new f ′ which satisfies the  desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Any mapping class [f] of a stable curve S has a lifting to a mapping class  [ �f] of a Riemann surface Σg such that (πS)∗([ �f]) = [f] holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  By identifying the Riemann surface S(Φ) with Σg = S(Φ(o)) as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, the  assertion follows from Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The lifted class [ �f] is not uniquely determined by [f].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This ambiguity comes from the  ambiguity of the choice of the map h near a node P as in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  By the same argument as (10), this ambiguity is cancelled modulo integral Dehn twists  along the exceptional circles and the lifting [ �f] is uniquely determined as an element of  Γg/Γ(σπS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' On the other hand, since f permutes the nodes P1, · · · , Pk of S, [ �f] permutes  the ambient isotopy classes [C1], · · · , [Ck] of exceptional circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  By [54, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5], the  subgroup of Γg consisting the elements which permute [C1], · · · , [Ck] is nothing but the  normalizer NΓ(σπS) of Γ(σπS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We set  W(σπS) = NΓ(σπS)/Γ(σπS)  and call it the Weyl group of the exceptional simplex σπS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Note that our W(σπS) is not  necessarily a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ([54, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7])  The mapping class group Γ(S) of a stable curve S is  isomorphic to the Weyl group  Γ(S) ∼= W(σπS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  14  3  Kuranshi families of stable curves and log struc-  tures  Here we review well-known facts about the Kuranishi families of stable curves which will  be used afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We mainly refer to Arbarello–Cornalba–Griffiths [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, we first review the formal properties of the standard Kuranishi families of  stable curves, and secondly the cohomological properties of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In particular, we discuss  the parameter spaces of the smoothing (plumbing) deformations by deforming the nodes  to annuli, and those of variable deformations by deforming the complex structures of  irreducible components and shifting the positions of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2, we first briefly review the notion of log geometry following Kato–Nakayama  [42] and Usui [72], and then review in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1 that the boundary of the log lifting  of the Kuranishi family of stable curves parametrises the possible Fenchel-Nielsen twist  at infinity which was discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The method of [9] does not use the terminology  of log geometry and shows this fact direcly by real blowing ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1  Deformations of stable curves and Kuranishi families  In this subsection, we review the standard results of the Kuranishi families of stable  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We recall Harvey’s curve complex Cg of a Riemann surface Σg (see [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By definition,  Cg is a (3g−3−1)-dimensional simplicial complex whose (k−1)-simplex σ = ⟨C1, · · · , Ck⟩  consists of mutually disjoint and non-homotopic isotopy classes of simple closed curves  Cj (1 ≤ j ≤ k) on Σg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' An (ℓ − 1)-simplex ρ = ⟨Ci1, · · · , Ciℓ⟩ is a face of σ, denoted by  ρ < σ, if {Ci1, · · · , Ciℓ} is a subset of {C1, · · · , Ck} as isotopy classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We denote by  contσ : Σg −→ Σg(σ)  (14)  the contraction map of σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the continuous map which contracts each Cj to a point and  is homeomorphic on the complement of the Cj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Here Σg(σ) is a nodal Riemann surface  with k nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The map (14) is topologically identified with the map (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let S be a stable curve whose topological type coincides with Σg(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the Kuranishi  family of S, we mean the fibration  ψ : X −→ B  (15)  which has the following properties (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [9, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='XI, §4, §6]):  (i) B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' X) is a (3g − 3)-dimensional (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (3g − 2)-dimensional) complex manifold  with S = ψ−1(b0) (b0 ∈ B), b0 being a fixed point,  15  (ii) ψ has the universal property with respect to local deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' for  any deformation ϕ : V −→ Z with ϕ−1(e0) = S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the restricted family ϕW : VW −→ W  to any sufficiently small connected neighborhood W (⊂ Z) of e0 has unique holomorphic  maps f : W −→ B and �f : VW −→ X with f(e0) = b0 such that ψ ◦ �f = f ◦ ϕW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (iii) For any b ∈ B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the Kodaira-Spencer map  TB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='b −→ Ext1  OXb(Ω1  Xb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' OXb)  is an isomorphism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' where TB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='b is the tangent space of B at b and Ω1  Xb is the sheaf of K¨ahler  differentials of Xb = ψ−1(b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (iv) The discriminant locus D of ψ is a normal crossing divisor on B such that the fibers of  ψ over the irreducible component Di1···iℓ of ℓ-codimensional open strata of D are the stable  curves whose topological types are Σg(ρ) corresponding to the face ρ = ⟨Ci1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ciℓ⟩ < σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Moreover, if ψ satisfies the additional condition (v), ψ is said to be a standard Kuran-  ishi family of S:  (v) The action of an analytic automorphism of S extends to a compatible action on X  and B (the totality of which will be denoted by Aut(X/B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Conversely, any isomorphism  between fibers of ψ is induced by an analytic automorphism of S (the totality of which  will be denoted by G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' That is to say, if there exists an isomorphism of fiberes hb1b2 :  ψ−1(b1) → ψ−1(b2) (for two points b1, b2 ∈ B), then there exists some h0 ∈ G and  its extension h0 ∈ Aut(X/B) (h0|ψ−1(b0) = h0) such that the restriction map h0|ψ−1(b1)  coincides with hb1b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (vi) A Kuranishi family and a standard Kuranishi family of S exist up to isomorphisms  of families and up to shrinking the base B near b0 ([9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Now we summarize well-known facts about the characterization of smoothing (or  plumbing) and non-smoothing deformations of S for ψ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [9, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='XI], [29, §1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  S = �r  i=1 Si be a stable curve, and let ( ˆSi, Pi) and Pj (1 ≤ j ≤ k) be as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since the  tangent space at b0 of B is isomorphic to Ext1  OS(Ω1  S, OS), we can choose a local coordinate  neighborhood Bloc at b0 of B as an open neighborhood of the origin of this vector space ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Bloc ⊂ Ext1  OS(Ω1  S, OS) ∼= C3g−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (16)  From the spectral sequence Hq(S, Extp(Ω1  S, OS)) =⇒ Extp+q(Ω1  S, OS), we have the exact  sequence  0 −→ H1(S, HomOS(Ω1  S, OS)) −→ Ext1  OS(Ω1  S, OS) −→ H0(S, Ext1(Ω1  S, OS)) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (17)  By using the Grothendieck–Serre duality, the dual vector spaces of each space in (17) are  H1(S, HomOS(Ω1  S, OS))∗ ∼=  r  �  i=1  H1( ˆSi, T ˆSi(−Pi))∗ ∼=  r  �  i=1  H0( ˆSi, 2K ˆSi + Pi),  (18)  16  Ext1  OS(Ω1  S, OS)∗ ∼= H0(S, Ω1  S ⊗ ωS),  H0(S, Ext1(Ω1  S, OS))∗ ∼=  k  �  j=1  τPj,  where ωS is the dualizing sheaf of S, T ˆSi and K ˆSi are the tangent sheaf and the canonical  sheaf of ˆSi respectively, and τPj is the torsion sheaf supported on Pj which is described as  follows (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [29, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='33]): If S is locally defined by xy = 0 near the node Pj, the differentials  ω1 = dx⊗2/x, ω2 = dy⊗2/y have the relation yω1 = xω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore  yω1 = ydx⊗2  x  = xdy⊗2  y  (19)  generates a one-dimensional submodule over C, which is nothing but the generator of τPj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Hence the dual exact sequence of (17) is  0 −→  k  �  j=1  τPj −→ H0(S, Ω1  S ⊗ ωS) −→  r  �  i=1  H0( ˆSi, 2K ˆSi + Pi) −→ 0  (20)  Then the deformation-theoretic meaning of (17) and (20) is the following;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (I) The space H1( ˆSi, T ˆSi(−Pi)) or H0( ˆSi, 2K ˆSi + Pi) parametrizes the variable deforma-  tions for varying the complex structures of ˆSi without smoothing the attaching nodes,  (II) The torsion sheaf τPj parametrizes the smoothing deformations of the nodes Pj to  annuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In particular, τPj is generated by the plumbing at Pj (Note that here the meaning  of “plumbing” is different from that in differential topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  For the meaning in the  present context, see [47, §2, §3], [9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='184–186]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The above facts (I) and (II) also follow from the general theory of deformations of  varieties of normal crossing (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2  Log structure and real blowing up of Kuranishi families  In this subsection, we apply a part of the standard arguments of log geometry[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='] to the  Kuranishi family ψ : X → B of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For the teminology, see [42, §1], [72, §1], [43, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2]  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Since the discriminant locus D is a normal crossing divisor on B, the sheaf of finitely  generated and saturated monoid M = {f ∈ OB | f is invertible outside D} embedded in  OB defines the log structure as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We set  Blog =  �  (b, h) | b ∈ B, h ∈ Hom(M gp  B,b, S1), h(f) = f(x)  |f(x)|, ∀f ∈ O∗  B,b  �  ,  where M is embedded in the abelian group M gp = {a/b | a, b ∈ M} as a sheaf on B, and  O∗  B,b is the stalk at b of the non-vanishing holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The first projection  (b, h) �→ b induces a map  τB : Blog −→ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (21)  17  Then Blog has the structure of a real analytic manifold with corners such that the map  (21) may be considered as the real blowing up of B along D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Namely, let (U, z1, · · · , z3g−3)  be a system of local coordinates of B around b0 so that zi = 0 (1 ≤ i ≤ k) defines locally  an irreducible component Di of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A component Di1,··· ,iℓ of the ℓ-codimensional open  strata of D on U is defined by zi1 = · · · zil = 0, zj ̸= 0 (j ∈ {1, · · · , 3g − 3} \\ {i1, · · · , iℓ}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then Blog is defined near a point of Di1,··· ,iℓ ∩ U as  {(z1, · · · , z3g−3, θ1, · · · , θℓ) ∈ U × (S1)ℓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' zij = |zij|e  √−1θj, 1 ≤ j ≤ ℓ}  by identifying S1 = R/2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The exceptional set E of Blog has the stratification so that  the component Ei1,··· ,iℓ = τ −1  B (Di1,··· ,iℓ) is locally written by  {zi1 = · · · zil = 0, (zj1, · · · , zj3g−3−ℓ, θ1, · · · , θℓ) ∈ (C∗)3g−3−ℓ × (S1)ℓ}  (22)  where j1, · · · , j3g−3−ℓ ∈ {1, · · · , 3g − 3} \\ {i1, · · · , iℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In particular, the restriction  Ei1,··· ,iℓ −→ Di1,··· ,iℓ of τB is an (S1)ℓ-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  On the other hand, since ψ−1(D) is a normal crossing divisor on X, we have the  construction τX : Xlog −→ X similar to (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then from [42, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3], we have the  log lifting ψlog : Xlog → Blog of ψ with which the diagram  Xlog  τX  �  ψlog  �  X  ψ  �  Blog  τB  � B  (23)  is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The geometric meaning of the log lifting is clarified by [72] in a more  general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In the case of the Kuranishi family of a stable curve, it is a real analytic  family of Riemann surfaces including the possible Fenchel–Nielsen twists at infinity ([9,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='149–156]) :  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ([9], [72], [42]) Let ψ : X −→ B be a Kuranishi family of a stable curve  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then ψlog : Xlog → Blog is a real analytic family of Riemann surfaces such that  (i)  The restriction (τB ◦ ψlog)−1(B \\ D) −→ τ −1  B (B \\ D) of ψlog is isomorphic to the  restriction ψ−1(B \\ D) −→ B \\ D of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) Let Q be a point of the fiber τ −1  B (P) of (S1)ℓ-bundle Ei1,··· ,iℓ −→ Di1,··· ,iℓ (P ∈ Di1,··· ,iℓ)  such that Θ = (θ1, · · · , θℓ) is the fiber coordinates of Q given in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the fiber  (ψlog)−1(Q) is isomorphic to the Riemann surface which is the lifting of the stable curve  ψ−1(P) by rotation angle Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' That is to say, in the notation (6), we have  (ψlog)−1(Q) ∼= ψ−1(P)(Φ(Θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (24)  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The lifting map of a real analytic map between real analytic manifolds  via real blowing ups is also constructed by Hubbard–Papadopol–Veselov [38, §5] and is  discussed for other purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  18  4  Construction of the universal degenerating family  Starting from Hubbard-Koch’s result [37], we constructed in [54] a new orbifold structure  on the Deligne-Mumford compactification , which we will denote here by M  orb  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In this  section, using this structure, we will construct an orbifold fiber space πorb : Y  orb  g  −→ M  orb  g  which is a family of stable curves with some universal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, we review the orbifold structure of M  orb  g  with some comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The “biggest  chart” of M  orb  g  is the moduli space Mg which is the quotient of Teichmuller space by  the mapping class group (Mg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Tg, Γg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The “boundary charts” are indexed by σ ∈ Cg  (Harvey’s curve complex), each of which is an open set Mϵ(σ) containing the locus V (σ)  of stable curves with topological type Σg(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Mϵ(σ) is the quotient of the controlled  deformation space Dϵ(σ) by the Weyl group W(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Here Dϵ(σ) is the space consisting of  σ-marked stable curves whose generalized Fenchel–Nielsen coordinates are “controlled”  so that W(σ) acts on Dϵ(σ) properly discontinuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2, we show that Dϵ(σ) is expressed by patching the bases of the Kuranishi  families of σ-marked stable curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' From this, we construct a family of stable curves  πσ : Xσ −→ Dϵ(σ) by patching the Kuranishi families of stable curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This family is a  refinement of Hubbard–Koch’s family [37, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1], and also is an extension of Arbarello–  Cornalba’s theorem [8] which says that the universal curve Cg −→ Tg over Teichm¨uller  space is expressed by patching the Kuranishi families of Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' As the main  tool of the proof of our theorem, we use the log lifting of the Kuranishi families discussed  in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3, by patching the families πσ naturally, we construct the desired orbifold fiber  space πorb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call it the universal degenerating family of Riemann surfaces for fibered  complex surfaces, because any fibered complex surface admitting unstable fibers can be  pulled back from πorb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This point will be discussed in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1  Weyl marking and controlled deformation spaces  In this subsection, we introduce the notions of the Weyl marking and controlled defor-  mation space from [54, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2], and review related results proved in [54] by adding some  comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We fix a simplex σ = ⟨C1, · · · , Ck⟩ of Cg, and let S be a stable curve whose topological  type coincides with Σg(σ) in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By a pre-marking of S, we mean the isotopy class [w]  of an oriented homeomorphism  w : Σg(σ) −→ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (25)  Two such pre-marked stable curves (S1, [w1]) and (S2, [w2]) are said to be equivalent  (and denoted by (S1, [w1]) ∼ (S2, [w2]) ) if there exists an analytic isomorphism f : S1 −→  19  S2 such that f ◦ w1 is isotopic to w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We denote the equivalence class by  [S, w] = (S, [w])/ ∼  and call it a stable curve with Weyl marking (or σ-marked stable curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This notion of  the marking is different from the ones given in [37, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='270] and [9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='490], since the action  of Γ(σ) is neglected in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the rigidity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the automorphism-freeness of a  stable curve with Weyl marking, is spelled out as follows:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let w : Σg(σ) → S be a Weyl marking, and f : S → S be an analytic  automorphism such that [S, f ◦ w] = [S, w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then f is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  If f permutes a proper subset of nodes of S non-trivially, then a lifting of f to  Γg would permute a non-trivial subset of the isotopy classes of C1, · · · , Ck, and f◦w cannot  be isotopic to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore, f stabilizes each irreducible component of S, and stabilizes  each normalized component of it as a pointed Riemann surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Hence the assertion  follows from the usual rigidity of the Teichm¨uller marking (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [36, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The following is a criterion of the equivalence of the marking:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Two oriented homeomorphisms wi : Σg(σ) −→ Si (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 2) give the same  σ-Weyl marking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' namely [S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' w1] = [S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' w2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' if and only if there exist an oriented homeo-  morphism �f : S1(Φ1) −→ S2(Φ2) which is a lift of an analytic isomorphism f : S1 −→ S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  oriented homeomorphisms �wi : Σg −→ Si(Φi) (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 2) and an action α of the Dehn twists  Σg −→ Σg (α ∈ Γ(σ)) such that the following diagram is homotopically commutative  Σg  �w1 �  α  �  S1(Φ1)  �f  �  π(Φ1) � S1  f  �  Σg  �w2 � S2(Φ2)  π(Φ2) � S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  The assertion is clear from (10) and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We denote by T(σ) = �  w:σ-Weyl marking[S, w] the set of σ-marked stable curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For a  face ρ < σ, we denote by T(ρ) = �  w:ρ-Weyl marking[S, w] the set of ρ-marked stable curves  similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If ρ = ∅, then T(∅) = Tg = �  w:∅-Weyl marking[S, w] is the Teichm¨uller space, since  a ∅-Weyl marking [S, w] is nothing but a Riemann surface S with a usual Teichm¨uller  marking w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We have the following real structure on Tg  �  ρ<σ T(ρ) as a subspace of the augmented  Teichm¨uller space �Tg (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [77, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We fix a maximal simplex  ˜σ = ⟨C1, · · · , Ck, Ck+1, · · · , C3g−3⟩  20  containing σ = ⟨C1, · · · , Ck⟩, and let (ℓ1, · · · , ℓ3g−3, τ1, · · · , τ3g−3) be the associated Fenchel-  Nielsen coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Wolpert [76] proves that these coordinates, except for the first k twist  coordinates τ1, · · · , τk, extend to Tg ∪ T(σ) continuously as  ((ℓj, τj), ℓi) : Tg  �  ρ<σ  T(ρ) −→  3g−3  �  j=k+1  (R>0 × R) ×  k  �  j=1  (R≥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (26)  Moreover, if we consider a point p = [S, w] ∈ T(ρ) for a face ρ = ⟨Ci1, · · · , Cis⟩, then  the nodes of S correspond to ℓij(p) = 0 for 1 ≤ j ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let M be a universal constant such that two distinct simple closed geodesics on any  Riemann surface of genus g are disjoint if their lengths are smaller than M ([44], [1, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3]),  which is sometimes called a 2-dimensional Margulis constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Take a number 0 < ϵ ≤ M  and fix it throughout the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By using the extended Fenchel–Nielsen coordinates  (26), we define the subspace Uϵ(σ) of Tg  �  ρ<σ T(ρ) by  Uϵ(σ) = { 0 ≤ ℓi < ϵ (1 ≤ i ≤ k), max{ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' · · · , ℓk} < ℓj (k + 1 ≤ j ≤ 3g − 3) } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (27)  Since T(σ) is defined by ℓj = 0 (1 ≤ j ≤ k), we have a natural inclusion T(σ) ⊂ Uϵ(σ)  and Γ(σ) naturally acts on Uϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ([54, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2])  The quotient space  Dϵ(σ) = Uϵ(σ)/Γ(σ)  is called the controlled deformation space with respect to the simplex σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  These spaces are considered as refinements of Bers’ deformation spaces ([15]), and  have the following properties:  (I) (topological properties) Dϵ(σ) is a Hausdorff topological space and the Weyl group  W(σ) acts on Dϵ(σ) as the change of the marking  [S, w] −→ [S, w ◦ ϕ−1]  for ϕ ∈ W(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (28)  This action is properly discontinuous ([54, Lemmata 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1 ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (II) (analytic properties) Dϵ(σ) has a complex structure on which W(σ) acts holomorphi-  cally, and the quotient space Mϵ(σ) = Dϵ(σ)/W(σ) has an analytic open embedding into  M g ([54, Lemmata 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5 ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The property (I) depends on the real analytic augumented Teichm¨uller theory (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [1])  and some Weil–Petersson geometry (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [76], [78]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Note that the delicate condition (27)  for the geodesic lengths ℓi comes from the determination of the region on which W(σ) acts  properly discontinuously (see [54, Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The property (II) is essentially comes  from Hubbard–Koch’s theorem [37, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In the next subsection, we add a stronger  analytic property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  21  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2  Kuranishi families over controlled deformation spaces  In this subsection, we first intoduce the complex structure on Dϵ(σ) by a method which is  independent of Hubbard-Koch’s theorem [37, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1], and construct an analytic family  over Dϵ(σ) by patching Kuranishi families of stable curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The basic idea of the method  here is due to Arbarello–Cornalba [8, §§3,4] and Arbarello–Cornalba–Griffiths [9, §8], but  our proof of the following theorem will be rather direct using the topological properties  (I) of Dϵ(σ) in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' There exist a (3g − 2)-dimensional complex manifold Xϵ(σ), a complex  structure Dϵ(σ) = �  i∈I Bi and a holomorphic map  πσ : Xϵ(σ) −→ Dϵ(σ)  such that each restricted family πσ|Xi : Xi = (πσ)−1(Bi) −→ Bi coincides with a standard  Kuranishi family of a stable curve (πσ)−1(bi) for some bi ∈ Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The Weyl group W(σ) acts on πσ holomorphically and properly discontinuously so that  Xϵ(σ)  Dϵ(σ)  Yϵ(σ) ∼= Xϵ(σ)/W(σ)  Mϵ(σ) ∼= Dϵ(σ)/W(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  πσ  πσ  ϕσ  ψσ  is a commutative diagram, where ϕσ and ψσ are the quotient holomorphic maps to the  normal analytic spaces Yϵ(σ) and Mϵ(σ), and πσ is the induced holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Moreover Mϵ(σ) has an holomorphic open embedding into M g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We need the following:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let p = [S, w] ∈ Dϵ(σ)∩T(σ) be a point, and let ψ : X −→ B be a standard  Kuranishi family of ψ−1(b0) = S with a sufficiently small base B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then there exists a  natural topological open embedding  ιB : B −→ Dϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  Step 1  We set S = ψ−1(b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let �w : Σg −→ S(Φ0) be the lift of w with a  certain twist Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, there exists a point e0 ∈ τ −1  B (b0) such that S(Φ0) is  canonically identified with the fiber  (ψlog)−1(e0) = S(Φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (29)  We consider the stable curve XP = ψ−1(P) for any P ∈ B, and choose a point eP ∈  τ −1  B (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then we have  (ψlog)−1(eP) = XP(ΦP)  (30)  22  with a certain twist ΦP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the connectivity of Blog, we can choose a smooth path Le0eP  connecting the points e0 and eP on Blog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The family ψlog : Xlog −→ Blog is a differentiable  family of Riemann surfaces over a manifold with corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Usui’s theorem [72] which is  an extension of Ehresmann’s theorem ([24]) to this situation states that there exists a  diffeomorphism  ϕe0eP : (ψlog)−1(e0) −→ (ψlog)−1(eP)  (31)  by connecting diffeomorphisms of fibers along Le0eP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' From (29), (30) and (31), we have a  diffeomorphism  �wP := ϕe0eP ◦ �w : Σg −→ XP(ΦP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (32)  Note that �wP is uniquely determined modulo the action of Γ(σ), independently of the  choices of Φ0, eP and Le0eP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore, from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2 and (32), we have a ρ-Weyl  marking  wP : Σg(ρ) −→ XP  (33)  where the face ρ (possibly, ρ = ∅, σ) is determined by the stratum of B which contains  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since the extended Fenchel–Nielsen coordinates (26) moves continuously ([76]), the  length condition (27) is satisfied for [XP, wP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence the continuous map  ιB : B ∋ P �−→ [XP, wP] ∈ Dϵ(σ)  (34)  is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Step 2  We will prove that the map ιB is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We assume ιB(P1) = ιB(P2)  for some P1, P2 ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By definition, the stable curves XPi = ψ−1(Pi) (i = 1, 2) have  ρ-Weyl markings wi : Σg(ρ) −→ XPi for some ρ such that there exists an isomorphism  gP1P2 : XP1 −→ XP2 satisfying w2 ≃ gP1P2 ◦ w1 (homotopic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  It follows from the property (v) of a standard Kuranishi family, stated in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, that  there exist an automorphism gb0 : Xb0 −→ Xb0 (Xb0 = S) and a relative automorphism of  the family  X  gX �  ψ  �  X  ψ  �  B  gB  � B  (35)  such that gb0 and gP1P2 are induced by the restriction to the fibers of the automorphism  gX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The stable curves XP1 and XP2 have the same topological type Σg(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let B(ρ) be  the stratum in B such that the fibers of ψ over B(ρ) are of type Σg(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let L1 be a real smooth path on B connecting the points P1 and b0 such that L∗  1 =  L1 \\ {b0} is contained in B(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the real proper transform �L1 of L1 on Blog via the  map τB : Blog → B is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  23  Assume that dim ρ = k0 − 1 and dim σ = k − 1 (k0 ≤ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' z1, · · · , z3g−3)  be local coordinates at b0 of B such that B(ρ) ∩ U = {z1 = · · · = zk0 = 0, zk0+1 ̸=  0, · · · , z3g−3 ̸= 0} and b0 = {(z1, · · · , zk, zk+1, · · · , z3g−3) = (0, · · · , 0, ck+1, · · · , c3g−3)} for  some ck1+1, · · · , c3g−3 ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Assume that L1 ∩ U is defined by  zi = 0,  zj = (1 − t)rj(t)exp(  √  −1αj(t)),  zℓ = cℓ  for 1 ≤ i ≤ k0, k0 + 1 ≤ j ≤ k, k + 1 ≤ ℓ ≤ 3g − 3, where rj(t)’s are non-vanishing  smooth functions and αj(t)’s are smooth functions with respect to a variable t ∈ [δ, 1]  (0 ≤ ∃δ < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' On the other hand, as stated in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2, τ −1  B (U) is defined by  {(z1, · · · , z3g−3, θ1, · · · , θk) ∈ U × (S1)k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' zj = |zj|e  √−1θj, 1 ≤ j ≤ k}  by identifying S1 = R/2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We define the smooth section (s1)|loc : L1 ∩ U → τ −1  B (U) by  θj = 0 (1 ≤ j ≤ k0),  θj = αj(t) (k0 + 1 ≤ j ≤ k)  using the fiber coordinates of U × (S1)k → U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the similar arguments for other charts  on B and patching them, we obtain the smooth section s1 : L1 → Blog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then we set  �L1 = s1(L1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The restriction map (τB)|�L1 : �L1 → L1 is a diffeomorphism by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Now we set  L2 = gB(L1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (36)  Then L2 is a real smooth path on B connecting P2 and b0 such that L∗  2 = L2 \\ {b0}  is contained in B(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let ϕi : [0, 1] → Li (i = 1, 2) be the parametrization such that  ϕi(0) = Pi, ϕi(1) = b0 and ϕ2(t) = gB◦ϕ1(t) (t ∈ [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the restriction of gX : X → X  in (35) to the fibers, we have the family of isomorphisms  {gt = (gX)|Xϕ1(t) : Xϕ1(t) −→ Xϕ2(t)}0≤t≤1  (37)  such that g0 = gP1P2 and g1 = gb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let �L2 = s2(L2) be the real proper transform of L2 via τB given by the smooth  section s2 : L2 → Blog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let �ϕi = si ◦ ϕi : [0, 1]  ϕi  −→ Li  si  −→ �Li be the parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  By the definition of the real proper transform, or the argument in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6 and  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, there exists the homeomorphism �gt : Xlog  �ϕ1(t) = (ψlog)−1(�ϕ1(t)) −→ Xlog  �ϕ2(t) =  (ψlog)−1(�ϕ2(t)) with the commutative diagram  Xlog  �ϕ1(t)  �gt  �  τX  �  Xlog  �ϕ2(t)  τX  �  Xϕ1(t)  gt  � Xϕ2(t)  (38)  where we use the same symbol τX as the restriction to the fibers of τX : Xlog → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  24  First we consider the case t = 0 in (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the same reasoning as above, the ρ-  marking wi : Σg(ρ) → XPi and the homotopy relation w2 ≃ g0 ◦ w1 are lifted to the  homeomorphism �wi : Σg → Xlog  �ϕi(0) and the homotopy relation �w2 ≃ �g0 ◦ �w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Secondly we consider (38) for any t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By applying [24] to the family ψlog, we  have the diffeomorphism f�ϕi(0)�ϕi(t) : Xlog  �ϕi(0) → Xlog  �ϕi(t) along the path �Li connecting �ϕi(0)  and �ϕi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let �wi(t) = f�ϕi(0)�ϕi(t) ◦ �wi : Σg → Xlog  �ϕi(t) be the composite homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By  the commutativity of (38), we obtain the family of the homotopy relations  { �w2(t) ≃ �gt ◦ �w1(t)}0≤t≤1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (39)  Lastly we consider the case t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The homeomorphism �wi(1) : Σg → Xlog  �ϕi(1) descends  to the homeomorphism wi(1) : Σg(σ) → Xb0 via the contraction maps contσ : Σg → Σg(σ)  and (τX)|Xlog  �  ϕi(1) : Xlog  �ϕi(1) → Xb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the relation (39) for t = 1 descends to  w2(1) ≃ gb0 ◦ w1(1)  (homotopic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (40)  From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1 and (40), the automorphism gb0 is the identity map of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence P1 = P2,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the injectivity of the map ιB is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Step 3  By shrinking B′ ⊂ B preserving the properties (i) ∼(v) in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, we may  assume that the map ιB′ : B′ −→ Dϵ(σ) for the closure B′ in B is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since B′  is compact and Dϵ(σ) is Hausdorff, ιB′ is homeomorphic onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore the  assertion of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5 follows for B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4 :  Step 1  Let p = [S, w] ∈ Dϵ(σ) be a point, and let ψ : X −→ B be a standard  Kuranishi family of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the same argument as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5, there exist an open  neighborhood Up of p in Dϵ(σ) and a homeomorphism ιB : B −→ Up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  Dϵ(σ) =  �  p∈Dϵ(σ)  Up  (41)  be the open covering by Up’s of these types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  By identifying Up with B, the complex  structures {B} induce a complex structure on Dϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In fact, the coordinate transformations are biholomorphic as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let Uq = B′  be another base of a standard Kuranishi family of S′ for q = [S′, w′] ∈ Dϵ(σ) such that  Up ∩ Uq ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since the Kodaira-Spencer map at any point r ∈ Up ∩ Uq = B ∩ B′ is an  isomorphism by the property (iii) of the Kuranishi family, stated in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, B ∩ B′ is a base  of a Kuranishi family of the stable curve ψ−1(r) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [9, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='XI, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the  uniqueness of the Kuranishi family modulo isomorphism, the coordinate transformations  are nothing but these isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Note that Dϵ(σ) is a Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore,  (41) defines the desired complex structure on Dϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  25  These standard Kuranishi families {ψp : Vp = X(p) −→ Up = B(p)}p∈Dϵ(σ) are patched  together, and define a complex manifold Xϵ(σ) and a holomorphic map  πσ = ∪ψp : Xϵ(σ) =  �  Vp −→  �  Up = Dϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  These are also well-patched by the uniqueness of Kuranishi families modulo isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Therefore, we have constructed the desired family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Since the action of ϕ ∈ W(σ) on Dϵ(σ) is given by (28), ϕ sends the open neighborhood  Up = B(p) of p = [S, w] which is the base of the standard Kuranishi family of S to an  open neighborhood of Uϕ(p) = B(ϕ(p)) of ϕ(p) = [S, w ◦ ϕ−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This is nothing but the  isomorphism of the bases of Kuranishi families, and is holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence ϕ acts on  Dϵ(σ) holomorphically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Step 2  It suffices to prove that W(σ) acts on Xϵ(σ) properly discontinuously, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (i) let x′, x′′ ∈ Xϵ(σ) be points such that g(x′) ̸= x′′ for an element g ∈ W(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then there  exist open neighborhoods U ′ and U ′′ of x′ and x′′ respectively such that g(U ′) ∩ U ′′ = ∅,  (ii) the isotropy sub-group Gx (⊂ W(σ)) of a point x ∈ Xϵ(σ) is finite,  (iii) there exists a Gx-invariant neighborhood U of x such that, if g(U) ∩ U ̸= ∅ for some  g ∈ W(σ), then g ∈ Gx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Since W(σ) acts on Dϵ(σ) properly discontinuously by [54, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4], the assertion  (i) is clear in the case where x′ and x′′ belong to distinct fibers of πσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Assume that x′ and  x′′ belong to the same fiber of πσ, say x′ = [[S], w′] and x′′ = [[S], w′′] with an isomorphism  class [S] ∈ M g and distinct markings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then, after forgetting the markings, W(σ) acts on  S as the analytic automorphism group of S which is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore the assertion (i)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The assertion (ii) is clear, since Gx is a subgroup of the isotropy sub-group of πσ(x)  by the action of W(σ) on Dϵ(σ), which is finite by [54, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  By the first half of this theorem, there exists an open neighborhood V of πσ(x) in Dϵ(σ)  such that π−1  σ (V ) −→ V is a standard Kuranishi family of the stable curve π−1  σ (πσ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Therefore the assertion (iii) follows from the property (v) stated in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence W(σ)  acts on Xϵ(σ) properly discontinuously, and the quotient space Xϵ(σ)/W(σ) is a normal  analytic space by Cartan’s theorem ([18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The holomorphic embedding of Dϵ(σ)/W(σ) into M g follows from the above holomor-  phy and the property (II) stated in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3  Orbifold fiber space over the Deligne-Mumford compactifi-  cation  In this subsection, as a globalization of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4, we construct an orbifold fiber space  (or orbifold fibration for short) πg : X  orb  g  −→ M  orb  g  so that the Bers fiber space ([14]) over  26  Teichm¨uller space is contained as an open chart of πg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  By a complex orbifold M, we mean here that M is a normal complex space such that M  is covered by (an atlas of) orbifold charts {(�Ui, Gi, ϕi, Ui)}i∈I which satisfy the following  conditions:  (i) Each chart consists of a complex manifold �Ui, a (not necessary finite) group Gi acting  on �Ui holomorphically and properly discontinuously (admitting non-effective action), an  open set Ui of M and a folding map ϕi : �Ui −→ Ui which induces a natural analytic  isomorphism �Ui/Gi −→ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) (compatibility condition) For x ∈ �Ui and y ∈ �Uj such that ϕi(x) = ϕj(y) ∈ Ui ∩ Uj,  there exists a biholomorphic map ϕx,y : �U ′  x −→ �U ′  y from an open neighborhood of x in �Ui  to an open neighborhood of y in �Uj such that ϕx,y(x) = y and ϕj ◦ ϕx,y = ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  For simplicity, we sometimes write this orbifold structure by M = {(Ui;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' �Ui, Gi)}i∈I or  M = �  i∈I �Ui/Gi if there is no fear of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let M ′ be another complex orbifold with its orbifold charts {(�Vk, Hk, φk, Vk)}k∈K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By  an orbifold map h : M ′ → M, we mean that h is a holomorphic map of normal complex  spaces which satisfies the following conditions:  (i) For any points p ∈ M ′ and h(p) ∈ M, let (�Vk, Hk, φk, Vk) be a small orbifold chart  of M ′ with p ∈ Vk and (�Ui, Gi, ϕi, Ui) be that of M with h(p) ∈ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then there exists a  holomorphic map �hki : �Vk → �Ui such that ϕi ◦ �hki = h|Vk ◦ φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) (compatibility condition) Assume that x ∈ �Vk, y ∈ �Vℓ,�hki(x) ∈ �Ui,�hℓj(y) ∈ �Uj such  that φk(x) = φℓ(y) and ϕi(�hki(x)) = ϕj(�hℓj(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then there exist open neighborhoods  �Vx, �Vy, �U�hki(x), �U�hℓj(y) of x, y,�hki(x),�hℓj(y) in �Vk, �Vℓ, �Ui, �Uj respectively and biholomorphic  maps φx,y : �Vx → �Vy and ϕh(x),h(y) : �U�hki(x) → �U�hℓj(y) such that ϕh(x),h(y) ◦ �hki|�Vx =  �hℓ,j|�Vy ◦ φx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Moreover, we define the following:  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let f : X −→ M be an orbifold map of complex orbifolds of relative  dimension ≥ 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' dimCX ≥ dimCM +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We say that f has a structure of an orbifold  fibration if the following conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (i) Let {(�Ui, Gi, ϕi, Ui)}i∈I be the orbifold charts of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For each i ∈ I, there exists a nor-  mal complex space �  Wi and a holomorphic map �fi : �  Wi −→ �Ui, and Gi acts holomorphically  and relatively on �fi such that the diagram  �  Wi  �Ui  f −1(Ui) ∼= �  Wi/Gi  Ui ∼= �Ui/Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  �fi  f  ϕi  ψi  is commutaive, where ψi is the projection map by Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  27  (ii)  Each �  Wi is an open complex sub-orbifold of X in the following sense: �  Wi has the  orbifold charts {(�  Wi,j, Hi,j, ϕi,j, �  Wi,j)}j∈J(i) (the set J(i) of suffixes depends on i), and  there exist a group Gi,j containing Hi,j as a normal subgroup and an exact sequence of  groups  1 −→ Hi,j −→ Gi,j −→ Gi −→ 1,  such that the orbifold charts of X are given by {(�  Wi,j, Gi,j, ψi◦ϕi,j, ψi◦ϕi,j(�  Wi,j))}i∈I,j∈J(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The typical example of an orbifold fibration will be given in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Note that the set  {(�  Wi, Gi, ψi, f −1(Ui))}i∈I  (42)  is not an atlas of orbifold charts of X in general, because �  Wi may have singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Nevertheless it is important in our discussion in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' See Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (i)  We call (42) the pseudo-orbifold charts of X with respect to the  orbifold fibration f : X → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) If �  Wi is nonsingular for each i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' if the set (42) gives an atlas of orbifold charts of  X, we call f : X → M a strong orbifold fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Now the theorem ([54, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='11]) says that the Deligne-Mumford compactification M g  has the orbifold charts  {(Dϵ(σ), W(σ), ϕσ, Mϵ(σ))}σ∈Cg/Γg,  (43)  where σ moves in the curve complex modulo the action of the mapping class group Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  From now on, since this orbifold structure is different from the usual one (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [9, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='XII])  of M g, we use the symbol M  orb  g  in order to specify it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Over this base structure, we have  the following strong orbifold fiber space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' There exists a strong orbifold fibration  π : Y  orb  g  −→ M  orb  g  (44)  such that the orbifold charts of M  orb  g  are given by (43), and those of Y  orb  g  are  {(Xϵ(σ), W(σ), ψσ, Yϵ(σ))}σ∈Cg/Γg  (45)  given in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  It suffices to prove the compatibility condition (ii) of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let [S] ∈ M g  be an isomorphism class, and let pi = [S, wi], wi : Σg(σ) −→ S (i = 1, 2) be two marked  stable curves such that the point pi belongs to Dϵ(ρi) for ρi ≤ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4, there  28  exists an open neighborhood Ui ⊂ Dϵ(ρi) of pi such that the restricted family πi : Xi −→  Ui of πσ : Xϵ(ρi) −→ Dϵ(ρi) over Ui coincides with the standard Kuranishi family of  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the uniqueness of the standard Kuranishi family modulo isomorphism, there exist  biholomorphic maps h : X1 −→ X2 and h : U1 −→ U2 which satisfy h ◦ π1 = π2 ◦ h after  a suitable shrinking of Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore, the desired compatibility condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The other required properties for the strong orbifold fibration are given in Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In particular, the normal analytic structure of Y  orb  g  is given by patching those of  Yϵ(σ)’s obtained by Cartan’s theorem via the above local biholomorphic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call π : Y  orb  g  −→ M  orb  g  the universal degenerating family of Riemann  surfaces of genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We believe that the family π is universal in the sense that every orbifold fibration  with a Riemann surface of genus g as a general fiber can be pulled back from this univer-  sal orbifold fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Our present achievement is, however, rather modest, and we have  proved the universality of π only for fibered complex surfaces, namely, for orbifold fibra-  tions whose base spaces are of dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' See §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For a precise definition of “orbifold  pull-back”, see the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let f : X → M and f ′ : X′ → M ′ be orbifold fibrations and h :  M ′ → M be an orbifold map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We say that f ′ is the orbifold pull back from f via h if the  following conditions are satisfied:  (i) Let {(�Ui, Gi, ϕi, Ui)}i∈I and {(�Vj, Hj, φj, Vj)}j∈J be the orbifold charts of M and M ′  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For any j ∈ J, there exist some i ∈ I depending on j and a holomorphic  map hji = h|Vj : Vj → Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let �hji : �Vj → �Ui be the lifting of hji and ϕ(j)  i  = ϕi|�U(j)  i  : �U (j)  i  =  �hji(�Vj) −→ U (j)  i  = hji(Vj) be the restriction map to the image, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  �Vj  �Vj/Hj ∼= Vj  �U (j)  i  ⊂ �Ui  U (j)  i  ⊂ Ui ∼= �Ui/Gi  φj  ϕ(j)  i  hji  �hji  is a commutative diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then there exists an injective group homomorphism  Hj �→ Gi  (46)  such that the subgroup Hj of Gi acts on �U (j)  i  holomorphically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii)  There exists a lifted orbifold map k : X′ → X of h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' f ◦ k = h ◦ f ′, which  has the following property: Let {(�  Wi, Gi, ψi, f −1(Ui))}i∈I and {( �Tj, Hj, ωj, (f ′)−1(Vj))}j∈J  be the pseudo-orbifold charts of X and X′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let �kji : �Tj → �  Wi be the lifting  of kji = k|(f′)−1(Vj) : (f ′)−1(Vj) −→ f −1(Ui), and ψ(j)  i  = ψi|�  W (j)  i  : �  W (j)  i  = �kji( �Tj) =  29  ψ−1  i (�U (j)  i ) −→ �U (j)  i  be the restriction map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the group Hj relatively acts on ψ(j)  i  such  that the diagram  �Tj  �Vj  �  W (j)  i  ⊂ �  Wi  �U (j)  i  ⊂ �Ui  ωj  ψ(j)  i  kji  �kji  expresses the fiber product compatible with the action of Hj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  �Tj is isomorphic to  �  W (j)  i  ×�U(j)  i  �Vj such that α ◦ �kji = kji ◦ α for any α ∈ Hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  5  Automorphisms of stable curves and cyclic equi-  symmetric strata on M  orb  g  We study an automorphism ϕ of a stable curve S and the associated cyclic branched  covering from several points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In particular, we define the numerical data Num(ϕ)  which consist of two kinds of data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the action on the dual graph of S and the system of  branch data of cyclic branched coverings which every irreducible component of S naturally  has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We study the locus T [ϕ]  σ  on the boundary charts of M  orb  g  consisting of marked stable  curves with the automorphisms of this type of numerical data, and describe its structure  in terms of pointed Teichm¨uller spaces of lower genera (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This result is an  extension of Harvey–Broughton’s theorem about the equisymmetric strata on Tg and Mg,  to the boundaries for cyclic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, we review some known results about automorphisms of Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  First, we review the notion of total valency essentially due to Nielsen [59] and Harvey  [30], whic provides precise information on the branch data of the cyclic covering associated  with the automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Secondly, we review Harvey–Broughton’s equisymmetric strata  on Tg and Mg ([31], [16]) in the case of cyclic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2, as a modified discussion of Eichler’s trace formula, we propose a method  to determine the characters of the representation of automorphisms of pointed Riemann  surfaces into the space of logarithmic quadratic differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In the terminology of augmented Teichm¨uller theory, the true boundary T(σ) on the  boundary chart of M  orb  g  should be called the little Teichm¨uller space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3, we interpret  this notion by the cohomological terminologies of Kuranishi spaces of stable curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The aim of §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4 is to define Num(ϕ) naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The point is to analyze the cyclic  branched covering πϕ : S → W associated with ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Although the base W is a nodal  Riemann surface, the dual graph of W is not a simple quotient graph of the dual graph of  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In fact, we extend the notion of graphs to those admitting open edges, and then define  30  the notion of compact quotient graphs as the desired one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By combining these data of  the graphs and the system of the total valencies which every component of S naturally  has, we have the definition of Num(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5, we prove the structure theorem of the equisymmetric strata T [ϕ]  σ  on T(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The  space T [ϕ]  σ  is described locally by the Kuranishi space of each normalized component of  W, and globally by the pointed Teichm¨uller spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Our basic method is, in the case of  Riemann surfaces, closely related to the one in [73] or [57, §§4,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In the case of stable  curves, the pioneering work of Terasoma [71] already shows the connectivity of the moduli  space M  [ϕ]  g  in our teminology by using the level structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6, we define a special system of local coordinates around a point of T [ϕ]  σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' These  coordinates consist of the eigenvectors for the action of ϕ on the standard chart of the  Kuranishi space of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Harris–Mumford [29, §1] intrinsically used this type of coordinates  systematically for a certain local analysis of M g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1  Automorphisms of Riemann surfaces and equisymmetric strata  In this subsection, we review some results about the automorphisms of Riemann surfaces,  the equisymmetric stratification on Teichm¨uller space Tg and the moduli space Mg ([31],  [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We consider a Fuchsian group F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' a discrete subgroup of Aut(H) ∼= PSL(2, R)  where H is the upper half plane, such that H/F is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' As an abstract group, F is  generated by a1, b1, · · · , ag, bg, x1, · · · , xs with the relations  x1 · · · xs  g�  i=1  [ai, bi] = 1,  xλi  i = 1 (1 ≤ i ≤ s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The ordered set (g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' λ1, · · · , λs) is called the signature of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We assume that there exists  a Fuchsian group K with signature (g, −) (where − means that {x1, · · · } is empty) and  exact sequence  1 → K  i∗−→ F  j∗  −→ Gϕ → 1  (47)  such that Gϕ = ⟨ϕ⟩ ∼= Z/nZ is the cyclic group of order n generated by ϕ and i∗(K) is  a normal subgroup of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In this case, Gϕ is geometrically characterized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The  Riemann surface S ∼= H/K of genus g has an analytic automorphism ϕ : S −→ S of order  n such that the associated branched covering πϕ : S −→ W = S/Gϕ has the following  properties: The genus of W coincides with g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let P1, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='Ps ∈ W be the branch points for  πϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then λi (1 ≤ i ≤ s) coincides with the ramification index at the point �Pi ∈ π−1  ϕ (Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The map ϕn/λi fixes �Pi and rotates the disc neighborhood of �Pi by the angle 2πδi/λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  1 ≤ σi ≤ λi − 1 be the natural number with σiδi ≡ 1 (mod λi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The triple (mi, λi, σi)  31  is called the valency of ϕ at �Pi ([59], [55, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5]), and sometimes is written by σi/λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then:  (a1) (Hurwitz formula)  2(g − 1)/n = 2(g − 1) + �s  i=1(1 − 1/λi),  (a2) (Nielsen [59, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6)])  �s  i=1 σi/λi is an integer,  (a3) (Wiman [74])  n ≤ 4g + 2,  (a4) (Harvey [30, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4])  We set M = lcm(λ1, · · · , λs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then  (a4-1)  lcm(λ1, · · · , �λi, · · · , λs) = M for all i, where �λi denotes the omission of λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (a4-2) M divides n, and if g = 0, then M = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (a4-3) s ̸= 1, and, if g = 0, then s ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (a4-4) If 2|M, the number of λ1, · · · , λs which are divisible by the maximal power of  2 dividing M is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We symbolically write these data by  TV(ϕ) =  �  g, g, n : σ1  λ1  + · · · + σs  λs  �  , TVc(ϕ) =  �  g, g, n :  � δ1  λ1  , · · · , δs  λs  ��  (48)  and call TV(ϕ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='TVc(ϕ)) the total valency (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the total co-valency) of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Note  that the total (co-)valency is determined by j∗ of (47) from [31, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (i) The valency was introduced by Nielsen [59] as the unique essential  conjugacy invariant of periodic maps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' finite automorphisms) in Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A periodic map  is realized as an analytic automorphism of a certain complex structure on Σg by a standard  augument or as a corollary of Kerchoff [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore, the total valency may be considered  as an invariant of both periodic maps in Γg and anlytic automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) Even if g = 0, 1, a finite automorphism ϕ satisfies the conditions (a1) ∼ (a3) and we  use the same terminology (48) in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (iii) For the classification of TV(ϕ) in the case where g = 2, 3, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  As discussed in [31], the choice of the inclusion map i∗ in (47) determines the Te-  ichm¨uller marking, and the choice of the surjective map j∗ determines the generators of  Gϕ, which determine the data (a2) in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The conditions (a1) ∼ (a4) are not only necessary conditions but also sufficient condi-  tions for the existence of automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This point is widely discussed from the moduli  theoretic viewpoint by Harvey and Broughton as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We fix the numerical data (48)  for g ≥ 2, and consider the locus in Tg (or Mg) of Riemann surfaces which have this type  of automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Note that, in the following results (I) and (II), the case where g = 2  and ϕ is the hyperelliptic involution is excluded as an exceptional case:  (I) ([31, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2 and (6) in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='392], [16, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5]) Let ϕ : S → S be a periodic map in  Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let T ϕ  g be the subset of Tg consisting of the points p = [S, f] ∈ Tg which are fixed  32  by the action of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We define two periodic maps ϕ and ψ to be equivalent if their total  valencies are the same: TV(ϕ) = TV(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' As we remarked in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1 (i), Nielsen [59]  proved that this is equivalent to saying that ϕ and ψ are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let [ϕ] denote the  equivalence class to which ϕ belongs, and we define T [ϕ]  g  as  T [ϕ]  g  =  �  ψ∈[ϕ]  T ψ  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Recall that πϕ : S → W = S/Gϕ is the branched covering associated with the analytic  action ϕ : S → S of order n, and g is the genus of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' P1, · · · , Ps ∈ W are the branch  points for πϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then T ϕ  g is real analytically isomorphic to the Teichm¨uller space Tg,s of  s-pointed Riemann surfaces of genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since T ϕ  g is a connected component (T [ϕ]  g )(0) of  T [ϕ]  g , we have  T ϕ  g ∼= (T [ϕ]  g )(0) ∼= Tg,s  (real analytically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Each connected component of T [ϕ]  g  is T ψ  g for some ψ ∈ [ϕ], and we can say the same thing  for T ψ  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Thus the space T [ϕ]  g  is a countable union of (3g − 3 + s)-dimensional complex  manifolds so that each of them is real analytically isomorphic to (T [ϕ]  g )(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (II) ([16, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2])  The locus M [ϕ]  g  of the isomorphism classes of Riemann surfaces which  have automorphisms whose total valencies coincide with TV(ϕ) is a closed irreducible  algebraic subvariety of Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The loci T [ϕ]  g  and M [ϕ]  g  are called the equisymmetric strata for [ϕ] of Tg and Mg respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Note that, in the excluded case where g = 2 and ϕ is the hyperelliptic involution,  we have T [ϕ]  2  = T2 and M [ϕ]  2  = M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For the study of the equisymmetric strata on Tg, Kuribayashi’s method  ([50]) using invariant quadratic differentials is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Recently, Takamura ([67]) and  Hirakawa and Takamura ([35]) developed this type of argument and gave a method for the  detailed analysis of the stratification for various group actions on Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2  Logarithmic quadratic representation of automorphisms  Here we discuss a representation of an automorphism of a pointed Riemann surface to  the logarithmic quadratic differentials for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let ϕ : S −→ S be an automorphism of a Riemann surface of genus g ≥ 0 with the  given total (co-)valency (48), and πϕ : S −→ W = S/Gϕ be the associated cyclic branched  covering (see also Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1 (ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We consider the vector space  V = H0(S, 2KS +  s′  �  i=1  π−1  ϕ (Pi) +  t  �  i=1  π−1  ϕ (Ps+i))  (49)  33  where {P1, · · · , Ps′} is a subset (it may be empty) of the set of branch points {P1, · · · , Ps′,  Ps′+1, · · · , Ps} (s′ ≤ s) for πϕ, and {Ps+1, · · · , Ps+t} is a subset of the set of un-branched  points for πϕ (which is intended to be the set of possible poles, and may be empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We  assume  2g − 2 + s + t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (50)  An element v ∈ V is considered as a logarithmic quadratic differenticial on S whose poles  are at most in �s′  i=1 π−1  ϕ (Pi) + �t  i=1 π−1  ϕ (Ps+i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since ϕ acts on each fiber of πϕ : S → W  as a permutation of points, ϕ naturally acts on the differential forms v by  ϕ(v) = v ◦ ϕ−1 ∈ V  (51)  (similarly to [25, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='269]), and ϕ induces a linear automorphism of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We call it the  representation of ϕ to the logarithmic quadratic forms V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Fundamental facts of the rep-  resentation to holomorphic differential forms discussed in [25, §V2] are easily extended  to this type of representation to logarithmic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For example, its eigenvalues are n-th  roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The proof of the following Proposition is a slight modification of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Guerrero’s argument  written in [25, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='274–277].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For a rational number x, we write [x] the maximal integer not  exceeding x and {x} = x − [x] its fractional part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We also use the symbol e(x) = e2πix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let ϕ : S −→ S be an automorphism of order n of a Riemann surface  S with the total (co-)valency (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then, under the assumption (50), the dimension of  the eigenspace of eigenvalue e(α/n) (0 ≤ α ≤ n − 1) for the the action ϕ on V defined by  (51) is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  h0  �  S, 2KS +  s′  �  i=1  π−1  ϕ (Pi) +  t  �  i=1  π−1  ϕ (Ps+i)  �  e(α/n)  = 3g − 3 + 2s + t −  s′  �  i=1  ��−ασi − 1  λi  �  + 1  λi  �  −  s  �  i=s′+1  ��−ασi − 2  λi  �  + 2  λi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  If 1 ≤ i ≤ s′ or s + 1 ≤ i ≤ s + t, then we set ri = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If s′ + 1 ≤ i ≤ s,  then we set ri = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We denote the eigenspace of eigenvalue e(α/n) by Vα = H0(S, 2KS +  �s+t  i=1 riπ−1  ϕ (Pi))e(α/n) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Moreover, we set  λi = 1, δi = σi = 0,  for s + 1 ≤ i ≤ s + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (52)  For 1 ≤ i ≤ s + t, we denote the fiber by  π−1  ϕ (Pi) =  �  1≤j≤n/λi  P (i)  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  34  Assuming Vα ̸= ∅, we fix an element v0 ∈ Vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then for any v ∈ Vα, the element v/v0  is ϕ-invariant, and it is a meromorphic function on W = S/Gϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore, there exits a  divisor Dα on W such that  Vα ∼= H0(W, Dα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (53)  We determine Dα explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By using a local coordinate z around P (i)  j , the Laurant  expansion of v is written as v = �  k≥0 Akzk−ridz2 for Ak ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since the map ϕ−n/λi is  written here by z �→ e(−δi/λi)z, we have  (ϕn/λi)∗v =  �  k≥0  Ake  �−δi(k + 2 − ri)  λi  �  zk−ridz2  which coincides with �  k≥0 e(α/λi)Akzk−ridz2 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore, Ak = 0 for −δi(k+  2 − ri) ̸≡ α (mod λi), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' k ̸≡ −ασi − 2 + ri (mod λi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' It follows that  v =  �  k∈ZK  Akzk−ridz2,  where ZK = {k = λia − ασi − 2 + ri ≥ 0 | ∃a ∈ Z } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let bi be the vanishing order of v0 at P (i)  j , which is automatically independent of j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then there exists some �bi ∈ Z such that  bi = λi�bi − ασi − 2 + ri ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (54)  Let Q1, · · · , Qu be the images by πϕ of zeros of v0 except for P1, · · · , Ps+t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Put π−1  ϕ (Qi) =  �  1≤j≤n Q(i)  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let βi be the vanishing order of v0 at Q(i)  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For a meromorphic function �h on  W, the element (�h◦πϕ)·v0 belongs to Vα if and only if �h satisfies ordPi�h ≥ −bi/λi, ordQi�h ≥  −βi and is holomorphic outside Pi’s and Qi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since −ασi − 2 + ri < 0, the integral  divisor which satisfies these conditions should be Dα = �s+t  i=1(�bi+[(−ασi − 2 + ri)/λi])Pi+  �u  i=1 βiQi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In particular,  deg Dα =  s+t  �  i=1  �  �bi +  �−ασi − 2 + ri  λi  ��  +  u  �  i=1  βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (55)  We rewrite the expression (55) of degDα so that it is independent of the choice of v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  First we clearly have deg v0 = �s+t  i=1(n/λi)bi + n �u  i=1 βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' On the other hand, since v0  belongs to Vα, we have degv0 = deg  �  2KS + �s+t  i=1 riπ−1  ϕ (Pi)  �  = 4(g − 1) + �s+t  i=1(n/λi)ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Therefore it follows from (54) that  degv0  n  =  s+t  �  i=1  �  �bi + −ασi − 2 + ri  λi  �  +  t  �  i=1  βi = 4(g − 1)  n  +  s+t  �  i=1  ri  λi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (56)  The Riemann–Hurwitz formula for the covering πϕ says that  2g − 2  n  = 2g − 2 +  s  �  i=1  �  1 − 1  λi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (57)  35  Therefore, from (52), (55), (56) and (57), we obtain  deg Dα =  s+t  �  i=1  �  �bi + −ασi − 2 + ri  λi  �  +  t  �  i=1  βi −  s+t  �  i=1  �−ασi − 2 + ri  λi  �  = 4g − 4 + 2  s  �  i=1  �  1 − 1  λi  �  +  s+t  �  i=1  � ri  λi  −  �−ασi − 2 + ri  λi  ��  = 4g − 4 + 2s + t −  s  �  i=1  ��−ασi − 2 + ri  λi  �  + 2 − ri  λi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (58)  By (50) and (58), we have deg(KW ⊗ D−1  α ) ≤ 2 − 2g − s − t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence  H0(W, KW ⊗ D−1  α ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (59)  From the Riemann–Roch formula, the Serre duality and (53), (58), (59), we have  dimVα = dimH0(W, Dα) = degDα−g+1 = 3g−3+2s+t−  s  �  i=1  ��−ασi − 2 + ri  λi  �  + 2 − ri  λi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  This equality coincides with the desired one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We consider (S, P) as a k′-pointed Riemann surface, where P = �s′  i=1 π−1  ϕ (Pi) +  �t  i=1 π−1  ϕ (Ps+i) in (49) and k′ = ♯(P) (cardinality), and write the set of the eigenvalues  for the action of ϕ on the (3g − 3 + k′)-dimensional vector space V = H0(S, 2KS + P) by  �  e  �θ1  n  �  , e  �θ2  n  �  , · · · , e  �θ3g−3+k′  n  ��  (0 ≤ θ1 ≤ θ2 ≤ · · · ≤ θ3g−3+k′ ≤ n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (60)  Here each eigenvalue is counted as many times as the dimension of its eigenspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then:  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By using (60), we define the ordered set of the log-quadratic characters  for the automorphism ϕ of the pointed Riemann surface (S, P) as  ChϕH0(S, 2KS + P) =  �θ1  n , θ2  n , · · · , θ3g−3+k′  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (61)  We also define the ordered set of the eigenbasis {v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' · · · , v3g−3+k′} of V as the basis con-  sisting of eigenvectors of each of ChϕH0(S, 2KS + P), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  ϕ∗(vi) = e  �θi  n  �  vi  (1 ≤ i ≤ 3g − 3 + k′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (62)  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since the dimension of the eigenspace for some eigenvalue might be strictly  greater than 1, the eigenbasis of V is not unique in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  36  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let ϕ : (S, P) → (S, P) be the automorphism of order 7 of the one-pointed  genus 3 Riemann surface with the total valency (6/7 + 6/7 + 2/7, ¯g = 0) where 6/7 is  attached to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since s = 3, s′ = 1 and t = 0, it follows from Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3 that  h0 (C, 2KC + P)e(ν/7) = 3−  ��−6ν − 1  7  �  + 1  7  �  −  ��−2ν − 2  7  �  + 2  7  �  −  ��−6ν − 2  7  �  + 2  7  �  = 0, 1, 2, 1, 1, 1, 1  (ν = 0, 1, 2, 3, 4, 5, 6, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Hence ChϕH0(S, 2KS + P) = {1/7, 2/7, 2/7, 3/7, 4/7, 5/7, 6/7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  For other examples, see Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='15 or [29, §1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3  The little Teichm¨uller space in an orbifold chart of M  orb  g  In this subsection, we interpret the little Teichm¨uller spaces in the augumented Te-  ichm¨uller theory (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [37, §7]) by the language of Kuranishi families, using the facts in  §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We follow the notation of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The maximal codimensional strata T(σ) ⊂ Dϵ(σ) \\ Tg  is written by the extended Fenchel–Nielsen coordinates (26) as  ℓ1 = · · · = ℓk = 0, ℓj > 0 (k + 1 ≤ j ≤ 3g − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The space T(σ) is called the little Teichm¨uller space for σ, which is isomorphic to the  product of lower-dimensional Teichm¨uller spaces of pointed Riemann surfaces complex  analytically (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [37, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='289]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Here we first review the real analytic structure of T(σ), and  then explain its complex analytic structure from the viewpoint of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let Σg(σ) = �r  i=1 Ri be the irreducible decomposition of the source stable curve and  ( ˆRi, ˆPi) (1 ≤ i ≤ r) be the pointed Riemann surfaces obtained from the normalization of  Σg(σ) as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let gi be the genus of ˆRi and ki = ♯(ˆPi) the number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' From  the count of the dimensions of the vector spaces in the exact sequence (20), we have  r  �  i=1  (3gi − 3 + ki) = 3g − 3 − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Now each member of the curve system ˜σ \\ σ = ⟨Ck+1, · · · , C3g−3⟩ may be considered as  a simple closed curve on a unique member of these pointed Riemann surfaces via the  normalization map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  {k + 1, · · · , 3g − 3} =  r�  i=1  Ii,  ♯(Ii) = 3gi − 3 + ki  be the decomposition of suffixes so that {Cij}ij∈Ii is a system of maximal simple closed  curves on ( ˆRi, ˆPi) which induces a pants decomposition of this surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let [S, w] be  37  a σ-marked stable curve with the marking w : Σg(σ) −→ S and let S = �r  i=1 Si be  the irreducible decomposition, ( ˆSi, Pi) (1 ≤ i ≤ r) being the pointed Riemann surfaces  obtained by the normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the restriction of w to each component is lifted via  the normalization to a Teichm¨uller marking  wi : ( ˆRi, ˆPi) −→ ( ˆSi, Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (63)  Then the Fenchel–Nielsen coordinates (lij(p), τij(p)) ∈ (R>0)3gi−3+ki × R3gi−3+ki are de-  fined for the point p = [( ˆSi, Pi), wi] of the pointed Teichm¨uller space Tgi,ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  They  are the hyperbolic length lij(p) of the unique geodesic in the isotopy class of wi(Cij)  (ij ∈ Ii) and the twisting parameter τij(p) along Cij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the real analytic isomorphism  T(σ) ∼= �  1≤i≤r Tgi,ki is induced via these Fenchel–Nielsen coordinates:  {ℓi([S, w]), τi([S, w])}k+1≤i≤3g−3 �−→  �  1≤i≤r  {(lij([( ˆSi, Pi), wi]), τij([( ˆSi, Pi), wi])}ij∈Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Over this real structure of T(σ), the complex structure is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By  shrinking the base spaces of standard Kuranishi families, it follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4  and (16) that there exists complex coordinate patching Dϵ(σ) = ∪α∈IBα, where each  Bα := BSα is the base space of a standard Kuranishi family for a σ-marked stable curve  [Sα, wα] which is embedded in Ext1OSα(Ω1  Sα, OSα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then T(σ) ∩ Bα is isomorphic to  H1(Sα, HomOSα(Ω1  Sα, OSα)) ∩ Bα by (17),(18) and the property (I) in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let Sα = �r  i=1 Sα,i be the irreducible decomposition, and ( ˆSα,i, Pα,i) be the pointed  Riemann surface of Sα,i via the normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the direct factor Tgi,ki of T(σ) comes  from the direct factor H1( ˆSα,i, T ˆSα,i(−Pα,i)) of H1(Sα, HomOSα(Ω1  Sα, OSα)) in (18), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Tgi,ki ∩ Bα is isomorphic to H1( ˆSα,i, T ˆSα,i(−Pα,i)) ∩ Bα ∼= H0( ˆSα,i, 2K ˆSα,i + Pα,i)∗ ∩ Bα by  the property (I) in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' These spaces are globally well-patched by the arguments in [9,  Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='XV,§2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Thus:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In the above notation, the complex analytic structures of the little Te-  ichm¨uller space T(σ) is described by the coordinate patchings  T(σ) =  �  α∈I  �  H1(Sα, HomOSα(Ω1  Sα, OSα)) ∩ Bα  �  ,  T(σ) ∼=  �  1≤i≤r  Tgi,ki (analytically),  Tgi,ki =  �  α∈I  �  H1( ˆSα,i, T ˆSα,i(−Pα,i)) ∩ Bα  �  =  �  α∈I  �  H0( ˆSα,i, 2K ˆSα,i + Pα,i)∗ ∩ Bα  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4  Automorphisms and cyclic branched coverings of stable curves  In this subsection, we study automorphisms of stable curves and the associated cyclic  branched coverings from stable curves to nodal Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  38  We start from the discussion of graphs and their automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  A graph G =  {vi,⃗ej}1≤i≤r,1≤j≤k is a 1-dimensional finite oriented “open” cell complex embedded in  Euclidian 3-space E3 in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A 0-cell vi is a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A 1-cell ⃗ej is an  oriented edge with one of the following two types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The first type is an ordinary edge  ⃗ej = ⃗ej(vh1(j), vh2(j)) which connects a vertex vh1(j) to a vertex vh2(j) in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (Note that vh1(j) = vh2(j) may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This case corresponds to a self-intersection of an  irreducible component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=') The second type is an open edge which emanates from a vertex  vh1(j) such that the end point is a point in E3 which is not contained in the set of vertices,  and we symbolically write it as ⃗ej = ⃗ej(vh1(j), ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The absolute edge |⃗ej| is the usual edge  obtained by neglecting the orientation of ⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If all the oriented edges of G are ordinary,  we call G a compact graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We define the contraction map of G  cont : G −→ Gc  as the identity map on the complement of open edges such that any open edge ⃗ej(vh1(j), ∗)  is contracted to the vertex vh1(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then Gc is a compact graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  An automorphism ¯ϕ : G → G of a compact graph G means that ¯ϕ is a homeomorphism  in the Euclidian topology which preserves the set of vertices and the set of absolute edges  such that ¯ϕn is the identity map for some n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The least natural number n which enjoys  the above property is called the order of ¯ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  For a vertex vi (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' an edge ⃗ej), there exists a minimal natural number m(vi)  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' m(⃗ej)) which satisfies ¯ϕm(vi)(vi) = vi (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ¯ϕm(⃗ej)(⃗ej) = ⃗ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  If m(⃗ej) is even  and ¯ϕm(⃗ej)/2 stabilizes |⃗ej| by reversing its orientation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ¯ϕm(⃗ej)/2(⃗ej) = −⃗ej, then ⃗ej is  said to be an amphidrome edge for ¯ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Otherwise, ⃗ej is said to be a non-amphidrome edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let V, NE, AE be the set of vertices, non-amphidrome edges and amphidrome edges of  G for ¯ϕ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  V =  �  1≤i≤r  �  0≤ℓ≤m(vi)−1  ¯ϕℓ(vi),  NE =  �  1≤j≤k1  �  0≤ℓ≤m(⃗ej)−1  ¯ϕℓ(⃗ej),  AE =  �  k1+1≤j≤k1+k2  �  0≤ℓ≤m(⃗ej)/2−1  ¯ϕℓ(⃗ej)  be the orbit decompositions for ¯ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Here {v1, · · · , vr}, {⃗e1, · · · ,⃗ek1} and {⃗ek1+1, · · · ,⃗ek1+k2}  are assumed to belong to mutually distinct orbits in V, NE and AE respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We  have r = �  1≤i≤r m(vi), k = �  1≤j≤k1 m(⃗ej) + �  k1+1≤j≤k1+k2 m(⃗ej)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let G ¯ϕ ∼= Z/nZ be the cyclic group generated by ¯ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (i) The quotient map and the quotient graph of G by G ¯ϕ  π ¯ϕ : G −→ W := G/G ¯ϕ  39  are defined by the following two conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ia) The vertices and the edges are given by W = {v♯  i, (⃗ej)♯}1≤i≤r,1≤j≤k1+k2 so that π ¯ϕ( ¯ϕℓ(vi))  = v♯  i and π ¯ϕ( ¯ϕℓ(⃗ej)) = (⃗ej)♯ for any i, j, ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ib) Suppose ⃗ej = ⃗ej(vh1(j), vh2(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If 1 ≤ j ≤ k1, then (⃗ej)♯ is an ordinary edge given by  (⃗ej)♯(π ¯ϕ(vh1(j)), π ¯ϕ(vh2(j))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If k1 + 1 ≤ j ≤ k1 + k2, then (⃗ej)♯ is an open edge given by  (⃗ej)♯(π ¯ϕ(vh1(j)), ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (64)  (ii) The compact quotient map and the compact quotient graph of G by G ¯ϕ are defined by  πc  ¯ϕ = cont ◦ π ¯ϕ : G −→ (W)c = (G/G ¯ϕ)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  With respect to (64), we may write (⃗ej)♯(π′  ¯ϕ(vh2(j)), ∗) because ⃗ej is amphidrome in  this case and vh1(j) and vh2(j) are on the same orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We consider the compact graph G = {vi,⃗ej(v1, v2)}1≤i,j≤2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let ¯ϕ, ¯ψ : G →  G be automorphisms of order 2 defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The first ¯ϕ interchanges v1 and v2, and  stablizes |⃗ej| by reversing their orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By using the usual symbols of permutations,  ¯ϕ is written by (v1, v2), (⃗e1, −⃗e1), (⃗e2, −⃗e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The second ¯ψ interchanges v1 and v2, and also  ⃗e1 and ⃗e2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ¯ψ is written by (v1, v2), (⃗e1, −⃗e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then ⃗e1 and ⃗e2 are amphidrome (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' non-amphidrome) for ¯ϕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' for ¯ψ), and  we have G/G ¯ϕ = {v♯  1, (⃗e1)♯(v♯  1, ∗), (⃗e2)♯(v♯  1, ∗)}, (G/G ¯ϕ)c = {v♯  1} (empty edge), G/G ¯ψ =  (G/G ¯ψ)c = {v♯  1, (⃗e1)♯(v♯  1, v♯  1)} as shown in Figure II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  v1  v2  ⃗e1  ⃗e2  G  v♯  1  (⃗e1)♯  (⃗e2)♯  G/G ¯ϕ  (G/G ¯ϕ)c  v♯  1  G/G ¯ψ = (G/G ¯ψ)c  v♯  1  (⃗e1)♯  (Figure II) The quotient and the compact quotient graphs in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9  Now let S be a stable curve of genus g ≥ 2 with its irreducible decomposition  S = �r  i=1 Si, and let P = {P1, · · · , Pk} be its set of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By this configuration, a com-  pact graph rg(S) = {vSi,⃗ePj}1≤i≤r,1≤j≤k is defined as follows: An irreducible component  Si corresponds to a vertex vSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If a node Pj is an intersection point of the irreducible com-  ponents Sh1(j) and Sh2(j), then it is expressed by an ordinary edge ⃗ePj = ⃗ePj(vSh1(j), vSh2(j))  (the orientation is arbitrary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call rg(S) the reduced dual graph of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let ϕ : S → S be an analytic automorphism of order N, and G = ⟨ϕ⟩ ∼= Z/NZ the  subgroup of Aut(S) generated by ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then ϕ clearly induces the automorphism ϕrg(S) :  rg(S) → rg(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We translate the terminologies defined for (rg(S), ϕrg(S)) into (S, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' That  40  is to say, m(Si) and m(Pj) are the minimal natural numbers which satisfy ϕm(Si)(Si) = Si  and ϕm(Pj)(Pj) = Pj, and Pj is an amphidrome node (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' a non-amphidrome node) for  ϕ if ⃗ePj is an amphidrome edge (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' a non-amphidrome edge) for ϕrg(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The orbit-  irreducible decomposition of S for ϕ is defined by  S =  r  �  i=1  m(Si)−1  �  j=0  ϕj(Si),  (65)  where each Si for 1 ≤ i ≤ r has mutually distinct orbits and r = �r  i=1 m(Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let h : �S = �r  i=1  �m(Si)−1  j=1  �  ϕj(Si) −→ S be the normalization map, and ϕi,j :  �  ϕj(Si) −→ �  ϕj(Si) the lifting of ϕm(Si)|ϕj(Si) : ϕj(Si) −→ ϕj(Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Since �  ϕj(Si) ∼= �Si  and ϕi,j is congruent to ϕi,0 for 0 ≤ j ≤ m(Si) − 1, we may consider  ϕi := ϕi,0 : �Si −→ �Si  (1 ≤ i ≤ r)  (66)  as the representatives of the ϕi,j’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let ni be the order of ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We have an ni-fold cyclic  branched covering of Riemann surfaces  πϕi : �Si −→ �  Wi ∼= �Si/Gϕi,  Gϕi = ⟨ϕi⟩ ∼= Z/niZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (67)  Note that ϕi also defines an automorphism of the pointed Riemann surface  ϕi : (�Si, Pi) −→ (�Si, Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (68)  We divide the set Pi and the set Qi = πϕi(Pi) on �  Wi into  Pi = PN  i  �  PA  i ,  Qi = QN  i  �  QA  i  (69)  where PN  i  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' PA  i ) consists of the points P such that the nodes h(P) in S are non-  amphidrome (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' amphidrome) for ϕ, and QN  i = πϕi(PN  i ), QA  i = πϕi(PA  i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The dual graph dg(S) = {(vSi, g(�Si)),⃗ePj}1≤i≤r,1≤j≤k is the weighted graph obtained  from the reduced dual graph rg(S) by attaching the weight g(�Si) to each vertex vSi, which  is the genus of �Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (If g(�Si) = 0, it is sometimes omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=') An automorphism ϕ : S → S  also induces an automorphism ϕdg(S) : dg(S) → dg(S), since ϕ preserves g(�Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The following two lemmata guarantee the existence of a natural quotient of S by G  as a nodal Riemann surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' There exists a finite holomorphic map  πϕ : S −→ W  (70)  to a nodal Riemann surface W which has the following properties:  41  (i) W has an irreducible decomposition �r  i=1 Wi such that Wi = πϕ(ϕj(Si)) for any j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) The normalizations of S and W naturally induce cyclic branched coverings  πϕi,j : �  ϕj(Si) −→ �  Wi  (1 ≤ i ≤ r, 0 ≤ j ≤ m(Si) − 1)  such that πϕi,j is isomorphic to πϕi in (67) for any j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (iii) The reduced dual graph rg(W) coincides with the compact quotient graph (rg(S)/Gϕrg(S))c  so that the dual graph is written as dg(W) = {(vWi, g(�  Wi)),⃗eQj}1≤i≤r,1≤j≤k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In partic-  ular, a non-amphidrome node in S is sent by πϕ to a node of W, while an amphidrome  node is sent to a non-singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  By using {(�  Wi, QN  i )}1≤i≤r in (67), (69) as the building blocks for patchings,  one can construct the desired nodal surface W so that QN  i are patched as the nodes of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  This patching process is constructed by identifying the two local components of xy = 0  at the origin of C2 with the disk neighborhoods of �  Wi’s at the suitable points in QN  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' W (in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10) is isomorphic to the quotient analytic space S/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  We consider the local ring OS,P at the node P of S, and let �  OS,P ∼= C[[x, y]]/(xy)  be its completion by the maximal ideal of OS,P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If P is non-amphidrome whose covalencies  at both banks are δ(1)/λ(1) and δ(2)/λ(2), then the action is written by  ϕm(P) : (x, y) �−→ (e(δ(1)/λ(1))x, (e(δ(2)/λ(2))y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (71)  We may assume λ(1) ≥ λ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the invariant subring of �  OS,P for ϕm(P) is given by  ( �  OS,P)ϕm(P ) ∼= C[[z, w]]/(zw),  where z = xλ(1), w = yλ(2) and zw = xλ(1)−λ(2)(xy)λ(2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If P is amphidrome for ϕm(P)  whose covalency is δ/λ, then  ϕm(P)/2 : (x, y) �−→ (e(δ/2λ)y, (e(δ/2λ)x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (72)  Hence  ( �  OS,P)ϕm(P ) ∼= C[[z]],  where z = xm(P) = ym(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If P is a nonsingular point of S, the similar invariant subring  is obviously regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  It follows that S/G exists as a complex curve with at most nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The natural mor-  phism S → S/G has the same properties as those in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Therefore, W is  isomorphic to S/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We write (70) as πϕ : S −→ W = S/G and call it the cyclic branched  covering associated with ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  42  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let S be a stable curve of genus 3 with two irreducible components and  two nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' S = �2  i=1 Si, S1 ∩ S2 = {P1, P2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Assume that each Si is isomorphic to an  elliptic curve with the period √−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Si ≃ C/(Z + √−1Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We consider the following two  automorphisms ϕ, ψ : S → S of order 8 so that the automorphisms ϕrg(S) and ψrg(S) are  given in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' First, ϕ fixes P1 and P2 and interchanges S1 and S2 such that the  total valencies of ϕ2|Si (i = 1, 2) are 3/4 + 3/4 + 1/2 (where 3/4 are attached to P1 and  P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Secondly, ψ interchanges P1 and P2 and interchanges S1 and S2 such that the total  valencies of ψ2|Si are 3/4 + 3/4 + 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then P1 and P2 are amphidrome nodes (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' non-amphidrome nodes) for ϕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let πϕ : S → Wϕ, πψ : S → Wψ be the cyclic branched covering associated with ϕ and  ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then Wϕ is isomorphic to P1, while Wψ is a stable curve of genus 1 with one node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Their reduced dual graphs are given in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' It may be natural to re-write πϕ : S −→ W as πϕ : S −→ W = �r  i=1 miWi  where W is a non-reduced scheme such that W  red = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This point will be also discussed  in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1 (see also [55, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let Bi ⊂ �  Wi be the set of branch points of πϕi : �Si → �  Wi in (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By comparing with  (69), we define the cardinalities of the sets by  ♯(Bi) = si, ♯(Bi∩Qi) = s′  i, ♯(Bi∩QN  i ) = (s′  i)(1), ♯(Bi∩QA  i ) = (s′  i)(2), s′  i = (s′  i)(1)+(s′  i)(2),  ♯(Qi \\ Bi) = ti, ♯(QN  i \\ Bi) = t(1)  i , ♯(QA  i \\ Bi) = t(2)  i ,  ti = t(1)  i  + t(2)  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (73)  We add the valency 1 for each non-branch point in Qi \\ Bi to the total valency (48) for  ϕi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' and define the dressed total valency for ϕi by  DTV(ϕi) =  �  �gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ni : σ(i)  1  λ(i)  1  + · · · +  σ(i)  (s′  i)(1)  λ(i)  (s′  i)(1)  +  ��σ(i)  (s′  i)(1)+1  λ(i)  (s′  i)(1)+1  ��  + · · · +  ��σ(i)  s′  i  λ(i)  s′  i  ��  +  σ(i)  (s′  i)+1  λ(i)  s′  i+1  + · · · σ(i)  si  λ(i)  si  + 1 + · · · + 1  �  ��  �  t(1)  i  + ((1)) + · · · + ((1))  �  ��  �  t(2)  i  �  �  �  �  (74)  where the valencies are ordered for Bi∩QN  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Bi∩QA  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Bi\\Qi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' QN  i \\Bi and QA  i \\Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' These  symbols (bold faced valency for a non-amphidrome branch point, soft double bracket  valency for an amphidrome branch point, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=') are borrowed from [11, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then:  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We define the numerical data of an automorphism ϕ : S −→ S of a  stable curve (of order N) by  Num(ϕ) =  �  N, ϕdg(S),  r�  i=1  DTV(ϕi)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (75)  43  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' These numerical data are essentially the same as the numerical data of  pseudo-periodic maps of negative twist (see §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' From this viewpoint, Num(ϕ) for g = 3  are classified in [11, Table 2, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5  Equisymmetric strata at the boundary charts of M  orb  g  We prove a structure theorem for equisymmetric strata consisting of marked stable curves  which have the same numerical invariants of automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let w : Σg(σ) −→ S be a σ-marking such that S has an automorphism ϕ with the  preassigned numerical data Num(ϕ) as in (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' From the orbit-irreducible decomposition  S = �r  i=1  �mi−1  j=0 ϕj(Si) (mi = m(Si)) as in (65), the little Teichm¨uller space T(σ) which  contains the point [S, w] is isomorphic to  T(σ) ∼=  r�  i=1  (Tgi,ki × · · · × Tgi,ki)  �  ��  �  mi  ,  (76)  where gi = g(�Si), ki = ♯(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' See Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, we defined, with the explanation (I), the subspaces T ϕ  g and T [ϕ]  g  for a periodic  map ϕ : S → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' There S was a Riemann suface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We consider here a similar definition  for a stable curve S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  As in the previous subsection §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4, let ϕ : S → S be an analytic automorphism of  order N of a stable curve S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let T ϕ  σ be the set of points p = [S, w] in T(σ) which is fixed  by the action of ϕ∗ : T(σ) → T(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (Through the σ-marking w : Σg(σ) → S, the analytic  automorphism ϕ is considered to be an element w−1 ◦ ϕ ◦ w of the Weyl group W(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The action ϕ∗ is the action as an element of W(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' See (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=') As in the case of Riemann  surfaces, we define the equivalence class [ϕ] to be the set of those analytic automorphisms  ψ : S → S which have the same numerical data (75) as ϕ : Num(ψ) = Num(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then  T [ϕ]  σ  is defined as follows:  T [ϕ]  σ  =  �  ψ∈[ϕ]  T ψ  σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The subspace M  [ϕ]  g  of M g is defined to be the quotient space of T [ϕ]  σ  by the Weyl group:  M  [ϕ]  g  = T [ϕ]  σ /W(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' As we will prove in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, an analytic automorphism ϕ of a stable curve  S is lifted to a pseudo-periodic map of negative twist ˜ϕ of a Riemann surface such that  Num(ϕ) may be identified with the invariants (a), (c) of ˜ϕ in Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence Num(ϕ)  determines the conjugacy class of the analytic automorphism ϕ of the stable curve S by  the results of [61], [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  44  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (i) There exist analytic embeddings of pointed Teichm¨uller spaces  ψi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='j : Tgi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='si+ti −→ Tgi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='ki (1 ≤ i ≤ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 0 ≤ j ≤ mi − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  where ti is given in (73),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' and also an analytic embedding  Φ :  r�  i=1  Tgi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='si+ti �→  r�  i=1  (Tgi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='ki × · · · × Tgi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='ki)  �  ��  �  mi  ∼= T(σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (77)  Φ : (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' xr) �−→ (ψ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='0(x1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ψ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='mi−1(x1)  �  ��  �  m1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ψr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='0(xr),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ψr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='mr−1(xr)  �  ��  �  mr  ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (78)  such that the connected component (T [ϕ]  σ )(0) of T [ϕ]  σ  containing the point [S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' w] is analyt-  ically isomorphic to the image Φ(�r  i=1 Tgi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='si+ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In particular, (T [ϕ]  σ )(0) is a �r  i=1(3gi −  3 + si + ti)-dimensional complex submanifold of T(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The space T [ϕ]  σ  itself is a countable  union of submanifolds of these types of connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) M  [ϕ]  g  is a locally closed irreducible subvariety of M g which is isomorphic to �r  i=1 Mgi,si+ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  Step 1  Let S be a stable curve with the irreducible decomposition S =  �r  i=1 Si such that S has an automorphism ϕ with the preassigned numerical data (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let ψ : X → B be a standard Kuransihi family of S = ψ−1(b0) (b0 ∈ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We may assume  that B is a small open ball around the origin (= b0) of the vector space Ext1  OS(Ω1  S, OS),  which we write B = Ext1  OS(Ω1  S, OS) ∩ B if we want to emphasize this ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  From (18) and (I) of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, we restrict B to the subspace which parematrizes the variable  deformation  H1(S, HomOS(Ω1  S, OS)) ∩ B ∼=  r  �  i=1  Vi, where Vi := H0( ˆSi, 2K ˆSi + Pi)∗ ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (79)  Now ϕ relatively acts on ψ, and let Bϕ = Ext1  OS(Ω1  S, OS)ϕ ∩ B be the invariant subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Since the action of ϕ on S preserves the set of nodes, the action of ϕ on B preserves the  subspace �r  i=1 Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Set mi = m(Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since the iterations of the action of ϕ on the direct  factor Vi isomorphically map Vi → ϕ(Vi) → ϕ2(Vi) → · · · and stabilize ϕmi(Vi) = Vi, the  ϕ-invariant subspace (�r  i=1 Vi)ϕ is isomorphic to the direct sum of ϕmi-invariant subspaces  �  r  �  i=1  Vi  �ϕ  ∼=  r  �  i=1  mi−1  �  j=0  �  ϕj(Vi)  �ϕi , where ϕi := ϕmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (80)  The direct factor V ϕi  i  of (80) is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We consider the covering πϕi : �Si −→  �  Wi in (67), and set Bi = {Q1, · · · , Qsi}, Qi \\ Bi = {Qsi+1, · · · , Qsi+ti} from (73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By  45  appling Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3 to the 0-eigenspace, the dimension of V ϕi  i  is equal to 3gi−3+si+ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  More precisely, the discussion in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3 says that  V ϕi  i  ∼= H0(�  Wi, 2K�  Wi +  si+ti  �  j=1  Qj)∗ ∩ B,  (81)  where B is a small open ball around the origin (= b0) of H0(�  Wi, 2K�  Wi + �si+ti  j=1 Qj)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Step 2  We show the existence of a natural family of pointed Riemann surfaces over  V ϕi  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The connected component of the normalization of the restricted family ψ−1(Vi) → Vi  naturally induces a family of pointed Riemann surfaces  ψi : (S, P) −→ Vi,  (82)  which should be a standard Kuranishi family of the pointed Riemann surface ψ−1  i (b0) =  (�Si, Pi) by H1(�Si, T�Si(−Pi)) ∼= H0(�Si, 2K�Si + Pi)∗ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='177]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore, the natural  action of the stabilizer Gϕi = Stab(Si) = ⟨ϕi⟩ on �Si is extended relatively to the family  ψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We consider the relative quotient map  ψi : W = S/Gϕi −→ Vi = Vi/Gϕi ∼= V ϕi  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (83)  The central fiber ψ  −1  i (b0) of (83) is isomorphic to �Si/Gϕi = �  Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For any b ∈ Vi, the fiber  ψ  −1  i (b) is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since (82) is a standard Kuranishi family, its fundamental  property (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [9, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='XI (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='13)]) says that the fiber ψ−1  i (b) (b ∈ Vi) admits a subgroup  Gϕi,b ⊂ Aut(ψ−1  i (b)) which is isomorphic to Gϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then  ψ  −1  i (b) ∼= ψ−1  i (b)/Gϕi,b := �  Wi,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In other words, the family (83) is a deformation of �  Wi = ψ  −1  i (b0) so that each fiber of ψi  is a quotient surface of each fiber of ψi in (82) by essentially the same Galois group Gϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Moreover, each point Qj ∈ {Q1, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='Qsi, Qsi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' · · · , Qsi+ti} on �  Wi is extended as a section  Qj : Vi → W of ψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In fact, we set πS : S → W = S/Gϕi and πS,b0 = πS|(πS)−1(b0) : �Si → �  Wi = �Si/Gϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then Qj satisfies one of the following two conditions:  (i) There exists a point Pi,j in the support of Pi such that Qj = πS,b0(Pi,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) Qj is a branch point of the covering πS,b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In case (i), since ψi is a Kuranishi family of (�Si, Pi), there exists a section s : Vi → S  such that s(Vi) passes through Pi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the image πS(s(Vi)) defines the desired section  Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In case (ii), since Gϕi and Gϕi,b have the same signature (λ(i)  1 , · · · , λ(i)  si ) by [31], the  components of the discriminant locus of ψi : W → Vi are sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then one of them is  the desired Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  46  Thus the family ψi is a deformation of pointed Riemann surfaces (�  Wi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Q1, · · · , Qsi+ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  More strongly, it follows from (81) that ψi is nothing but a standard Kuranishi family of  (�  Wi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Q1, · · · , Qsi+ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Step 3  We fix a point [S, w] ∈ T(σ) ⊂ Dϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We may assume that the source space  Σg(σ) of the Weyl marking w : Σg(σ) → S is the topological model of a stable curve  which has an automorphism ϕ with the numerical data (74), (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We consider the Teichm¨uller marking wi : ( ˆRi, ˆPi) −→ ( ˆSi, Pi) in (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then there  exists a branched covering ˆπi : ˆRi → Σgi over a Riemann surface Σgi of genus gi such that  the covering transformation group of ˆπi coincides with Gϕi in (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The cardinality of  the set of points consisting of the branch points for ˆπi and ˆπi(ˆPi) is si + ti, and we write  this set by { �Q1, · · · , �Qsi+ti}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the natural descent of wi, we have a pointed oriented  homeomorphism  ˆwi : (Σgi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' �Q1, · · · , �Qsi+ti) −→ (�  Wi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Q1, · · · , Qsi+ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (84)  We may consider (84) as a Teichm¨uller marking of the fiber ψ  −1  i (b0) of the family (83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Since (83) is a standard Kuranishi family, it follows from the discussion of Arbarello–  Cornalba–Griffiths [9, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='XV, §2] that this marking is extended to the whole family  (83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  That is to say, (83) induces a family of Teichm¨uller-marked pointed Riemann  surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Step 4  By applying the method of the expression of pointed Teichm¨uller spaces via  the patching of the base spaces of the standard Kuranishi families of pointed Riemann  surfaces due to [9, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='XV, §2] and also the fundamental property of the action on  Kuranishi families, we globalize the discussion in Steps 1 ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let T ϕ  σ (p) be the connected component of T ϕ  σ containing the point p = [S, w], where  the marking w is the one defined in Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This marking induces the marking of the  ϕj-image of the normalized componet ( ˆSi, Pi) of S as ϕj ◦wi : ( ˆRi, ˆPi) → (ϕj( ˆSi), ϕj(Pi)),  where wi is also given in Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the point  pi,j = [(ϕj( ˆSi), ϕj(Pi)), ϕj ◦ wi],  (1 ≤ i ≤ r, 0 ≤ j ≤ mi − 1)  is contained in the ϕi(= ϕmi)-invariant locus T ϕi  gi,ki of Tgi,ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For a fixed i, the connected  components T ϕi  gi,ki(pi,j) of T ϕi  gi,ki containing pi,j (0 ≤ j ≤ mi − 1) are clearly isomorphic  to each other, and we set (T ϕi  gi,ki)(0) := T ϕi  gi,ki(pi,0) for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then, from (80) and  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7, we have the composition of analytic embeddings  r�  i=1  (T ϕi  gi,ki)(0) −→  r�  i=1  T ϕi  gi,ki(pi,0) × · · · × T ϕi  gi,ki(pi,mi−1) −→  r�  i=1  Tgi,ki × · · · × Tgi,ki  �  ��  �  mi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (85)  47  From (83) and the discussions in Step 3 and in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, we have an isomorphism  (T ϕi  gi,ki)(0) ∼= Tgi,si+ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (86)  Then the first assertion of (i) follows from (85) and (86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since each of the set of connected  components of the T ϕi  gi,ki’s is countable by the same argument as in [31], the set of connected  components of T ϕ  σ is also countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence the assertion (i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Since the connected components of T ϕ  σ are isomorphic to each other so that these  isomorphisms are given by the change of markings, they are mapped by the forgetting map  T(σ) → M g of the markings onto the same image, which is isomorphic to �r  i=1 Mgi,si+ti  from the assertion (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This is irreducible since each M gi,si+ti is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the same  argument as in [16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='106] using the factorization of the forgetful map to the composition  of the infinite unramified covering and the finite Galois covering, it is a closed subvariety  on M g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence the assertion (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ([71])  M  ϕ  g is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The relation between the equisymmetric strata T ϕ  σ at the boundary and  their limits on Mg or Tg seems to be interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' See for instance [22], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6  Harris–Mumford coordinates around equisymmmetric strata  In this subsection, we define a special system of local coordinates on the controlled de-  formation space Dϵ(σ) around an arbitrarily chosen point p of the equisymmetric strata  according to the method of [29, §1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We fix a point p = [S, w] ∈ T ϕ  σ ⊂ T(σ) ⊂ Dϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since an open neighborhood of p in  Dϵ(σ) is the base B of a standard Kuranishi family of S, it follows form (16), (17) and  (18) that the dual bases of H0(S, Ω1  S ⊗ ωS) express a system of local coordinates at p in  Dϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' From (20), it can be decomposed into the parameters for the variable defomations  (I) and the smoothing deformations (II) as in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By considering the action of ϕ on  H0(S, Ω1  S ⊗ ωS), we choose the basis as follows:  (I) Let S = �r  i=1  �mi−1  j=0 ϕj(Si) be the orbit-irreducible decomposition in (65), and  �S = �¯r  i=1  �mi−1  j=0  �  ϕj(Si) be the natural decomposition of the normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' As the dual  to (79), we consider the parameter space �¯r  i=1  �mi−1  j=0 ϕj(V ∗  i ) of the variable deformations,  where V ∗  i = H0(�Si, 2K�Si + Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Now we choose the eigenbasis in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4 of the  vector space V ∗  i with respect to the ϕmi-action  {vi,1, · · · , vi,qi} ∈ V ∗  i ,  (ϕmi)∗(vi,j) = e  �θi,j  ni  �  vi,j,  (1 ≤ j ≤ qi)  (87)  where qi = dim V ∗  i and ni is the order of the action of ϕmi on �Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  48  With respect to the vector space ϕj(V ∗  i ) for 1 ≤ j ≤ mi − 1, we choose the basis  {ϕj(vi,1), · · · , ϕj(vi,qi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then we have (ϕmi)∗(ϕj(vi,j)) = e(θi,j/ni)ϕj(vi,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (II) Let �¯k  i=1  �m(Pi)−1  j=0  ϕj(Pi) be the set of nodes of S such that StabG(Pi) = ⟨ϕm(Pi)⟩  and k = �¯k  i=1 m(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let vPi be the generator of the torsion sheaf τPi given in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then  the basis of the parameter space of the smoothing deformations is given by  {ϕj(vPi)}1≤i≤¯k,0≤j≤m(Pi)−1 ∈  ¯k  �  i=1  m(Pi)−1  �  j=0  τ ϕj(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (88)  For the eigenvalues, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Here we write P = ϕj(Pi), vP = ϕj(vPi),  (δ(1)/λ(1))(P) = δ(1)/λ(1) and so on for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If P is a non-amphidrome node such that the covalencies of both sides are  δ(1)/λ(1) and δ(2)/λ(2), then (ϕm(P))∗vP = e  �  −δ(1)/λ(1) − δ(2)/λ(2)�  vP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  If P is an amphidrome node with covalency δ/λ, then (ϕm(P)/2)∗vP = e (−δ/λ) vP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  Assume P is non-amphidrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Note that vP is written as ydx⊗2/x = xdy⊗2/y  modulo xy = 0 by (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' From (71), we have  (ϕm(P))∗vP =  e  �  − δ(2)  λ(2)  �  y · e  �  − 2δ(1)  λ(1)  �  dx⊗2  e  �  − δ(1)  λ(1)  �  x  = e  �  − δ(1)  λ(1) − δ(2)  λ(2)  �  vP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Siminarly in the case where P is amphidrome, we have the desired result from (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Based on this discussion, we define the local coordinates at p of Dϵ(σ) by noticing that  k = �¯k  i=1 m(Pi) and 3g − 3 − k = �¯r  i=1 miqi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The system of Harris–Mumford coordinates (z1, · · · , z3g−3) of Dϵ(σ) at  p = [S, w] ∈ T �µ  σ is defined by the following:  (i) p = {(z1, · · · , z3g−3) = (0, · · · , 0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) We rewrite it by using the lexicographic order with respect to i, j, α, β, γ as  (z1, · · · , z3g−3) = (z(0)  1 , · · · , z(j)  i , · · · , z(m(Pk)−1)  ¯k  , z(0)  1,1, · · · , z(γ)  α,β, · · · , z(m¯r−1)  ¯r,q¯r  )  (89)  for 1 ≤ i ≤ ¯k, 0 ≤ j ≤ m(Pi) − 1, 1 ≤ α ≤ ¯r, 1 ≤ β ≤ qα, 0 ≤ γ ≤ mα − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the  coordinate z(j)  i  is the dual vector of ϕj(vPi) where vPi is given in (88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The coordinate z(γ)  α,β  is the dual vector of ϕγ(vα,β) which is given in (87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  For the coordinates (89) on B ⊂ Dϵ(σ) near p, the action of �µ on B is written as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='23 is obvious from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='21 and (87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The action of �µ on B near p is written via the coordinates (89) as  �µ : z(j)  i  �−→ z(j+1)  i  = e  � 1  N  � δ(1)  λ(1)(Pi) + δ(2)  λ(2)(Pi)  ��  z(j)  i , z(γ)  α,β �−→ z(γ+1)  α,β  = e  �  −θα,β  N  �  z(γ)  α,β  by identifying z(m(Pi))  i  = z(0)  i  and z(mα)  α,β  = z(0)  α,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  49  6  Monodromy and orbifold moduli maps of degener-  ations of Riemann surfaces  We discuss fundamental properties of the monodromy and the moduli maps of degenera-  tion of Riemann surfaces of genus g ≥ 2 from several points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, we study pseudo-periodic maps of negative twist µ, whose totality is denoted  by P(−)  g  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Our interest in P(−)  g  comes from the fact that the topological monodromy of  a degeneration of a Riemann surface belongs to this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In this subsection, we show  that µ is obtained from the lifting via the real blow-up of an analytic automorphism µan  of a stable curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' It follows that the fundamental invariants of µ ([61], [55]) essentially  come from those of µan except for the screw numbers ([55, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4]), which express  the fractional Dehn twists along the exceptional circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We also review the notion of  generalized quotient space Σg/µ consisting of cores and non-core components constructed  in [55, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This gives important information about the central fiber of a normally  minimal degeneration whose topological monodormy coincides with µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2, we define the orbifold model f : S → ∆ of a degeneration by contracting  the non-core components of the central fiber which is topologically identified with the  generalized quotient ([55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Here S is a normal complex space with at most quotient  singularities such that the orbifold structure of f is explicitly induced from the precise  stable reduction �f : S → �∆ given in [10, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Note that this type of orbifold model  historically originates from Imayoshi [39] from the viewpoint of Teichm¨uller theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The discussion in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3 is the main part of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We define the notion of orbifold  moduli map Jf : ∆ → M  orb  g  for an orbifold model f of a degeneration, and show that  Jf has the Kodaira-periodicity property in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For an elliptic fibration,  Kodaira [46] defined the functional invariant J as the map from the base to the upper  half-plane H (=Teichm˝uller space of g = 1) by means of the elliptic modular function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Since the monodormy in SL(2, Z) (=the mapping class group of g = 1) acts on J around  a degenerate fiber, the local expression of J has a certain “periodicity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This property is  extended to g ≥ 2 as the orbifold moduli maps Jf and the action of the Weyl groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4, we give two examples of degenerations of Riemann surfaces and their orbifold  structures together with their orbifold moduli maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1  Pseudo-periodic maps and automorphisms of stable curves  We compare a pseudo-periodic map of negative twist with an automorphism of a stable  curve via the lifting given in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For the basic terminologies, see [55], [10, §1], [40, §5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The isotopy class of an oriented homeomorphism µ : Σg → Σg is called a pseudo-  periodic map of negative twist with respect to a simplex σ = ⟨C1, · · · , Ck⟩ (of Harvey’s  50  curve complex, [33]) if  (i) µ preserves σ, and the restriction µ|B to the complement B = Σg \\ �  1≤i≤k Ci is a  periodic map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' a certain power (µ|B)N is isotopic to the identity map idB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) Let m(Ci) (1 ≤ i ≤ k) be the minimal natural number such that µm(Ci)(−→  Ci) = −→  Ci as  oriented curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then µm(Ci) acts on an annular neighborhood Ai of Ci as a right-handed  fractional Dehn twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Decomposing B = �r  i=1 Bi into connected components, we call Bi a body component  ([55, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8]), see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1 of [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We also call the minimal natural number N with  the property stated in (i) the pseudo-period of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The map µ with the property (i) is  called a pseudo-periodic map ([55, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1]), or in Bers’ terminology, a map of elliptic or  parabolic type (see [40, §3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We denote the subset of Γg consisting of pseudo-periodic  maps of negative twist with respect to σ by  P−  g (σ) ⊂ Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (90)  The set of congujacy classes of (90) is denoted by �  P−  g (σ) ⊂ �  Γg, which is characterized as  follows:  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ([61], [55]) An element µ ∈ �  P−  g (σ) is uniquely determined by  (a) the Nielsen valencies at the multiple points and at the boundary curves of {Bi},  (b) the screw numbers {s(Cj)} of {Aj} where Aj is a small annular neighborhood of Cj  such that |s(Cj)| (s(Cj) ≤ 0) is the fractional Dehn twist of Aj,  (c) the action of µ on the extended partition graph Γ(µ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the one-dimensional oriented  graph whose vertices correspond to {Bi} and whose edges correspond naturally to {Cj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The invariants (a) and (b) are due to [61], and (c) is due to [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call them the nu-  merical data (a), (b), (c) of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The pseudo-periodic map and the analytic automorphism  of a stable curve are related as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let µ : Σg → Σg be an element of P−  g (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then there exists an analytic  automorphism µσ : Σg(σ) → Σg(σ) with respect to some complex structure on the stable  curve Σg(σ) such that the following diagram is isotopically commutative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Σg  Σg(σ)  Σg  Σg(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  contσ  contσ  µσ  µ  Conversely, any analytic automorphism µσ of a stable curve Σg(σ) is lifted to an element  µ of P−  g (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  51  Proof (Step 1) If σ = ∅, then µ is a periodic map, and it is known that µ is isotopic  to an analytic automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This fact is a corollary of Kerchhoff’s theorem [45], or this  isotopy is explicitly constructed, see [60] or [55, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  If σ ̸= ∅, µ is isotopic on B = Σg\\�  1≤i≤k Ai to a direct sum of analytic automorphisms  of Riemann surfaces with real boundaries, and the restrictions to � Ai could shrink to  point maps (in Σg(σ)), just as in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Since Σg(σ) is constructed from Σg by  shrinking � Ai to nodes via isotopy, the analytic automorphism on B descends to an  analytic automorphism of Σg(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (Step 2)  Conversely, let µσ : Σg(σ) → Σg(σ) be an analytic automorphism with  respect to some complex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Assume that Pi = cont(Ci) is a non-amphidrome node with co-valencies δ(1)/λ(1)  and δ(2)/λ(2) at the disk neighborhoods U (1) and U (2) of the local components of both  sides, where Pi = U (1) ∩ U (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  By definition, µm(Pi)  σ  preserves U (j) and rotates it by  the angle 2πδ(j)/λ(j) clockwise, within the view from the insides of both components  (j = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let π(j)  P : �U (j) → U (j) be the real blowing up at Pi with the exceptional circle  C(j) = (π(j)  P )−1(P), and A a small annulus with boundary ∂A = ∂A(1) � ∂A(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We construct a part of a Riemann surface (Σg)loc = �U (1) ∪ A ∪ �U (2) from the building  blocks �U (1), �U (2) and A by pasting C(j) to ∂A(j) naturally, and define a homeomorphism  µloc : (Σg)loc → (Σg)loc which is a local lift of µm(Pi)  σ  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The restriction µloc|�U(j) :  �U (j) → �U (j) is defined to be the lifting of µm(Pi)  σ  via π(j)  P , and µloc|A : A → A is defined  to be the fractional Dehn twist, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the linear twist ([55, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3]) with screw number [55,  Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4]  s(Ci) = − δ(1)  i  λ(1)  i  − δ(2)  i  λ(2)  i  − K(Ci)  (K(Ci) ∈ Z, K(Ci) ≥ −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (91)  Since −δ(2)/λ(2) ≡ δ(1)/λ(1) + s(Ci) (mod Z), the map µloc is well-defined and expresses a  right-handed fractional Dehn twist with s(Ci) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  When Pi is an amphidrome node with co-valency δ/λ of the local components of both  sides, then by the same notations as above we define the homeomorphism µloc : (Σg)loc →  (Σg)loc which is the local lift of µm(Pi)/2  σ  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The restrictions µloc|�U(1) : �U (1) → �U (2)  and µloc|�U(2) : �U (2) → �U (1) are defined to be the liftings of µm(Pi)/2  σ  via π(1)  P  and π(2)  P ,  and µloc|A : A → A is defined to be the special twist ([55, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4]) which interchanges the  boundary components ∂A(1) and ∂A(2) with screw number s(Ci)/2, where  s(Ci) = −2δi  λi  − 2K(Ci)  (K(Ci) ∈ Z, K(Ci) ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (92)  The restrictions of µσ to parts of Σg(σ), other than the disk neighborhoods of the  nodes, have trivial liftings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Following the action of µσ on the dual graph of Σg(σ), we  52  patch these parts of Riemann surfaces and the homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then we obtain Σg and  the desired homeomorphism µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Note that (91) and (92) are the screw numbers in the data (b) of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By  the descent from µ to µσ, the data (a) and (c) are preserved, while the data (b) vanish by  this descent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the screw numbers are not the data of the analytic automorphism of the  stable curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (The screw numbers express essentially the fractional Dehn twist coefficients  along the exceptional circles, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Liu [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=')  With respect to this complex structure on Σg(σ), there exists a quotient holomorphic  map πµσ : Σg(σ) −→ Σg(σ)/Gµσ (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This quotient map is topologically lifted to  the generalized quotient map [55, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4]  πµ : Σg(σ) −→ W(µ) = Σg/⟨µ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (93)  Here W(µ) is a non-reduced nodal Riemann surface in general (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' a multiplicity is at-  tached to each component) and πµ is a pinched covering ([55], Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  a finite  unramified topological covering except over nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Each connected component of the  inverse image of a node is homeomorphic to a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The space W(µ) is decomposed into  the parts ([55], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='95, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  W(µ) =  �  i  Corei +  �  j  Tailj +  �  j  Archj +  �  j  Quasitailj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (94)  The orbit µα(Bi) (0 ≤ α ≤ m(Bi) − 1) of Bi is mapped to Corei by πµ except for small  disk neighborhoods of multiple points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The parts Tailj, Archj, Quasitailj are chains or  trees of P1’s which are the images under πµ of the neighborhoods of multiple points,  non-amphidrome annuli and amphidrome annuli, respectively, and these multiplicities  are explicitly determined from the data (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let cont(nc) : W(µ) → W(µ)♯ be the contraction of the non-cores (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' all the Tails,  Archs and Quasitails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then W(µ)♯ can be identified with Σg(σ)/Gµσ (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='14),  in other words, the following homotopically commutative diagram is the lifting of πµσ to  πµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Σg  Σg(σ)  W(µ)  W(µ)♯ ∼= Σg/Gµσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  contσ  cont(nc)  πµσ  πµ  (Diagram I) The lifting of the analytic quotient to the generalized quotient  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2  Orbifold structures of degenerations of Riemann surfaces  In this subsection, we propose the notion of orbifold model of a degeneration of Riemann  surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  53  Let M be a 2-dimensional normal complex space and f : M → ∆ = {t ∈ C | |t| ≤ ϵ0}  a proper surjective holomorphic map to a disc with a sufficiently small radius such that  any fiber f −1(t) over t ∈ ∆∗ = ∆ \\ {0} is a Riemann surface of genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call f a  degeneration of genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We always assume g ≥ 2 unless otherwise mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  First we assume that M is nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let F = f −1(0) = �r0  i=1 miFi be the irre-  ducible decomposition of the central fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If the reduced scheme F red = �r0  i=1 Fi has  at most nodal singularities such that any (−1)-spherical component of F red has at least  three intersection points with other components, then f is called the normally minimal  model which is uniquely determined in the local birational equivalence class of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Two degenerations of genus g are topologically (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' analytically) equivalent if, with  their normally minimal models f : M → ∆ and f ′ : M ′ → ∆′, there exist oriented  homeomorphisms (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' analytic isomorphisms) hM : M → M ′ and h∆ : ∆ → ∆′ such  that h∆ ◦ f = f ′ ◦ hM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The equivalence class is called the topological type (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the  analytic type) of the degeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Now we fix a point t0 ∈ ∆∗, and let w : Σg → f −1(t0) be a Teichm¨uller marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We  choose a smooth loop Lt0t0 in ∆∗ which starts from t0, goes around 0 ∈ ∆ once counter-  clockwise and comes back to t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let µ(Lt0t0) : f −1(t0) → f −1(t0) be the diffeomorphism  along Lt0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then w−1 ◦ µ(Lt0t0) ◦ w : Σg → Σg is an oriented homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For  another point t′  0 ∈ ∆∗, we define the Teichm¨uller marking of f −1(t′  0) coming from w, by  w′ = µ(Lt0t′  0) ◦ w : Σg −→ f −1(t′  0) where µ(Lt0t′  0) : f −1(t0) → f −1(t′  0) is the diffeomor-  phism along a path Lt0t′  0 in ∆∗ connecting t0 and t′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the map w′−1 ◦ µ(Lt′  0t′  0) ◦ w′ is  conjugate to w−1 ◦ µ(Lt0t0) ◦ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In other words, the conjugacy class of  µf(w) = [w−1 ◦ µ(Lt0t0) ◦ w]  (95)  is well-defined, independently of the choices of t0 and Lt0t′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The proof goes as follows: We take a “straight” path L′  t0t′  0 in ∆∗ connecting t0 and  t′  0, and we suppose that the loop L′  t0t′  0Lt′  0t′  0L′  t′  0t0 is isotopic to Lt0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We will denote the  diffeomorphisms µ(Lt0t′  0) and µ(L′  t0t′  0) by a and b : f −1(t0) → f −1(t′  0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then  by the asumption on L′  t0t′  0, we have b−1µ(Lt′  0t′  0)b = µ(Lt0t0), and  [w′−1 ◦ µ(Lt′  0t′  0) ◦ w′]  = [w−1a−1 ◦ µ(Lt′  0t′  0) ◦ aw]  = [w−1a−1bb−1 ◦ µ(Lt′  0t′  0) ◦ bb−1aw]  = [w−1a−1b ◦ (b−1µ(Lt′  0t′  0)b) ◦ b−1aw]  = [w−1a−1b ◦ µ(Lt0t0) ◦ b−1aw]  = [w−1(a−1b)ww−1 ◦ µ(Lt0t0) ◦ ww−1(b−1a)w]  = [c−1w−1 ◦ µ(Lt0t0) ◦ wc]  (96)  54  where we put c = w−1(b−1a)w : Σg → Σg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The equation (96) shows that the cojugacy  classes [w′−1 ◦ µ(Lt′  0t′  0) ◦ w′] and [w−1 ◦ µ(Lt0t0) ◦ w] coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call this conjugacy class  µf(w) the topological monodromy of f with respect to the marking w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Note that µf(w) is  determined by the fibering structure of f −1(∆∗) → ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  For the same complex structure of f −1(t0), we choose another Teichm¨uller marking  �w : Σg → f −1(t0) and consider the topological monodormy µf( �w) with respect to �w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then  µf(w) and µf( �w) are obviously conjugate to each other in Γg, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  µf( �w) = [w−1 ◦ �w]−1µf(w)[w−1 ◦ �w] ∈ Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Therefore, the conjugacy class �  µf(w) of µf(w) is uniquely determined by f, independently  of the choice of the marking w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We write  µf = �  µf(w) ∈ �Γg (conjugacy classes of Γg),  (97)  and call µf the topological monodromy of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then:  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (i) ([20], [2], [39], [65], [23]) The topological monodromy belongs to the  conjugacy classes of pseudo-periodic maps of negative twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) ([55, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2]) The topological structure of a degeneration of genus g is uniquely de-  termined by its topological monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The set of topological monodromies and the set of  conjugacy classes of pseudo-periodic maps of negative twist correspond bijectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The relation between the normally minimal model and the generalized quotient (93)  of the topological monodormy µf is the following:  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ([55, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9]) The central fiber F = f −1(0) of the normally minimal  model f : M → ∆ of a degeneration of genus g and the generalized quotient space W(µf)  of µf topologically coincide with each other as non-reduced nodal Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In  paticular, F is decomposed into cores and non-cores as in (94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Now we change the model in the birational equivalence class of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since any proper  subset of irreducible components of the central fiber F of the normally minimal model f  has negative intersection form, there exists the analytic contraction map τ : M → M ♯ of  the non-cores of F by Grauert’s theorem [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let f ♯ : M ♯ → ∆ be the natural map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The  restriction of τ to F coincides with the contraction map  τ|F = cont(nc) : F −→ F ♯ = (f ♯)−1(0)  given in Diagram I of §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The total space M ♯ is a normal complex space with cyclic and  dihedral quotient singularities whose supports are on the contraction points of the non-  cores, and the types of singularities are explicitly given by the data of µf ([10, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1])  55  such that the non-cores in F are the exceptional set of the minimal resolution of these  singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since M ♯ has at most quotient singularities, we may consider it as a complex  orbifold in the sense of Satake [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call f ♯ the orbifold model of the degeneration f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The explicit orbifold structure of f ♯ is described as follows ([10, §§2,3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let π : �∆ → ∆  be the covering map of disks defined by u �→ t = uN where N is the pseudo-period of µf,  and let �  M be the normalization of the fiber product of M ♯ and �∆ over ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let �f : �  M → �∆  be the natural map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the covering transformation group G := ⟨�µ⟩ ≃ Z/NZ acting  on �f −1(�∆∗) → �∆∗ = �∆ \\ {0} is extended holomorphically over �f by the normality of �  M  such that the following commutative diagram holds:  ∆ ≃ �∆/G  M ♯ ≃ �  M/G  M  f  �  M  �∆  �f  π  �π  τ  f ♯  (Diagram II) The process of the precise stable reduction  Since the topological monodromy µ �f ≃ µN  f is trivial on the body B (as defined before  Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1), the non-cores of the generalized quotient W(µ �f) consists of archs ([55, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7])  of multiplicity 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' W(µ �f) is a semi-stable curve of genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the central fiber  �F = �f −1(0) is a stable curve with the topological type Σg(σ) such that �  M has rational  double points of type A at the nodes of �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' From the viewpoint of the semi-stable curve  W(µ �f), �F is the image of the contraction of the archs on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The restriction to the fiber  �π| �F : �F → F ♯ of the map �π in Diagram II essentially coincides with the quotient map  πµσ in Diagram I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call �f the precise stable reduction of f ([10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This stable reduction  is minimal in the sense that the degree N of the base change is minimal among all the  stable reductions of f, because the generalized quotient for µn  f with n < N cannot be a  semi-stable curve of genus g by the algorithms given in [55, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The action of the generator �µ of G on �f near �F is an extension of the automorphism  �µ| �F : �F → �F of the stable curve �F to that of the family �f, and is locally described as  follows:  (i) Assume that P is a nonsingular point of �F which belongs to an irreducible compo-  nent �Fi with StabG( �Fi) = ⟨�µm( �Fi)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The automorphism �µm( �Fi)| �Fi : �Fi → �Fi of order n( �Fi)  and the associated n( �Fi)-fold cyclic covering �π| �Fi : �Fi → �π( �Fi) ⊂ F ♯ are given in (66) and  (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let (x, u) be the local coordinates at P = {(x, u) = (0, 0)} of �  M such that x is the  fiber coordinate and u is the lift of the base coordinate �∆ = {u ∈ C | |u| ≤ ϵ1/N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then  the map �µm( �Fi) locally acts as  �µm( �Fi) : (x, u) �−→  �  �µm( �Fi)| �Fi(x), e  �  m( �Fi)  N  �  u  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (98)  56  Moreover if P is the ramification point of �π| �Fi with co-valency δ/λ, then StabG(P) =  ⟨�µm( �Fi)n( �Fi)/λ⟩ and the map �µm( �Fi)n( �Fi)/λ acts near P as  �µm( �Fi)n( �Fi)/λ : (x, u) �−→  �  e  � δ  λ  �  x, e  �  m( �Fi)n( �Fi)  λN  �  u  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (99)  (ii) Assume P is a non-amphidrome node of �F with StabG(P) = ⟨�µm(P)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By (91),  the screw number at the curve CP = Contσ  −1(P) via the map Contσ : Σg → Σg(σ) ≃ �F  is given by s(CP) = −δ(1)/λ(1) − δ(2)/λ(2) − K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The monodromy map µ �f ≃ µN  f behaves  on an anular neighborhood of C as the right-handed integral Dehn twist of times  n(P) := N|s(CP)|  m(P)  = ℓ(δ(1)λ(2) + δ(2)λ(1) + Kλ(1)λ(2))  gcd(λ(1), λ(2))  ,  (100)  where ℓ = N/  �  lcm(λ(1), λ(2))m(P)  �  ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The restriction of this map to the tubular neigh-  borhood of P gives the Milnor fibration of the singularity P whose monodormy map is  the one described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This means that �  M is defined locally near P by the equation  xy = un(P) where x and y are local parameters of the components of both sides of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Here n(P) coincides with the Milnor number of the singularity P of �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The map �µm(P)  is given locally near P from (71) by  �µm(P) : (x, y, u) �−→  �  e  � δ(1)  λ(1)  �  x, e  � δ(2)  λ(2)  �  y, e  �m(P)  N  �  u  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (101)  (iii) Assume P is an amphidrome node with StabG(P) = ⟨�µm(P)/2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The screw number  is given by s(CP) = −2(δ/λ) − 2K from (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then �  M is defined locally near P by the  equation xy = un(P) such that the Milnor number n(P) is given by  n(P) = N|s(CP)|  m(P)  = 2ℓ  �  δ + Kλ  �  ,  (102)  where ℓ = N/(λm(P)) ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The map �µm(P)/2 is locally given from (72) by  �µm(P)/2 : (x, y, u) �−→  �  e  � δ  2λ  �  y, e  � δ  2λ  �  x, e  �m(P)  2N  �  u  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (103)  Depending on the above arguments, we define the following;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The marked orbifold structure of a degeneration f is defined by;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (i) the set consisting of the orbifold model, the precise stable reduction and the group  { �f : �  M → �∆, f ♯ : M ♯ → ∆, G = Z/NZ}  (104)  satisfying the Diagram II so that the action of G on �f is locally induced from (98) ∼  (103), and  57  (ii) the lifting of the Weyl marking of the stable curve �F = �f −1(0) given by  �w = cont(nc) ◦ πµ �  f : Σg −→ W(µ �f) −→ �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (105)  We sometimes write this marked orbifold structure by f orb = { �f, f ♯, G, �w} for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (i) Since �  M has singularities, ( �f, G) does not define directly the complex  orbifold structure on f ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' However this point is easily recovered by Takamura’s method  [66]: the actions (101) and (103) near the nodes are lifted to the local linear actions of  type A of C2 explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence the orbifold charts of M ♯ in the neighborhoods of the points  of the contraction of the non-cores are defined as the open neighborhoods at the origin  of C2 and the above lifted actions (see also [10, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For other orbifold charts of M ♯,  we could use the open sets of �  M and the restricted actions of G on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In conclusion,  f ♯ : M ♯ −→ ∆ has the structure of an orbifold fibration (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) The marking �w (105) trivially decends to the Weyl marking  w : Σg(σ) −→ �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (106)  If one compares the marking �w with w, �w is determined through the lifting Σg → Σg(σ)  which includes the data of the screw numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This is the reason why we use �w instead of  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Note that this type of marking for a stable curve is used by Hubbard–Koch [37, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1]  for the construction of the augmented Teichm¨uller space (see also [9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='490]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3  Local orbifold moduli maps and Kodaira-periodicity  In this subsection, we define the local orbifold moduli map of a degenetaion and show that  it has the Kodaira-periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  For a given degeneration f : M → ∆ of genus g with topological monodormy µf ∈  P(−)  g  (σ), the restricted holomorphic family f −1(∆∗) → ∆∗ induces the moduli map ∆∗ →  Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the classical stable reduction theorem and the valuative criterion algebraically,  or by Imayoshi [39, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2,Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4] analytically, this map has a holomorphic extension to the  Deligne–Mumford compactification  Jf : ∆ −→ M g,  (107)  which is called the canonically extended moduli map ([58]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This map is determined by  f −1(∆∗) → ∆∗, and is independent of the choice of the local birational model of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Now we consider the marked orbifold structure f orb = { �f, f ♯, G, �w} of f in Definition  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5 and the orbifold M  orb  g  = {(Dϵ(σ), W(σ), ϕσ, Mϵ(σ))}σ∈Cg/Γg in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The map Jf is  lifted to the map of these spaces as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the descent of the marking �w to w by (106),  we consider the σ-Weyl marked stable curve [ �F, w] ( �F = �f −1(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since the generator  58  �µ of G induces an analytic automorphism of �F, the point p = [ �F, w] is contained in the  equisymmetric strata T �µ  σ in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='18:  p = [ �F, w] ∈ T �µ  σ ⊂ T(σ) ⊂ Dϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (108)  We consider the universal family π : Y  orb  g  −→ M  orb  g  of (44), (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4, there  exists an open neighborhood B of p in Dϵ(σ) such that the restricted family πX = πσ|X :  X = π−1  σ (B) ⊂ Xϵ(σ) −→ B ⊂ Dϵ(σ) is a standard Kuranishi family of �F with marking  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since �f : �  M → �∆ is a local deformation of �F, it follows from the universality of the  Kuranishi family that there uniquely exists a holomorphic map  �J �f : �∆ −→ B  (109)  such that �f is the pull back of πX by �J �f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then we have the commutative diagram  �∆  ∆  �J �f(�∆) ⊂  B  ⊂ Dϵ(σ)  Jf(∆) ⊂ ϕσ(B) ⊂ Mϵ(σ) ⊂ M g  π  ϕσ,�∆  ϕσ  �J �f  Jf  (Diagram III) The local orbifold moduli map  where ϕσ,�∆ is the restriction to �J �f(�∆) of the orbifold structure map ϕσ : Dϵ(σ) −→  Mϵ(σ) = Dϵ(σ)/W(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' More precisely, the space B in Diagram III is really the Kuranishi  space with Weyl marking (B, w) and the image of the restriction of the forgetting map  (B, w) → B → ϕσ(B) is nothing but the quotient of B by G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ϕσ(B) = B/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the  construction of �f in Diagram II in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2, the group G also acts on the analytic subspace  �J �f(�∆) of B such that Jf(∆) = �J �f(�∆)/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In this sense, �J �f is the lifting of Jf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Considering  these points, we define the following:  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For a degeneration f : M → ∆ of Riemann surfaces with topological  monodormy µf ∈ P(−)  g  (σ) and orbifold structure f orb = { �f, f ♯, G, �w}, we define the local  orbifold moduli map by  { �J �f : �∆ → Dϵ(σ), Jf : ∆ → Mϵ(σ), G}  (110)  which satisfies Diagram III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For simplicity, we sometimes write (110) merely by �J �f : �∆ →  Dϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In order to describe �J �f : �∆ → Dϵ(σ) explicitly, we will define a special class of  holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  59  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A holomorphic function ϕ(t) of a variable t at the origin in C is called  a pseudo-periodic function with multiplicity γ ∈ N and period L ∈ N if there exists a  holomorphic function �ϕ(t) = �∞  i=0 citi (ci ∈ C) with  ϕ(t) = tγ �ϕ(tL) = c0tγ + c1tγ+L + c2tγ+2L + · · ·  (c0 ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (111)  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The notion of pseudo-periodic function in this framework is inspired by  Kodaira [46, §8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' As we already explained, the functional invariant of a degeneration of  elliptic curves defined in [46, §7] is the orbifold moduli map in our terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The func-  tional invariant belongs to the class of pseudo-periodc functions in the present terminology  around each degenerate fiber germ in [46, §8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Pseudo-periodic functions are characterized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let ϕ(t) be a holomorphic function at the origin with ϕ(0) = 0 and ϕ(t) ̸≡  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the following conditions (i) and (ii) are equivalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (i) ϕ(t) is a pseudo-periodic function with multiplicity γ and period L,  (ii) ϕ(t) admits an action C → C given by t �→ e(1/L)t with the charactor e(γ/L) such  that the derivations of ϕ(t) at the origin satisfy the following;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  ϕ  �  e  � 1  L  �  t  �  = e  � γ  L  �  ϕ(t),  ϕ(γ)(0) ̸= 0, ϕ(j)(0) = 0 for ∀j < γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (112)  Proof  We assume (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' From the expression (111) of ϕ(t), we have  ϕ  �  e  � 1  L  �  t  �  = c0e  � γ  L  �  tγ +  �  i≥1  cie  �γ + iL  L  �  tγ+iL = e  � γ  L  �  ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Since c0 ̸= 0, the conditions of the derivatives in (ii) are also satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Conversely we assume (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We decompose ϕ(t) into  ϕ(t) = ϕ1(t) + ϕ2(t),  where ϕ1(t) =  �  i≡γ mod L  citi,  ϕ2(t) =  �  j̸≡γ mod L  cjtj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then we have  ϕ  �  e  � 1  L  �  t  �  = e  � γ  L  �  ϕ1(t) +  �  j̸≡γ mod L  cje  � j  L  �  tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Thus the condition (ii) says that  cje  � j  L  �  = cje  � γ  L  �  , for ∀j ̸≡ γ (mod L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  This means that cj = 0, ∀j ̸≡ γ (mod L), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ϕ2(t) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence we have ϕ(t) = ϕ1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By  the conditions of the derivatives in (ii), the leading term of ϕ1(t) should be the non-zero  constant multiple of tγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Thus we obtain (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  60  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let ϕ(t) be a pseudo-periodic function with multiplicity γ and the period  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let γ′ and K be integers which satify γ = γ′ + KL, K ≥ 0,  0 ≤ γ′ ≤ L − 1,  γ′ ≡ γ (mod L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We define the analytic screw number of ϕ(t) by  s(ϕ) = γ  L = γ′  L + K,  (113)  and call K and γ′/L the integral term and the fractional term of s(ϕ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The geometric meaning of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='11 will be clarified by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='12 (iv) and Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Now we express the map �J �f : �∆ → Dϵ(σ) in (110) around p = [ �F, w] ∈ T �µ  σ explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  From the discussions in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6, the orbit-irreducible decomposition �F = �¯r  i=1  �m( �Fi)−1  j=0  �µj( �Fi)  and the decomposition �¯k  i=1  �m(Pi)−1  j=0  �µj(Pi) of nodes of �F induce the Harris–Mumford  coordinates (z(0)  1 , · · · , z(j)  i , · · · , z(γ)  α,β, · · · , z(m( �F¯r)−1)  ¯r,q¯r  ) of Dϵ(σ) around p as in (89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the  map �J �f is expressed by 3g−3 holomorphic functions ϕ(j)  i (u) (1 ≤ i ≤ ¯k, 0 ≤ j ≤ m(Pi)−1),  ψ(γ)  α,β(u) (1 ≤ α ≤ ¯r, 1 ≤ β ≤ qα, 0 ≤ γ ≤ m( �Fα) − 1) with ϕ(j)  i (0) = ψ(γ)  α,β(0) = 0 sending  to each of these coordinates as  �J �f :  z(j)  i  = ϕ(j)  i (u),  z(γ)  α,β = ψ(γ)  α,β(u)  (u ∈ �∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (114)  The following theorem is an extension of Kodaira’s result from the viewpoint of Remark  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let ( �J �f : �∆ → Dϵ(σ), G) be the chart of the orbifold moduli map (110)  of the marked orbifold structure (104),(105) of a degeneration f : M → ∆ of genus g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let ϕ(j)  i (u), ψ(γ)  α,β(u) be the holomorphic functions in (114) expressing �J �f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then  (i) ϕ(0)  i (u) is a pseudo-periodic function with period N/m(Pi) and with multiplicity  n(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Here n(Pi) is the Milnor number described in (100) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (102)) in the case  where Pi is a non-amphidrome node (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' an amphidrome node).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) If ψ(0)  α,β(u) ̸≡ 0, then ψ(0)  α,β(u) is a pseudo-periodic function with period N/m( �Fα)  and multiplicity  γα,β :=  �  n( �Fα) − θα,β  n( �Fα)  + Kα,β  �  N  m( �Fα)  (115)  where Kα,β is a non-negative integer and θα,β/n( �Fα) is given in (87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (iii) We have ϕ(j)  i (u) = ϕ(0)  i (u) for any j, and ψ(γ)  α,β(u) = ψ(0)  α,β(u) for any γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (iv) The analytic screw numbers of the pseudo-periodic functions ϕ(j)  i (u) and ψ(γ)  α,β(u)  are given by  s(ϕ(j)  i ) = |s(CPi)|,  s(ψ(γ)  α,β) = n( �Fα) − θα,β  n( �Fα)  + Kα,β  (116)  for any j and γ, where |s(CPi)| is the absolute value of the screw number of the cut curve  CPi given in (91) and (100), or (92) and (102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  61  Proof  We prove (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' First we assume that Pi is a non-amphidrome node of �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We  set L = N/m(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since z(0)  i  is the dual vector of the generator vPi of the torsion sheaf  τPi, it follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='21 and (100) that �µm(Pi) acts on z(0)  i  by  �µm(Pi) : z(0)  i  �−→ e  � δ(1)  λ(1) + δ(2)  λ(2)  �  z(0)  i  = e  �n(Pi)  L  �  z(0)  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (117)  Since �µ(�∆) is naturally the G-invariant subspace in H0( �F, Ω1  �F ⊗ ω �F)∗ ⊂ B ⊂ Dϵ(σ)  and �µm(Pi) acts on �∆ by u �→ e(1/L)u, the function ϕ(0)  i (u) satisfies ϕ(0)  i (e(1/L)u) =  e  �  n(Pi)/L  �  ϕ(0)  i (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10, ϕ(0)  i (u) is a pseudo-periodic function with  period L, and its multiplicity is congruent to n(Pi) modulo L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  On the other hand, as we explained in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1 for the comment (II) of (20), the Kuranishi  family of �F has a local structure {xy = t|(x, y, t) ∈ C3, |x|, |y|, |t| < 1} at a node ([9,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='184–186]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' it is a plumbing variety (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='[47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since �  M has singularity xy = un(Pi)  at Pi and �f : �  M → �∆ should be pulled back from the Kuranishi family by �J �f, the map  �J �f is locally given by  �J �f : u �→ t = un(Pi)ψ(u)  where ψ(u) is a holomorphic function with ψ(0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Therefore the multiplicity of ϕ(0)  i (u) exactly coincides with n(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Hence we obtain the assertion (i) for non-amphidrome case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The case where Pi is an  amphidrome node, the discussion is similar and is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We prove (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We consider a component �Fα and set L′ = N/m( �Fα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since z(0)  α,β is the  dual vector of vα,β in (87), the map �µm( �Fα) acts by  �µm( �Fα) : z(0)  α,β �−→ e  �  − θα,β  n( �Fα)  �  z(0)  α,β = e  �  1  L′  �  −θα,βL′  n( �Fα)  ��  z(0)  α,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Therefore we similarly obtain the assertion (ii) by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Since the iterations of the map �µ permute cyclically and isomorphically each neigh-  borhood of the orbit of a node and also each neighborhood of the orbit of a component  of �F, the assertion (iii) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The assertion (iv) is clear from (i), (ii), (iii) and Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In the property (ii) of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='12, the trivial function ψ(0)  α,β(u) ≡ 0 may  occure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In particular, in the case where ¯k = 0 and ψ(0)  1,β(u) ≡ 0 for all 1 ≤ β ≤ 3g − 3, �J �f  is the constant map �J �f(�∆) = p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' �  M ≃ �F × �∆ and �f is the projection �F × �∆ −→ �∆  such that f is obtained from the resolution of ( �F × �∆)/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The fractional term (n( �Fα) − θα,β)/n( �Fα) of the analytic screw number  s(ψ(γ)  α,β) in (116) is a topological invariant, because it is determined by the total valency  62  using Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' On the other hand, the integral term Kα,β of s(ψ(γ)  α,β) is not a topological  invariant, and is determined purely by the analytic structure of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By comparison with  the usual screw numbers (91) and (92), the number Kα,β seems to be an “analytic analog  of the number of integral Dehn twists”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' See also Kuno’s paper [49, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In a certain situation, Kα,β is related to the modular invariant ([63],[68]) of the fiber  germ (f, F) from the viewpoint of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' This point will be discussed in a forthcoming  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4  Examples of degenerations and their invariants  Here we give two examples of degenerations of Riemann surfaces and show their orbifold  structures and the properties of their orbifold moduli maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let µ : Σ8 −→ Σ8 be the element of P(−)  8 (σ) with σ = ⟨C1, · · · , C5⟩ and  the pseudo-period N = 84 described in Figure III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The total valencies of B = �4  i=1 Bi =  Σ8 \\ �5  j=1 Cj are (g = 3, ¯g = 0, n = 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 2/7 + 6/7 + 6/7) on B1, (g = 1, ¯g = 0, n =  4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 1/4 + 1/4 + 1/2) on B2, (g = 1, ¯g = 0, n = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 2/3 + 2/3 + 2/3) on B3 and B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Here  the valencies written by boldface are attached to the boundary curves and those by roman  are to the multiple points assigned by the cross symbols inside the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The action on the graph has order 2 with µ(B3) = B4, µ(C4) = C5 such that C1, C4,  C5 are non-amphidrome and C2, C3 are amphidrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The screw numbers are s(C1) =  −6/7 − 1/4 − K1 (K1 ≥ −1), s(C2) = −2(2/3 + K2) (K2 ≥ 0), s(C3) = −2(2/3 + K3)  (K3 ≥ 0), s(C4) = s(C5) = −1/2 − 2/3 − K4 (K4 ≥ −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let f : M −→ ∆ be the normally minimal model of a degeneration with topological  monodormy µf = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The central fiber F = f −1(0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the generalized quotient W(µf)  is given as in Figure IV by the algorithm in [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Here the circles mean P1’s and the  numbers in the circles mean their multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (The case of Ki = −1 for some i is  B1  C1  2  7 + 6  7 + 6  7  − 6  7 − 1  4 − K1  B2  C4  C5  1  4 + 1  4 + 1  2  − 1  2 − 2  3 − K4  − 1  2 − 2  3 − K4  B3  B4  C3  C2  µ  −2( 2  3 + K3)  2  3 + 2  3 + 2  3  2  3 + 2  3 + 2  3  −2( 2  3 + K2)  (Figure III) The data of the topological monodromy of Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='15  63  7  2  1  6  5  4  3  2  1  6 5 4 3 2 1  K1  1 1 4 2 2  K4  2 4  1  6  4 2  4 2  K2  2  2  K3  2  2  1  1  1  1  (Figure IV) The central fiber F of the normally minimal model of Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='15  omitted in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=') F has three core components of multiplicities 7, 4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  By the contraction M −→ M ♯ of the non-core of F, we obtain the orbifold model  f ♯ : M ♯ −→ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The fiber F ♯ = (f ♯)−1(0) is described in Figure V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Here M ♯ has five  isolated cyclic quotient singularities and two dihedral quotient singularities whose supports  are indicated by the cross symbols in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let �∆ −→ ∆ be the cover u �→ t = u84, and �  M be the normalization of M ♯ ×∆ �∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We  obtain the precise stable reduction �f : �  M −→ �∆, and the orbifold structure (f ♯, �f, G =  ⟨�µ⟩ ≃ Z/84Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The stable fiber �F = �f −1(0) is as in Figure V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The topological monodromy  µ �f ≃ µ84  f  is trivial on B and acts as ni = 84|s(Ci)|-right Dehn twist at an annular  neighborhood of Cj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  n1 = 84K1 + 93, n2 = 84K2 + 56, n3 = 84K3 + 56, n4 = n5 = 42K4 + 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (118)  The marking is defined by the composition w : Σ8 → Σ8(σ) ≃ W(µ �f) → �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let �F =  �2  i=1 �Fi + �1  j=0 �µj( �F3) be the orbit-irreducible decomposition such that w(Bi) = �Fi (1 ≤  i ≤ 3), w(B4) = �µ( �F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let P = �3  i=1 Pi + �1  j=0 �µ(P4) be the decomposion of the set of  nodes on �F such that w(Ci) = Pi (1 ≤ i ≤ 4) and w(C5) = �µ(P4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' �  M has the singularity  of type xy = uni for (118) at each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The action of G to �  M is not effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In fact,  the order of StabG( �F1) = G is 86 while the order of the automorphism �µ| �F1 of �F1 is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  64  Fi  4  42K2+48  A84Ks+55  F  F3  84 : 1  4  6  X  F#C M#~M/G  A84K1+92  A42K2+48A84K4+55  u(F3)  (Figure V)The central fibers of the precise stable reduction of Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='15We set V1 = H0( �F1, 2K �F1 + P1), V2 = H0( �F2, 2K �F2 + P1 + P4 + �µ(P4)), V3 =  H0( �F3, 2K �F3 +P2 +P3 +P4) and �µ(V3) = H0(�µ( �F3), 2K �F3 +P2 +P3 + �µ(P4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By Example  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6 and the similar argument using Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3, the log-quadratic characters are  Ch�µV1 =  �1  7, 2  7, 2  7, 3  7, 4  7, 5  7, 6  7  �  , Ch�µV2 =  �1  4, 1  2, 3  4  �  , Ch�µ2�µj(V3) =  �1  3, 1  3, 2  3  �  ,  (119)  for j = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The little Teichm¨uller space T(σ) is locally an open set of �2  i=1 Vi  �1  j=0 �µj(V3)  and dim T(σ) = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The equisymmetric strata T �µ  σ consists of a unique point p = [ �F, w],  since the 0-eigen space is 0-dimensional by (119) and T �µ  σ is connected by Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  (z1, z2, z3, z(0)  4 , z(1)  4 , z1,1, · · · , z1,7, z2,1, z2,2, z2,3, z(0)  3,1, z(0)  3,2, z(0)  3,3, z(1)  3,1, z(1)  3,2, z(1)  3,3)  (120)  be the system of Harris–Mumford coordinates at p on Dϵ(σ) which are ordered as the dual  vectors of the ordered torsion sheaf at the nodes and the eigenvectors corresponding to the  characters in (119).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' From (119), the multiplicities given in (115) are  γ1,j = 84K1,j + 12γ′  1,j, γ′  1,j = 6, 5, 5, 4, 3, 2, 1 (j = 1, 2, 3, 4, 5, 6, 7), K1,j ∈ Z≥0,  (121)  γ2,j = 84K1,j + 21γ′  2,j, γ′  2,j = 3, 2, 1 (j = 1, 2, 3), K2,j ∈ Z≥0,  (122)  γ3,j = 42K3,j + 14γ′  3,j, γ′  3,j = 2, 2, 1 (j = 1, 2, 3), K3,j ∈ Z≥0,  (123)  where the notations mean that γ′  1,1 = 6, γ′  1,2 = 5, · · · , γ′  3,3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' From Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='12, (120),  (118) and (121)∼(123), the the orbifold moduli map �J �f : �∆ → Dϵ(σ) (u ∈ ∆) has the  Kodaira-periodicity given by  zi =  ∞  �  k=0  ci,kuni+84k  (i = 1, 2, 3, ci,0 ̸= 0, ci,k ∈ C),  z(j)  4  =  ∞  �  k=0  c4,kun4+42k  (j = 0, 1, c(k)  4,0 ̸= 0, c4,k ∈ C),  zi,j =  ∞  �  k=0  ci,j,kuγi,j+84k  (i = 1, 2, j = 1, 2, 3, ci,j,k ∈ C),  z(j)  3,i =  ∞  �  k=0  c3,i,kuγ3,i+42k  (i = 1, 2, 3, j = 0, 1, c3,i,k ∈ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let µ : Σ5 −→ Σ5 be the element of P(−)  5 (σ) with σ = ⟨C1, · · · , C5⟩  and N = 15 as in Figure VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The total valencies of B = �5  i=0 Bi = Σ5 \\ �5  j=1 Cj are  (g = 0, ¯g = 0, n = 5, 2/5+3/5+1) on B0 and (g = 1, ¯g = 0, n = 3, 2/3+2/3+2/3) on Bi  (1 ≤ i ≤ 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The action on the graph has order 5 with permutations (B1, B3, B5, B2, B4),  (C1, C3, C5, C2, C4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the 2/5-turn in the terminology of [55, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='148] such that Cj is  non-amphidrome with the screw number s(Cj) = −2/3 − K (K ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  65  Let f : S −→ ∆ be the degeneration with topological monodormy µf = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The fiber  F = f −1(0) and the process to obtain its precise stable reduction �f : �S −→ �∆ is shown  in Figure VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Here �F = �f −1(0) = �5  i=0 �F (i) is the irreducible decomposition such that  �F (i) = �µi−1( �F (1)) (1 ≤ i ≤ 5) for the induced automorphism �µ : �F −→ �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then  g( �F (0)) = 0, g( �F (i)) = 1 and �S has A3K+1-singularity at the node Pi (1 ≤ i ≤ 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  15  10  5  10  5  10 5  5  K  5  3 1  2 1  contr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  15  5  15 : 1  P1  P2  P5  P3  P4  �F (1)  �F (2)  �F (5)  �F (3)  �F (4)  �F (0)  A3K+1(∀Pi)  2  5-turn  F ⊂ S  F ♯ ⊂ S♯  �F ⊂ �S  (Figure VII) The central fibers of the precise stable reduction of Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='16  We set V0 = H0( �F0, 2K �F0 + �5  i=1 Pi), V1 = H0( �F1, 2K �F1 + P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A calculation by using  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3 shows that the log-quadratic characters are  Ch�µV0 =  �1  5, 4  5  �  , Ch�µV1 =  �2  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (124)  Then T(σ) is locally an open set of V0  �4  j=0 �µj(V1) ≃ C7, and T �µ  σ consists of a unique  point p = [ �F, w] by (124).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' From (124), the multiplicities given in (115) are  γ0,1 = 15K0,1 + 12, γ0,2 = 15K0,2 + 3, γ1,1 = 3K1,1 + 1 (K0,1, K0,2, K1,1 ∈ Z≥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let (z(0)  1 , z(1)  1 , z(2)  1 , z(3)  1 , z(4)  1 , z0,1, z0,2, z(0)  1,1, z(1)  1,1, z(2)  1,1, z(3)  1,1, z(4)  1,1) be the system of Harris–Mumford  coordinates at p on Dϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the orbifold moduli map �J �f : �∆ → Dϵ(σ) (u ∈ ∆) has  the Kodaira-periodicity as  z(j)  1  =  ∞  �  k=0  c1,ku2+3K+3k (0 ≤ j ≤ 4, c1,0 ̸= 0, c1,k ∈ C),  66  (Bo  ("A)-  2-turn  Bo  网B1  /5  2  (Bi,i≠0)  3  3  B:  (FigureVI)The data of topological monodromy ofEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='16z0,j =  ∞  �  k=0  c0,j,kuγ0,j+15k (j = 1, 2, c0,j,k ∈ C), z(j)  1,1 =  ∞  �  k=0  c1,1,kuγ1,1+3k (0 ≤ j ≤ 4, c1,1,k ∈ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  7  Recovery of fibered complex surfaces from the uni-  versal degenerating family  The goal of this section is to show that any fibered complex surface can be pulled back  from the universal degenerating family of Riemann surfaces π : Y  orb  g  −→ M  orb  g  given in  §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By globalizing the notions discussed in §6, we define the global monodromy µ and the  global orbifold moduli map Jorb for a fibered complex surface f : M → B of genus g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Conversely, starting from the invariants (µ, Jorb), we construct a fibered complex surface  f realizing these invariants, by pulling back the universal family π via Jorb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Note that Kodaira [46] constructed an elliptic surface with given (µ, Jorb) (the homo-  logical invariant and the functional invariant in his terminology), and called it the basic  member of the elliptic surface ([46, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='603]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Our Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8 may be considered as the  construction of the basic members of the fibered complex surfaces of genus g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1, our discussion is from the local point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For a given pseudo-periodic map  µloc ∈ P−  g (σ), we define the set P∗(�∆, ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Dϵ(σ), Mϵ(σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' T µσ  σ ) of the dual pseudo-periodic  maps of µloc, which consists of the orbifold maps Jorb  loc from the orbifold disk to the chart  Dϵ(σ) of M  orb  g  at the equisymmetric strata T µσ  σ  which have the Kodaira-periodicity given in  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Concisely speaking, for a given (µloc, Jorb  loc ), we construct the degeneration  f : M → ∆ which realizes these invariants, by pulling back in the orbifold theoretic  sense via Jorb  loc the local structure of the universal family π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Moreover, we show that  P∗(�∆, ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Dϵ(σ), Mϵ(σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' T µσ  σ ) is the classifying space of analytic structures over the fixed  topological structure of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2, we define the global monodormy and the global orbifold moduli map of a  fibered complex surface f : M → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Over the restricted holomorphic family f (0) : M(0) →  B(0) outsides the discriminant locus of f, these notions are the usual ones, namely, the  monodromy representation to the mapping class group µ : π1(B(0), b0) → Γg and the  pair (J(0))orb = (�J(0), J(0)) consisting of the moduli map J(0) : B(0) → Mg and its lift  �J(0) : �B(0) → Tg from the universal cover of B(0) to the Teichm˝uller space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We prove  that (µ, (J(0))orb) and (µloc, Jorb  loc )’s around the critical set given in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1 are well-patched  globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3, we achieve our main purpose by proving Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  67  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1  Local recovery of degenerations from the universal family  We discuss the recovery of a degeneration f : M → ∆ from the local structure of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We fix an element µ ∈ P−  g (σ) with pseudo-period N (see (90)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We set G = Z/NZ,  and let (�∆, G, π�∆, ∆) be the orbifold disk defined by π�∆ : �∆ ∋ u �→ t = uN ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  µσ : Σg(σ) → Σg(σ) be the analytic automorphism which is a descent of µ (see Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We consider the equisymmetric strata T µσ  σ  ⊂ Dϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We fix a point p = [S, w] ∈ T µσ  σ , and  let (· · · , z(j)  i , · · · , z(γ)  α,β, · · · ) be the system of Harris–Mumford coordinates at p on Dϵ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' An orbifold map  ( �J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' G) : (�∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' π�∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ∆) −→ (Dϵ(σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' W(σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' ϕσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Mϵ(σ))  to the chart (43) of M  orb  g  is said to be a dual pseudo-periodic map of µ at p if the following  conditions are satisfied: A holomorphic map �J : �∆ → Dϵ(σ) with �J(0) = p is expressed  by (3g − 3) pseudo-periodic functions z(j)  i  = ϕ(j)  i (u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' z(γ)  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='β = ψ(γ)  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='β(u) such that  (d-i) ϕ(0)  i (u) has period N/m(Pi) and multiplicity n(Pi) as in (100) or (102),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (d-ii) ψ(0)  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='β(u) has period N/m( �Fα) and multiplicity γα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='β as in (115),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' or ψ(0)  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='β(u) ≡ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (d-iii) ϕ(j)  i (u) = ϕ(0)  i (u) for j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 0 ≤ j ≤ m(Pi) − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' and ψ(γ)  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='β(u) = ψ(0)  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='β(u) for γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 0 ≤ γ ≤  m( �Fα) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The holomorphic map J = ϕσ ◦ �J ◦ (π�∆)−1 : ∆ −→ Mϵ(σ) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The motivation of this definition comes from Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  All the invariants in (d-i)  ∼ (d-iii) except for Kα,β are numerically determined from the data (a),(b),(c) of µ in  Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' But the Kα,β’s can be any non-negative integers provided that γα,β > 0 (see  (115)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since there are infinitely many choices of such Kα,β’s (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='12 (ii)), there are  infinitely many choices of ( �J, J, G) for a given µ and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let P∗(�∆, ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Dϵ(σ), Mϵ(σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' p) be  the set of the dual pseudo-periodic maps ( �J, J, G) of µ at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By varying p over T µσ  σ , we  have the following definition:  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The set of dual pseudo-periodic maps of µ for T µσ  σ  is defined by  P∗(�∆, ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Dϵ(σ), Mϵ(σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' T µσ  σ ) =  �  p∈T µσ  σ  P∗(�∆, ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Dϵ(σ), Mϵ(σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (i) For any µ ∈ P−  g (σ) and ( �J, J, G) ∈ P∗(�∆, ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Dϵ(σ), Mϵ(σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' T µσ  σ ), there  uniquely exists a degeneration f : M → ∆ of Riemann surfaces of genus g such that the  marked topological monodormy of f coincides with µ, and the chart map of the orbifold  moduli map of f coincides with ( �J, J, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) For elements ( �J(i), J(i), G) ∈ P∗(�∆, ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Dϵ(σ), Mϵ(σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' T µσ  σ ) (i = 1, 2), let f (i) :  M (i) → ∆ be the degenerations given in (i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then f (1) is analytically equivalent to f (2) if  and only if ( �J(1), J(1), G) coincides with ( �J(2), J(2), G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  68  Proof  We prove (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We consider an element ( �J, J, G) ∈ P∗(�∆, ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Dϵ(σ), Mϵ(σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' p)  for p = [S, w] ∈ T µ  σ , Harris-Mumford coordinates (· · · , z(j)  i , · · · , z(γ)  α,β, · · · ) of p on Dϵ(σ),  and the expression of �J via the pseudo-periodic functions ϕ(j)  i (u), and ψ(γ)  α,β(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let πσ : Xϵ(σ) → Dϵ(σ) be the family in Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4, and let �f : �  M → �∆ be the pulled  back family (in the sense of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10) from πσ by the map �J : �∆ → Dϵ(σ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' �f is the  second projection of the fiber product �  M = π−1  σ ( �J(�∆)) × �  J(�∆) �∆ −→ �∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the central  fiber �f −1(0) is isomorphic to S by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Here we prove the following claim:  Claim �  M is normal with A-type singularities and any fiber �f −1(u) for u ̸= 0 is non-  singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  In fact, since z(j)  i ’s are the smoothing coordinates at the nodes and the ϕ(j)  i (u)’s are not  identically zero, all the nodes of S vanish via the deformation �f of S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the fiber �f −1(u)  for u ̸= 0 is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' the total space of the restricted family π−1  σ ( �J(�∆)) −→  �J(�∆) has non-normal singularities along the central fiber which is the non-isolated cusp  of the complex space curve locally given in C3g−2 by the image of the map  �J : �∆ ∈ u �−→ (· · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' z(j)  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' z(γ)  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' x) = (· · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' un(Pi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' uγα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' x)  (125)  modulo units under the conditions n(Pi) ≥ 2 and γα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='β ≥ 2 for any i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' where x is a local  fiber coordinate of πσ at a nonsingular point of π−1  σ (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Fortunately, these codimension-  one singularities are resolved along the open locus of the central fiber automatically by  the pull back by �J, for u may be considered via (125) as the local uniformization pa-  rameter of this singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' On the other hand, the germ (�Yg, P (j)  i  ) at the node P (j)  i  in  �Yg coincides with the germ at the origin given by xy = z(j)  i  in the ambient coordinates  (· · · , z(j)  i , · · · , z(γ)  α,β, · · · , x, y) of C3g−1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [9, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='XI, §3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the germ (�  M, P (j)  i  ) is  the A-type singularity xy = un(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The above claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Now by the definitions (d-i) ∼ (d-iii), Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10 and (88), the complex curve �J(�∆) ⊂  B ⊂ Dϵ(σ) is invariant under the action of G = ⟨µ⟩ on B compatible with respect to the  action on �∆, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  �J  �  e  � 1  N  �  u  �  = �J (µ(u)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Therefore G acts relatively on the family �f, which is a natural extension of the automor-  phism of the central fiber S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The explicit descriptions of this action on �f are essentially  the same as (98) ∼ (103) given for Diagram II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let f ♯ : M ♯ = �  M/G −→ ∆ = �∆/G be the  quotient family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' By the composition of f ♯ and the minimal resolution M → M ♯ of the  singularities on M ♯, we obtain the normally minimal model f : M → ∆ of a degeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  69  The marked topological monodormy of f coincides with the preassigned µ ∈ P−  g (σ),  since it is determined by the action of G as in [10, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The orbifold structure of f is  nothing but the above {f ♯, �f, G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let Jf : ∆ → Mσ be the canonically extended moduli  map of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then �J is clearly the lift of Jf by G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' �J is the chart map of the orbifold  moduli map of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Thus we obtain the desired unique degeneration f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We prove (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Assume that f (i) are analytically equivalent to each other for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then Jf(i) : ∆ −→ Mσ ⊂ M g coincides with each other (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' [58]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore there exists  an element g ∈ G such that �J �f(2) = g ◦ �J �f(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If g ̸= id, then µf(1) ̸= µf(2), contradicting the  assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence g = id and ( �J(1), J(1), G) = ( �J(2), J(2), G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The converse is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let f : M → ∆ be a degeneration of genus g with a marking w : Σg →  f −1(t0) (t0 ∈ ∂∆), and µf ∈ P−  g (σ) be the marked topological monodromy of (f, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let ( �J �f, Jf, G) : (�∆, G, π�∆, ∆) −→ (Dϵ(σ), W(σ), ϕσ, Mϵ(σ)) be the orbifold moduli  map of (f, w), and πσ : Xϵ(σ) → Dϵ(σ) be the family in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the orbifold  model of f is isomorphic to the orbifold pull back (in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10) of πσ via ( �J �f, Jf, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let AS(µ) be the set of (complex) analytic structures of degenerations  f : M → ∆ of genus g under a fixed marking w whose topological monodromies �µf  coincide with a fixed µ ∈ P−  g (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then AS(µ) is in a bijective correspondence with  P∗(�∆, ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Dϵ(σ), Mϵ(σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' T µσ  σ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2  Global orbifold moduli maps for fibered complex surfaces  Here we define the notion of orbifold moduli map for a fibered complex surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let f : M → B be a proper surjective holomorphic map from a 2-dimensional com-  plex manifold M to a compact Riemann surface B of genus h ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let Discf(B) =  {Q1, · · · , Qs} ⊂ B be the discriminant locus, and set B(0) = B \\ Discf(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call f a  fibered complex surface of genus g ≥ 2 if any fiber of f over B(0) is a Riemann surface of  genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Each degenerate fiber Fi = f −1(Qi) is assumed to be normally minimal in the  sense of §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let f (0) : M(0) → B(0) be the restricted holomorphic family of f over B(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We fix a point b0 ∈ B(0), and consider the fiber f −1(b0) := Σg as the base Riemmann  surface of the marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The fundamental group π1(B(0), b0) is generated by the loops β1, · · · , β2h, α1, · · · , αs  on B(0) starting and ending at b0 with the unique relation  β1β2β−1  1 β−1  2 β3β4 · · · β−1  2h−1β−1  2h α1 · · · αs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Here each αi is a loop which goes around Qi once counterclockwise and β1, · · · , β2h are  canonical generators of π1(B, b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since f (0) is a differentiable fiber bundle, we have the  70  monodromy representation to the mapping class group  µf : π1(B(0), b0) −→ Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (126)  Let ∆i = {ti ∈ C | |ti| < ϵ0} be a local disk coordinate at Qi on B, and fi = f|Mi :  Mi = f −1(∆i) −→ ∆i be the degeneration over ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then we may identify µf(αi) as  the marked topological monodromy µfi of fi defined in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore, there exists an  element σ(i) ∈ Cg such that µf(αi) = µfi ∈ P−  g (σ(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let Ni denote the pseudo-period of  µfi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let ϕ�B(0) : �B(0) −→ B(0) be the universal covering of B(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then π1(B(0), b0) acts on  �B(0), and we have B(0) ∼= �B(0)/π1(B(0), b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' On the other hand, the cyclic group Z/NiZ  acts on �∆i = {ui ∈ C | |ui| < ϵ1/Ni} by ui �→ e(1/Ni)ui, and the projection ϕ�∆i : �∆i → ∆i  is defined by ui �→ ti = uNi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We define the complex orbifold structure over B by  Borb =  �  �B(0), π1(B(0), b0), ϕ�B(0), B(0)� �  1≤i≤s  �  �∆i, Z/NiZ, ϕ�∆i, ∆i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (127)  Note that, although π1(B(0), b0) is an infinite group and Z/NiZ is a finite group, the  orbifold structure Borb is well-defined in the sense of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let �f (0) : �  M(0) = M(0) ×B(0) �B(0) −→ �B(0) be the pull back of f (0) over �B(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  M → M♯ be the contraction map of all the non-cores of Fi (1 ≤ i ≤ s), and f ♯ : M♯ → B  be the natural holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let {f ♯  i : M ♯  i → ∆i, �fi : �  Mi → �∆i, Gi = Z/NiZ} be the  orbifold structure of fi in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='5 and Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We define the complex orbifold  structure over M♯ by  (M♯)orb =  �  �  M(0), π1(B(0), b0), ϕ�  M(0), M(0)� �  1≤i≤s  �  �  Mi, Z/NiZ, ϕ�  Mi, M ♯  i  �  (128)  where ϕ�  M(0) and ϕ�  Mi are natural projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Then we have the orbifold fibration of  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6 by  (f ♯)orb : (M♯)orb −→ Borb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (129)  Now let  J(0) : B(0) −→ Mg  (130)  be the moduli map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  The lifting of J(0) over �B(0) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  For a point  x ∈ B(0), let γ(b0, x) be an arc on B(0) starting from b0 and ending at x, and [γ(b0, x)] be  its homopoty class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then �B(0) consists of all the pairs (x, [γ(b0, x)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let  wγ(b0,x) : Σg = f −1(b0) −→ f −1(x)  be the oriented homeomorphism of fibers of f (0) along γ(b0, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since the isotopy class of  wγ(b0,x) defines a Teichm¨uller marking of f −1(x), the map  �J(0) : �B(0) −→ Tg  (131)  71  is defined by �J(0) ((x, [γ(b0, x)])) = [f −1(x), wγ(b0,x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then �J(0) is a holomorphic map which  satisfies ϕTg ◦ �J(0) = J(0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' it is a lifting of J(0) in (130).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  On the other hand, it follows from (107) that J(0) is holomorphically extended to  J : B −→ M g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (132)  By (110), the restricted map Jfi = J|∆i : ∆i −→ M g is lifted to  �J �fi : �∆i −→ Dϵ(σ(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (133)  We assure the compatibility condition for the patching of (131) and (133) as an orbifold  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' If J is a constant map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' J(B) = b0, then the maps (131) and (133) are clearly  well-patched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore we assume that J(B) is an analytic curve on M g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let x ∈ B(0) ∩  (∆i\\Qi) be a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let �x ∈ �B(0) and x♯ ∈ �∆i be the points with ϕ�B(0)(�x) = ϕ�∆i(x♯) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let Vx be a sufficiently small open neighborhood of x of B(0) ∩(∆i\\Qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let �V�x (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' V ♯  x♯)  be the open neighborhood of �x of �B(0) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' of x♯ of �∆i) with ϕ�B(0)(�V�x) = ϕ�∆i(V ♯  x♯) = Vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We set �y = �J(0)(�x) ∈ Tg = Dϵ(∅) and y♯ = �J �fi(x♯) ∈ Dϵ(σ(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' There exist open neighborhoods �U�y of �y in Tg, and U ♯  y♯ of y♯ in Dϵ(σ(i))  respectively such that  (i) �J(0)(�V�x) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' �J �fi(V ♯  x♯)) is an analytic curve in �U�y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' U ♯  y♯),  (ii) there exists a biholomorphic map ψ : �U�y −→ U ♯  y♯ so that the restriction map of ψ to  �J(0)(�V�x) induces an isomorphism onto �J �fi(V ♯  x♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  By Arbarello–Cornalba’s theorem [8], we may assume that �U�y is the base of  a sufficiently small Kuranishi family of the Riemann surface f −1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' On the other hand,  Dϵ(σ(i)) is a union of the bases of standard Kuranishi families of stable curves by Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The Kodaira–Spencer map is an isomorphism at any point, particularly at y♯, of the  base of the standard Kuranishi family by the property (iii) in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Therefore we also may  assume that U ♯  y♯ is the base of a sufficiently small Kuranishi family of f −1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Hence we may assume that there exists a biholomorphic map ψ′ : �U�y −→ U ♯  y♯ such  that ψ′ is induced from the extension g0 to the Kuranishi family of an automorphism  g0 ∈ Aut(f −1(x)) by the property (v) in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Since ϕTg : Tg → Mg and ϕDϵ(σ(i)) : Dϵ(σ(i)) → Mϵ(σ(i)) are the forgetting maps of the  Teichm¨uller marking and the Weyl marking respectively, we have ϕTg(�U�y) = ϕDϵ(σ(i))(U ♯  y♯)  as an open set (in the classical topology) of Mg containing J(0)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Now we consider the analytic curves �J(0)(�V�x) and �J �fi(V ♯  x♯) on the above spaces �U�y and  U ♯  y♯ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' It is clear that ϕTg(�J(0)(�V�x)) = ϕDϵ(σ(i))( �J �fi(V ♯  x♯)) = J(0)(Vx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence there  exists an element g0 ∈ Aut(f −1(x)) such that the restriction of the biholomorphic map  g0 ◦ ψ′ : �U�y −→ U ♯  y♯ to �J(0)(�V�x) sends isomorphically onto �J �fi(V ♯  x♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  72  We extend the canonically extended moduli map J : B −→ M g in (132) to the orbifold  map as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='6 assures its well-definedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The maps �J(0) : �B(0) −→ Tg in (131) and �J �fi : �∆i −→ Dϵ(σ(i)) in (133)  (1 ≤ i ≤ s) are well-patched and define the orbifold map  Jorb  f  : Borb −→ M  orb  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We call Jorb  f  the (global) orbifold moduli map of a fibered complex surface f : M → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3  Global recovery of basic members of fibered complex sur-  faces  We construct all the fibered complex surfaces for possible monodromy representations and  orbifold moduli maps by pulling back from our universal degenerating family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We let D(B) = {Q1, · · · , Qs} be a finite set of a compact Riemann surface B, and  set B(0) = B \\ D(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' For a fixed point b0 ∈ B(0), we consider a representation of the  fundamental group to the mapping class group of genus g ≥ 2  µ : π1(B(0), b0) −→ Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (134)  We assume that µ satisfies the following condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let αi be a loop which goes around  Qi once counterclockwise for 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then there exists an element σ(i) ∈ Cg such that  µ(αi) ∈ P−  g (σ(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We call such a µ a pseudo-periodic representaion on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Let Ni be the pseudo-period of µ(αi) and Borb be the orbifold structure over B as in  (127).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Let J : B → M g be a holomorphic map, and  Jorb : Borb −→ M  orb  g  (135)  be an orbifold map over J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We assume that Jorb satisfies the following conditions:  (i) the restricted chart map on �B(0) maps �B(0) to Tg, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' �J : �B(0) −→ Tg = Dϵ(∅),  (ii) the restricted chart map on each �∆i belongs (in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1) to  �J|�∆i ∈ P∗(�∆i, ∆i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Dϵ(σ(i)), Mϵ(σ(i));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' T  µσ(i)  σ(i) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  We call such Jorb a dual pseudo-periodic map associated with µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (i)  For any (µ, Jorb) ∈ F(B), there uniquely exists (modulo analytc  equivalence) a fibered complex surface of genus g ≥ 2  f : M −→ B  (136)  73  such that the monodromy representation of f coincides with µ, and the orbifold moduli  map of f coincides with Jorb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii)  The orbifold fibration (f ♯)orb : (M♯)orb −→ Borb associated with (136) (see (129))  coincides with the orbifold pull-back (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='10) from the universal degenerating family  (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9) π : Y  orb  g  −→ M  orb  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Proof  The restricted family of π to Dϵ(∅) = Tg is nothing but the universal family  ([14]) of Teichm¨uller-marked Riemann surfaces Yg → Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since �J(�B(0)) is contained in Tg  by assumption, the family over �B(0) obtained by the pull-back via �J from this universal  family is a holomorphic family �f (0) : �  M(0) → �B(0) of Teichm¨uller-marked Riemann sur-  faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Since B(0) is the quotient space of �B(0) by π1(B(0), b0), the forgetting map of the  markings induces a holomorphic family f (0) : M(0) → B(0) of Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  On the other hand, it follows from Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3 that the pull-back via �J|�∆i of π induces  a family of stable curves �fi : �  Mi → �∆i and its quotient family f ♯  i : M ♯  i → ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  By the same argument as that of the well-definedness of Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7, the maps  (�f (0), f (0)) and ( �fi, f ♯  i ) (1 ≤ i ≤ s) are well-patched as orbifold maps and define an orbifold  fibration (f ♯)orb : (M♯)orb → Borb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hence by the argument in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2, we have a fibered  complex surface f : M → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  It is clear from the construction that the monodromy representation of f coincides  with µ, and the orbifold moduli map of f coincides with Jorb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The uniqueness of f modulo  analytic equivalence is also clear since it has the fixed orbifold moduli map Jorb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The construction of the fibered complex surface (136) of genus g ≥ 2 is  analogous to that of the basic members of elliptic surfaces (basic elliptic surfaces, for  short) due to Kodaira [46, §8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' nevertheless they are different in the following points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (i)  Basic elliptic surfaces are assumed to have no multiple fibers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='e fibers of types  mIb, m ≥ 2 given in [46, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='565]), while our fibered surfaces (136) are admitted to have  any singular fibers including multiple fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (ii) Let F(J , G) be the set of elliptic surfaces without multiple fibers which have J-  invariant J and homological invariant G (see [46, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Then the basic elliptic  surface in F(J , G) is characterized as the unique member which admits a global section  ([46, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2]), and other members in F(J , G) are obtained from the basic elliptic surface  by twisting defined in [46, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='2] (see [46, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  On the other hand, a fibered complex surface of genus g ≥ 2 is uniquely determined by  the data (µ, Jorb) by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  This difference essentially comes from the fact that an elliptic curve has translation  automorphisms, while a Reimann surface of genus g ≥ 2 has no such infinite automor-  phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  74  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' We thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Sampei Usui for his advice and discussions on log  geometry, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Toshiyuki Akita for his advice on representations of automorphisms of  Riemann surfaces, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kunio Obitsu for his advice on augmented Teichm¨uller spaces,  Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kazuhiro Konno for discussions on fibered complex surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Finally we thank  Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Athanase Papadopoulos for his interest in our results and for many comments which  improved our presentation very much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' The second named author is partially supported  by JSPS Grant KAKENHI 17H01091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  References  [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Abikoff, The real analytic theory of Teichm¨uller space, Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 820  (1980), Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A’Campo, Sur la monodromie des singularit´es isol´ees d’hypersurfaces complexes, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 20 (1973), 147–169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A’Campo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ji, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Papadopoulos, On Grothendieck’s construction of Teichm¨uller space,  in Handbook of Teichm¨uller theory, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' VI (2016), 35–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A’Campo-Neuen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A’Campo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ji, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Papadopoulos, A commentary on Teichm¨uller’s  paper “Ver¨anderliche Riemannsche Fl¨achen”, in Handbook of Teichm¨uller theory, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' IV  (2014), 805–814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [5] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ahlfors, Some remarks on Teichm¨uller’s space of Riemann surfaces, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (2) 74 (1961), 171–191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [6] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ahlfors and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Bers, Riemann’s mapping theorem for variable metrics, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  (2) 72 (1960), 385–404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Alexander, On the deformation of a n-cell, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Nath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 9 (1923), 406–407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [8] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Arbarello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Cornalba, Teichm¨uller space via Kuranishi families, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Scuola  Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Pisa Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (5) Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' VIII (2009), 89–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [9] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Arbarello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Cornalba, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Griffiths, Geometry of Algebraic Curves, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' II, Springer,  2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ashikaga, Local signature defect of fibered complex surfaces via monodromy and stable  reduction, Comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 85 (2010), 417-461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [11] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ashikaga and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ishizaka, Classification of degenerations of curves of genus 3 via  Matsumoto-Montesinos’ theorem, Tohoku Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 54 (2002), 195–226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ashikaga and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Konno, Global and local properties of pencils of algebraic curves, in  Algebraic Geometry 2000 Azumino, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Usui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=', Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' in Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 36 (2002),  1–49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [13] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ashikaga and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Yoshikawa, A divisor on the moduli space of curves associated to the  signature of fibered surfaces, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' in Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 56 (2009), 1–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [14] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Bers, Fiber space over Teichm¨uller spaces, Acta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 130 (1973), 89–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Bers, Space of degenerating Riemann surfaces with nodes, in Discontinuous Groups and  Riemann surfaces, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Studies 79, Princeton U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (1974), 43–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  75  [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Broughton, The equisymmetric stratification of the moduli space and the Krull di-  mension of mapping class group, Topology and its Applications 37 (1990), 101–113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [17] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Cartan, Quotient d’un espace analytique par un group d’automorphismes, Algebraic  geometry and topology, A Symposium in honor of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Lefschetz, Edited by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Fox, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Spencer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Tucker, Princeton Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Series, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Press, Princeton,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=', 1957, 90–102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [18] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Cartan, Quotients of complex analytic spaces, in Contributions to Function Theory,  Tata Institute, Bombay (1960), 1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [19] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Catanese, A superficial working guide to deformation and moduli, in Handbook of Moduli,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' I (2013), 161–215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Clemens, Picard-Lefschetz theorem for families of nonsingular algebraic varieties ac-  quiring ordinary singularities, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 136 (1969), 93–108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [21] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Deligne and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Mumford, The irreducibility of the space of curves of given genus, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 36 (1969), 75–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [22] Dias Raquel and Gonz´ales-Aguilera Victor, Limit points of the branch locus of Mg,  Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 19 (2019), 505–526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [23] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Earle and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Sipe, Families of Riemann surfaces over the punctured disk, Pacific J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  150 (1991), 79–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [24] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ehresmann, Sur les sepaces fib´es diff´erentiables, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Paris 224 (1947),  1611-1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [25] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Farkas and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kra, Riemann surfaces, 1979, Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [26] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Friedman, Global smoothings of varieties with normal crossings, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  118  (1983), 75–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [27] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Grauert, ¨Uber Modifikationen und exzeptionelle analytische Mengen, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ann, 146  (1962), 331–368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Grothendieck, Techniques de construction en g´eom´etrie analytique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' I∼X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' S´eminaire  Henri Cartan, tome 13, (1960-1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Harris and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Mumford, On the Kodaira dimension of the moduli space of curves, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 67 (1982), 23–88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [30] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Oxford Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (2) 17 (1966), 86–97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [31] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Harvey, On branch loci in Teichm¨uller space, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 153 (1971),  387–399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [32] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Harvey, Chabauty spaces of discrete groups, Discontinuous groups and Riemann  surfaces (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=', Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Maryland, College Park, Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=', 1973), Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Press,  Princeton, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=', 1974, 239–246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Studies, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 79, MR 0364629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [33] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Harvey, Geometric structure of surface mapping class groups, in Homological gourp  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Proceedings of a Symposium on Homological and Combinatorial Techniques in  Group Theory held at Durham, September, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Lecture Note Series, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Cambridge University Press, 1979, 255–269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  76  [34] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hinich and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Vaintrob, Augmented Teichm˝uller spaces and orbifolds, Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' New  Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 16 (2010), 533–629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hirakawa and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Takamura, Atlas for Moduli Geometry of Riemann Surfaces, Kyoto  University, preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hubbard, Teichm¨uller theory and applications to geometry, topology, and dynamics,  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' I, Matrix Editions, Ithaca, NY, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [37] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hubbard and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Koch, An analytic construction of the Deligne-Mumford compactifi-  cation of the moduli space of curves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 98 (2014), 261–313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Hubbard, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Papadopol and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Veselov, A compactification of H´eron mapping in C2  as dynamical systems, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 184 (2000), 203–270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [39] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Imayoshi, Holomorphic families of Riemann surfaces and Teichm¨uller spaces, in Riemann  surfaces and related topics, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Studies 97 (1981), 277–300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [40] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Imayoshi, A construction of holomorphic families of Riemann surfaces over punctured  disk with given monodromy, in Handbook of Teichm¨uller theory, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' II (2009), 93–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [41] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Imayoshi and Taniguchi, An introduction to Teichm¨uller spaces, Springer, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [42] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kato and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Nakayama, Log Betti cohomology, log ´etale cohomology, and log de Rham  cohomology of log schemes over C, Kodai Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 22 (1999), 161–189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [43] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kato and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Usui, Classifying spaces of degeneration polarized Hodge structures, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Studies 169, Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Press, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [44] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Keen, Collars on Riemann surfaces, In Discontinuous groups and Riemann surfaces,  Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Press (1974), 263–268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [45] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kerchhoff, The Nielsen realization problem, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 117 (1983), 235–265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [46] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kodaira, On compact analytic surfaces II, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 77 (1963), 563–626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [47] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kra, Horocyclic Coordinates for Riemann Surfaces and Moduli Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' I: Teichm¨uller  and Riemann Spaces of Kleinian Groups, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 3 (1990), 499–578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [48] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kra, Book review of : S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Donaldson, Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=') 49  (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='3, 455–463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' MR 2952711 Zbl 1321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='00036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [49] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kuno, Meyer functions and the signature of fibered 4-manifolds, in Handbook of Te-  ichm¨uller theory, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' V (2016), 75–96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [50] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kuribayashi, On analytic families of compact Riemann surfaces with non-trivial auto-  morphisms, Nagoya J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 28 (1966), 119–165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [51] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Liu, Fractional Dehn twists and modular invariants, Science China Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 64 (2021),  1735 –1744.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [52] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Masai, Compactification and distance on Teichm¨uller space via renormalized volume,  arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='06059v3 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='GT] 27Sep 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [53] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Matsumoto, On the universal degenerating family of Riemann surfaces, in Singularities  in Geometry and Topology–Strasbourg 2009, Euro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' (2012), 71–102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [54] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Matsumoto, The Deligne-Mumford compactification and crystallographic groups, in  Handbook of Teichm¨uller theory, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' VII (2020), 21–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  77  [55] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Matsumoto and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Montesinos-Amilibia, Pseudo-periodic maps and degeneration of  Riemann surfaces, Lecture Note in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 2030 (2011), Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [56] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Miyachi, Lipschitz algebras and compactifications of Teichm¨uller space, in Handbook of  Teichm¨uller theory, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' IV (2014), 375–413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [57] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Mizuta and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Namba, Greenberg’s theorem and equivalence problem on compact Rie-  mann surfaces, Osaka J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 43 (2006), 137–178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [58] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Namikawa, Studies on degeneraions, in Classification of Algebraic Varieties and Compact  Complex Manifolds, Lecture Note in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 412 (1974), Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [59] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Nielsen,  Die  Structur  periodischer  Transformationen  von  Fl¨achen,  Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='-  Fys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Medd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Danske Vid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Selsk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 15 (1937), 77 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' English translation:  in Collected  Papers 2, Birkh¨auser, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [60] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Nielsen, Abbildungsklassen endlicher Ordnung, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=', 75 (1942), 23–115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' English  translation: in Collected Papers 2, Birkh¨auser, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [61] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Nielsen, Surface transformation classes of algebraically finite type, Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='-Fys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Medd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Danske Vid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Selsk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 21 (1944), 3–89 : in Collected Papers 2, Birkh¨auser, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [62] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Ohshika, Compactifications of Teichm¨uller spaces, in Handbook of Teichm¨uller theory,  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' IV (2014), 235–254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [63] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Reid, Problems on pencils of small genus, Preprint (1990) (see his home page).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [64] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Satake, The Gauss-Bonnet Theorem for V-manifolds, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Japan, 9 (1957),  464–492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [65] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Shiga and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Tanigawa, On the Maskit coordinates of Teichm¨uller spaces and modular  transformations, Kodai Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 12 (1989), 437–443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [66] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Takamura, Towards the classification of atoms of degenerations II, (Cyclic quotient con-  struction of degenerations of complex curves), Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kyoto Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Preprint  series No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 1344 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [67] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Takamura, Linearization of quotient families, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Tokyo 26 (2019), 361–  389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [68] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Tan, Chern numbers of a singular fiber, modular invariants and isotrivial families of  curves, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Vietnamica 35 (2010), 159 –172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [69] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Teichm¨uller, Extremale quasikonforme Abbildungen und quadratische Differentiale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Abh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Preuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=', Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='-Naturw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 22 (1940), 1–197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' English translation by  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Th´eret, Extremal quasiconformal mappings and quadratic differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In Handbook of  Theichm¨uller theory (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Papadopoulos, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=') Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' V, EMS Publishing House, Z¨urich, 2016,  321–483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [70] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Teichm¨uller, Ver¨anderliche Riemannsche Fl¨achen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Deutsche Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 7, 344–359 (1944).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  English translation by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' A’Campo Neuen, Variable Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' In Handbook of  Teichm¨uller theory (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Papadopoulos, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=') Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' IV, EMS Publishing House, Z¨urich, 2014,  787–803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [71] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Terasoma, Degeneration of curves and analytic deformations, arXiv:math/9801118v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='AG] 26 Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  78  [72] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Usui, Recovery of vanishing cycles by log geometry, Tohoku Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 53 (2001), 1–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [73] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' V¨olklein, Moduli spaces for covers of the Riemann sphere, Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 85 (1994),  407–430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [74] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Wiman, ¨Uber die hyperelliptischen Curven und diejenigen vom Geschlechte p=3,  welche eindeutige Transformationen in sich zulassen, Bihang Kongl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Svenska Vetenskaps-  Akademiens Handligar (1985), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 21 (1), 1–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [75] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Wolpert, Geometry of Weil-Petersson completion of Teichm¨uller space, In Surveys in  Differential Geometry VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  357–393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Press, Cambridge, MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [76] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Wolpert, Behaviour of geodesic-length function on Teichm¨uller space, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Differential  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 79 (2008), 277–334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [77] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Wolpert, Families of Riemann surfaces and Weil-Petersson geometry, CBMS Regional  Conference Series in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 113, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  [78] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Yamada, On the geometry of Weil-Petersson completion of Teichm¨uller spaces, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' 11 (2004), 327–344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='  Tadashi Ashikaga, Faculty of Engineering, Tohoku-Gakuin University, Tagajo,  Miyagi 985-8537, Japan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' e-mail: ashikaga@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='tohoku-gakuin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='jp  Yukio Matsumoto, Department of Mathematics, Gakushuin University, Mejiro,  Toshima-ku, Tokyo 171-8588, Japan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content=' e-mail: yukiomat@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
+page_content='gakushuin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfhvgZ/content/2301.00381v1.pdf'}
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+Ultra-narrowband interference circuits
+enable low-noise and high-rate photon counting
+for InGaAs/InP avalanche photodiodes
+YUANBIN FAN,1,† TINGTING SHI,1,2,3,† WEIJIE JI,1LAI ZHOU,1
+YANG JI 2,3 AND ZHILIANG YUAN1,*
+1Beijing Academy of Quantum Information Sciences, Beijing 100193, China
+2State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese
+Academy of Sciences, Beijing 100083, China
+3College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences,
+Beijing 100049, China
+†These authors contributed equally.
+*yuanzl@baqis.ac.cn
+Abstract: Afterpulsing noise in InGaAs/InP single photon avalanche photodiodes (APDs) is
+caused by carrier trapping and can be suppressed successfully through limiting the avalanche
+charge via sub-nanosecond gating. Detection of faint avalanches requires an electronic circuit
+that is able to effectively remove the gate-induced capacitive response while keeping photon
+signals intact. Here we demonstrate a novel ultra-narrowband interference circuit (UNIC) that
+can reject the capacitive response by up to 80 dB per stage with little distortion to avalanche
+signals. Cascading two UNIC’s in a readout circuit, we were able to enable high count rate of
+up to 700 MC/s and low afterpulsing of 0.5 % at a detection efficiency of 25.3 % for 1.25 GHz
+sinusoidally gated InGaAs/InP APDs. At -30 ◦C, we measured 1 % afterpulsing at a detection
+efficiency of 21.2 %.
+Introduction
+Semiconductor avalanche photodiodes (APD’s) are versatile for weak light detection, with
+applications from remote ranging [1,2], quantum communication [3] and fluorescence lifetime
+imaging [4] to optical time-domain reflectometry [5, 6].
+For practical fiber quantum key
+distribution (QKD), InGaAs/InP APD’s are the detector of choice because they are compact and
+low cost, and allow cryogenic-free or even room-temperature operation [3]. However, they suffer
+from spurious afterpulsing arising from carrier trapping by defects in the multiplication layer,
+especially at high detection efficiencies [7,8]. To minimise afterpulsing, an APD can be biased
+on for a sub-nanosecond duration only when a photon arrival is expected. In doing so, charge per
+avalanche can be reduced to the order of 10 fC [9–11], corresponding to a transient current of
+less than 0.1 mA. Such weak avalanches have to be discriminated through use of a readout circuit
+that removes the strong capacitive response to the applied gates. Gated InGaAs detectors are
+capable of counting photons at up to 60% efficiencies [12] and 1 GHz rate [13] and with photon
+number resolution [14]. Thanks to this success, gating approach has been applied to traditionally
+free-running Si devices for performance enhancement [15,16].
+Existing readout circuits include band stop [8,11,17] or low-pass [12,18,19] filtering under
+sine-wave gating [11], self-differencing [7, 20], and transient reference cancellation [10, 21].
+While simple for implementation, frequency filtering distorts the avalanche signals due to its
+rejection of a sizeable portion of frequency components, thus increasing time jitter and temporal
+errors in photon registrations [18]. Self-differencing [20] and reference cancellation methods [10]
+are able to maintain avalanche signal fidelity but may suffer operational complexities. The former
+requires wideband performance for the entire circuitry and thus inconvenience an adjustable
+delayline [9] for frequency alignment, while the latter can be unstable because the transient
+reference is derived separately from the capacitive response.
+1
+arXiv:2301.01570v1  [quant-ph]  4 Jan 2023
+
+Here we propose and experimentally demonstrate a simple, low-distortion ultra-narrowband
+interference circuit (UNIC) that can suppress the capacitive response for a 1.25 GHz gated
+InGaAs/InP APD single photon detector. The circuit is an asymmetric radio-frequency (RF)
+interferometer. One of its arms contains a narrow band pass filter (BPF) based on surface acoustic
+wave resonator (SAW) to retrieve the fundamental wave of the gating signal. The filtered wave
+then interferes destructively with the same frequency component transmitted via the other arm
+through a coupling module, thereby eliminating the capacitive response. This interference occurs
+over the narrow band, so it can provide a broad and continuous pass band in frequency domain to
+maintain the avalanche signal with little distortion. This allows to achieve ultra-low afterpulsing
+probabilities and excellent jitter performance at high detection efficiencies from two InGaAs
+APD’s that exhibit capacitive responses of very different amplitudes.
+Detector characterisation setup
+35
+40
+45
+101
+102
+103
+104
+105
+106
+0
+20
+40
+60
+80
+Count
+Time (ns)
+650 ps
+Count
+Time (ns)
+SG
+Laser
+DC bias
+TDC
+AMP
+UNIC 1
+BSF
+AMP
+Start    Click
+Out
+ 10 MHz
+REF IN
+DISC
+Threshold 
+Level
+AMP
+50Ω
+50Ω
+APD
+1.25 GHz
+UNIC 2
+(a) 
+(b) 
+Illuminated
+Non-Illuminated
+VOA
+Fig. 1. (a) Single-photon characterisation setup for 1.25 GHz sinusoidally gated
+InGaAs/InP APDs using UNICs for avalanche impulse readout; (b) Histogram of the
+photon detection events measured by the characterisation setup (a) on an InGaAs APD
+detector that was regulated at a temperature of 30 ◦C. The photon detection peak exhibits
+a 30 dB width of 650 ps. AMP: amplifier; APD: avalanche photodiode; BSF: band
+stop filter; DISC: discriminator; SG: signal generator; TDC: time-to-digital converter;
+UNIC: ultra-narrowband interference circuit; VOA: variable optical attenuator.
+Figure 1(a) shows our single photon characterisation setup for InGaAs APDs. A 1550 nm
+passively mode-locked laser serves as the light source and provides stable short pulses of 0.5 ps
+duration at a repetition rate of 10 MHz. The laser output power is monitored by an optical power
+meter (EXFO FTB-1750) and its pulse intensity is set by a variable optical attenuator (VOA,
+EXFO FTB-3500) to 0.1 photon/pulse at the fiber input of APD under test. It provides a 10 MHz
+reference to a signal generator (SG) for synthesising a 1.25 GHz sinusoidal wave with up to 27 V
+voltage swing. In combination of a suitable DC bias, this AC signal periodically gates the APD
+above its breakdown voltage (60 − 70 V) to achieve single photon sensitivity. The APD output is
+processed by the readout module consisting of two identical 1.25 GHz UNIC’s, one 2.5 GHz
+band stop filter (BSF), and three RF amplifiers (AMPs). Amplification of the raw APD signalis
+2
+
+is useful as it prevents weak avalanche signals from falling below thermal noise by attenuation
+of the first UNIC. The readout signal is discriminated by a discriminator for avalanches before
+feeding to a time-digital-converter (TDC) with a dead time of 2 ns for time-resolved photon
+counting. Figure 1(b) is a typical histogram obtained with this setup.
+APD under test is temperature-regulated using their integrated thermal-electric cooler, which
+is driven by a temperature controller (Thorlabs TED200C). A source-measure unit (Keithley
+2635B) provides the DC bias and simultaneously monitors the current flowing through the APD.
+In characterising the maximum count rate, we replace the 10 MHz laser with a continuous-wave
+distributed feedback laser (DFB) laser, the output of which is carved into 1.25 GHz, 50 ps pulse
+train using an intensity modulator. We use a high speed digital oscilloscope to record the detector
+output and extract the count rate through digital discrimination in software. The oscilloscope
+method is carefully calibrated at low count rate regimes to be consistent with the hardware
+discriminated result using the photon counter (Stanford Research SR400).
+The setup is able to measure dark count probability, afterpulsing probability, detection
+efficiency, maximum count rate, avalanche charge and time jitter. With no performance screening,
+two fiber-pigtailed APDs from different manufacturers were used in this study, named APD#1
+and APD#2 respectively.
+Ultra-narrowband interference circuit (UNIC)
+0
+1
+2
+3
+4
+5
+-90
+-60
+-30
+0
+1.24
+1.25
+1.26
+-90
+-60
+-30
+0
+-1
+0
+1
+-10
+-5
+0
+5
+10
+-1
+0
+1
+0
+1
+-10
+-5
+0
+5
+10
+0
+1
+Insertion  Loss (dB)
+Frequency (GHz)
+Loss [dB]
+Freq. (GHz)
+Vp-p = 0.42 V 
+Vp-p  = 1.75 V 
+APD#1
+APD#2
+Signal (V)
+Time (ns)
+APD#1
+  
+APD#2
+Signal (a.u.)
+Time (ns)
+(a) 
+(c) 
+(b)
+(d)
+SAW
+ BPF
+IN         
+         OUT
+50Ω
+50Ω
+ATT
+Fig. 2. (a) Schematic for ultranarrow interference circuit (UNIC); (b) Transmission
+spectrum of a heroic UNIC PCB; Inset: Magnified view for region of 1.24 – 1.26 GHz.
+(c) Raw capacitive responses from APD#1 (top) and APD#2 (bottom) under identical
+27.0 V V𝑝−𝑝 gating; (d) Recovered avalanche impulses. ATT: attenuator; SAW BPF:
+surface acoustic wave band pass filter.
+With sub-nanosecond gating, a photon induced avalanche is an impulse and has a wide spectrum.
+On the other hand, the capacitive response is periodic and has its most energy concentrated at the
+3
+
+gating frequency or its higher harmonics. This spectral difference allows frequency-dependent
+signal processing to remove the capacitive response and keep the wide-band impulses intact.
+Figure 2(a) shows a circuit diagram of UNIC. It is an RF interferometer containing two couplers
+of 9:1 power splitting ratio, a 𝜋-resistive attenuator (ATT) and surface acoustic wave band pass
+filter. Two of the ports are terminated by 50 Ω resistors. The SAW BPF features a central
+frequency of 1.25 GHz, 20-dB passband of 35 MHz, insertion loss of 3 dB, and group delay of
+34 ns. It filters out the fundamental wave of the gating frequency, which then interferes with the
+APD signal transmitted through the other arm. The attenuation and differential delay are set to
+enable destructive interference for the 1.25 GHz frequency component at the UNIC output port.
+The UNIC differential delay (Δ𝑡) meets the condition below
+Δ𝑡 = 𝑇𝑆𝐴𝑊
+𝑔
++ 𝛿𝑡 = (𝑁 + 1/2)/ 𝑓𝑔,
+(1)
+where 𝑇𝑆𝐴𝑊
+𝑔
+is the group delay of the SAW BPF, 𝛿𝑡 the delay caused by the track length difference
+between two arms, 𝑓𝑔 = 1.25 GHz the APD gating frequency, and 𝑁 is an integer number. For
+a compact circuit, we choose 𝛿𝑡 to be less than the half-wave of the gating signal. With the
+SAW device used, 𝑁 = 42 and 𝛿𝑡 = 155 ps. The resulting UNIC unit has a small footprint of
+38 × 15 mm2 on printed circuit boards (PCBs).
+The large 𝑇𝑆𝐴𝑊
+𝑔
+brings two additional benefits. Firstly, it substantially increases the PCB
+manufacturing tolerance, as a 0.5 mm deviation in the RF track length will just alter the circuit
+central frequency by less than 10−4. This eliminates the requirement of an adjustable delayline
+which is required in a self-differencing circuit for precise frequency alignment. Secondly, it
+helps to produce an ultra-narrow band rejection at its designed frequency. Figure 2(b) shows the
+measured transmission spectrum (S21 parameter) of our heroic UNIC PCB, and its inset expands
+the frequency section of 1.24 – 1.26 GHz to show the narrowness of the insertion loss dip in the
+close proximity of the resonance frequency of 1.25 GHz. The dip of the heroic (typical) PCB
+features a loss of -95 dB (-80 dB), representing a suppression of 80 dB (65 dB) as compared with
+the background loss for other frequencies under 2 GHz. The dip has a 30 dB linewidth of merely
+30 kHz, thus ensuring crucial suppression of the APD gating signal without overly distorting the
+avalanche signals. The background loss of about 14 dB is caused mainly by the 9:1 couplers and
+can be reduced in future with more balanced splitters.
+Cascading two UNIC’s enables a stable 100 dB suppression of the primary gating frequency
+and thus provides sufficient performance redundancy. Their attenuation to the avalanche signal is
+compensated by using RF amplifiers (Fig. 1(a)). Second order harmonics (2.5 GHz) is suppressed
+by a band stop filter of conventional LC design. Figure 2(c) shows raw outputs from two different
+APD’s under identical sinusoidal gating. Their respective capacitive responses are measured
+to be 0.42 V and 1.75 V. Despite their 4 times differences, UNIC’s can successfully reject the
+sinusoidal responses and retrieve avalanches with excellent signal-to-background ratio, as shown
+in Fig. 2(d). For APD#2, we just adjusted the gain of the first AMP to avoid saturation and
+distortion.
+Results and discussion
+Time-resolved photon counting allows precise extraction of the net photon detection efficiency
+(𝜂𝑛𝑒𝑡) and the afterpulsing probability (𝑃𝐴), which is defined as the ratio of the total afterpulses
+per photon counting event. Figure 1(b) shows a histogram of avalanche events measured for
+APD#1 under 10 MHz pulsed excitation of 0.1 photon/pulse. The illuminated peak has a
+full-width of 1/1000 maximum (30 dB width) of just 650 ps, which is shorter than the gating
+period of 800 ps and thus allows low-error clock number assignment that is essential for high
+speed QKD. The count at non-illuminated gates arise from detector dark count and afterpulses
+and is more than 3 orders of magnitude lower than that of the illuminated gate. We extract
+quantities of 𝑃𝐼 and 𝑃𝑁 𝐼, i.e., the respective counting probabilities for each illuminated and
+4
+
+10-7
+10-6
+10-5
+10-4
+10-3
+10
+20
+30
+40
+50
+0
+2
+4
+6
+10-7
+10-6
+10-5
+10-4
+10-3
+10
+20
+30
+40
+50
+0
+2
+4
+6
+PD (%)
+   30 °C
+     0 °C
+  -30 °C
+PA (%)
+ηnet (%)
+APD#1
+APD#2
+PD (%)
+   30 °C
+     0 °C
+  -20 °C
+PA (%)
+ηnet (%)
+(a) 
+(b) 
+Fig. 3. Dark count probability (top) and afterpulse probability (bottom) as a function of
+photon detection efficiency of (a) APD#1 and (b) APD#2 measured for several different
+temperatures.
+non-illuminated gate. With a separate measurement of the detector dark count probability (𝑃𝐷),
+we calculate the afterpulsing probability using the standard method [17,20],
+𝑃𝐴 = (𝑃𝑁 𝐼 − 𝑃𝐷) · 𝑅
+𝑃𝐼 − 𝑃𝑁 𝐼
+,
+(2)
+where 𝑅 = 125 here is the ratio of the gating frequency (1.25 GHz) to the laser illumination
+(10 MHz). Excluding the dark and afterpulse count probabilities, the net single photon detection
+efficiency is given by [7]
+𝜂𝑛𝑒𝑡 = 1
+𝜇ln1 − 𝑃𝑁 𝐼
+1 − 𝑃𝐼
+,
+(3)
+where 𝜇 is the average incident photon number per illumination pulse.
+Figure 3 shows the characterisation results for APD#1 and APD#2. We fixed the amplitude of
+the 1.25 GHz sinusoidal signal at 27.0 V, and measured the relevant parameters as a function
+of the applied direct current (DC) bias, but for clarity the results are plotted as a function of
+the net detection efficiency (𝜂𝑛𝑒𝑡). Each device was measured at several different temperatures,
+while APD#2 could reach only a narrower temperature range due to its cooler compatibility with
+the temperature control driver. Qualitatively, two devices behave similarly. Both dark count
+and afterpulsing probabilities increase with photon detection efficiency, and exhibit opposite
+dependencies on temperature. For both APDs at 𝜂net = 30 %, the afterpulsing probabilities
+are less than 2.3 % at their lowest measurement temperatures with corresponding dark count
+probabilities of 1.25 × 10−6 and 1.6 × 10−6 for APD#1 (-30 ◦C) and APD#2 (-20 ◦C), respectively.
+Moreover, our UNIC-APDs can offer record low afterpulsing probabilities, as summarised for
+APD#1 in Figure 4. At -30 ◦C, APD#1 is able to achieve 5 % and 21.2 % detection efficiencies at
+0.5 % and 1.0 % afterpulsing probabilities. At these afterpulsing probabilities, the maximum
+detection efficiency increases with temperature and reaches 25.3 % and 34.2 % at 30 ◦C. At 5.9 %
+𝑃𝐴, APD#2 has a detection efficiency of 50 % efficiency at 30 ◦C and dark count probability of
+1.1 × 10−4.
+5
+
+-30
+-15
+0
+15
+30
+0
+10
+20
+30
+40
+ PA = 1 %
+ PA = 0.5 %
+T (°C)
+ηnet (%)
+-30
+-15
+0
+15
+30
+0
+10
+20
+30
+40
+ PA = 1 %
+ PA = 0.5 %
+T (°C)
+ηnet (%)
+Fig. 4. Temperature dependencies of photon detection efficiency for APD#1 at the
+given afterpulsing probabilities of 0.5 % (blue) and 1 % (red).
+The maximum count rate is a crucial parameter for a number of applications, for example,
+high bit rate QKD [3] and rapid phase tracking in twin-field QKD [22,23]. To determine their
+maximum count rates, we used a DFB laser transmitting at 1.25 GHz as the illumination source
+and measure the count rate as a function of photon flux. Figure 5 shows an exemplar result
+obtained from APD#1 at a temperature of 30 ◦C with its detection efficiency set to 25.3 % in
+the low flux regime. The detector maintains a linear dependence with incident flux for count
+rates exceeding 100 MHz, while a maximum count rate of 700 MHz is obtained at the few
+photons/pulse regime. We attribute the high count rate to the UNIC’s ability of removing the
+capacitive response and thus allowing discrimination of faint avalanches. From the accompanying
+current measurement, we extract an avalanche charge of 38 fC, comparable to the best value of
+35 fC [9] obtained with the photocurrent measurement method. The ability to detect such weak
+avalanches ensures low afterpulsing probabilities in our UNIC-APDs. APD#2 was measured to
+10-3
+10-2
+10-1
+100
+101
+102
+105
+106
+107
+108
+109
+ Count Rate
+ Photocurrent
+Photon Flux (photon/pulse)
+Count Rate (Hz)
+700MHz
+10-8
+10-7
+10-6
+10-5
+10-4
+ Photocurrent (A)
+Fig. 5. Maximum count rate (blue) and photoncurrent (red) vs incident flux for APD#1.
+6
+
+have a similar avalanche charge as that of APD#1. When setting its efficiency to 50 %, APD#2’s
+avalanche charge rose to 65 fC due to stronger bias applied. Nevertheless, it was still able to
+achieve a maximum count rate of 600 MHz.
+Table 1. Performance comparison of sub-nanosecond gated InGaAs detectors using
+different types of readout circuits.
+𝑃A(%) 𝜂net (%) 𝑃D (gate−1) T (◦C) 𝑓𝑔 (GHz)
+Readout Method
+This work
+1.0
+21.2
+5.4×10−7
+-30
+1.25
+UNIC
+He et al [19]
+1.0
+20.7
+7.6×10−7
+-30
+1.00
+low-pass filter +
+variable width discriminator
+Tada et al [8]
+1.8
+27.7
+8×10−7
+-35
+1.27
+band stop filter
+Fang et al [12]
+2.5
+20
+1.1×10−6
+-30
+1.25
+low-pass filter
+Comandar et al [7]
+2.9
+20
+1.0×10−6
+-30
+1.00
+self-differencing
+Liang et al [21]
+4.5
+20
+3.2×10−6
+-30
+1.25
+reference subtraction
+Table 1 compares our results with those gigahertz-gated detectors equipped with different
+readout circuits. For impartiality, we list just data measured at a fixed temperature of -30 ◦C
+whenever possible. Here, our UNIC-APD achieved an impressive 1% afterpulsing probability
+at 𝜂net = 21.2 %, considerably outperforming most other methods among filtering [8, 12],
+self-differencing [7] and reference subtraction [21]. In terms of detection efficiency, our result
+improves marginally over the previous best [19], but which was achieved with help of an
+uncommon variable width discriminator to mitigate signal distortion by excessive filtering. We
+attribute the outstanding performance of our detectors to low-distortion signal processing by
+UNIC’s.
+It is useful to compare our UNIC-APDs with detectors deployed in QKD systems. In the
+QKD system optimised for secure key rates (SKRs) [3], the room-temperature self-differencing
+detectors featured 𝑓𝑔 = 1 GHz, 𝜂net = 31 %, 𝑃𝐴 = 4.4% and 𝑃𝐷 = 2.25 × 10−4 and a SKR of
+13.72 Mb/s over a 2 dB channel was obtained. Our UNIC-APD could outperform in all these
+parameters. At 30 ◦C and with 𝑃𝐴 = 4.4 %, APD#2 offers a higher efficiency of 49 % efficiency
+and twice lower dark count probability of 9.4 × 10−5, see Fig. 3b. Combined with its high
+count capability, UNIC detectors are expected to allow a SKR exceeding 25 Mb/s over the same
+channel loss. This provides an interesting technological path towards 100 Mb/s via wavelength
+multiplexing.
+Conclusion
+To summarise, we have developed a novel approach of using UNICs for reading out avalanche
+signals from 1.25 GHz sinusoidally gated InGaAs APDs. UNIC-APDs were characterised
+to exhibit excellent performance across the temperature range of −30 – 30 ◦C, and can offer
+>20 % detection efficiency at an ultra low afterpulsing probability of 1 %. This performance,
+together with the circuit’s compactness and manufacturing tolerance, will allow UNIC-APDs a
+considerable potential in QKD applications.
+Disclosures.
+The authors declare that there are no conflicts of interest related to this article.
+Data availability.
+Data underlying the results presented in this paper are not publicly available at this
+time but may be obtained from the authors upon reasonable request.
+7
+
+References
+1.
+A. Wehr and U. Lohr, “Airborne laser scanning-an introduction and overview,” ISPRS J. Photogramm. & Remote.
+Sens. 54, 68–82 (1999).
+2.
+U. Schreiber, A. Schlicht, and K.-H. Haufe, “Systematic biases in laser ranging measurements,” Laser Radar Ranging
+Atmospheric Lidar Tech. II 3865, 64–73 (1999).
+3.
+Z. L. Yuan, A. Plews, R. Takahashi, K. Doi, W. Tam, A. W. Sharpe, A. R. Dixon, E. Lavelle, J. F. Dynes, A. Murakami,
+M. Kujiraoka, M. Lucamarini, Y. Tanizawa, H. Sato, and A. J. Shields, “10 Mb/s quantum key distribution,” J. Light.
+Technol. 36, 3627–3433 (2018).
+4.
+L. Damalakiene, V. Karabanovas, S. Bagdonas, and R. Rotomskis, “Fluorescence-lifetime imaging microscopy for
+visualization of quantum dots’ endocytic pathway,” Int. J. Mol. Sci. 17 (2016).
+5.
+P. Healey, “Optical time domain reflectometry-a performance comparison of the analogue and photon counting
+techniques,” Opt. Quant. Electron. 16, 267–276 (1984).
+6.
+P. Eraerds, M. Legré, J. Zhang, H. Zbinden, and N. Gisin, “Photon counting OTDR: Advantages and limitations,” J.
+Light. Technol. 28, 952–964 (2010).
+7.
+L. C. Comandar, B. Fröhlich, J. F. Dynes, A. W. Sharpe, M. Lucamarini, Z. L. Yuan, R. V. Penty, and A. J. Shields,
+“Gigahertz-gated InGaAs/InP single-photon detector with detection efficiency exceeding 55% at 1550 nm,” J. Appl.
+Phys. 117, 083109 (2015).
+8.
+A. Tada, N. Namekata, and S. Inoue, “Saturated detection efficiency of single-photon detector based on an InGaAs/InP
+single-photon avalanche diode gated with a large-amplitude sinusoidal voltage,” Jpn. J. Appl. Phys. 59, 072004
+(2020).
+9.
+Z. L. Yuan, A. W. Sharpe, J. F. Dynes, A. R. Dixon, and A. J. Shields, “Multi-gigahertz operation of photon counting
+InGaAs avalanche photodiodes,” Appl. Phys. Lett. 96, 071101 (2010).
+10. A. Restelli, J. C. Bienfang, and A. L. Migdall, “Single-photon detection efficiency up to 50% at 1310 nm with an
+InGaAs/InP avalanche diode gated at 1.25 GHz,” Appl. Phys. Lett. 102, 141104 (2013).
+11. N. Namekata, S. Sasamori, and S. Inoue, “800 MHz single-photon detection at 1550-nm using an InGaAs/InP
+avalanche photodiode operated with a sine wave gating,” Opt. Express 51, 10043–10049 (2006).
+12. Y. Q. Fang, W. Chen, T. H. Ao, C. Liu, L. Wang, X. J. Gao, J. Zhang, and J. W. Pan, “InGaAs/InP single-photon
+detectors with 60% detection efficiency at 1550 nm,” Rev. Sci. Instr. 91, 083102 (2020).
+13. K. A. Patel, J. F. Dynes, A. W. Sharpe, Z. L. Yuan, R. V. Penty, and A. J. Shields, “Gigacount/second photon detection
+with InGaAs avalanche photodiodes,” Electron. Lett. 48, 111–113 (2012).
+14. B. E. Kardynal, Z. L. Yuan, and A. J. Shields, “An avalanche-photodiode-based photon-number-resolving detector,”
+Nat. Photon. 2, 425–428 (2008).
+15. O. Thomas, Z. L. Yuan, J. F. Dynes, A. W. Sharpe, and A. J. Shields, “Efficient photon number detection with silicon
+avalanche photodiodes,” Appl. Phys. Lett. 97, 031102 (2010).
+16. M. A. Wayne, J. C. Bienfang, and A. L. Migdall, “Low-noise photon counting above 100 ×106 counts per second
+with a high-efficiency reach-through single-photon avalanche diode system,” Appl. Phys. Lett. 118, 134002 (2021).
+17. N. Namekata, S. Adachi, and S. Inoue, “1.5 GHz single-photon detection at telecommunication wavelengths using
+sinusoidally gated InGaAs/InP avalanche photodiode,” Opt. Express 17, 6275–6282 (2009).
+18. N. Walenta, T. Lunghi, O. Guinnard, R. Houlmann, H. Zbinden, and N. Gisin, “Sine gating detector with simple
+filtering for low-noise infra-red single photon detection at room temperature,” J. Appl. Phys. 112, 063106 (2012).
+19. D. Y. He, S. Wang, W. Chen, Z. Q. Yin, Y. J. Qian, Z. Zhou, G. C. Guo, and Z. F. Han, “Sine-wave gating InGaAs/InP
+single photon detector with ultralow afterpulse,” Appl. Phys. Lett. 110, 111104 (2017).
+20. Z. L. Yuan, B. E. Kardynal, A. W. Sharpe, and A. J. Shields, “High speed single photon detection in the near infrared,”
+Appl. Phys. Lett. 91, 041114 (2007).
+21. Y. Liang, Q. Fei, Z. Liu, K. Huang, and H. Zeng, “Low-noise InGaAs/InP single-photon detector with widely tunable
+repetition rates,” Photon. Res. 7, A1–A6 (2019).
+22. M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate-distance limit of quantum key
+distribution without using quantum repeater,” Nature 557, 400–403 (2018).
+23. L. Zhou, J. Lin, Y. Jing, and Z. Yuan, “Twin-field quantum key distribution without optical frequency dissemination,”
+(2022). ArXiv:2208.09347.
+8
+
diff --git a/Y9AzT4oBgHgl3EQfmv3V/content/tmp_files/load_file.txt b/Y9AzT4oBgHgl3EQfmv3V/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..e47c900dd5228a5828b154c3cb06939f1dcd7b32
--- /dev/null
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf,len=524
+page_content='Ultra-narrowband interference circuits  enable low-noise and high-rate photon counting  for InGaAs/InP avalanche photodiodes  YUANBIN FAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='† TINGTING SHI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='† WEIJIE JI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='1LAI ZHOU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='1  YANG JI 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='3 AND ZHILIANG YUAN1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='*  1Beijing Academy of Quantum Information Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Beijing 100193,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' China  2State Key Laboratory for Superlattices and Microstructures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Institute of Semiconductors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Chinese  Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Beijing 100083,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' China  3College of Materials Science and Opto-Electronic Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' China  †These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  yuanzl@baqis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='cn  Abstract: Afterpulsing noise in InGaAs/InP single photon avalanche photodiodes (APDs) is  caused by carrier trapping and can be suppressed successfully through limiting the avalanche  charge via sub-nanosecond gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Detection of faint avalanches requires an electronic circuit  that is able to effectively remove the gate-induced capacitive response while keeping photon  signals intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Here we demonstrate a novel ultra-narrowband interference circuit (UNIC) that  can reject the capacitive response by up to 80 dB per stage with little distortion to avalanche  signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Cascading two UNIC’s in a readout circuit, we were able to enable high count rate of  up to 700 MC/s and low afterpulsing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='5 % at a detection efficiency of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='3 % for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz  sinusoidally gated InGaAs/InP APDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' At -30 ◦C, we measured 1 % afterpulsing at a detection  efficiency of 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='2 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Introduction  Semiconductor avalanche photodiodes (APD’s) are versatile for weak light detection, with  applications from remote ranging [1,2], quantum communication [3] and fluorescence lifetime  imaging [4] to optical time-domain reflectometry [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  For practical fiber quantum key  distribution (QKD), InGaAs/InP APD’s are the detector of choice because they are compact and  low cost, and allow cryogenic-free or even room-temperature operation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' However, they suffer  from spurious afterpulsing arising from carrier trapping by defects in the multiplication layer,  especially at high detection efficiencies [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' To minimise afterpulsing, an APD can be biased  on for a sub-nanosecond duration only when a photon arrival is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' In doing so, charge per  avalanche can be reduced to the order of 10 fC [9–11], corresponding to a transient current of  less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='1 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Such weak avalanches have to be discriminated through use of a readout circuit  that removes the strong capacitive response to the applied gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Gated InGaAs detectors are  capable of counting photons at up to 60% efficiencies [12] and 1 GHz rate [13] and with photon  number resolution [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Thanks to this success, gating approach has been applied to traditionally  free-running Si devices for performance enhancement [15,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Existing readout circuits include band stop [8,11,17] or low-pass [12,18,19] filtering under  sine-wave gating [11], self-differencing [7, 20], and transient reference cancellation [10, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  While simple for implementation, frequency filtering distorts the avalanche signals due to its  rejection of a sizeable portion of frequency components, thus increasing time jitter and temporal  errors in photon registrations [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Self-differencing [20] and reference cancellation methods [10]  are able to maintain avalanche signal fidelity but may suffer operational complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The former  requires wideband performance for the entire circuitry and thus inconvenience an adjustable  delayline [9] for frequency alignment, while the latter can be unstable because the transient  reference is derived separately from the capacitive response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='01570v1  [quant-ph]  4 Jan 2023  Here we propose and experimentally demonstrate a simple, low-distortion ultra-narrowband  interference circuit (UNIC) that can suppress the capacitive response for a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz gated  InGaAs/InP APD single photon detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The circuit is an asymmetric radio-frequency (RF)  interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' One of its arms contains a narrow band pass filter (BPF) based on surface acoustic  wave resonator (SAW) to retrieve the fundamental wave of the gating signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The filtered wave  then interferes destructively with the same frequency component transmitted via the other arm  through a coupling module, thereby eliminating the capacitive response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' This interference occurs  over the narrow band, so it can provide a broad and continuous pass band in frequency domain to  maintain the avalanche signal with little distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' This allows to achieve ultra-low afterpulsing  probabilities and excellent jitter performance at high detection efficiencies from two InGaAs  APD’s that exhibit capacitive responses of very different amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Detector characterisation setup  35  40  45  101  102  103  104  105  106  0  20  40  60  80  Count  Time (ns)  650 ps  Count  Time (ns)  SG  Laser  DC bias  TDC  AMP  UNIC 1  BSF  AMP  Start    Click  Out  10 MHz  REF IN  DISC  Threshold  Level  AMP  50Ω  50Ω  APD  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz  UNIC 2  (a)  (b)  Illuminated  Non-Illuminated  VOA  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' (a) Single-photon characterisation setup for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz sinusoidally gated  InGaAs/InP APDs using UNICs for avalanche impulse readout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' (b) Histogram of the  photon detection events measured by the characterisation setup (a) on an InGaAs APD  detector that was regulated at a temperature of 30 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The photon detection peak exhibits  a 30 dB width of 650 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' AMP: amplifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' APD: avalanche photodiode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' BSF: band  stop filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' DISC: discriminator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' SG: signal generator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' TDC: time-to-digital converter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  UNIC: ultra-narrowband interference circuit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' VOA: variable optical attenuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Figure 1(a) shows our single photon characterisation setup for InGaAs APDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' A 1550 nm  passively mode-locked laser serves as the light source and provides stable short pulses of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='5 ps  duration at a repetition rate of 10 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The laser output power is monitored by an optical power  meter (EXFO FTB-1750) and its pulse intensity is set by a variable optical attenuator (VOA,  EXFO FTB-3500) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='1 photon/pulse at the fiber input of APD under test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' It provides a 10 MHz  reference to a signal generator (SG) for synthesising a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz sinusoidal wave with up to 27 V  voltage swing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' In combination of a suitable DC bias, this AC signal periodically gates the APD  above its breakdown voltage (60 − 70 V) to achieve single photon sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The APD output is  processed by the readout module consisting of two identical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz UNIC’s, one 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='5 GHz  band stop filter (BSF), and three RF amplifiers (AMPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Amplification of the raw APD signalis  2  is useful as it prevents weak avalanche signals from falling below thermal noise by attenuation  of the first UNIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The readout signal is discriminated by a discriminator for avalanches before  feeding to a time-digital-converter (TDC) with a dead time of 2 ns for time-resolved photon  counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Figure 1(b) is a typical histogram obtained with this setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  APD under test is temperature-regulated using their integrated thermal-electric cooler, which  is driven by a temperature controller (Thorlabs TED200C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' A source-measure unit (Keithley  2635B) provides the DC bias and simultaneously monitors the current flowing through the APD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  In characterising the maximum count rate, we replace the 10 MHz laser with a continuous-wave  distributed feedback laser (DFB) laser, the output of which is carved into 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz, 50 ps pulse  train using an intensity modulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' We use a high speed digital oscilloscope to record the detector  output and extract the count rate through digital discrimination in software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The oscilloscope  method is carefully calibrated at low count rate regimes to be consistent with the hardware  discriminated result using the photon counter (Stanford Research SR400).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  The setup is able to measure dark count probability, afterpulsing probability, detection  efficiency, maximum count rate, avalanche charge and time jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' With no performance screening,  two fiber-pigtailed APDs from different manufacturers were used in this study, named APD#1  and APD#2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Ultra-narrowband interference circuit (UNIC)  0  1  2  3  4  5  90  60  30  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='24  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='26  90  60  30  0  1  0  1  10  5  0  5  10  1  0  1  0  1  10  5  0  5  10  0  1  Insertion  Loss (dB)  Frequency (GHz)  Loss [dB]  Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' (GHz)  Vp-p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='42 V  Vp-p  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='75 V  APD#1  APD#2  Signal (V)  Time (ns)  APD#1  APD#2  Signal (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=')  Time (ns)  (a)  (c)  (b)  (d)  SAW  BPF  IN  OUT  50Ω  50Ω  ATT  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' (a) Schematic for ultranarrow interference circuit (UNIC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' (b) Transmission  spectrum of a heroic UNIC PCB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Inset: Magnified view for region of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='24 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='26 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  (c) Raw capacitive responses from APD#1 (top) and APD#2 (bottom) under identical  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='0 V V𝑝−𝑝 gating;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' (d) Recovered avalanche impulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' ATT: attenuator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' SAW BPF:  surface acoustic wave band pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  With sub-nanosecond gating, a photon induced avalanche is an impulse and has a wide spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  On the other hand, the capacitive response is periodic and has its most energy concentrated at the  3  gating frequency or its higher harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' This spectral difference allows frequency-dependent  signal processing to remove the capacitive response and keep the wide-band impulses intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Figure 2(a) shows a circuit diagram of UNIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' It is an RF interferometer containing two couplers  of 9:1 power splitting ratio, a 𝜋-resistive attenuator (ATT) and surface acoustic wave band pass  filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Two of the ports are terminated by 50 Ω resistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The SAW BPF features a central  frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz, 20-dB passband of 35 MHz, insertion loss of 3 dB, and group delay of  34 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' It filters out the fundamental wave of the gating frequency, which then interferes with the  APD signal transmitted through the other arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The attenuation and differential delay are set to  enable destructive interference for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz frequency component at the UNIC output port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  The UNIC differential delay (Δ𝑡) meets the condition below  Δ𝑡 = 𝑇𝑆𝐴𝑊  𝑔  + 𝛿𝑡 = (𝑁 + 1/2)/ 𝑓𝑔,  (1)  where 𝑇𝑆𝐴𝑊  𝑔  is the group delay of the SAW BPF, 𝛿𝑡 the delay caused by the track length difference  between two arms, 𝑓𝑔 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz the APD gating frequency, and 𝑁 is an integer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' For  a compact circuit, we choose 𝛿𝑡 to be less than the half-wave of the gating signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' With the  SAW device used, 𝑁 = 42 and 𝛿𝑡 = 155 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The resulting UNIC unit has a small footprint of  38 × 15 mm2 on printed circuit boards (PCBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  The large 𝑇𝑆𝐴𝑊  𝑔  brings two additional benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Firstly, it substantially increases the PCB  manufacturing tolerance, as a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='5 mm deviation in the RF track length will just alter the circuit  central frequency by less than 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' This eliminates the requirement of an adjustable delayline  which is required in a self-differencing circuit for precise frequency alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Secondly, it  helps to produce an ultra-narrow band rejection at its designed frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Figure 2(b) shows the  measured transmission spectrum (S21 parameter) of our heroic UNIC PCB, and its inset expands  the frequency section of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='24 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='26 GHz to show the narrowness of the insertion loss dip in the  close proximity of the resonance frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The dip of the heroic (typical) PCB  features a loss of -95 dB (-80 dB), representing a suppression of 80 dB (65 dB) as compared with  the background loss for other frequencies under 2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The dip has a 30 dB linewidth of merely  30 kHz, thus ensuring crucial suppression of the APD gating signal without overly distorting the  avalanche signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The background loss of about 14 dB is caused mainly by the 9:1 couplers and  can be reduced in future with more balanced splitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Cascading two UNIC’s enables a stable 100 dB suppression of the primary gating frequency  and thus provides sufficient performance redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Their attenuation to the avalanche signal is  compensated by using RF amplifiers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Second order harmonics (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='5 GHz) is suppressed  by a band stop filter of conventional LC design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Figure 2(c) shows raw outputs from two different  APD’s under identical sinusoidal gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Their respective capacitive responses are measured  to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='42 V and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='75 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Despite their 4 times differences, UNIC’s can successfully reject the  sinusoidal responses and retrieve avalanches with excellent signal-to-background ratio, as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' For APD#2, we just adjusted the gain of the first AMP to avoid saturation and  distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Results and discussion  Time-resolved photon counting allows precise extraction of the net photon detection efficiency  (𝜂𝑛𝑒𝑡) and the afterpulsing probability (𝑃𝐴), which is defined as the ratio of the total afterpulses  per photon counting event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Figure 1(b) shows a histogram of avalanche events measured for  APD#1 under 10 MHz pulsed excitation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='1 photon/pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The illuminated peak has a  full-width of 1/1000 maximum (30 dB width) of just 650 ps, which is shorter than the gating  period of 800 ps and thus allows low-error clock number assignment that is essential for high  speed QKD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The count at non-illuminated gates arise from detector dark count and afterpulses  and is more than 3 orders of magnitude lower than that of the illuminated gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' We extract  quantities of 𝑃𝐼 and 𝑃𝑁 𝐼, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=', the respective counting probabilities for each illuminated and  4  10-7  10-6  10-5  10-4  10-3  10  20  30  40  50  0  2  4  6  10-7  10-6  10-5  10-4  10-3  10  20  30  40  50  0  2  4  6  PD (%)  30 °C  0 °C  30 °C  PA (%)  ηnet (%)  APD#1  APD#2  PD (%)  30 °C  0 °C  20 °C  PA (%)  ηnet (%)  (a)  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Dark count probability (top) and afterpulse probability (bottom) as a function of  photon detection efficiency of (a) APD#1 and (b) APD#2 measured for several different  temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  non-illuminated gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' With a separate measurement of the detector dark count probability (𝑃𝐷),  we calculate the afterpulsing probability using the standard method [17,20],  𝑃𝐴 = (𝑃𝑁 𝐼 − 𝑃𝐷) · 𝑅  𝑃𝐼 − 𝑃𝑁 𝐼  ,  (2)  where 𝑅 = 125 here is the ratio of the gating frequency (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz) to the laser illumination  (10 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Excluding the dark and afterpulse count probabilities, the net single photon detection  efficiency is given by [7]  𝜂𝑛𝑒𝑡 = 1  𝜇ln1 − 𝑃𝑁 𝐼  1 − 𝑃𝐼  ,  (3)  where 𝜇 is the average incident photon number per illumination pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Figure 3 shows the characterisation results for APD#1 and APD#2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' We fixed the amplitude of  the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz sinusoidal signal at 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='0 V, and measured the relevant parameters as a function  of the applied direct current (DC) bias, but for clarity the results are plotted as a function of  the net detection efficiency (𝜂𝑛𝑒𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Each device was measured at several different temperatures,  while APD#2 could reach only a narrower temperature range due to its cooler compatibility with  the temperature control driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Qualitatively, two devices behave similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Both dark count  and afterpulsing probabilities increase with photon detection efficiency, and exhibit opposite  dependencies on temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' For both APDs at 𝜂net = 30 %, the afterpulsing probabilities  are less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='3 % at their lowest measurement temperatures with corresponding dark count  probabilities of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 × 10−6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='6 × 10−6 for APD#1 (-30 ◦C) and APD#2 (-20 ◦C), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Moreover, our UNIC-APDs can offer record low afterpulsing probabilities, as summarised for  APD#1 in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' At -30 ◦C, APD#1 is able to achieve 5 % and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='2 % detection efficiencies at  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='5 % and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='0 % afterpulsing probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' At these afterpulsing probabilities, the maximum  detection efficiency increases with temperature and reaches 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='3 % and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='2 % at 30 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' At 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='9 %  𝑃𝐴, APD#2 has a detection efficiency of 50 % efficiency at 30 ◦C and dark count probability of  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='1 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  5  30  15  0  15  30  0  10  20  30  40  PA = 1 %  PA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='5 %  T (°C)  ηnet (%)  30  15  0  15  30  0  10  20  30  40  PA = 1 %  PA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='5 %  T (°C)  ηnet (%)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Temperature dependencies of photon detection efficiency for APD#1 at the  given afterpulsing probabilities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='5 % (blue) and 1 % (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  The maximum count rate is a crucial parameter for a number of applications, for example,  high bit rate QKD [3] and rapid phase tracking in twin-field QKD [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' To determine their  maximum count rates, we used a DFB laser transmitting at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz as the illumination source  and measure the count rate as a function of photon flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Figure 5 shows an exemplar result  obtained from APD#1 at a temperature of 30 ◦C with its detection efficiency set to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='3 % in  the low flux regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The detector maintains a linear dependence with incident flux for count  rates exceeding 100 MHz, while a maximum count rate of 700 MHz is obtained at the few  photons/pulse regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' We attribute the high count rate to the UNIC’s ability of removing the  capacitive response and thus allowing discrimination of faint avalanches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' From the accompanying  current measurement, we extract an avalanche charge of 38 fC, comparable to the best value of  35 fC [9] obtained with the photocurrent measurement method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' The ability to detect such weak  avalanches ensures low afterpulsing probabilities in our UNIC-APDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' APD#2 was measured to  10-3  10-2  10-1  100  101  102  105  106  107  108  109  Count Rate  Photocurrent  Photon Flux (photon/pulse)  Count Rate (Hz)  700MHz  10-8  10-7  10-6  10-5  10-4  Photocurrent (A)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Maximum count rate (blue) and photoncurrent (red) vs incident flux for APD#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  6  have a similar avalanche charge as that of APD#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' When setting its efficiency to 50 %, APD#2’s  avalanche charge rose to 65 fC due to stronger bias applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Nevertheless, it was still able to  achieve a maximum count rate of 600 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Performance comparison of sub-nanosecond gated InGaAs detectors using  different types of readout circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  𝑃A(%) 𝜂net (%) 𝑃D (gate−1) T (◦C) 𝑓𝑔 (GHz)  Readout Method  This work  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='0  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='4×10−7  30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25  UNIC  He et al [19]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='0  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='6×10−7  30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='00  low-pass filter +  variable width discriminator  Tada et al [8]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='8  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='7  8×10−7  35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='27  band stop filter  Fang et al [12]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='5  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='1×10−6  30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25  low-pass filter  Comandar et al [7]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='9  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='0×10−6  30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='00  self-differencing  Liang et al [21]  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='5  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='2×10−6  30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25  reference subtraction  Table 1 compares our results with those gigahertz-gated detectors equipped with different  readout circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' For impartiality, we list just data measured at a fixed temperature of -30 ◦C  whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Here, our UNIC-APD achieved an impressive 1% afterpulsing probability  at 𝜂net = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='2 %, considerably outperforming most other methods among filtering [8, 12],  self-differencing [7] and reference subtraction [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' In terms of detection efficiency, our result  improves marginally over the previous best [19], but which was achieved with help of an  uncommon variable width discriminator to mitigate signal distortion by excessive filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' We  attribute the outstanding performance of our detectors to low-distortion signal processing by  UNIC’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  It is useful to compare our UNIC-APDs with detectors deployed in QKD systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' In the  QKD system optimised for secure key rates (SKRs) [3], the room-temperature self-differencing  detectors featured 𝑓𝑔 = 1 GHz, 𝜂net = 31 %, 𝑃𝐴 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='4% and 𝑃𝐷 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 × 10−4 and a SKR of  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='72 Mb/s over a 2 dB channel was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Our UNIC-APD could outperform in all these  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' At 30 ◦C and with 𝑃𝐴 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='4 %, APD#2 offers a higher efficiency of 49 % efficiency  and twice lower dark count probability of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='4 × 10−5, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Combined with its high  count capability, UNIC detectors are expected to allow a SKR exceeding 25 Mb/s over the same  channel loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' This provides an interesting technological path towards 100 Mb/s via wavelength  multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Conclusion  To summarise, we have developed a novel approach of using UNICs for reading out avalanche  signals from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz sinusoidally gated InGaAs APDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' UNIC-APDs were characterised  to exhibit excellent performance across the temperature range of −30 – 30 ◦C, and can offer  >20 % detection efficiency at an ultra low afterpulsing probability of 1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' This performance,  together with the circuit’s compactness and manufacturing tolerance, will allow UNIC-APDs a  considerable potential in QKD applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  The authors declare that there are no conflicts of interest related to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Data underlying the results presented in this paper are not publicly available at this  time but may be obtained from the authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  7  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Wehr and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lohr, “Airborne laser scanning-an introduction and overview,” ISPRS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Photogramm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' & Remote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 54, 68–82 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Schreiber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Schlicht, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Haufe, “Systematic biases in laser ranging measurements,” Laser Radar Ranging  Atmospheric Lidar Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' II 3865, 64–73 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Yuan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Plews, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Takahashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Doi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Tam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Sharpe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Dixon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lavelle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Dynes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Murakami,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Kujiraoka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lucamarini, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Tanizawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Sato, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Shields, “10 Mb/s quantum key distribution,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 36, 3627–3433 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Damalakiene, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Karabanovas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Bagdonas, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Rotomskis, “Fluorescence-lifetime imaging microscopy for  visualization of quantum dots’ endocytic pathway,” Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 17 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Healey, “Optical time domain reflectometry-a performance comparison of the analogue and photon counting  techniques,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 16, 267–276 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Eraerds, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Legré, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Zbinden, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Gisin, “Photon counting OTDR: Advantages and limitations,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 28, 952–964 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Comandar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Fröhlich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Dynes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Sharpe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lucamarini, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Yuan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Penty, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Shields,  “Gigahertz-gated InGaAs/InP single-photon detector with detection efficiency exceeding 55% at 1550 nm,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 117, 083109 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Tada, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Namekata, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Inoue, “Saturated detection efficiency of single-photon detector based on an InGaAs/InP  single-photon avalanche diode gated with a large-amplitude sinusoidal voltage,” Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 59, 072004  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Yuan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Sharpe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Dynes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Dixon, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Shields, “Multi-gigahertz operation of photon counting  InGaAs avalanche photodiodes,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 96, 071101 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Restelli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Bienfang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Migdall, “Single-photon detection efficiency up to 50% at 1310 nm with an  InGaAs/InP avalanche diode gated at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='25 GHz,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 102, 141104 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Namekata, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Sasamori, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Inoue, “800 MHz single-photon detection at 1550-nm using an InGaAs/InP  avalanche photodiode operated with a sine wave gating,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Express 51, 10043–10049 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Fang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Ao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Pan, “InGaAs/InP single-photon  detectors with 60% detection efficiency at 1550 nm,” Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Instr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 91, 083102 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Patel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Dynes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Sharpe, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Yuan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Penty, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Shields, “Gigacount/second photon detection  with InGaAs avalanche photodiodes,” Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 48, 111–113 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Kardynal, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Yuan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Shields, “An avalanche-photodiode-based photon-number-resolving detector,”  Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 2, 425–428 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Thomas, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Dynes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Sharpe, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Shields, “Efficient photon number detection with silicon  avalanche photodiodes,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 97, 031102 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Wayne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Bienfang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Migdall, “Low-noise photon counting above 100 ×106 counts per second  with a high-efficiency reach-through single-photon avalanche diode system,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 118, 134002 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Namekata, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Adachi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Inoue, “1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='5 GHz single-photon detection at telecommunication wavelengths using  sinusoidally gated InGaAs/InP avalanche photodiode,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Express 17, 6275–6282 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Walenta, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lunghi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Guinnard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Houlmann, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Zbinden, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Gisin, “Sine gating detector with simple  filtering for low-noise infra-red single photon detection at room temperature,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 112, 063106 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' He, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Yin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Qian, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Zhou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Guo, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Han, “Sine-wave gating InGaAs/InP  single photon detector with ultralow afterpulse,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 110, 111104 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Yuan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Kardynal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Sharpe, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Shields, “High speed single photon detection in the near infrared,”  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 91, 041114 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Liang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Fei, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Huang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Zeng, “Low-noise InGaAs/InP single-photon detector with widely tunable  repetition rates,” Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' 7, A1–A6 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lucamarini, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Dynes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Shields, “Overcoming the rate-distance limit of quantum key  distribution without using quantum repeater,” Nature 557, 400–403 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Jing, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' Yuan, “Twin-field quantum key distribution without optical frequency dissemination,”  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content=' ArXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='09347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
+page_content='  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfmv3V/content/2301.01570v1.pdf'}
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+arXiv:2301.04663v1  [astro-ph.SR]  11 Jan 2023
+ACTA ASTRONOMICA
+Vol. 72 (2022) pp. 1–12
+The OGLE Collection of Variable Stars.
+Over 2600 δ Scuti Stars in the Small Magellanic Cloud*
+I. S o s z y ´n s k i1, A. U d a l s k i1, J. S k o w r o n1, P. P i e t r u k o w i c z1 ,
+M. K. Szyma ´nski1 , R. Poleski1, D. M. Skowron1 , S. Kozłowski1 ,
+P. Mróz1, P. Iwanek1, M. Wrona1, K. Ulaczyk2,1, K. Rybicki3.1,
+and M. G r o m a d z k i1
+1Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa,
+Poland
+2 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry,
+CV4 7AL, UK
+3Department of Particle Physics and Astrophysics, Weizmann Institute of Science,
+Rehovot 76100, Israel
+Received January 13, 2023
+ABSTRACT
+We present the first-ever collection of δ Scuti stars found over the entire area of the Small Mag-
+ellanic Cloud (SMC). The sample consists of 2810 variables of which over 2600 objects belong to
+the SMC while the remaining stars are most likely members of the Milky Way’s halo. The sample
+has been divided into 2733 singlemode and 77 multimode pulsators. We provide observational pa-
+rameters (pulsation periods, mean magnitudes, amplitudes, Fourier coefficients) of all δ Sct stars and
+the long-term I- and V-band time-series photometric measurements collected during the fourth phase
+of the Optical Gravitational Lensing Experiment (OGLE-IV).
+Key words: Stars: variables: delta Scuti – Stars: oscillations – Magellanic Clouds – Catalogs
+1.
+Introduction
+δ Scuti variables are intermediate-mass stars pulsating in low-order radial or
+non-radial modes driven by the κ mechanism operating in the He II ionization
+zone (Chevalier 1971). This class includes young and intermediate-age stars on
+the pre-main sequence, main sequence, and post-main sequence, as well as stars
+belonging to the old population that are probably merged binaries (Breger 2000).
+The latter group is referred to as SX Phoenicis stars. The pulsation periods of δ Sct
+*Based on observations obtained with the 1.3-m Warsaw telescope at the Las Campanas Observa-
+tory of the Carnegie Institution for Science.
+
+2
+A. A.
+variables are shorter than 0.3 d, while the amplitudes reach 1 mag in the V band,
+although most stars of this type exhibit much smaller amplitudes, down to the sub-
+millimagnitude level (Balona and Dziembowski 2011).
+δ Sct stars, like other classical pulsators, follow period–luminosity (PL) and
+period–luminosity–color (PLC) relations. The Large and Small Magellanic Cloud
+(LMC and SMC) play a crucial role in studying the PL relations obeyed by various
+types of pulsating stars (e.g., Leavitt and Pickering 1912, Glass and Lloyd Evans
+1981, Madore 1982, Udalski et al. 1999, Muraveva et al. 2015). The PL relations
+for δ Sct stars were also investigated based on variables detected in the Magellanic
+Clouds (e.g., Poleski et al. 2010, McNamara 2011, Martínez-Vázquez 2022), but
+these studies were limited by a small number of known δ Sct stars in the Clouds,
+especially in the SMC.
+The first δ Sct variables in the SMC were discovered as a by-product of a
+search for RR Lyr stars in the OGLE-II photometric database. Soszy´nski et al.
+(2002) published a list of 19 short-period variables (with periods in the range 0.16–
+0.57 d) suggesting that most of them are δ Sct stars. In the following years, about
+half of these stars were re-classified as RR Lyr stars or Cepheids (see Section 4 for
+the definition of the boundary between classical Cepheids and δ Sct stars), but eight
+of them turned out to be bona fide δ Sct variables and are included in the present
+work. One more δ Sct star from the OGLE project was reported by Pietrukowicz
+(2018). The second, and so far the last, sample of δ Sct stars in the SMC was
+published by Martínez-Vázquez et al. (2021). They used deep photometry of the
+SMC globular cluster NGC 419 obtained by the Gemini South telescope to detect
+54 δ Sct stars of which six objects are probable cluster members and 48 are field
+variables. The Gaia DR3 catalog of main-sequence oscillators (Gaia Collaboration
+et al. 2022) contains over 50 δ Sct stars in the region of the sky occupied by the
+SMC, but all these pulsators belong to the Milky Way halo and are located in the
+foreground of the SMC.
+Thus, only 63 δ Sct stars in the SMC have been known until now. With this
+paper, we release the first catalog of δ Sct variables found over the entire area of
+the SMC. Our sample consists of 2810 pulsators of which over 2600 objects are
+likely members of the SMC. This is part of the OGLE Collection of Variable Stars
+(OCVS), currently containing over a million manually classified variable stars in
+the Milky Way and Magellanic Clouds. In the next paper in this series, we will
+present the OGLE collection of over 14,000 δ Sct pulsators in the LMC.
+This work is structured as follows. In Section 2, we present a summary of the
+OGLE survey of the Magellanic Clouds. Section 3 is devoted to a brief review of
+the variable star selection and classification procedures. In Section 4, we discuss the
+criteria that can be used to distinguish between δ Sct stars and classical Cepheids.
+The OGLE collection of δ Sct variables in the SMC is described in Section 5. The
+completeness of our catalog is discussed in Section 6. We close the paper with a
+summary in Section 7.
+
+Vol. 72
+3
+2.
+OGLE Observations of the Magellanic System
+The OGLE survey uses a dedicated 1.3-m Warsaw Telescope equipped with a
+large format mosaic CCD camera with a field of view of 1.4 square degrees and
+a pixel scale of 0.26 arcsec. The telescope is located at Las Campanas Observa-
+tory, Chile, operated by the Carnegie Institution for Science. Our collection of
+δ Sct stars is based on photometric data from the fourth phase of the OGLE project
+(OGLE-IV, Udalski et al. 2015), which has been operating since 2010 until today.
+In this work, we use the observations obtained before March 2020, when the War-
+saw Telescope was temporarily closed due to the COVID-19 pandemic.
+The OGLE observations are carried out in the I and V photometric bands,
+closely resembling those from the Cousins-Johnson standard systems. From several
+dozen to over 1000 observations per star (typically 600-700) have been acquired in
+the I-band and from several to over 300 (typically 40-60) data points have been
+gathered in the V-band light curves. The uniform exposure time of 150 s resulted
+in the saturation limit of the OGLE photometry at the level of about I ≈ 13 mag
+and the sensitivity limit at about I ≈ 21.5 mag.
+The total sky area monitored by the OGLE survey in the region of the Mag-
+ellanic System is 765 square degrees. The OGLE footprint covers both galaxies
+together with the Magellanic Bridge and vast regions on their periphery. For the
+purposes of this work, we adopted the celestial meridian of 2.h8 as an arbitrary on-
+sky boundary between the LMC and SMC regions. The same value was selected
+to separate the LMC and SMC RR Lyr stars in the OCVS (Soszy´nski et al. 2016).
+In the SMC area of about 280 square degrees, OGLE regularly observes about 14
+million stars belonging to both the SMC and the halo of the Milky Way.
+3.
+Selection and Classification of δ Sct Stars
+We began our search for δ Sct variables in the SMC with a massive period
+analysis for all 14 million I-band light curves stored in the OGLE-IV databases.
+We used the FNPEAKS code† to span the frequency domain from 0 to 100 cycles
+per day with a step of 5 · 10−5 cycles per day. For each light curve, the period
+with the highest signal-to-noise ratio was considered to be dominant. Then, this
+dominant frequency and its harmonics were subtracted from the light curve and the
+period search was repeated on the residual data.
+In the next step of the procedure, we used both I- and V-band OGLE-IV photo-
+metric data to select and classify δ Sct stars in the SMC. The much smaller number
+of the OGLE observations in the V-band was compensated by the lower noise of
+the V-band light curves compared to the I-band ones. Fig. 1 shows the I- and V-
+band light curves of nine example δ Sct variables arranged in descending pulsation
+periods. Note that the shortest-period (and therefore the faintest) stars have a signif-
+†http://helas.astro.uni.wroc.pl/deliverables.php?lang=en&active=fnpeaks
+
+4
+A. A.
+Fig. 1. OGLE-IV I-band (red) and V-band (blue) light curves of nine example δ Sct stars in the SMC.
+icant scatter of the data points in the I-band, while the V-band light curves of these
+objects are much less dispersed. However, every δ Sct variable identified based
+on the V-band data was later confirmed with the I-band observations, although in
+some cases the I-band light curves are very noisy.
+Our variability classification was primarily based on the shapes of the I- and V-
+band light curves. For single-periodic variables, we filtered out stars with sinusoidal
+light curves and obvious eclipsing binaries. For multi-periodic objects, we took into
+account the characteristic period ratios of multimode pulsators.
+In the final stage of our classification procedure, we examined the positions of
+the candidate pulsation stars in the color–magnitude diagram depicted in Fig. 2,
+
+Vol. 72
+5
+Fig. 2. (V − I)0 vs. I0 color–magnitude diagram for δ Sct stars in the SMC (blue points). The
+background gray points show stars from the subfield SMC719.11.
+where the I-band magnitudes and V − I color index were corrected for the inter-
+stellar extinction using the most accurate reddening maps of the Magellanic Clouds
+recently published by Skowron et al. (2021). Most of the stars with the dereddened
+color index (V − I)0 < 0.2 mag or (V − I)0 > 0.7 mag, i.e., outside the δ Sct in-
+stability strip, were removed from the list. However, as can be seen in Fig. 2, we
+decided to keep some too-blue or too-red stars on the list, because their light curve
+morphology suggested that these objects could be genuine δ Sct variables blended
+with nearby unresolved stars. Nonetheless, the objects with atypical colors should
+be treated with caution as uncertain candidates for δ Sct stars.
+Finally, our collection of δ Sct stars detected toward the SMC consists of 2810
+objects. In addition, the OCVS has been enriched with nine new classical Cepheids
+and 123 RR Lyr stars identified as a by-product of the present search for δ Sct
+stars or a result of cross-matching the OGLE database with the Gaia DR3 catalog
+of Cepheids (Ripepi et al. 2022) and RR Lyr stars (Clementini et al. 2022). A more
+
+6
+A. A.
+detailed comparison of the Gaia DR3 and OGLE catalogs in the area of the Mag-
+ellanic System will be discussed in the forthcoming paper presenting the OGLE
+collection of δ Sct stars in the LMC (Soszy´nski et al. in prep.).
+4.
+Boundary Between δ Sct Stars and Classical Cepheids
+Population I δ Sct stars and classical Cepheids occupy common sequences
+in the PL (Fig. 3), color–magnitude, period–radius (e.g., Fernie 1992), Petersen
+(e.g., Poleski et al. 2010) and other diagrams. In fact, it is a matter of convention
+what value of the pulsation period is adopted to separate both types of variable
+stars. Various authors proposed different maximum periods of δ Sct stars. For ex-
+ample, in the General Catalogue of Variable Stars (Samus’ et al. 2017) it is 0.2 d,
+Breger (2000) defined δ Sct variables as pulsators with periods shorter than 0.25 d,
+Rodríguez et al. (2000) adopted 0.3 d, Handler (2009) – 0.33 d (8 hr), while Catelan
+and Smith (2015) – 0.42 d.
+One might ask if it is possible to establish a “natural” boundary period sepa-
+rating δ Sct stars from classical Cepheids. Both types of variables may pulsate,
+among others, in the fundamental mode (F), first-overtone (1O), second-overtone
+(2O), or in the mix of these radial modes. All but one of these combinations form
+continuity in the vicinity of the Cepheid–δ Sct border. The exception is a class
+of single-mode F pulsators that show a natural gap between δ Sct stars and clas-
+sical Cepheids. OGLE-SMC-CEP-1899 (PF = 0.842 d; Soszy´nski et al. 2010) is
+a classical Cepheids oscillating solely in the F mode with the shortest known pe-
+riod, while OGLE-LMC-DSCT-0617 (PF = 0.299 d; Poleski et al. 2010) is the
+longest-period single-mode δ Sct star. Currently, no known Population I single-
+mode F-mode star from the Cepheid instability strip pulsates with a period in the
+range of 0.3–0.8 d‡. For this reason, all Population I radially oscillating stars (both
+single- and multimode pulsators) with an F-mode period below 0.3 d are classified
+in our collection as δ Sct variables. Consequently, the 1O boundary period between
+δ Sct stars and Cepheids was fixed at P1O = 0.23 d (corresponding to the period
+ratio P1O/PF of about 0.77).
+With the strict definition of different types of pulsating stars, we could extend
+the OCVS by nine short-period classical Cepheids in the SMC. One F/1O double-
+mode and eight 1O single-mode pulsators have 1O periods in the range 0.23–0.9 d.
+The OCVS (Soszy´nski et al. 2019) currently contains 4954 carefully selected clas-
+sical Cepheids in the SMC.
+‡Of course, the Population II pulsators, like RR Lyr stars or anomalous Cepheids, may have
+periods in this range, but these stars can be isolated using other properties, e.g., in the PL diagram,
+they lie below the classical Cepheid–δ Sct ridge.
+
+Vol. 72
+7
+Fig. 3. Period–luminosity (upper panel) and period–Wesenheit index (lower panel) diagrams for
+classical pulsators in the SMC. Green points show classical Cepheids, violet points – anomalous
+Cepheids, red points – type II Cepheids, orange points – RR Lyr stars, and blue points – δ Sct stars.
+Lighter colors indicate stars oscillating in higher modes. Wesenheit index is a reddening-free function
+defined as WI = I −1.55(V −I) (Madore 1982).
+
+8
+A. A.
+5.
+The OGLE Collection of δ Sct Stars in the SMC
+Our collection of 2810 δ Sct variables in the SMC is available at the OGLE
+Internet Archive:
+https://ogle.astrouw.edu.pl → OGLE Collection of Variable Stars
+https://www.astrouw.edu.pl/ogle/ogle4/OCVS/smc/dsct/
+The sample was divided into 2733 singlemode and 77 multimode pulsators.
+The latter group contains objects with well-defined secondary or tertiary periods
+that may be associated with additional radial or non-radial pulsation modes. At
+least 160 stars in our sample are members of our Galaxy – these are likely SX Phe
+variables in the halo of the Milky Way located in front of the SMC. Fig. 4 shows
+on-sky maps of δ Sct stars from our collection. The left panel presents the spatial
+distribution of probable members of the SMC, while the right panel shows the
+positions of the putative Milky Way SX Phe stars. The line separating these two
+populations was arbitrarily set at 1.5 mag above the mean PL relation for the SMC
+δ Sct variables (Fig. 3). More sophisticated methods would probably divide the
+Galactic and SMC populations more efficiently, however, one should remember
+that this separation cannot be done unequivocally because the Magellanic Clouds
+are submerged in the Milky Way halo – old stellar populations in both galaxies
+smoothly transition into each other.
+The full list of δ Sct stars with their equatorial coordinates (J2000.0), classifi-
+cation (singlemode, multimode), internal designations in the OGLE-IV, OGLE-III,
+and OGLE-II databases (if available), and the cross-matches with the International
+Variable Star Index (VSX; Watson et al. 2006) are provided in the ident.dat file.
+In total, we identified only 11 common stars between our collection and the VSX
+catalog – all of them are unquestionable members of the Milky Way.
+The file dsct.dat contains observation parameters of the stars: their intensity-
+averaged mean magnitudes in the I- and V-bands, up to three pulsation periods
+derived with the TATRY code (Schwarzenberg-Czerny 1996), epochs of the maxi-
+mum light, I-band peak-to-peak amplitudes, and Fourier coefficients. The periods
+are very precisely determined thanks to the long timespan of the OGLE-IV light
+curves, however, it cannot be ruled out that in the case of some the most noisy light
+curves our approach produced daily aliases of the true periods.
+The OGLE-IV time-series photometry in the I- and V-bands are stored in the
+directory phot/. Obvious outlying points (deviating by more than ±4 sigma from
+the fitted model) were removed from the light curves, but it was verified before-
+hand whether these points are due to, for example, additional eclipses. We did not
+find any δ Sct stars with eclipses in the SMC. Finding charts can be found in the
+directory fcharts/. These are 60′′ ×60′′ subframes of the I-band reference images,
+oriented with North at the top and East to the left.
+
+Vol. 72
+9
+Fig. 4. On-sky distributions of δ Sct stars toward the SMC. Left panel shows positions of over 2600 probable members of the SMC. Right panel shows δ Sct
+variables that likely belong to the Milky Way halo. The gray area shows the OGLE footprint in the Magellanic System region.
+
+10
+A. A.
+Fig. 5. Luminosity–amplitude diagram for δ Sct stars in the SMC.
+6.
+Completeness of the Catalog
+The completeness of our collection is strongly limited by the brightness, am-
+plitudes, and light curve shapes of δ Sct stars. Figs. 2 and 3 show that the number
+of variables in our sample drops sharply for mean magnitudes I > 21.2 mag due to
+the sensitivity limit of the OGLE photometry.
+We cross-matched our collection with the list of 54 δ Sct variables discovered
+with the Gemini South telescope in the field surrounding the SMC globular cluster
+NGC 419 (Martínez-Vázquez et al. 2021). We found only one common object
+in both catalogs – V46 = OGLE-SMC-DSCT-2067 – which is the brightest δ Sct
+star detected by Martínez-Vázquez et al. (2021). The remaining δ Sct variables
+identified with the 8.1-meter Gemini telescope are below the detection limit of the
+1.3-meter Warsaw telescope used by the OGLE survey.
+The vast majority of δ Sct variables are small-amplitude pulsators (Breger
+2000, Balona and Dziembowski 2011), so our collection must be the tip of the
+iceberg because we had the opportunity to discover only the high-amplitude δ Sct
+stars in the SMC. Fig. 5 shows the I-band peak-to-peak amplitudes plotted against
+the mean I-band magnitudes of our sample. It is obvious that the amplitude de-
+tection limits in the OGLE data are strongly correlated with the brightness of the
+pulsating candidates: for stars with the luminosities of about I = 19 mag, the small-
+est detectable amplitudes are of the order of 0.04 mag, for I = 20 mag stars, the
+amplitude detection limit rises to 0.1 mag, and for the faintest stars in our collec-
+
+Vol. 72
+11
+tion (I = 21 mag), the light curve amplitude should be larger than about 0.2 mag
+to be recognized as a δ Sct variable. On the other hand, our sample of the Galactic
+δ Sct stars (I < 18.5 mag) should be much more complete because we could detect
+variables with amplitudes as small as 0.01 mag in the I-band.
+We used the δ Sct stars with double entries in the OGLE-IV database to es-
+timate the completeness of our collection within the brightness and amplitude de-
+tection limits. These objects are located in the overlapping parts of the adjacent
+OGLE fields, so we had the opportunity to identify them twice during the selection
+and classification process. We found that 311 out of 2810 δ Sct stars included in
+the final version of our collection have their duplicates in the OGLE-IV database,
+so we had a chance to detect 622 counterparts. In fact, 449 objects from this list
+were independently classified as δ Sct variables which correspond to the catalog
+completeness slightly above 60%. However, if we consider only stars brighter than
+I = 19 mag, the formal completeness of our collection increases to about 85%.
+Nonetheless, one should remember that the bulk of δ Sct stars in the SMC have
+luminosities and amplitudes below the detection limits of the OGLE project.
+7.
+Summary
+We presented the first-ever catalog of δ Sct stars found in the entire area of the
+SMC. The number of known δ Sct variables in this galaxy has increased from about
+60 to over 2600. Additionally, our collection contains at least 160 Galactic SX Phe
+stars located in the foreground of the SMC. The most obvious application of our
+collection includes the examination of the PL and PLC relations obeyed by δ Sct
+stars in the metal-poor environment of the SMC compared to the more metal-rich
+variables in the LMC and Milky Way. The three-dimensional distribution of δ Sct
+variables can be analyzed to better understand the structure of the SMC. Moreover,
+the long-term photometric time series produced by the OGLE project can be used
+to study the stability of stellar pulsations over long time scales.
+This work demonstrates the great versatility of OGLE photometric data. The
+OCVS contains variable stars with periods ranging from minutes to years, lumi-
+nosities from 12 mag to over 22 mag in the V-band, and amplitudes from milli-
+magnitudes to several magnitudes. We expect the next breakthrough in the field
+of variable stars in the Magellanic Clouds to come after the launch of the LSST
+project on the Rubin Observatory (Hambleton et al. 2022).
+Acknowledgements. This work has been supported by the National Science
+Centre, Poland, grant no. 2022/45/B/ST9/00243. MG is supported by the EU Hori-
+zon 2020 research and innovation programme under grant agreement no. 101004719.
+This research has made use of the International Variable Star Index (VSX) database,
+operated at AAVSO, Cambridge, Massachusetts, USA.
+
+12
+A. A.
+REFERENCES
+Balona, L.A., and Dziembowski, W.A. 2011, MNRAS, 417, 591. 2011MNRAS.417..591B
+Breger M. 2000, in Breger M., Montgomery M., eds, ASP Conf. Ser. Vol. 210, Delta Scuti and Related
+Stars. Astron. Soc. Pac., San Francisco, p. 3.
+Catelan, M., and Smith, H.A. 2015, Pulsating Stars (New York: Wiley).
+Chevalier, C. 1971, A&A, 14, 24.
+Clementini, G., Ripepi, V., Garofalo, A., et al. 2022, arXiv:2206.06278.
+Fernie, J.D. 1992, AJ, 103, 1647.
+Gaia Collaboration, De Ridder, J., Ripepi, V., et al. 2022, arXiv e-prints, arXiv:2206.06075.
+Glass, I.S., and Lloyd Evans, T. 1981, Nature, 291, 303.
+Hambleton, K.M., Bianco, F.B., Street, R., et al. . 2022, arXiv:2208.04499.
+Handler, G. 2009, in AIP Conf. Ser. 1170, Stellar Pulsation: Challenges for Theory and Observation,
+ed. J. A. Guzik & P. A. Bradley (Melville, NY: AIP), 403.
+Leavitt, H.S., and Pickering, E.C. 1912, Harvard Coll. Obs. Circ., 173, 1.
+Madore, B.F. 1982, ApJ, 253, 575.
+Martínez-Vázquez, C.E., Salinas, R., and Vivas, A.K. 2021, AJ, 161, 120.
+Martínez-Vázquez, C.E., Salinas, R., Vivas, A.K., and Catelan, M. 2022, ApJ, 940, L25.
+McNamara, D.H. 2011, AJ, 142, 110.
+Muraveva, T., Palmer, M., Clementini, G., et al. 2015, ApJ, 807, 127.
+Pietrukowicz, P. 2018, in The RR Lyrae 2017 Conference. Revival of the Classical Pulsators: from
+Galactic Structure to Stellar Interior Diagnostics., Vol. 6., ed. R. Smolec, K. Kinemuchi, &
+R.I. Anderson (Poland: Proc. Polish Astronomical Soc.), 258.
+Poleski, R., Soszy´nski, I., Udalski, A., et al. 2010, Acta Astron., 60, 1.
+Ripepi, V., Clementini, G., Molinaro, R., et al. 2022, arXiv:2206.06212.
+Rodríguez, E., López-González, M.J., and López de Coca, P. 2000, A&AS, 144, 469.
+Samus’, N.N., Kazarovets, E.V., Durlevich, O.V., Kireeva, N.N., and Pastukhova, E.N. 2017,
+Astronomy Reports, 61, 80.
+Schwarzenberg-Czerny, A. 1996, ApJ, 460, L107.
+Skowron, D.M., Skowron, J., Udalski, A., et al. 2021, ApJS, 252, 23.
+Soszy´nski, I., Udalski, A., Szyma´nski, M., et al. 2002, Acta Astron., 52, 369.
+Soszy´nski, I., Poleski, R., Udalski, A., et al. 2010, Acta Astron., 60, 17.
+Soszy´nski, I., Udalski, A., Szyma´nski, M.K., et al. 2016, Acta Astron., 66, 131.
+Soszy´nski, I., Udalski, A., Szyma´nski, M.K., et al. 2019, Acta Astron., 69, 87.
+Udalski, A., Szyma´nski, M., Kubiak, M., Pietrzy´nski, G., Soszy´nski, I., Wo´zniak, P., and ˙Zebru´n, K.
+1999, Acta Astron., 49, 201.
+Udalski, A., Szyma´nski, M.K., and Szyma´nski, G. 2015, Acta Astron., 65, 1.
+Watson, C.L., Henden, A.A., and Price, A. 2006, Soc. Astron. Sci. Annu. Symp., 25, 47.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf,len=485
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='04663v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='SR]  11 Jan 2023  ACTA ASTRONOMICA  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 72 (2022) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 1–12  The OGLE Collection of Variable Stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Over 2600 δ Scuti Stars in the Small Magellanic Cloud*  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' S o s z y ´n s k i1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' U d a l s k i1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' S k o w r o n1, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' P i e t r u k o w i c z1 ,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Szyma ´nski1 , R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Poleski1, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Skowron1 , S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Kozłowski1 ,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Mróz1, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Iwanek1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Wrona1, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Ulaczyk2,1, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Rybicki3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='1,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' G r o m a d z k i1  1Astronomical Observatory, University of Warsaw, Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Ujazdowskie 4, 00-478 Warszawa,  Poland  2 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry,  CV4 7AL, UK  3Department of Particle Physics and Astrophysics, Weizmann Institute of Science,  Rehovot 76100, Israel  Received January 13, 2023  ABSTRACT  We present the first-ever collection of δ Scuti stars found over the entire area of the Small Mag-  ellanic Cloud (SMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The sample consists of 2810 variables of which over 2600 objects belong to  the SMC while the remaining stars are most likely members of the Milky Way’s halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The sample  has been divided into 2733 singlemode and 77 multimode pulsators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' We provide observational pa-  rameters (pulsation periods, mean magnitudes, amplitudes, Fourier coefficients) of all δ Sct stars and  the long-term I- and V-band time-series photometric measurements collected during the fourth phase  of the Optical Gravitational Lensing Experiment (OGLE-IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Key words: Stars: variables: delta Scuti – Stars: oscillations – Magellanic Clouds – Catalogs  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Introduction  δ Scuti variables are intermediate-mass stars pulsating in low-order radial or  non-radial modes driven by the κ mechanism operating in the He II ionization  zone (Chevalier 1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' This class includes young and intermediate-age stars on  the pre-main sequence, main sequence, and post-main sequence, as well as stars  belonging to the old population that are probably merged binaries (Breger 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  The latter group is referred to as SX Phoenicis stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The pulsation periods of δ Sct  Based on observations obtained with the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='3-m Warsaw telescope at the Las Campanas Observa-  tory of the Carnegie Institution for Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  variables are shorter than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='3 d, while the amplitudes reach 1 mag in the V band,  although most stars of this type exhibit much smaller amplitudes, down to the sub-  millimagnitude level (Balona and Dziembowski 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  δ Sct stars, like other classical pulsators, follow period–luminosity (PL) and  period–luminosity–color (PLC) relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The Large and Small Magellanic Cloud  (LMC and SMC) play a crucial role in studying the PL relations obeyed by various  types of pulsating stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Leavitt and Pickering 1912, Glass and Lloyd Evans  1981, Madore 1982, Udalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 1999, Muraveva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The PL relations  for δ Sct stars were also investigated based on variables detected in the Magellanic  Clouds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Poleski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2010, McNamara 2011, Martínez-Vázquez 2022), but  these studies were limited by a small number of known δ Sct stars in the Clouds,  especially in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  The first δ Sct variables in the SMC were discovered as a by-product of a  search for RR Lyr stars in the OGLE-II photometric database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Soszy´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  (2002) published a list of 19 short-period variables (with periods in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='16–  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='57 d) suggesting that most of them are δ Sct stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' In the following years, about  half of these stars were re-classified as RR Lyr stars or Cepheids (see Section 4 for  the definition of the boundary between classical Cepheids and δ Sct stars), but eight  of them turned out to be bona fide δ Sct variables and are included in the present  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' One more δ Sct star from the OGLE project was reported by Pietrukowicz  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The second, and so far the last, sample of δ Sct stars in the SMC was  published by Martínez-Vázquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' They used deep photometry of the  SMC globular cluster NGC 419 obtained by the Gemini South telescope to detect  54 δ Sct stars of which six objects are probable cluster members and 48 are field  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The Gaia DR3 catalog of main-sequence oscillators (Gaia Collaboration  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2022) contains over 50 δ Sct stars in the region of the sky occupied by the  SMC, but all these pulsators belong to the Milky Way halo and are located in the  foreground of the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Thus, only 63 δ Sct stars in the SMC have been known until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' With this  paper, we release the first catalog of δ Sct variables found over the entire area of  the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Our sample consists of 2810 pulsators of which over 2600 objects are  likely members of the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' This is part of the OGLE Collection of Variable Stars  (OCVS), currently containing over a million manually classified variable stars in  the Milky Way and Magellanic Clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' In the next paper in this series, we will  present the OGLE collection of over 14,000 δ Sct pulsators in the LMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  This work is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' In Section 2, we present a summary of the  OGLE survey of the Magellanic Clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Section 3 is devoted to a brief review of  the variable star selection and classification procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' In Section 4, we discuss the  criteria that can be used to distinguish between δ Sct stars and classical Cepheids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  The OGLE collection of δ Sct variables in the SMC is described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The  completeness of our catalog is discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' We close the paper with a  summary in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 72  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  OGLE Observations of the Magellanic System  The OGLE survey uses a dedicated 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='3-m Warsaw Telescope equipped with a  large format mosaic CCD camera with a field of view of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='4 square degrees and  a pixel scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='26 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The telescope is located at Las Campanas Observa-  tory, Chile, operated by the Carnegie Institution for Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Our collection of  δ Sct stars is based on photometric data from the fourth phase of the OGLE project  (OGLE-IV, Udalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2015), which has been operating since 2010 until today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  In this work, we use the observations obtained before March 2020, when the War-  saw Telescope was temporarily closed due to the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  The OGLE observations are carried out in the I and V photometric bands,  closely resembling those from the Cousins-Johnson standard systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' From several  dozen to over 1000 observations per star (typically 600-700) have been acquired in  the I-band and from several to over 300 (typically 40-60) data points have been  gathered in the V-band light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The uniform exposure time of 150 s resulted  in the saturation limit of the OGLE photometry at the level of about I ≈ 13 mag  and the sensitivity limit at about I ≈ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='5 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  The total sky area monitored by the OGLE survey in the region of the Mag-  ellanic System is 765 square degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The OGLE footprint covers both galaxies  together with the Magellanic Bridge and vast regions on their periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' For the  purposes of this work, we adopted the celestial meridian of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='h8 as an arbitrary on-  sky boundary between the LMC and SMC regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The same value was selected  to separate the LMC and SMC RR Lyr stars in the OCVS (Soszy´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  In the SMC area of about 280 square degrees, OGLE regularly observes about 14  million stars belonging to both the SMC and the halo of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Selection and Classification of δ Sct Stars  We began our search for δ Sct variables in the SMC with a massive period  analysis for all 14 million I-band light curves stored in the OGLE-IV databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  We used the FNPEAKS code† to span the frequency domain from 0 to 100 cycles  per day with a step of 5 · 10−5 cycles per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' For each light curve, the period  with the highest signal-to-noise ratio was considered to be dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Then, this  dominant frequency and its harmonics were subtracted from the light curve and the  period search was repeated on the residual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  In the next step of the procedure, we used both I- and V-band OGLE-IV photo-  metric data to select and classify δ Sct stars in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The much smaller number  of the OGLE observations in the V-band was compensated by the lower noise of  the V-band light curves compared to the I-band ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 1 shows the I- and V-  band light curves of nine example δ Sct variables arranged in descending pulsation  periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Note that the shortest-period (and therefore the faintest) stars have a signif-  †http://helas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='wroc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='pl/deliverables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='lang=en&active=fnpeaks  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' OGLE-IV I-band (red) and V-band (blue) light curves of nine example δ Sct stars in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  icant scatter of the data points in the I-band, while the V-band light curves of these  objects are much less dispersed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' However, every δ Sct variable identified based  on the V-band data was later confirmed with the I-band observations, although in  some cases the I-band light curves are very noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Our variability classification was primarily based on the shapes of the I- and V-  band light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' For single-periodic variables, we filtered out stars with sinusoidal  light curves and obvious eclipsing binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' For multi-periodic objects, we took into  account the characteristic period ratios of multimode pulsators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  In the final stage of our classification procedure, we examined the positions of  the candidate pulsation stars in the color–magnitude diagram depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2,  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 72  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' (V − I)0 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' I0 color–magnitude diagram for δ Sct stars in the SMC (blue points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The  background gray points show stars from the subfield SMC719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  where the I-band magnitudes and V − I color index were corrected for the inter-  stellar extinction using the most accurate reddening maps of the Magellanic Clouds  recently published by Skowron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Most of the stars with the dereddened  color index (V − I)0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='2 mag or (V − I)0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='7 mag, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', outside the δ Sct in-  stability strip, were removed from the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' However, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2, we  decided to keep some too-blue or too-red stars on the list, because their light curve  morphology suggested that these objects could be genuine δ Sct variables blended  with nearby unresolved stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Nonetheless, the objects with atypical colors should  be treated with caution as uncertain candidates for δ Sct stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Finally, our collection of δ Sct stars detected toward the SMC consists of 2810  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' In addition, the OCVS has been enriched with nine new classical Cepheids  and 123 RR Lyr stars identified as a by-product of the present search for δ Sct  stars or a result of cross-matching the OGLE database with the Gaia DR3 catalog  of Cepheids (Ripepi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2022) and RR Lyr stars (Clementini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' A more  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  detailed comparison of the Gaia DR3 and OGLE catalogs in the area of the Mag-  ellanic System will be discussed in the forthcoming paper presenting the OGLE  collection of δ Sct stars in the LMC (Soszy´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Boundary Between δ Sct Stars and Classical Cepheids  Population I δ Sct stars and classical Cepheids occupy common sequences  in the PL (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 3), color–magnitude, period–radius (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Fernie 1992), Petersen  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Poleski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2010) and other diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' In fact, it is a matter of convention  what value of the pulsation period is adopted to separate both types of variable  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Various authors proposed different maximum periods of δ Sct stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' For ex-  ample, in the General Catalogue of Variable Stars (Samus’ et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2017) it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='2 d,  Breger (2000) defined δ Sct variables as pulsators with periods shorter than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='25 d,  Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' (2000) adopted 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='3 d, Handler (2009) – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='33 d (8 hr), while Catelan  and Smith (2015) – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='42 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  One might ask if it is possible to establish a “natural” boundary period sepa-  rating δ Sct stars from classical Cepheids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Both types of variables may pulsate,  among others, in the fundamental mode (F), first-overtone (1O), second-overtone  (2O), or in the mix of these radial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' All but one of these combinations form  continuity in the vicinity of the Cepheid–δ Sct border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The exception is a class  of single-mode F pulsators that show a natural gap between δ Sct stars and clas-  sical Cepheids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' OGLE-SMC-CEP-1899 (PF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='842 d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Soszy´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2010) is  a classical Cepheids oscillating solely in the F mode with the shortest known pe-  riod, while OGLE-LMC-DSCT-0617 (PF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='299 d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Poleski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2010) is the  longest-period single-mode δ Sct star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Currently, no known Population I single-  mode F-mode star from the Cepheid instability strip pulsates with a period in the  range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='3–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='8 d‡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' For this reason, all Population I radially oscillating stars (both  single- and multimode pulsators) with an F-mode period below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='3 d are classified  in our collection as δ Sct variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Consequently, the 1O boundary period between  δ Sct stars and Cepheids was fixed at P1O = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='23 d (corresponding to the period  ratio P1O/PF of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  With the strict definition of different types of pulsating stars, we could extend  the OCVS by nine short-period classical Cepheids in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' One F/1O double-  mode and eight 1O single-mode pulsators have 1O periods in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='23–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='9 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  The OCVS (Soszy´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2019) currently contains 4954 carefully selected clas-  sical Cepheids in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  ‡Of course, the Population II pulsators, like RR Lyr stars or anomalous Cepheids, may have  periods in this range, but these stars can be isolated using other properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', in the PL diagram,  they lie below the classical Cepheid–δ Sct ridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 72  7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Period–luminosity (upper panel) and period–Wesenheit index (lower panel) diagrams for  classical pulsators in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Green points show classical Cepheids, violet points – anomalous  Cepheids, red points – type II Cepheids, orange points – RR Lyr stars, and blue points – δ Sct stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Lighter colors indicate stars oscillating in higher modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Wesenheit index is a reddening-free function  defined as WI = I −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='55(V −I) (Madore 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  The OGLE Collection of δ Sct Stars in the SMC  Our collection of 2810 δ Sct variables in the SMC is available at the OGLE  Internet Archive:  https://ogle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='astrouw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='pl → OGLE Collection of Variable Stars  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='astrouw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='pl/ogle/ogle4/OCVS/smc/dsct/  The sample was divided into 2733 singlemode and 77 multimode pulsators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  The latter group contains objects with well-defined secondary or tertiary periods  that may be associated with additional radial or non-radial pulsation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' At  least 160 stars in our sample are members of our Galaxy – these are likely SX Phe  variables in the halo of the Milky Way located in front of the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 4 shows  on-sky maps of δ Sct stars from our collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The left panel presents the spatial  distribution of probable members of the SMC, while the right panel shows the  positions of the putative Milky Way SX Phe stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The line separating these two  populations was arbitrarily set at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='5 mag above the mean PL relation for the SMC  δ Sct variables (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' More sophisticated methods would probably divide the  Galactic and SMC populations more efficiently, however, one should remember  that this separation cannot be done unequivocally because the Magellanic Clouds  are submerged in the Milky Way halo – old stellar populations in both galaxies  smoothly transition into each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  The full list of δ Sct stars with their equatorial coordinates (J2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='0), classifi-  cation (singlemode, multimode), internal designations in the OGLE-IV, OGLE-III,  and OGLE-II databases (if available), and the cross-matches with the International  Variable Star Index (VSX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2006) are provided in the ident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='dat file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  In total, we identified only 11 common stars between our collection and the VSX  catalog – all of them are unquestionable members of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  The file dsct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='dat contains observation parameters of the stars: their intensity-  averaged mean magnitudes in the I- and V-bands, up to three pulsation periods  derived with the TATRY code (Schwarzenberg-Czerny 1996), epochs of the maxi-  mum light, I-band peak-to-peak amplitudes, and Fourier coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The periods  are very precisely determined thanks to the long timespan of the OGLE-IV light  curves, however, it cannot be ruled out that in the case of some the most noisy light  curves our approach produced daily aliases of the true periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  The OGLE-IV time-series photometry in the I- and V-bands are stored in the  directory phot/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Obvious outlying points (deviating by more than ±4 sigma from  the fitted model) were removed from the light curves, but it was verified before-  hand whether these points are due to, for example, additional eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' We did not  find any δ Sct stars with eclipses in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Finding charts can be found in the  directory fcharts/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' These are 60′′ ×60′′ subframes of the I-band reference images,  oriented with North at the top and East to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 72  9  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' On-sky distributions of δ Sct stars toward the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Left panel shows positions of over 2600 probable members of the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Right panel shows δ Sct  variables that likely belong to the Milky Way halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The gray area shows the OGLE footprint in the Magellanic System region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Luminosity–amplitude diagram for δ Sct stars in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Completeness of the Catalog  The completeness of our collection is strongly limited by the brightness, am-  plitudes, and light curve shapes of δ Sct stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2 and 3 show that the number  of variables in our sample drops sharply for mean magnitudes I > 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='2 mag due to  the sensitivity limit of the OGLE photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  We cross-matched our collection with the list of 54 δ Sct variables discovered  with the Gemini South telescope in the field surrounding the SMC globular cluster  NGC 419 (Martínez-Vázquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' We found only one common object  in both catalogs – V46 = OGLE-SMC-DSCT-2067 – which is the brightest δ Sct  star detected by Martínez-Vázquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The remaining δ Sct variables  identified with the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='1-meter Gemini telescope are below the detection limit of the  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='3-meter Warsaw telescope used by the OGLE survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  The vast majority of δ Sct variables are small-amplitude pulsators (Breger  2000, Balona and Dziembowski 2011), so our collection must be the tip of the  iceberg because we had the opportunity to discover only the high-amplitude δ Sct  stars in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 5 shows the I-band peak-to-peak amplitudes plotted against  the mean I-band magnitudes of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' It is obvious that the amplitude de-  tection limits in the OGLE data are strongly correlated with the brightness of the  pulsating candidates: for stars with the luminosities of about I = 19 mag, the small-  est detectable amplitudes are of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='04 mag, for I = 20 mag stars, the  amplitude detection limit rises to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='1 mag, and for the faintest stars in our collec-  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 72  11  tion (I = 21 mag), the light curve amplitude should be larger than about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='2 mag  to be recognized as a δ Sct variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' On the other hand, our sample of the Galactic  δ Sct stars (I < 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='5 mag) should be much more complete because we could detect  variables with amplitudes as small as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='01 mag in the I-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  We used the δ Sct stars with double entries in the OGLE-IV database to es-  timate the completeness of our collection within the brightness and amplitude de-  tection limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' These objects are located in the overlapping parts of the adjacent  OGLE fields, so we had the opportunity to identify them twice during the selection  and classification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' We found that 311 out of 2810 δ Sct stars included in  the final version of our collection have their duplicates in the OGLE-IV database,  so we had a chance to detect 622 counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' In fact, 449 objects from this list  were independently classified as δ Sct variables which correspond to the catalog  completeness slightly above 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' However, if we consider only stars brighter than  I = 19 mag, the formal completeness of our collection increases to about 85%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Nonetheless, one should remember that the bulk of δ Sct stars in the SMC have  luminosities and amplitudes below the detection limits of the OGLE project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Summary  We presented the first-ever catalog of δ Sct stars found in the entire area of the  SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The number of known δ Sct variables in this galaxy has increased from about  60 to over 2600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Additionally, our collection contains at least 160 Galactic SX Phe  stars located in the foreground of the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The most obvious application of our  collection includes the examination of the PL and PLC relations obeyed by δ Sct  stars in the metal-poor environment of the SMC compared to the more metal-rich  variables in the LMC and Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The three-dimensional distribution of δ Sct  variables can be analyzed to better understand the structure of the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Moreover,  the long-term photometric time series produced by the OGLE project can be used  to study the stability of stellar pulsations over long time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  This work demonstrates the great versatility of OGLE photometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' The  OCVS contains variable stars with periods ranging from minutes to years, lumi-  nosities from 12 mag to over 22 mag in the V-band, and amplitudes from milli-  magnitudes to several magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' We expect the next breakthrough in the field  of variable stars in the Magellanic Clouds to come after the launch of the LSST  project on the Rubin Observatory (Hambleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' This work has been supported by the National Science  Centre, Poland, grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2022/45/B/ST9/00243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' MG is supported by the EU Hori-  zon 2020 research and innovation programme under grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 101004719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  This research has made use of the International Variable Star Index (VSX) database,  operated at AAVSO, Cambridge, Massachusetts, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  REFERENCES  Balona, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', and Dziembowski, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2011, MNRAS, 417, 591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2011MNRAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='.591B  Breger M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2000, in Breger M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Montgomery M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', eds, ASP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 210, Delta Scuti and Related  Stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Pac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', San Francisco, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Catelan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', and Smith, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2015, Pulsating Stars (New York: Wiley).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Chevalier, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 1971, A&A, 14, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Clementini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Ripepi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Garofalo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2022, arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='06278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Fernie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 1992, AJ, 103, 1647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Gaia Collaboration, De Ridder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Ripepi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2022, arXiv e-prints, arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='06075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Glass, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', and Lloyd Evans, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 1981, Nature, 291, 303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Hambleton, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Bianco, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Street, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2022, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='04499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Handler, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2009, in AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 1170, Stellar Pulsation: Challenges for Theory and Observation,  ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Guzik & P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Bradley (Melville, NY: AIP), 403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Leavitt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', and Pickering, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 1912, Harvard Coll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Circ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', 173, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Madore, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 1982, ApJ, 253, 575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Martínez-Vázquez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Salinas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', and Vivas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2021, AJ, 161, 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Martínez-Vázquez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Salinas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Vivas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', and Catelan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2022, ApJ, 940, L25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  McNamara, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2011, AJ, 142, 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Muraveva, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Palmer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Clementini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2015, ApJ, 807, 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Pietrukowicz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2018, in The RR Lyrae 2017 Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Revival of the Classical Pulsators: from  Galactic Structure to Stellar Interior Diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Smolec, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Kinemuchi, &  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Anderson (Poland: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Polish Astronomical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' ), 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Poleski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Soszy´nski, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Udalski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2010, Acta Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', 60, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Ripepi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Clementini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Molinaro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2022, arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='06212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Rodríguez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', López-González, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', and López de Coca, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2000, A&AS, 144, 469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Samus’, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Kazarovets, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Durlevich, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Kireeva, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', and Pastukhova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2017,  Astronomy Reports, 61, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Schwarzenberg-Czerny, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 1996, ApJ, 460, L107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Skowron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Skowron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Udalski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2021, ApJS, 252, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Soszy´nski, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Udalski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Szyma´nski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2002, Acta Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', 52, 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Soszy´nski, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Poleski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Udalski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2010, Acta Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', 60, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Soszy´nski, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Udalski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Szyma´nski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2016, Acta Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', 66, 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Soszy´nski, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Udalski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Szyma´nski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2019, Acta Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', 69, 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Udalski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Szyma´nski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Kubiak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Pietrzy´nski, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Soszy´nski, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Wo´zniak, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', and ˙Zebru´n, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  1999, Acta Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', 49, 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Udalski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Szyma´nski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', and Szyma´nski, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2015, Acta Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', 65, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='  Watson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', Henden, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', and Price, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' 2006, Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
+page_content=', 25, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE3T4oBgHgl3EQfrwrw/content/2301.04663v1.pdf'}
diff --git a/_tFJT4oBgHgl3EQfqizo/content/tmp_files/2301.11605v1.pdf.txt b/_tFJT4oBgHgl3EQfqizo/content/tmp_files/2301.11605v1.pdf.txt
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+EUROPEAN ORGANISATION FOR NUCLEAR RESEARCH (CERN)
+Submitted to: Phys. Rev. D
+CERN-EP-2022-044
+January 30, 2023
+Search for flavor-changing neutral-current
+couplings between the top quark and the 𝒁 boson
+with LHC Run 2 proton–proton collisions at
+√𝒔 = 13 TeV with the ATLAS detector
+The ATLAS Collaboration
+A search for flavor-changing neutral-current couplings between a top quark, an up or charm
+quark and a 𝑍 boson is presented, using proton–proton collision data at √𝑠 = 13 TeV collected
+by the ATLAS detector at the Large Hadron Collider. The analyzed dataset corresponds to an
+integrated luminosity of 139 fb−1. The search targets both single-top-quark events produced
+as 𝑔𝑞 → 𝑡𝑍 (with 𝑞 = 𝑢, 𝑐) and top-quark-pair events, with one top quark decaying through
+the 𝑡 → 𝑍𝑞 channel. The analysis considers events with three leptons (electrons or muons),
+a 𝑏-tagged jet, possible additional jets, and missing transverse momentum. The data are
+found to be consistent with the background-only hypothesis and 95% confidence-level limits
+on the 𝑡 → 𝑍𝑞 branching ratios are set, assuming only tensor operators of the Standard
+Model effective field theory framework contribute to the 𝑡𝑍𝑞 vertices. These are 6.2 × 10−5
+(13 × 10−5) for 𝑡 → 𝑍𝑢 (𝑡 → 𝑍𝑐) for a left-handed 𝑡𝑍𝑞 coupling, and 6.6 × 10−5 (12 × 10−5)
+in the case of a right-handed coupling. These results are interpreted as 95% CL upper limits
+on the strength of corresponding couplings, yielding limits for |𝐶 (13)∗
+𝑢𝑊 | and |𝐶 (13)∗
+𝑢𝐵
+| (|𝐶 (31)
+𝑢𝑊 |
+and |𝐶 (31)
+𝑢𝐵 |) of 0.15 (0.16), and limits for |𝐶 (23)∗
+𝑢𝑊 | and |𝐶 (23)∗
+𝑢𝐵
+| (|𝐶 (32)
+𝑢𝑊 | and |𝐶 (32)
+𝑢𝐵 |) of 0.22
+(0.21), assuming a new-physics energy scale ΛNP of 1 TeV.
+© 2023 CERN for the benefit of the ATLAS Collaboration.
+Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.0 license.
+arXiv:2301.11605v1  [hep-ex]  27 Jan 2023
+
+CERNContents
+1
+Introduction
+2
+2
+ATLAS detector
+3
+3
+Data and samples of simulated events
+4
+4
+Object reconstruction
+8
+5
+Event reconstruction and selection
+9
+6
+Background estimation and separation from signal
+13
+7
+Systematic uncertainties
+15
+8
+Results
+17
+9
+Conclusions
+23
+1 Introduction
+The top quark is the heaviest elementary particle known and it decays almost exclusively into Wb [1]. In
+the Standard Model of particle physics (SM), flavor-changing neutral-current (FCNC) processes involving
+a top quark, an up-type quark and a Z boson are forbidden at tree level and are strongly suppressed by the
+GIM mechanism [2] at higher orders, leading to branching ratios for top-quark decays via FCNC processes
+of the order of 10−14 [3]. However, several SM extensions predict such branching ratios to be between 10−4
+and 10−7. Examples of SM extensions are the quark-singlet model [4], the two-Higgs-doublet model [5],
+the Minimal Supersymmetric Standard Model (MSSM) [6], the MSSM with R-parity violation [7], models
+with warped extra dimensions [8], and extended mirror fermion models [9].
+FCNC couplings can be described by an effective field theory (EFT) [10, 11] that extends the SM Lagrangian
+LSM with higher-dimensional operators suppressed by the scale of new physics, ΛNP, as shown in Eq. (1).
+At order Λ−2
+NP the strength of the anomalous couplings is given by the Wilson coefficients 𝐶𝑘 that multiply
+dimension-six operators O𝑘,
+Leff = LSM +
+1
+Λ2
+NP
+∑︁
+𝑘
+𝐶𝑘O𝑘.
+(1)
+The relevant operators for an FCNC process with a top quark and a Z boson, following the notation in
+Ref. [12], are the operators O(𝑖 𝑗)
+𝑢𝐵 and O(𝑖 𝑗)
+𝑢𝑊 with 𝑖 ≠ 𝑗. The indices 𝑖 and 𝑗 of the operators refer to the
+flavor indices of the quark generations. One index is always equal to 3 as a top quark must be involved,
+while the other one is either 1 or 2, corresponding to an up or charm quark. The FCNC tZq interactions
+can be introduced by vector and tensor couplings, but only the latter are considered in this analysis because
+they would produce most of the “FCNC-in-single-top-production” signal [11]. The FCNC operators can be
+left-handed (LH) or right-handed (RH). The order of the indices 𝑖 and 𝑗 in Eq. (1) defines the chirality
+2
+
+of the FCNC operators. A linear combination of the 𝐶 (13)
+𝑢𝐵 and 𝐶 (13)
+𝑢𝑊 coefficients corresponds to the tZu
+LH coupling while a linear combination of the 𝐶 (31)
+𝑢𝐵 and 𝐶 (31)
+𝑢𝑊 coefficients defines the tZu RH coupling.
+Similarly, the tZc couplings are defined by the 𝐶 (23)
+𝑢𝐵 and 𝐶 (23)
+𝑢𝑊 coefficients for the LH case, while the 𝐶 (32)
+𝑢𝐵
+and 𝐶 (32)
+𝑢𝑊 coefficients describe the RH case. For each linear combination, the two coefficients assume the
+same value with an opposite sign [11].
+Experimental limits on the branching ratio of FCNC t → Zq decays were previously established by
+experiments at the Large Electron–Positron Collider (LEP) [13–16], the Hadron–Electron Ring Accelerator
+(HERA) [17], the Tevatron [18, 19] and the Large Hadron Collider (LHC) [20–23]. The most stringent
+observed limits, B(t → Zu) < 17 × 10−5 and B(t → Zc) < 24 × 10−5 [21], were set by ATLAS in a
+search for FCNC processes in tt decays only, using 36.1 fb−1 of 𝑝𝑝 collision data at √𝑠 = 13 TeV. The
+quoted limits apply to both the left- and right-handed couplings, as the analysis is not sensitive to the
+chirality.
+This paper presents a search for FCNC tZq couplings, using 𝑝𝑝 collision data at √𝑠 = 13 TeV collected
+by the ATLAS experiment at the LHC and corresponding to an integrated luminosity of 139 fb−1. The
+search is performed by analyzing the top-quark decays in tt events as well as the production of single top
+quarks, as illustrated in Figure 1. In the former channel, one of the top quarks decays through an FCNC
+process (t → Zq) and the other through the dominant mode (t → Wb). In contrast, in the latter channel the
+production of a single top quark proceeds through an FCNC process (gq → tZ). Single top production
+with FCNC decay contributes negligibly and is not considered in this analysis. While single top-quark
+production gives the analysis more sensitivity to the FCNC tZu coupling, the tt decay mode provides
+almost equal sensitivity to the FCNC tZu and tZc couplings. Since the FCNC production and decay
+processes are induced by the same couplings, the production cross-section and decay branching ratio are
+connected. Therefore, the FCNC single-top production cross-section can be interpreted as the branching
+ratio of the corresponding FCNC decay. Thus, the analysis results for the numbers of production and decay
+signal events are translated into branching ratios for t → Zq. For both of the considered channels, only
+the trileptonic final state is selected, in which the Z boson decays into charged leptons and the W boson
+from the top quark decays leptonically. The final states where either the Z boson or the W boson decays
+hadronically are not considered because of the larger backgrounds. Therefore, the analysis selects events
+with three leptons (electrons or muons), a b-tagged jet, possible additional jets, and missing transverse
+momentum. After the selection, the main background sources are diboson, ttZ and tZ production. To
+improve the separation of signal from background events, a multivariate technique is used, which was not
+employed in the previous analysis. The statistical analysis uses a binned profile likelihood fit to the data.
+2 ATLAS detector
+The ATLAS experiment [24] at the LHC is a multipurpose particle detector with a forward–backward sym-
+metric cylindrical geometry and a near 4𝜋 coverage in solid angle.1 It consists of an inner tracking detector
+surrounded by a thin superconducting solenoid providing a 2 T axial magnetic field, electromagnetic and
+1 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the center of the detector
+and the 𝑧-axis along the beam pipe. The 𝑥-axis points from the IP to the center of the LHC ring, and the 𝑦-axis points
+upwards. Cylindrical coordinates (𝑟, 𝜙) are used in the transverse plane, 𝜙 being the azimuthal angle around the 𝑧-axis. The
+pseudorapidity is defined in terms of the polar angle 𝜃 as 𝜂 = − ln tan(𝜃/2). Distances in the 𝜂–𝜙 plane are measured in units of
+Δ𝑅 ≡
+√︃
+(Δ𝜂)2 + (Δ𝜙)2.
+3
+
+g
+g
+W
+b
+Z
+u / c
+g
+t
+t
+(a)
+u / c
+g
+Z
+b
+W
+u / c
+t
+(b)
+Figure 1: Examples of the lowest-order Feynman diagrams for (a) tt production, with one top quark decaying through
+the dominant mode in the SM and the other via an FCNC process and for (b) single top-quark production via an
+FCNC process in the 𝑠-channel.
+hadron calorimeters, and a muon spectrometer. The inner tracking detector (ID) covers the pseudorapidity
+range |𝜂| < 2.5. It consists of silicon pixel, silicon microstrip, and transition radiation tracking detectors.
+Lead/liquid-argon (LAr) sampling calorimeters provide electromagnetic (EM) energy measurements
+with high granularity. A steel/scintillator-tile hadron calorimeter covers the central pseudorapidity range
+(|𝜂| < 1.7). The endcap and forward regions are instrumented with LAr calorimeters for both the EM and
+hadronic energy measurements up to |𝜂| = 4.9. The muon spectrometer surrounds the calorimeters and is
+based on three large superconducting air-core toroidal magnets with eight coils each. The field integral of
+the toroids ranges between 2.0 and 6.0 T m across most of the detector. The muon spectrometer includes a
+system of precision tracking chambers and fast detectors for triggering. A two-level trigger system [25] is
+used to select events. The first-level trigger is implemented in hardware and uses a subset of the detector
+information to accept events at a rate below 100 kHz. This is followed by a software-based trigger that
+reduces the accepted event rate to 1 kHz on average depending on the data-taking conditions. An extensive
+software suite [26] is used in data simulation, in the reconstruction and analysis of real and simulated data,
+in detector operations, and in the trigger and data acquisition systems of the experiment.
+3 Data and samples of simulated events
+The data sample used in this analysis corresponds to 139 fb−1 of pp collisions at √𝑠 = 13 TeV collected by
+the ATLAS detector during 2015–2018, after requiring stable LHC beams and that all detector subsystems
+were operational [27].
+Candidate events were required to satisfy one of the single-electron triggers or one of the single-muon
+triggers [25, 28, 29]. Single-lepton triggers with low transverse momentum (𝑝T) thresholds and isolation
+requirements were combined in a logical OR with higher-threshold triggers that had a looser identification
+criterion and did not have any isolation requirement. The lowest 𝑝T threshold used for electrons was 24 GeV
+(26 GeV) in 2015 (2016–2018), while for muons the corresponding threshold was 20 GeV (26 GeV).
+To evaluate the effects of the detector resolution and acceptance on the signal and background, and to
+estimate the SM backgrounds, simulated event samples were produced using a Geant4-based Monte Carlo
+(MC) detector simulation [30, 31]. Some of the samples used for evaluating systematic uncertainties did
+not use the full Geant4 simulation but instead relied on parameterized showers in the calorimeter [31].
+4
+
+The top-quark mass in the event generators described below was set to 𝑚𝑡 = 172.5 GeV. In all samples, the
+decays of bottom and charm hadrons were performed by EvtGen 1.2.0 [32], unless stated otherwise.
+The simulated data must account for the fact that significantly more than one inelastic pp collision occurs
+per bunch crossing. The average number of collisions per bunch crossing ranged from 13 to 38 for the
+2015–2018 data-taking periods. Inelastic collisions were simulated using Pythia 8.186 [33] with the A3
+set of tuned parameters [34] and the NNPDF2.3lo [35] set of parton distribution functions (PDFs), and
+overlaid on the signal and background MC samples. These simulated events were reweighted to match
+the conditions of the collision data, specifically the number of additional pp interactions in the same and
+neighboring bunch crossings (pileup).
+Several MC signal event samples were generated at next-to-leading order (NLO) in QCD with
+MadGraph5_aMC@NLO 2.7.2 [36], using the NNPDF3.0nlo [37] PDF set. Parton showering and
+hadronization were modeled with Pythia 8.302 with the NNPDF2.3lo PDF set and the A14 set of
+tuned parameters [38]. Only events with leptonic decays (including 𝜏-leptons) of the W and Z bosons
+were generated. The TopFCNC Universal FeynRules Output (UFO) model [11, 39, 40] was used for the
+computation of top-quark FCNC production and decay processes at NLO in QCD. Since FCNC processes
+in both production and decay are considered in this analysis, separate samples for each mode and for tZu
+and tZc couplings were generated. In order to study the chirality of these couplings, separate samples with
+LH and RH couplings were produced.
+Additional signal samples generated with the same version of MadGraph5_aMC@NLO were interfaced
+to Herwig 7.1.6 [41, 42] instead of Pythia 8.302 to assess the uncertainty related to the choice of
+parton-shower model. The Herwig 7.1 default set of tuned parameters [42, 43] was used together with the
+MMHT2014lo PDF set [44]. The decays of bottom and charm hadrons were performed by EvtGen 1.7.0.
+For the normalization, the branching ratios are set to the best observed limits reported in Section 1,
+constraining B(𝑡
+→
+𝑞′𝑊) = 1 − B(𝑡
+→
+𝑢𝑍/𝑐𝑍), with 𝑞′ = 𝑑, 𝑠, 𝑏. The FCNC tt decay signal
+is normalized using the tt cross-section prediction at next-to-next-to-leading order (NNLO) in QCD
+including the resummation of next-to-next-to-leading logarithmic (NNLL) soft-gluon terms calculated
+using Top++ 2.0 [45–51]. The FCNC single top-quark production signal normalization cross-section is
+calculated at NLO using the TopFCNC model as implemented in MadGraph5_aMC@NLO.
+The background is estimated using simulated samples that contain at least two leptons and at least two jets.
+These samples include the production of tt, ttH, ttZ, ttW, tZ, tW, tWZ, Z + jets, diboson, triboson, ttt, tttt,
+ttWW, ZH and WH events.
+The production of tt and ttH events was modeled using the Powheg Box v2 [52–56] generator at NLO with
+the NNPDF3.0nlo PDF set and the ℎdamp parameter2 set to 1.5 𝑚𝑡 for tt [57] and to 0.75 × (2 𝑚𝑡 + 𝑚H)
+for ttH, with 𝑚𝐻 = 125 GeV. The events were interfaced to Pythia 8.230 [58] to model the parton
+shower, hadronization, and underlying event, with parameters set according to the A14 tune and using the
+NNPDF2.3lo set of PDFs. The decays of bottom and charm hadrons were performed by EvtGen 1.6.0.
+Additional tt simulated samples are used to assess modeling uncertainties [59]. The impact of using a
+different parton shower and hadronization model is evaluated by comparing the nominal “Powheg+Pythia
+” tt sample with another event sample produced with the Powheg Box v2 generator, but interfaced with
+Herwig 7.1.3, which used the Herwig 7.1 default set of tuned parameters and the MMHT2014lo PDF set.
+2 The ℎdamp parameter is a resummation damping factor and one of the parameters that controls the matching of Powheg matrix
+elements to the parton shower and thus effectively regulates the high-𝑝T radiation against which the tt system recoils.
+5
+
+To estimate the systematic uncertainty in the choice of the ℎdamp parameter, a sample generated in the same
+way as the nominal one but with the ℎdamp parameter set to 3.0 𝑚𝑡 was produced.
+The production of ttZ and ttW events was modeled using the MadGraph5_aMC@NLO 2.3.3 generator at
+NLO with the NNPDF3.0nlo PDF set. The events were interfaced to Pythia 8.210, which used the A14
+tune and the NNPDF2.3lo PDF set.
+Additional ttZ simulated samples are used to assess modeling uncertainties. The impact of using a different
+parton shower and hadronization model is evaluated by comparing the nominal ttZ sample with an event
+sample produced with the MadGraph5_aMC@NLO 2.6.2 generator interfaced with Herwig 7.0.4, which
+used the Herwig 7.0 default set of tuned parameters and the MMHT2014lo PDF set. The decays of bottom
+and charm hadrons were performed by EvtGen 1.6.0. The uncertainty due to initial-state radiation (ISR) is
+estimated by comparing the nominal event sample with two samples where the Var3c [38] up and down
+variations of the A14 tune were employed.
+The SM production of a single top quark in association with a Z boson (tZ) was modeled using the
+MadGraph5_aMC@NLO 2.3.3 generator at NLO with the NNPDF3.0nlo PDF set. The events were
+interfaced with Pythia 8.230, which used the A14 tune and the NNPDF2.3lo PDF set.
+Similarly to ttZ, additional tZ simulated samples are used to assess modeling uncertainties. The impact
+of using a different parton shower and hadronization model is evaluated by comparing the nominal tZ
+sample with an event sample produced with the MadGraph5_aMC@NLO 2.8.1 generator interfaced with
+Herwig 7.2.1, which used the Herwig 7.1 default set of tuned parameters and the MMHT2014lo PDF set.
+The decays of bottom and charm hadrons were performed by EvtGen 1.7.0. The uncertainty due to ISR is
+estimated by comparing the nominal tZ sample with two additional samples, which had the same settings
+as the nominal one, but employed the Var3c up and down variations of the A14 tune.
+The associated production of a single top quark with a W boson (tW) was modeled by the Powheg Box v2 [60]
+generator at NLO in QCD using the five-flavor scheme and the NNPDF3.0nlo set of PDFs. The diagram
+removal (DR) scheme [61] was used to remove interference and overlap with tt production. The events
+were interfaced to Pythia 8.230, which used the A14 tune and the NNPDF2.3lo set of PDFs.
+The production of tWZ events was modeled using the MadGraph5_aMC@NLO 2.3.3 generator at NLO
+with the NNPDF3.0nlo PDF set. The events were interfaced with Pythia 8.212, which used the A14 tune
+and the NNPDF2.3lo PDF set. The DR scheme was employed to handle the interference between the tWZ
+and ttZ processes. A sample with an alternative scheme described in Ref. [62] was produced to assess the
+associated systematic uncertainty.
+The Powheg Box v1 MC generator [63] was used to simulate at NLO accuracy the hard-scattering processes
+of Z boson production and decay in the electron, muon, and 𝜏-lepton channels. It was interfaced to
+Pythia 8.186 for the modeling of the parton shower, hadronization, and underlying event, with parameters
+set according to the AZNLO tune [64]. The CT10nlo [65] PDF set was used for the hard-scattering
+processes, whereas the CTEQ6L1 [66] PDF set was used for the parton shower. The effect of QED
+final-state radiation was simulated with Photos++ 3.52 [67, 68].
+Samples of diboson final states (𝑉𝑉, with 𝑉 = W, Z) were simulated with the Sherpa 2.2.1 or 2.2.2 [69]
+generator depending on the process, including off-shell effects and Higgs boson contributions where
+appropriate. Fully leptonic final states and semileptonic final states, where one boson decays leptonically
+and the other hadronically, were generated using matrix elements at NLO accuracy in QCD for up to
+one additional parton and at LO accuracy for up to three additional parton emissions. Samples for the
+loop-induced processes 𝑔𝑔 → 𝑉𝑉 were generated using LO-accurate matrix elements for up to one
+6
+
+additional parton emission for both the cases of fully leptonic and semileptonic final states. The matrix
+element calculations were matched and merged with the Sherpa parton shower based on Catani–Seymour
+dipole factorization [70, 71] using the MEPS@NLO prescription [72–75]. The virtual QCD corrections
+were provided by the OpenLoops library [76–78]. The NNPDF3.0nnlo set of PDFs was used, along
+with the dedicated set of tuned parton-shower parameters developed by the Sherpa authors. Electroweak
+production of a diboson in association with two jets (𝑉𝑉 𝑗 𝑗) was simulated with the Sherpa 2.2.2 generator.
+The LO-accurate matrix elements were matched to a parton shower based on Catani–Seymour dipole
+factorization using the MEPS@LO prescription. Samples were generated using the NNPDF3.0nnlo PDF
+set, along with the dedicated set of tuned parton-shower parameters developed by the Sherpa authors. The
+decays of bottom and charm hadrons are performed with built-in Sherpa features. An invariant mass of
+𝑚ℓℓ > 4 GeV was required at matrix-element level for any pair of same-flavor charged leptons.
+To assess the uncertainty that the generator contributes to the simulation of diboson final states, alternative
+samples are employed. For these, the Powheg Box v2 [79] generator was used instead of Sherpa. The effect
+of singly resonant amplitudes and interference effects due to 𝑍/𝛾∗ and same-flavor lepton combinations in
+the final state were included where appropriate. Interference effects between 𝑊𝑊 and 𝑍𝑍 for same-flavor
+charged leptons and neutrinos were ignored. Events were interfaced to Pythia 8.186 for the modeling of
+the parton shower, hadronization, and underlying event, with parameters set according to the AZNLO tune.
+The CT10 PDF set was used for the hard-scattering processes, whereas the CTEQ6L1 PDF set was used
+for the parton shower. The factorization and renormalization scales were set to the invariant mass of the
+boson pair. The same invariant mass selection as for the Sherpa samples was applied.
+The production of triboson (𝑉𝑉𝑉, with 𝑉 = W, Z) events was simulated with the Sherpa 2.2.2 generator.
+Matrix elements, accurate to NLO for the inclusive process and to LO for up to two additional parton
+emissions, were matched and merged with the Sherpa parton shower based on Catani–Seymour dipole
+factorization using the MEPS@NLO prescription. The virtual QCD corrections for matrix elements at NLO
+accuracy were provided by the OpenLoops library. Samples were generated using the NNPDF3.0nnlo
+PDF set, along with the dedicated set of tuned parton-shower parameters developed by the Sherpa authors.
+The decays of bottom and charm hadrons are performed with built-in Sherpa features.
+The production of tttt events was modeled using the MadGraph5_aMC@NLO 2.6.2 generator at NLO
+with the NNPDF3.1nlo [37] PDF set. The events were interfaced with Pythia 8.230, which used the A14
+tune and the NNPDF2.3lo PDF set. The decays of bottom and charm hadrons were simulated using the
+EvtGen 1.6.0 program.
+Other rare top-quark processes, namely the production of ttWW and ttt events, were modeled using the
+MadGraph5_aMC@NLO generator at LO interfaced with Pythia 8, which used the A14 tune. The
+associated production of a Higgs boson with a W or Z boson, 𝑉H, was modeled using Pythia 8.186 with
+the A14 tune and the NNPDF2.3lo PDF set.
+Throughout the paper the various MC samples are merged or split as follows. The ttZ and tWZ backgrounds
+are combined. The diboson contribution is split according to the origin of the associated jets using
+generator-level information. Their origin is determined by matching, within a cone of size Δ𝑅 = 0.3, jets
+to hadrons with 𝑝T > 5 GeV. If one of the jets contains a b- or c-hadron, then it is classified as diboson
++ heavy flavor (𝑉𝑉 + HF), otherwise the event is classified as diboson + light flavor (𝑉𝑉 + LF). The tt,
+tW, Z + jets, 𝑉𝑉 and tt𝑉 processes with two prompt3 leptons and one nonprompt or fake lepton (a jet
+3 Prompt leptons are leptons from the decay of W or Z bosons, either directly or through an intermediate 𝜏 → ℓ𝜈𝜈 decay, or from
+the semileptonic decay of top quarks.
+7
+
+misidentified as a lepton) are shown together and called “Fakes”. The other minor backgrounds, namely
+ttW, ttH, 𝑉H, ttWW, triboson, ttt and tttt, are merged and called “Other bkg.”.
+4 Object reconstruction
+The reconstruction of the basic objects used in the analysis is described in the following. The primary
+vertex [80] is selected as the pp vertex candidate with the highest sum of the squared transverse momenta
+of all associated tracks with 𝑝T > 500 MeV.
+Electron candidates are reconstructed from energy clusters in the EM calorimeter that match a reconstructed
+track [81].
+The clusters are required to be within the range |𝜂| < 2.47, excluding the transition
+region between the barrel and endcap calorimeters at 1.37 < |𝜂| < 1.52. Each electron candidate’s
+transverse impact parameter relative to the beam axis, 𝑑0, divided by its estimated uncertainty must satisfy
+|𝑑0|/𝜎(𝑑0) < 5, while the longitudinal distance 𝑧0 from the reconstructed primary vertex to the point
+where 𝑑0 is measured must satisfy |𝑧0 sin(𝜃)| < 0.5 mm. Electron candidates must also satisfy a transverse
+momentum requirement of 𝑝T > 15 GeV. A likelihood-based discriminant is constructed from a set of
+variables that enhance the electron selection, while rejecting photon conversions and hadrons misidentified
+as electrons. An 𝜂- and 𝑝T-dependent selection on the likelihood discriminant is applied, and the “Medium”
+identification [81] is used. Electrons are also required to be isolated using criteria based on ID tracks.
+Nonprompt leptons are rejected using a boosted decision tree (BDT) discriminant based on isolation and
+b-tagging variables, referred to as the nonprompt-lepton BDT [82]. The efficiency at the chosen working
+point for electrons satisfying the isolation criteria is about 70% for a 𝑝T of 20 GeV and reaches a plateau of
+95% at a 𝑝T of 100 GeV. The corresponding rejection factor for leptons from the decay of b-hadrons is
+about 50, estimated from a simulated tt sample. Correction factors are applied to simulated electrons to
+take into account the small differences in trigger, reconstruction, identification and isolation efficiencies
+between data and MC simulation.
+Muon candidates are reconstructed by combining a reconstructed track from the inner detector with
+one from the muon spectrometer, and are required to have 𝑝T > 15 GeV and |𝜂| < 2.5 and to meet the
+“Medium” identification [83] criteria. Similarly to electrons, muon candidates must have |𝑑0|/𝜎(𝑑0) < 3
+and |𝑧0 sin(𝜃)| < 0.5 mm. To reject misidentified muon candidates, several quality requirements are
+imposed on the muon candidate. An isolation requirement based on ID tracks is imposed, and a threshold is
+set for the nonprompt-lepton BDT output. The efficiency at the chosen working point for muons satisfying
+the isolation criteria is about 80% for a 𝑝T of 20 GeV and reaches a plateau of 99% at a 𝑝T of 100 GeV.
+The corresponding rejection factor for leptons from the decay of b-hadrons is about 20, estimated from a
+simulated tt sample. Like for electrons, correction factors are applied to simulated muons to account for
+the small differences between data and simulation.
+Jets are reconstructed from the particle-flow objects [84] using the anti-𝑘𝑡 algorithm [85, 86] with the
+radius parameter set to 𝑅 = 0.4. Their calibration follows the methodology described in Ref. [87]. Jets are
+required to have 𝑝T > 25 GeV and |𝜂| < 2.5. To suppress jets arising from pileup, a discriminant called the
+“jet vertex tagger” (JVT) is constructed using a two-dimensional likelihood method [88]. The jet energy
+scale and resolution are corrected with 𝜂- and 𝑝T-dependent scale factors.
+To identify jets containing a b-hadron (b-jets), the “DL1r” multivariate algorithm is employed [89]. It uses
+impact parameter and secondary and tertiary vertex information from tracks contained in the jet as input.
+Operating points are defined by a threshold value for the 𝑏-tagging discriminant output and are chosen
+8
+
+to provide a specific b-jet efficiency in an inclusive tt sample. Candidate b-jets must have a 𝑏-tagging
+discriminant value that exceeds a threshold corresponding to a 70% b-jet selection efficiency. With this
+criterion, 0.25% of light-jets, containing neither a b- nor a c-hadron, are misidentified as b-jets, as are 10%
+of jets initiated by c-quarks. Correction factors are derived and applied to correct for differences in b-jet
+selection efficiency and the mistagging rates between data and MC simulation [89].
+The missing transverse momentum, with magnitude 𝐸miss
+T
+, is calculated as the negative of the vector sum of
+the transverse momenta of all reconstructed objects. To account for soft hadronic activity, a term including
+tracks associated with the primary vertex but not with any of the reconstructed objects is added to the 𝐸miss
+T
+calculation [90, 91].
+To avoid cases where the detector response to a single physical object is reconstructed as two separate
+final-state objects, an overlap removal procedure is used. If electron and muon candidates share a track, the
+electron candidate is removed. After that, if the Δ𝑅𝑦,𝜙 distance4 between a jet and an electron candidate is
+less than 0.2, the jet is discarded. If multiple jets satisfy this requirement, only the closest jet is removed.
+For jet–electron distances between 0.2 and 0.4, the electron candidate is removed. If the distance between a
+jet and a muon candidate is less than 0.2, and the jet has less than three associated tracks, the jet is removed.
+Any muon subsequently found at a distance of less than 0.4 from a jet is removed.
+5 Event reconstruction and selection
+The analysis searches for effects of FCNC tZq couplings both in tt decay and in single-top-quark production
+processes. In the first process, one of the top quarks decays through the dominant mode into a W boson
+and a b-quark (hereafter called the “SM top quark”, denoted by 𝑡SM), while the other top quark (hereafter
+called the “FCNC top quark”, denoted by 𝑡FCNC) decays into a Z boson and a 𝑢- or 𝑐-quark. In the second
+process, the production of a single top quark proceeds through an FCNC interaction in association with a
+Z boson, while its decay is through the dominant mode. In each channel, only the trilepton final state is
+targeted, in which the Z and W bosons decay leptonically. Therefore, the final state of the FCNC process in
+tt decays is characterized by the presence of three leptons, at least two jets, one of which is a b-jet, and
+missing transverse momentum from the escaping neutrino. The final state of the FCNC process in single
+top-quark production is instead characterized by the presence of three leptons, a b-jet, up to one additional
+jet, and missing transverse momentum. Due to the different final states, two separate signal regions (SRs)
+are defined, targeting the two processes: SR1 targets FCNC processes in tt decays while SR2 targets FCNC
+processes in single top-quark production. The SRs share common selections for the leptons and they differ
+in their top-quark reconstruction and jet multiplicity requirements.
+In both SRs, exactly three leptons (electrons or muons) that do not all have the same charge are required.
+One of the leptons must have 𝑝T > 27 GeV, because of the trigger thresholds, and must be matched, with
+Δ𝑅 < 0.15, to the lepton reconstructed by the trigger. Events with a fourth reconstructed lepton are
+vetoed. At least one opposite-sign same-flavor lepton pair (OSSF) with an invariant mass in the range
+|𝑚ℓℓ − 91.2 GeV| < 15 GeV is required. In the 𝜇ee and e𝜇𝜇 channels the pair is uniquely identified,
+whereas in the eee and 𝜇𝜇𝜇 channels both of the possible combinations are considered and the pair with
+the invariant mass closer to the Z boson mass is chosen. The lepton not used to reconstruct the Z boson is
+4 Δ𝑅𝑦,𝜙 is the Lorentz-invariant distance in the rapidity–azimuthal-angle plane, defined as Δ𝑅𝑦,𝜙 =
+√︃
+(Δ𝑦)2 + (Δ𝜙)2, where 𝑦
+is the rapidity, defined as 𝑦 = (1/2) ln [(𝐸 + 𝑝𝑧)/(𝐸 − 𝑝𝑧)].
+9
+
+assumed to be the one coming from the W boson, ℓW. In SR2, to help reject background sources with a
+third nonprompt lepton, events are required to have 𝑚T(ℓW, 𝜈) > 40 GeV.5
+In SR1 the selected events have at least two jets, with exactly one b-tagged. In SR2 the selected events have
+one or two jets, with exactly one b-tagged. For events with exactly two jets, orthogonality between SR1
+and SR2 is ensured by using an invariant mass cut on reconstructed top-quark candidates. An additional
+SR targeting the FCNC tZc coupling in tt decay, based on the presence of a c-jet, was considered. The
+c-tagging was done using the soft-muon tagging technique employed in Ref. [92]. With the current dataset,
+this SR was found to bring only marginal improvements to the final limits.
+In the events having at least two jets with one of them being b-tagged, the reconstruction of FCNC and
+SM top-quark candidates is based on the “FCNC-in-tt-decay” signal hypothesis. The kinematics of the
+top-quark candidates are reconstructed from the corresponding decay particles by minimizing the following
+expression:
+𝜒2
+tt =
+�
+𝑚reco
+𝑗𝑎ℓℓ − 𝑚𝑡FCNC
+�2
+𝜎2
+tFCNC
++
+�
+𝑚reco
+𝑗𝑏ℓW 𝜈 − 𝑚𝑡SM
+�2
+𝜎2
+tSM
++
+�
+𝑚reco
+ℓW 𝜈 − 𝑚W
+�2
+𝜎2
+W
+,
+(2)
+where 𝑚reco
+𝑗𝑎ℓℓ, 𝑚reco
+𝑗𝑏ℓW 𝜈, and 𝑚reco
+ℓW 𝜈 are the reconstructed masses of the Zq, Wb, and ℓW𝜈 systems, respectively.
+The minimization has two independent parts. The first is the jet permutation, where any non-b-tagged jet
+can be assigned to 𝑗𝑎, while 𝑗𝑏 must correspond to a b-tagged jet. The second is the minimization of the 𝜒2
+tt
+for each permutation by varying the longitudinal component of the neutrino momentum, 𝑝𝜈
+𝑧, to determine
+the most probable value while its transverse component is set to the missing transverse momentum in the
+event.
+This procedure assigns a reconstructed jet to the q-quark from the decay of the FCNC top quark and
+determines the 𝑝𝜈
+𝑧 value to reconstruct the four-momenta of the two top-quark candidates.
+In Eq. (2), the central values (𝑚𝑡FCNC, 𝑚𝑡SM and 𝑚W) and the widths (𝜎tFCNC, 𝜎tSM and 𝜎W) of the distributions
+of the reconstructed masses of the top quark and W boson candidates are taken from reconstructed simulated
+FCNC-in-tt-decay signal events that undergo the common object selection procedure just described. This
+is done by matching the true q- and b-quarks in the simulated events to the reconstructed jets, setting
+the longitudinal momentum of the neutrino to the 𝑝𝑧 of the true generated neutrino, and the transverse
+component to the missing transverse momentum in the event, and then performing a likelihood fit with a
+Bukin function6 [93] to the masses of the reconstructed top quarks and W boson. The mass values for
+the LH coupling are reported in Table 1. Compatible mass values are obtained for the RH coupling. The
+fraction of reconstructed top-quark candidates that are matched to the true simulated particles within a
+cone of size Δ𝑅 = 0.4 is 𝜖𝑡FCNC = 75% for the FCNC top-quark candidates and 𝜖𝑡SM = 54% for the SM
+top-quark candidates, where the difference comes from the fact that for the SM top-quark decay the match
+of the missing transverse momentum with the generated neutrino is less efficient.
+5 The transverse mass is calculated using the momentum of the lepton associated with the W boson, the 𝐸miss
+T
+and the azimuthal
+angle, 𝜙, between them: 𝑚T(ℓW, 𝜈)=
+√︃
+2𝑝ℓ
+T𝐸miss
+T
+(1 − cos Δ𝜙).
+6 These fits use a generalization of the Gaussian function to allow for asymmetric tails in the distribution. The overall normalization
+is fixed to the yield and the shape of the function is determined by five parameters: the peak position, the width of the core, the
+asymmetry, the size of the lower tail, and the size of the higher tail. From these parameters, only the peak position and the
+width enter the 𝜒2.
+10
+
+Table 1: Summary of the mean values and standard deviations of the invariant mass distributions for the top-quark
+candidates and the W boson. These values are obtained from the Bukin fits using the FCNC-in-tt-decay signal
+samples with the LH coupling. The two FCNC tZu and tZc coupling samples are combined.
+FCNC top quark
+SM top quark
+W boson
+𝑚𝑡FCNC [GeV]
+𝜎tFCNC [GeV]
+𝑚𝑡SM [GeV]
+𝜎tSM [GeV]
+𝑚W [GeV]
+𝜎W [GeV]
+FCNC in tt decay (LH)
+171.0
+11.1
+166.5
+23.2
+80.5
+15.4
+Under the FCNC-in-single-top-quark-production signal hypothesis, the SM top-quark candidate is instead
+reconstructed in events having one or two jets, with exactly one 𝑏-tagged. The missing transverse
+momentum is assumed to be the transverse component of the neutrino momentum, while the most probable
+value of 𝑝𝜈
+𝑧 is determined by minimizing the following expression:
+𝜒2
+tZ =
+�
+𝑚reco
+𝑗𝑏ℓW 𝜈 − 𝑚𝑡SM
+�2
+𝜎2
+tSM
++
+�
+𝑚reco
+ℓW 𝜈 − 𝑚W
+�2
+𝜎2
+W
+,
+(3)
+where 𝑚reco
+𝑗𝑏ℓW 𝜈 and 𝑚reco
+ℓW 𝜈 are the reconstructed masses of the Wb and ℓW𝜈 systems, respectively. In Eq. (3),
+the central values for the masses and widths of the top quark and W boson are taken from reconstructed
+simulated FCNC-in-tt-decay signal events, as is done in Eq. (2).7 Therefore, in the events with two
+jets, the four-momentum of the SM top-quark candidate reconstructed under the FCNC-in-single-top-
+quark-production signal hypothesis is the same as that reconstructed under the FCNC-in-tt-decay signal
+hypothesis. In this case, the fraction of reconstructed top-quark candidates that are matched to the true
+simulated particles within a cone of size Δ𝑅 = 0.4 is 𝜖𝑡SM = 71%.
+In SR1, the mass of the FCNC top-quark candidate, 𝑚reco
+𝑗𝑎ℓℓ, is required to be within 2𝜎tFCNC of 172.5 GeV,
+while no requirement is placed on the mass of the SM top-quark candidate, 𝑚reco
+𝑗𝑏ℓW 𝜈. In SR2, the mass of the
+SM top-quark candidate is required to be within 2𝜎tSM of 172.5 GeV. In addition, to ensure orthogonality
+with SR1, for events with exactly two jets the mass of the FCNC top-quark candidate is required to be
+more than 2𝜎tFCNC from 172.5 GeV. Table 2 summarizes the selection criteria applied to the signal regions
+considered. With these criteria, 496 data events are selected in SR1 and 460 are selected in SR2.
+Figure 2 shows the distributions of the masses of the two top-quark candidates in SR1, and the mass of the
+top-quark candidate and the 𝑝T of the reconstructed Z boson in SR2. These kinematic distributions are
+some of the key features that distinguish signal events from the backgrounds and they are utilized in the
+multivariate analysis described in Section 6. In SR1, the dominant signal is the FCNC-in-tt-decay events
+(shown with solid lines in Figure 2 separately for the tZu and tZc couplings), while the FCNC-in-single-
+top-quark-production contribution (shown with dashed lines) is smaller. In contrast, SR2 is more sensitive
+to the tZu FCNC-in-single-top-quark-production signal, with similar smaller contributions from the other
+three signals. After the event selection the main background sources are ttZ, tZ and diboson production.
+7 Using the central values for the masses and widths extracted from the FCNC single-top production signal sample does not have
+a significant effect on the final results.
+11
+
+100
+150
+200
+250
+300
+350
+400
+450
+ [GeV]
+reco
+ν
+W
+l
+bj
+m
+0.5
+0.75
+1
+1.25
+ 
+Data / Bkg.
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Events / 10 GeV
+ATLAS   
+-1
+ = 13 TeV, 139 fb
+s
+SR1, Pre-Fit
+Data
+Z+tWZ
+tt
+VV+LF
+VV+HF
+tZ
+Fake lep.
+Other bkg.
+Bkg. uncertainty
+ 5
+×
+FCNC (u)tZ 
+ 5
+×
+(uZ) 
+t
+FCNC t
+ 5
+×
+FCNC (c)tZ 
+ 5
+×
+(cZ) 
+t
+FCNC t
+(a)
+155
+160
+165 170
+175
+180
+185
+190
+ [GeV]
+reco
+ll
+aj
+m
+0.5
+0.75
+1
+1.25
+ 
+Data / Bkg.
+0
+20
+40
+60
+80
+100
+120
+140
+Events / 5 GeV
+ATLAS   
+-1
+ = 13 TeV, 139 fb
+s
+SR1, Pre-Fit
+Data
+Z+tWZ
+tt
+VV+LF
+VV+HF
+tZ
+Fake lep.
+Other bkg.
+Bkg. uncertainty
+ 5
+×
+FCNC (u)tZ 
+ 5
+×
+(uZ) 
+t
+FCNC t
+ 5
+×
+FCNC (c)tZ 
+ 5
+×
+(cZ) 
+t
+FCNC t
+(b)
+130 140 150 160 170 180 190 200 210
+ [GeV]
+reco
+ν
+W
+l
+bj
+m
+0.5
+0.75
+1
+1.25
+ 
+Data / Bkg.
+0
+20
+40
+60
+80
+100
+120
+Events / 5 GeV
+ATLAS   
+-1
+ = 13 TeV, 139 fb
+s
+SR2, Pre-Fit
+Data
+Z+tWZ
+tt
+VV+LF
+VV+HF
+tZ
+Fake lep.
+Other bkg.
+Bkg. uncertainty
+ 5
+×
+FCNC (u)tZ 
+ 5
+×
+(uZ) 
+t
+FCNC t
+ 5
+×
+FCNC (c)tZ 
+ 5
+×
+(cZ) 
+t
+FCNC t
+(c)
+0
+50
+100 150 200 250 300 350 400 450
+ [GeV]
+T
+p
+ boson 
+Z
+0.5
+0.75
+1
+1.25
+ 
+Data / Bkg.
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Events / 20 GeV
+ATLAS   
+-1
+ = 13 TeV, 139 fb
+s
+SR2, Pre-Fit
+Data
+Z+tWZ
+tt
+VV+LF
+VV+HF
+tZ
+Fake lep.
+Other bkg.
+Bkg. uncertainty
+ 5
+×
+FCNC (u)tZ 
+ 5
+×
+(uZ) 
+t
+FCNC t
+ 5
+×
+FCNC (c)tZ 
+ 5
+×
+(cZ) 
+t
+FCNC t
+(d)
+Figure 2: Comparison between data and background prediction before the fit (“Pre-Fit”) for some kinematic
+distributions in the SRs. The distributions are: (a) the mass of the SM top-quark candidate in SR1, (b) the mass of
+the FCNC top-quark candidate in SR1, (c) the mass of the SM top-quark candidate in SR2 and (d) the transverse
+momentum of the Z boson candidate in SR2. The uncertainty band includes both the statistical and systematic
+uncertainties in the background prediction. The four FCNC LH signals are also shown separately, normalized to five
+times the cross-section corresponding to the most stringent observed branching ratio limits [21]. The first (last) bin
+in all distributions includes the underflow (overflow). The lower panels show the ratios of the data (“Data”) to the
+background prediction (“Bkg.”).
+12
+
+Table 2: Overview of the requirements applied to select the events in the signal regions. OSSF is an opposite-sign
+same-flavor lepton pair, 𝑚Z = 91.2 GeV and 𝑚t = 172.5 GeV.
+Common selections
+Exactly 3 leptons with 𝑝T(ℓ1) > 27 GeV
+≥ 1 OSSF pair, with |𝑚ℓℓ − 𝑚Z| < 15 GeV
+SR1
+SR2
+≥ 2 jets
+1 jet
+2 jets
+1 b-jet
+1 b-jet
+1 b-jet
+–
+𝑚T(ℓW, 𝜈)> 40 GeV
+𝑚T(ℓW, 𝜈)> 40 GeV
+|𝑚reco
+𝑗𝑎ℓℓ − 𝑚t| < 2𝜎tFCNC
+–
+|𝑚reco
+𝑗𝑎ℓℓ − 𝑚t| > 2𝜎tFCNC
+–
+|𝑚reco
+𝑗𝑏ℓ𝑊 𝜈 − 𝑚t| < 2𝜎tSM
+|𝑚reco
+𝑗𝑏ℓ𝑊 𝜈 − 𝑚t| < 2𝜎tSM
+6 Background estimation and separation from signal
+Two classes of backgrounds are considered: processes in which three or more prompt leptons are produced,
+such as diboson production or the associated production of top quarks (ttZ, tWZ, tZ, ttW, ttH) and
+processes with two prompt leptons in the final state along with one additional nonprompt or fake lepton that
+satisfies the selection criteria, such as tt, tW, and Z + jets. Such nonprompt or fake leptons can originate
+from decays of bottom or charm hadrons, jets misidentified as electrons, leptons from kaon or pion decays,
+or electrons from photon conversions.
+All background contributions are estimated by using MC samples that are normalized to their respective
+SM predicted cross-sections calculated at NLO in QCD. The cross-section of the ttH background includes
+NLO+NLL soft-gluon resummation [94]. For the tt + tW nonprompt lepton backgrounds the normalization
+is extracted from data, as described later.
+After applying the event selection requirements, diboson, ttZ and tZ production constitute the largest
+backgrounds. For SR1, the dominant backgrounds are ttZ and 𝑉𝑉 + HF production. Monte Carlo
+simulation indicates that these represent more than 65% of the total number of selected background events
+in this region, with the two processes contributing equally. For SR2, 𝑉𝑉 + HF and tZ are the dominant
+backgrounds, giving 70% of background events. The processes with nonprompt leptons constitute a minor
+background, with their contribution being at most 10% of the total selected events.
+Four control regions (CRs) are defined and used in the fit that is described in Section 8. The CRs are used
+to adjust the normalization and to reduce the associated systematic uncertainties in the main backgrounds.
+The selections applied to define the CRs are summarized in Table 3 and described in the following.
+A tt CR is designed to control the tt background. The tt CR is constructed by requiring the presence of
+three leptons, with one of the possible pairs having opposite charge, as in the SRs. To veto the presence
+of a Z boson, the opposite-sign lepton pair is also required to consist of different flavors. Events with at
+least one jet, with exactly one b-tagged, are considered. This region is dominated by tt events with 40%
+contamination from other backgrounds, mainly ttW and ttH. A total of 157 data events are selected for the
+tt CR.
+To control the ttZ background, a ttZ CR is defined. The requirements on the leptons are the same as for
+the SRs, while at least four jets, with exactly two b-tagged, are required. This region is dominated by ttZ
+13
+
+events with 25% contamination from other backgrounds, mainly tZ and 𝑉𝑉 + HF. A total of 286 data
+events are selected for the ttZ CR.
+Two mass sideband CRs are also included. These CRs are designed to contain a mixture of the main
+background sources (ttZ and diboson). The mass sideband CR1 is defined with almost the same event
+selection as SR1, with the differences being that the mass of the FCNC top-quark candidate must be more
+than 2𝜎tFCNC from 172.5 GeV, and the mass of the SM top-quark candidate must also be more than 2𝜎tSM
+from 172.5 GeV. The mass sideband CR2 is defined with almost the same event selection as SR2, with
+the differences being that only events with one jet are considered and that the mass of the SM top-quark
+candidate must be more than 2𝜎tSM from 172.5 GeV. Totals of 343 and 104 data events are selected for the
+mass sidebands CR1 and CR2 respectively.
+Table 3: Overview of the requirements applied to select the events in the control regions. OSSF is an opposite-sign
+same-flavor lepton pair, 𝑚Z = 91.2 GeV and 𝑚t = 172.5 GeV.
+Common selections
+Exactly 3 leptons with 𝑝T(ℓ1) > 27 GeV
+tt CR
+ttZ CR
+Sideband CR1
+Sideband CR2
+≥ 1 OS pair, no OSSF
+≥ 1 OSSF pair
+≥ 1 OSSF pair
+≥ 1 OSSF pair
+with |𝑚ℓℓ − 𝑚Z| < 15 GeV
+with |𝑚ℓℓ − 𝑚Z| < 15 GeV
+with |𝑚ℓℓ − 𝑚Z| < 15 GeV
+–
+–
+–
+𝑚T(ℓW, 𝜈) > 40 GeV
+≥ 1 jet
+≥ 4 jets
+≥ 2 jets
+1 jet
+1 b-jet
+2 b-jets
+1 b-jet
+1 b-jet
+–
+–
+|𝑚reco
+𝑗𝑎ℓℓ − 𝑚t| > 2𝜎tFCNC
+–
+–
+–
+|𝑚reco
+𝑗𝑏ℓ𝑊 𝜈 − 𝑚t| > 2𝜎tSM
+|𝑚reco
+𝑗𝑏ℓ𝑊 𝜈 − 𝑚t| > 2𝜎tSM
+To better separate the signal from the backgrounds, a multivariate analysis (MVA) technique is used. The
+chosen MVA is the gradient boosted decision tree (GBDT) method implemented with TMVA [95, 96].
+Decision trees [97] recursively partition the parameter space into regions where signal or background
+purities are enhanced. Gradient boosting is a method which improves the performance and stability of
+decision trees and involves the combination of many trees into a single final discriminant. After boosting,
+the final score undergoes a transformation to map the scores onto the interval −1 to +1. The most signal-like
+events have scores near +1 while the most background-like events have scores near −1. A 𝑘-fold cross
+validation is employed.
+The GBDT training is done separately for the LH and RH samples and in each SR as follows. In SR1, for
+both the FCNC tZu and tZc coupling searches, the expected contribution from FCNC processes in tt decay
+is significantly higher than the one from single-top-quark production. Therefore, the GBDT is trained with
+only the FCNC-in-tt-decay signal against all backgrounds. Since the kinematics of FCNC-in-tt-decay
+events for tZu and tZc couplings are similar, the FCNC-in-tt-decay signal samples with the two couplings
+are combined to train the GBDT. Therefore, in SR1 a single MVA discriminant, 𝐷1, is built for both the
+FCNC tZu and tZc coupling searches. In contrast, SR2 is particularly sensitive to the FCNC tZu coupling
+in single-top-production events. Thus, the corresponding MVA discriminant, 𝐷𝑢
+2, is built by training the
+GBDT with the the tZu-coupling FCNC-in-single-top-production sample against all backgrounds. Despite
+the lower sensitivity to the FCNC tZc coupling in SR2, this region is used in combination with SR1 in
+the search for a FCNC tZc coupling signal. In the total expected FCNC tZc signal yield, the contribution
+from the FCNC processes in tt decay events is comparable to the one from the single-top-quark-production
+14
+
+events. Therefore, in SR2 the MVA discriminant for the search for a FCNC tZc coupling signal, 𝐷𝑐
+2, is built
+using both the FCNC-in-tt-decay and FCNC-in-single-top-production samples against all backgrounds.
+For the training of each of the three discriminants, a total of six variables is used. These variables are
+chosen from a larger set. Only variables that provide good separation and are well modeled are used in
+the final training. For the 𝐷1 discriminant the six variables are: the reconstructed masses of the SM and
+FCNC top-quark candidates, the Δ𝑅 separation between them, the Δ𝑅 separation between the lepton from
+the SM top-quark decay and the reconstructed Z boson, the number of jets, and the transverse momentum
+of the jets associated with the u/c-quark from the FCNC top-quark candidate’s decay. For both the 𝐷𝑢
+2 and
+𝐷𝑐
+2 discriminants the following six variables are used: the 𝑝T of the Z boson and of the b-tagged jet, the
+Δ𝑅 separation between them, the SM top-quark candidate’s mass, the Δ𝑅 separation between the lepton
+from the SM top-quark candidate decay and the reconstructed Z boson, and the 𝜒2 from the kinematic fit
+under the signal hypothesis of an FCNC process in single-top-quark production.
+7 Systematic uncertainties
+Systematic uncertainties in the signal acceptance and in the normalization of the individual backgrounds,
+as well as uncertainties in the shape of the fitted distributions, are taken into account. These are treated as
+being correlated between the different regions, unless stated otherwise. The uncertainties are classified
+into the following categories:
+Reconstruction efficiency and calibration uncertainties:
+Systematic uncertainties affecting the recon-
+struction efficiency and energy calibration of electrons, muons, jets and b-jets are propagated through the
+analysis.
+The differences between the electron (muon) trigger, reconstruction, selection and isolation efficiencies
+in data and those in MC simulation are corrected for by scale factors derived from dedicated Z → e+e−
+( Z → 𝜇+𝜇− ) enriched control samples using a tag-and-probe method [81, 83]. Uncertainties in these
+scale factors are taken into account. Moreover, uncertainties are included for the electron (muon) energy
+(momentum) scale and resolution [81, 83].
+For the jets, an uncertainty for the JVT requirement is considered. The jet energy scale was derived using
+information from test-beam data, LHC collision data and simulation, as described in Ref. [98]. The impact
+of the uncertainty in the jet energy resolution is also evaluated.
+The 𝑏-tagging efficiencies and mistagging rates are measured in data using the same methods as described
+in Refs. [99–101], with the systematic uncertainties due to 𝑏-tagging efficiency and the mistagging rates
+calculated separately. The impact of the uncertainties on the 𝑏-tagging calibration is evaluated separately
+for b-, c- and light-jets in the MC samples.
+The uncertainty in 𝐸miss
+T
+due to a possible miscalibration of the soft-track component of the 𝐸miss
+T
+is derived
+from data–MC comparisons of the 𝑝T balance between the hard and soft 𝐸miss
+T
+components [90]. The
+uncertainty associated with the leptons and jets is propagated from the corresponding uncertainties in the
+energy/momentum scales and resolutions, and is classified together with the uncertainty associated with
+the corresponding objects.
+15
+
+Signal and background modeling:
+The systematic uncertainties due to MC modeling of the signal and
+the main backgrounds are estimated by comparing samples from different MC generators and PDF sets
+and by varying the parameters associated with the renormalization and factorization scales, and additional
+radiation. For some processes, some of these uncertainties are found to be negligible and therefore they are
+not mentioned in the following.
+For the signal, the effects of the systematic uncertainty in the renormalization and factorization scales, 𝜇r
+and 𝜇f, are taken into account by varying these parameters by factors of 2 and 0.5 with respect to their
+default values and comparing the results of these variations with the nominal prediction. The uncertainty in
+the modeling of the parton shower is estimated by comparing the nominal signal sample with one generated
+with Herwig 7 instead of Pythia 8. PDF uncertainties are found to be negligible and are not included for
+the signal.
+For the ttZ and tZ backgrounds, the following uncertainties are included. The effect of changing the parton
+shower is considered as an uncertainty, following the same strategy used for the signal. The uncertainty
+due to ISR is estimated by comparing the nominal event sample with two samples where the Var3c up and
+down variations of the A14 tune were employed. Uncertainties from the variation of 𝜇r and 𝜇f are also
+included.
+For the tWZ background, the effect of changing the modeling of the interference with ttZ is included by
+comparing two different diagram removal predictions.
+The effect of changing the MC generator for the modeling of the diboson background is considered as an
+uncertainty. It is evaluated by comparing the nominal Sherpa sample with one generated with Powheg Box.
+This uncertainty is split into the two light- and heavy-flavor components and evaluated separately for each
+jet multiplicity. Uncertainties in the 𝜇r and 𝜇f scales, as well as in the PDF and in 𝛼s are also included for
+the diboson background.
+For the tt background, several sources of uncertainty are taken into account. The effect of changing the
+parton shower is included as an uncertainty. The Var3c A14 tune variations, as well as variations of 𝜇r and
+𝜇f are also included. Additionally, the uncertainty associated with the ℎdamp parameter is evaluated by
+using the alternative sample with the ℎdamp value increased to 3 𝑚𝑡. The NNPDF3.0lo replicas are used to
+evaluate the PDF uncertainties for the nominal PDF. Finally, an uncertainty is added to take into account
+the differences in tt background composition between the SRs and the tt CR, which is used to control the tt
+background in the fit to data. In particular, the fractions of nonprompt leptons originating from each source
+are computed, separately for photon conversions and b-hadron decays, in the SRs and in the tt CR, for each
+jet multiplicity. Then the maximum variation of the fractions between the control region and the signal
+regions is taken as an uncertainty.
+Signal and background rate uncertainty:
+The tt cross-section uncertainties due to the PDF and 𝛼s are
+calculated using the PDF4LHC15 prescription [102] with the MSTW2008nnlo [103, 104], CT10nnlo [65,
+105] and NNPDF2.3lo PDF sets, and are added in quadrature to the effect of the scale uncertainty, resulting
+in a total uncertainty of 5.5% that is assigned to the FCNC-in-tt-decay signal.
+For the ttZ background, a 12% rate uncertainty is included [106], and for the ttH process the normalization
+uncertainty is 15% [106], while for ttW a more conservative 50% is used [107]. For the tZ process,
+an uncertainty of 15% in the normalization is applied [108, 109], while for the tWZ process a more
+conservative 30% is used. For 𝑉𝑉 + LF production, the normalization uncertainty is taken to be 20% [110]
+and for 𝑉𝑉 + HF production it is 30% [111]. Concerning the Z + jets process, a rate uncertainty of 100% is
+16
+
+applied, due to the presence of a nonprompt lepton. A conservative overall normalization uncertainty of
+50% is applied to the remaining minor backgrounds (ttt, tttt, 𝑉𝑉𝑉, 𝑉H and ttWW). These background
+components are typically well below 1% in the SRs.
+Luminosity:
+The uncertainty in the combined 2015–2018 integrated luminosity is 1.7% [112], obtained
+using the LUCID-2 detector [113] for the primary luminosity measurements.
+Uncertainty in pileup modeling:
+The uncertainty in pileup modeling is accounted for by varying the
+reweighting of the MC samples to the data pileup conditions, using the uncertainty in the average number
+of interactions per bunch crossing.
+8 Results
+A simultaneous binned profile likelihood fit to the data in the SRs and the CRs is performed using MC
+distributions of both the signal and background predictions. Four separate fits are performed to extract LH
+and RH results for the FCNC tZu and tZc couplings. Only the relevant signal templates are used in each fit.
+The templates are binned distributions of the 𝐷1 discriminant in SR1 and in the mass sideband CR1; the
+𝐷𝑢
+2 discriminant in SR2 and in the mass sideband CR2; and the total event yields in the tt CR and the ttZ
+CR. When fitting to extract limits on the FCNC tZc coupling, the 𝐷𝑐
+2 discriminant is used instead of 𝐷𝑢
+2.
+The fitted SRs are defined from the SRs described in Section 5 after removing events that constitute
+two validation regions (VRs) that are not included in the fit, but the fit results are propagated to those
+regions. The VRs are used to check the stability of the fit and they are obtained by applying a selection
+on the GBDT discriminants. VR1 is defined by selecting events with 𝐷1 < −0.6 from the SR1, while
+VR2 contains events from SR2 with 𝐷𝑢
+2 < −0.7 and 𝐷𝑐
+2 < −0.4. With the given normalization of
+the signal samples, the fraction of signal events that is selected from the SRs to enter the VRs ranges
+from 2% to 5%, depending on the SR. The signal contamination in the VRs is at most 2%. The signal
+selection efficiency for the FCNC-in-tt-decay signal in SR1 ranges between 4% and 5%, while that for the
+FCNC-in-single-top-production signal in SR2 ranges between 3% and 4%. In contrast, the signal selection
+efficiency for the FCNC-in-tt-decay signal in SR2 and the FCNC-in-single-top-production signal in SR1 is
+around 1%.
+The statistical analysis used to extract the signal is based on a binned likelihood function L(𝜇, �𝜃) constructed
+as a product of Poisson probability terms over all bins in each considered distribution, and Gaussian
+constraint terms for �𝜃, a set of nuisance parameters that parameterize effects of MC statistical and systematic
+uncertainties in the signal and background expectations. The signal strength parameter 𝜇 is a multiplicative
+factor applied to the number of signal events normalized to a reference branching ratio. For that, the
+most stringent limits mentioned in Section 1 are used. The nuisance parameters are allowed to vary in
+the combined fit to adjust the expectations for signal and background according to the corresponding
+systematic uncertainties, and their final values are the adjustment that best fits the data. The normalization
+of the tt + tW backgrounds is unconstrained in the fit.
+17
+
+A test statistic, ˜𝑞𝜇, is constructed according to the profile likelihood ratio:
+˜𝑞𝜇 =
+�����������
+�����������
+−2 ln ��
+�
+L
+�
+𝜇, ˆˆ�𝜃 (𝜇)
+�
+L
+�
+0, ˆˆ�𝜃 (0)
+� ��
+�
+if ˆ𝜇 < 0,
+−2 ln ��
+�
+L
+�
+𝜇, ˆˆ�𝜃 (𝜇)
+�
+L
+�
+ˆ𝜇, ˆ�𝜃
+� ��
+�
+if 0 ≤ ˆ𝜇 ≤ 𝜇,
+0
+if ˆ𝜇 > 𝜇,
+(4)
+where ˆ𝜇 and ˆ�𝜃 are the parameters that maximize the likelihood, and ˆˆ�𝜃 are the nuisance parameter values
+that maximize the likelihood for a given 𝜇 hypothesis. This test statistic is used to determine the probability
+for accepting the background-only hypothesis for the observed data.
+Table 4 shows the pre- and post-fit predictions for the signal and background event yields along with the
+observed numbers of events in the VRs. The post-fit yields refer to the fit for the FCNC tZu LH coupling
+extraction. The data and background expectation are in better agreement after the fit, with an increase
+of the 𝑉𝑉 + HF background normalization within its pre-fit uncertainty. The post-fit level of agreement
+between data and the background prediction in the VRs shows no significant mismodeling.
+Table 4: Predicted and observed yields in the two VRs considered in the fit. The signal and background predictions
+are shown before (“Pre-fit”) and after the fit to data for the FCNC tZu LH coupling extraction (“Post-fit”). The quoted
+uncertainties include the statistical and systematic uncertainties of the yields. For the post-fit predictions, they are
+computed taking into account correlations among nuisance parameters and among processes. For the backgrounds
+with a nonprompt or fake lepton, the contribution from tt + tW is shown separately from “Other fakes”. For the
+minor backgrounds, the contribution from ttW and ttH are shown separately from “Other bkg.”.
+Pre-fit
+Post-fit
+VR1
+VR2
+VR1
+VR2
+ttZ + tWZ
+70
+± 10
+2.2
+± 0.6
+70
+± 7
+2.4
+± 0.6
+𝑉𝑉 + LF
+10
+± 5
+9.8
+± 3.4
+10
+± 5
+9.7
+± 3.0
+𝑉𝑉 + HF
+56
+± 28
+36
+± 14
+60
+± 14
+47
+± 8
+tZ
+6.5 ± 1.6
+13.5
+± 2.7
+6.6 ± 1.5
+14.7
+± 2.6
+tt + tW fakes
+5.4 ± 2.6
+4.5
+± 1.7
+4.8 ± 2.1
+3.8
+± 1.4
+Other fakes
+0.0 ± 0.6
+1.4
+± 1.9
+0.03 ± 0.24
+0.8
+± 1.1
+ttW
+2.3 ± 1.2
+0.48 ± 0.26
+2.3 ± 1.2
+0.48 ± 0.25
+ttH
+3.0 ± 0.5
+0.101 ± 0.032
+3.0 ± 0.5
+0.108 ± 0.033
+Other bkg.
+0.8 ± 0.4
+0.5
+± 0.7
+0.8 ± 0.4
+0.5
+± 0.6
+Total background
+154
+± 31
+69
+± 15
+158
+± 13
+79
+± 7
+Data
+151
+80
+151
+80
+Data / Bkg.
+0.98 ± 0.22
+1.16 ± 0.29
+0.96 ± 0.11
+1.01 ± 0.15
+Tables 5 and 6 show the observed number of events in data and the post-fit predictions for the signal and
+background event yields in the SRs and CRs. The yields refer to the fit for the FCNC tZu LH coupling
+extraction. Good agreement between data and the SM expectation is observed. The normalization factor
+for the tt + tW backgrounds, which is an unconstrained fit parameter, agrees with unity within uncertainties.
+The variations of the post-fit background normalizations are within pre-fit uncertainties. All post-fit values
+of the nuisance parameters are less than one standard deviation from the pre-fit values. The statistical
+component is the dominant contribution in the total uncertainty. The same conclusions are obtained from
+the fits for the other FCNC couplings.
+18
+
+Table 5: Predicted and observed yields in the two SRs considered in the fit. The signal and background predictions are
+shown after the fit to data for the FCNC tZu LH coupling extraction. The quoted uncertainties include the statistical
+and systematic uncertainties of the yields, computed taking into account correlations among nuisance parameters and
+among processes. For the backgrounds with a nonprompt or fake lepton, the contribution from tt + tW is shown
+separately from “Other fakes”. For the minor backgrounds, the contribution from ttW and ttH are shown separately
+from “Other bkg.”.
+SR1
+SR2
+(𝐷1 > −0.6)
+(𝐷𝑢
+2 > −0.7 or 𝐷𝑐
+2 > −0.4)
+ttZ + tWZ
+137
+± 12
+36
+±
+6
+𝑉𝑉 + LF
+18
+± 7
+24
+±
+8
+𝑉𝑉 + HF
+114
+± 19
+162
+± 26
+tZ
+46
+± 7
+108
+± 18
+tt + tW fakes
+14
+± 4
+27
+±
+8
+Other fakes
+7
+± 8
+5
+±
+6
+ttW
+4.2 ± 2.1
+3.1 ±
+1.6
+ttH
+4.8 ± 0.7
+0.89 ±
+0.17
+Other bkg.
+2.0 ± 1.0
+2.5 ±
+2.9
+FCNC (𝑢)𝑡𝑍
+0.9 ± 1.7
+4
+±
+8
+FCNC tt(𝑢𝑍)
+5
+± 9
+0.8 ±
+1.5
+Total background
+348
+± 15
+369
+± 21
+Data
+345
+380
+Table 6: Predicted and observed yields in the four CRs considered in the fit. The signal and background predictions
+are shown after the fit to data for the FCNC tZu LH coupling extraction. The quoted uncertainties include the
+statistical and systematic uncertainties of the yields, computed taking into account correlations among nuisance
+parameters and among processes. For the backgrounds with a nonprompt or fake lepton, the contribution from
+tt + tW is shown separately from “Other fakes”. For the minor backgrounds, the contribution from ttW and ttH are
+shown separately from “Other bkg.”.
+Sideband CR1
+Sideband CR2
+ttZ CR
+tt CR
+ttZ + tWZ
+102
+± 14
+8.2 ± 1.4
+230
+± 18
+15.4
+± 1.5
+𝑉𝑉 + LF
+27
+± 11
+12
+± 4
+0.23 ± 0.19
+0.38 ± 0.25
+𝑉𝑉 + HF
+166
+± 25
+64
+± 9
+17
+± 8
+2.9
+± 0.5
+tZ
+22
+± 4
+6.8 ± 1.4
+21
+± 5
+0.96 ± 0.19
+tt + tW fakes
+9.3 ± 2.6
+7.2 ± 2.1
+4.0 ± 1.3
+93
+± 19
+Other fakes
+2
+± 4
+2.0 ± 2.8
+0.15 ± 0.18
+0.08 ± 0.09
+ttW
+4.5 ± 2.3
+2.3 ± 1.2
+3.0 ± 1.5
+27
+± 13
+ttH
+2.6 ± 0.4
+0.33 ± 0.07
+7.5 ± 1.2
+14.1
+± 2.2
+Other bkg.
+3.3 ± 2.5
+0.8 ± 0.4
+1.9 ± 0.9
+3.2
+± 1.5
+FCNC (𝑢)𝑡𝑍
+0.4 ± 0.7
+0.17 ± 0.33
+0.09 ± 0.18
+0.05 ± 0.10
+FCNC tt(𝑢𝑍)
+0.14 ± 0.27
+0.04 ± 0.07
+0.11 ± 0.20
+0.018 ± 0.035
+Total background
+338
+± 18
+104
+± 8
+284
+± 16
+157
+± 13
+Data
+343
+104
+286
+157
+19
+
+The 𝜇 parameters are shown in Table 7.
+Table 7: Summary of the signal strength 𝜇 parameters obtained from the fits to extract LH and RH results for the
+FCNC tZu and tZc couplings. For the reference branching ratio, the most stringent limits are used [21].
+Vertex
+Coupling
+𝜇
+tZu
+LH
+0.08 ± 0.12 (stat.) ± 0.08 (syst.)
+tZu
+RH
+0.10 ± 0.12 (stat.) ± 0.08 (syst.)
+tZc
+LH
+0.10 ± 0.17 (stat.) ± 0.14 (syst.)
+tZc
+RH
+0.06 ± 0.16 (stat.) ± 0.13 (syst.)
+Figure 3 shows the distributions of the fitted variables in the CRs and SRs after the fit for the FCNC tZu
+LH coupling extraction. For the FCNC tZc LH coupling extraction, the fitted distributions are presented in
+Figure 4, where 𝐷𝑐
+2 is used in SR2 and in the mass sideband CR2. For the tt and ttZ CRs, only the event
+yields are used. The data and background prediction agree within the uncertainties.
+Limits on each FCNC t → Zq branching ratio are computed with the CLs method [114] using the asymptotic
+properties of 𝑞𝜇 [115] and assuming that only the corresponding FCNC coupling contributes. The observed
+and expected 95% confidence-level (CL) limits on the branching ratios are shown in Table 8, where the
+limits on the relevant Wilson coefficients are also reported. The expected limits on the branching ratios
+calculated without systematic uncertainties are lower by 20% and 25% for the tZu and tZc couplings,
+respectively. The leading systematic uncertainties include the uncertainty in the SM tZ background
+normalization and the diboson modeling uncertainties.
+Table 8 also shows limits on the FCNC tZu LH and RH couplings obtained when considering only one
+SR, either SR1 or SR2, and all CRs in the likelihood. The results show that SR2, targeting the FCNC
+single-top-production signal, contributes more strongly than SR1 to the combined limits. Separate results
+for the FCNC tZc coupling are not shown, since the limits are dominated by the FCNC-tt-in-decay signal.
+20
+
+1
+−
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+1
+1
+D
+0.5
+0.75
+1
+1.25
+ 
+Data / Bkg.
+0
+50
+100
+150
+200
+250
+Events / 0.1
+ATLAS   
+-1
+ = 13 TeV, 139 fb
+s
+Sideband CR1
+Post-Fit
+Data
+Z+tWZ
+tt
+VV+LF
+VV+HF
+tZ
+Fake lep.
+Other bkg.
+Bkg. uncertainty
+ 500
+×
+FCNC (u)tZ 
+ 500
+×
+(uZ) 
+t
+FCNC t
+(a)
+1
+−
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+1
+u
+2
+D
+0.5
+0.75
+1
+1.25
+ 
+Data / Bkg.
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Events / 0.2
+ATLAS   
+-1
+ = 13 TeV, 139 fb
+s
+Sideband CR2
+Post-Fit
+Data
+Z+tWZ
+tt
+VV+LF
+VV+HF
+tZ
+Fake lep.
+Other bkg.
+Bkg. uncertainty
+ 500
+×
+FCNC (u)tZ 
+ 500
+×
+(uZ) 
+t
+FCNC t
+(b)
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+1
+1
+D
+0.5
+0.75
+1
+1.25
+ 
+Data / Bkg.
+0
+20
+40
+60
+80
+100
+120
+140
+Events / 0.18
+ATLAS   
+-1
+ = 13 TeV, 139 fb
+s
+SR1
+ > -0.6
+1
+D
+Post-Fit
+Data
+Z+tWZ
+tt
+VV+LF
+VV+HF
+tZ
+Fake lep.
+Other bkg.
+Bkg. uncertainty
+ 50
+×
+FCNC (u)tZ 
+ 50
+×
+(uZ) 
+t
+FCNC t
+(c)
+1
+−
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+1
+u
+2
+D
+0.5
+0.75
+1
+1.25
+ 
+Data / Bkg.
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+Events / 0.2
+ATLAS   
+-1
+ = 13 TeV, 139 fb
+s
+SR2
+ > -0.4
+c
+2
+ > -0.7 or D
+u
+2
+D
+Post-Fit
+Data
+Z+tWZ
+tt
+VV+LF
+VV+HF
+tZ
+Fake lep.
+Other bkg.
+Bkg. uncertainty
+ 50
+×
+FCNC (u)tZ 
+ 50
+×
+(uZ) 
+t
+FCNC t
+(d)
+Figure 3: Comparison between data and background prediction after the fit to data (“Post-Fit”) for the FCNC tZu LH
+coupling extraction for the fitted distributions in the CRs and SRs. The distributions are: (a) the 𝐷1 discriminant in
+the mass sideband CR1, (b) the 𝐷𝑢
+2 discriminant in the mass sideband CR2, (c) the 𝐷1 discriminant in SR1 and (d)
+the 𝐷𝑢
+2 discriminant in SR2. The uncertainty band includes both the statistical and systematic uncertainties in the
+background prediction. The FCNC tZu LH signals are also separately shown, normalized to 500 or 50 times the best
+fit of the signal yield. The lower panels show the ratios of the data (“Data”) to the background prediction (“Bkg.”).
+21
+
+1
+−
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+1
+1
+D
+0.5
+0.75
+1
+1.25
+ 
+Data / Bkg.
+0
+50
+100
+150
+200
+250
+Events / 0.1
+ATLAS   
+-1
+ = 13 TeV, 139 fb
+s
+Sideband CR1
+Post-Fit
+Data
+Z+tWZ
+tt
+VV+LF
+VV+HF
+tZ
+Fake lep.
+Other bkg.
+Bkg. uncertainty
+ 500
+×
+FCNC (c)tZ 
+ 500
+×
+(cZ) 
+t
+FCNC t
+(a)
+1
+−
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+1
+c
+2
+D
+0.5
+0.75
+1
+1.25
+ 
+Data / Bkg.
+0
+10
+20
+30
+40
+50
+60
+Events / 0.2
+ATLAS   
+-1
+ = 13 TeV, 139 fb
+s
+Sideband CR2
+Post-Fit
+Data
+Z+tWZ
+tt
+VV+LF
+VV+HF
+tZ
+Fake lep.
+Other bkg.
+Bkg. uncertainty
+ 500
+×
+FCNC (c)tZ 
+ 500
+×
+(cZ) 
+t
+FCNC t
+(b)
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+1
+1
+D
+0.5
+0.75
+1
+1.25
+ 
+Data / Bkg.
+0
+20
+40
+60
+80
+100
+120
+140
+Events / 0.18
+ATLAS   
+-1
+ = 13 TeV, 139 fb
+s
+SR1
+ > -0.6
+1
+D
+Post-Fit
+Data
+Z+tWZ
+tt
+VV+LF
+VV+HF
+tZ
+Fake lep.
+Other bkg.
+Bkg. uncertainty
+ 50
+×
+FCNC (c)tZ 
+ 50
+×
+(cZ) 
+t
+FCNC t
+(c)
+1
+−
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+1
+c
+2
+D
+0.5
+0.75
+1
+1.25
+ 
+Data / Bkg.
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Events / 0.12
+ATLAS   
+-1
+ = 13 TeV, 139 fb
+s
+SR2
+ > -0.4
+c
+2
+ > -0.7 or D
+u
+2
+D
+Post-Fit
+Data
+Z+tWZ
+tt
+VV+LF
+VV+HF
+tZ
+Fake lep.
+Other bkg.
+Bkg. uncertainty
+ 50
+×
+FCNC (c)tZ 
+ 50
+×
+(cZ) 
+t
+FCNC t
+(d)
+Figure 4: Comparison between data and background prediction after the fit to data (“Post-Fit”) for the FCNC tZc LH
+coupling extraction for the fitted distributions in the CRs and SRs. The distributions are: (a) the 𝐷1 discriminant in
+the mass sideband CR1, (b) the 𝐷𝑐
+2 discriminant in the mass sideband CR2, (c) the 𝐷1 discriminant in SR1 and (d)
+the 𝐷𝑐
+2 discriminant in SR2. The uncertainty band includes both the statistical and systematic uncertainties in the
+background prediction. The FCNC tZc LH signals are also separately shown, normalized to 500 or 50 times the best
+fit of the signal yield. The lower panels show the ratios of the data (“Data”) to the background prediction (“Bkg.”).
+22
+
+Table 8: Observed and expected 95% CL limits on the FCNC t → Zq branching ratios and the effective coupling
+strengths for different vertices and couplings (top eight rows). For the latter, the energy scale is assumed to be
+ΛNP = 1 TeV. The bottom rows show, for the case of the FCNC t → Zu branching ratio, the observed and expected
+95% CL limits when only one of the two SRs, either SR1 or SR2, and all CRs are included in the likelihood.
+Observable
+Vertex
+Coupling
+Observed
+Expected
+SRs+CRs
+B(t → Zq)
+tZu
+LH
+6.2×10−5
+4.9 +2.1
+−1.4 × 10−5
+B(t → Zq)
+tZu
+RH
+6.6×10−5
+5.1 +2.1
+−1.4 × 10−5
+B(t → Zq)
+tZc
+LH
+13×10−5
+11 +5
+−3 × 10−5
+B(t → Zq)
+tZc
+RH
+12×10−5
+10 +4
+−3 × 10−5
+|𝐶 (13)∗
+𝑢𝑊 | and |𝐶 (13)∗
+𝑢𝐵
+|
+tZu
+LH
+0.15
+0.13 +0.03
+−0.02
+|𝐶 (31)
+𝑢𝑊 | and |𝐶 (31)
+𝑢𝐵 |
+tZu
+RH
+0.16
+0.14 +0.03
+−0.02
+|𝐶 (23)∗
+𝑢𝑊 | and |𝐶 (23)∗
+𝑢𝐵
+|
+tZc
+LH
+0.22
+0.20 +0.04
+−0.03
+|𝐶 (32)
+𝑢𝑊 | and |𝐶 (32)
+𝑢𝐵 |
+tZc
+RH
+0.21
+0.19 +0.04
+−0.03
+SR1+CRs
+B(t → Zq)
+tZu
+LH
+9.7×10−5
+8.6 +3.6
+−2.4 × 10−5
+B(t → Zq)
+tZu
+RH
+9.5×10−5
+8.2 +3.4
+−2.3 × 10−5
+SR2+CRs
+B(t → Zq)
+tZu
+LH
+7.8×10−5
+6.1 +2.7
+−1.7 × 10−5
+B(t → Zq)
+tZu
+RH
+9.0×10−5
+6.6 +2.9
+−1.8 × 10−5
+9 Conclusions
+A search for FCNC processes involving a top quark, an up-type quark and a Z boson is presented. FCNC
+tZq couplings are searched for both in tt decay events, where one top quark decays according to the SM
+and the other one decays as t → Zq, and in single top-quark production through the gq → tZ FCNC
+process, followed by SM top-quark decay. The analysis uses 139 fb−1 of pp collision data collected by the
+ATLAS experiment at the LHC between 2015 and 2018 at a center-of-mass energy of 13 TeV. Events
+with three leptons, a b-tagged jet, possible additional jets and missing transverse momentum are selected.
+Multivariate discriminants are used to distinguish signal events from background events.
+The data are in good agreement with the SM expectations, and no evidence of a signal is found. Limits at
+95% CL are placed on the t → Zq branching ratios for both the tZu and tZc vertices and for both the RH
+and LH couplings. Assuming a LH coupling, the observed limits on the branching ratios are 6.2 × 10−5 for
+t → Zu and 13 × 10−5 for t → Zc. These results for t → Zu (t → Zc) improve on the previous observed
+limits from ATLAS by a factor of 3 (2), and on the previous expected limits by a factor of 5 (3). These are
+the most stringent limits to date. The improvement relative to the previous results comes from the inclusion
+of the FCNC-in-single-top-quark-production signal and the usage of a multivariate analysis in addition to
+the higher integrated luminosity. These results also constrain the values of Wilson coefficients for effective
+field theory operators contributing to the t → Zu and t → Zc FCNC decays of the top quark.
+23
+
+References
+[1]
+Particle Data Group, Review of Particle Physics, PTEP 2020 (2020) 083C01.
+[2]
+S. L. Glashow, J. Iliopoulos, and L. Maiani, Weak Interactions with Lepton-Hadron Symmetry,
+Phys. Rev. D 2 (1970) 1285.
+[3]
+J. A. Aguilar-Saavedra,
+Top flavour-changing neutral interactions: theoretical expectations and experimental detection,
+Acta Phys. Polon. B 35 (2004) 2695, arXiv: hep-ph/0409342.
+[4]
+J. A. Aguilar-Saavedra, Effects of mixing with quark singlets, Phys. Rev. D 67 (2003) 035003,
+arXiv: hep-ph/0210112, [Erratum: Phys. Rev. D 69 (2004) 099901].
+[5]
+D. Atwood, L. Reina, and A. Soni,
+Phenomenology of two Higgs doublet models with flavor-changing neutral currents,
+Phys. Rev. D 55 (1997) 3156, arXiv: hep-ph/9609279.
+[6]
+J. J. Cao et al., Supersymmetry-induced flavor-changing neutral-current top-quark processes at the
+CERN Large Hadron Collider, Phys. Rev. D 75 (2007) 075021, arXiv: hep-ph/0702264.
+[7]
+J. M. Yang, B.-L. Young, and X. Zhang,
+Flavor-changing top quark decays in R-parity-violating supersymmetric models,
+Phys. Rev. D 58 (1998) 055001, arXiv: hep-ph/9705341.
+[8]
+K. Agashe, G. Perez, and A. Soni,
+Collider signals of top quark flavor violation from a warped extra dimension,
+Phys. Rev. D 75 (2007) 015002, arXiv: hep-ph/0606293.
+[9]
+P. Q. Hung, Y.-X. Lin, C. S. Nugroho, and T.-C. Yuan,
+Top quark rare decays via loop-induced FCNC interactions in extended mirror fermion model,
+Nucl. Phys. B 927 (2018) 166, arXiv: 1709.01690 [hep-ph].
+[10]
+J. A. Aguilar-Saavedra, A minimal set of top anomalous couplings, Nucl. Phys. B 812 (2009) 181,
+arXiv: 0811.3842 [hep-ph].
+[11]
+G. Durieux, F. Maltoni, and C. Zhang, Global approach to top-quark flavor-changing interactions,
+Phys. Rev. D 91 (2015) 074017, arXiv: 1412.7166 [hep-ph].
+[12]
+B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek,
+Dimension-six terms in the Standard Model Lagrangian, JHEP 10 (2010) 085,
+arXiv: 1008.4884 [hep-ph].
+[13]
+OPAL Collaboration, Search for single top quark production at LEP2,
+Phys. Lett. B 521 (2001) 181, arXiv: hep-ex/0110009.
+[14]
+ALEPH Collaboration, Search for single top production in 𝑒+𝑒− collisions at √𝑠 up to 209 GeV,
+Phys. Lett. B 543 (2002) 173, arXiv: hep-ex/0206070.
+[15]
+L3 Collaboration, Search for single top production at LEP, Phys. Lett. B 549 (2002) 290,
+arXiv: hep-ex/0210041.
+[16]
+DELPHI Collaboration, Search for single top production via FCNC at LEP at √𝑠 = 189-208 GeV,
+Phys. Lett. B 590 (2004) 21, arXiv: hep-ex/0404014.
+[17]
+ZEUS Collaboration, Search for single-top production in 𝑒𝑝 collisions at HERA,
+Phys. Lett. B 708 (2012) 27, arXiv: 1111.3901 [hep-ex].
+24
+
+[18]
+CDF Collaboration,
+Search for the Flavor-Changing Neutral-Current Decay 𝑡 → 𝑍𝑞 in 𝑝 ¯𝑝 Collisions at √𝑠 = 1.96 TeV,
+Phys. Rev. Lett. 101 (2008) 192002, arXiv: 0805.2109 [hep-ex].
+[19]
+D0 Collaboration, Search for flavor changing neutral currents in decays of top quarks,
+Phys. Lett. B 701 (2011) 313, arXiv: 1103.4574 [hep-ex].
+[20]
+ATLAS Collaboration, Search for flavour-changing neutral current top-quark decays to 𝑞𝑍 in 𝑝𝑝
+collision data collected with the ATLAS detector at √𝑠 = 8 TeV, Eur. Phys. J. C 76 (2016) 12,
+arXiv: 1508.05796 [hep-ex].
+[21]
+ATLAS Collaboration, Search for flavour-changing neutral current top-quark decays 𝑡 → 𝑞𝑍 in
+proton–proton collisions at √𝑠 = 13 TeV with the ATLAS detector, JHEP 07 (2018) 176,
+arXiv: 1803.09923 [hep-ex].
+[22]
+CMS Collaboration, Search Search for Flavor-Changing Neutral Currents in Top-Quark Decays
+𝑡 → 𝑍𝑞 in 𝑝𝑝 Collisions at √𝑠 = 8 TeV, Phys. Rev. Lett. 112 (2014) 171802,
+arXiv: 1312.4194 [hep-ex].
+[23]
+CMS Collaboration, Search for associated production of a 𝑍 boson with a single top quark and for
+𝑡𝑍 flavour-changing interactions in 𝑝𝑝 collisions at √𝑠 = 8 TeV, JHEP 07 (2017) 003,
+arXiv: 1702.01404 [hep-ex].
+[24]
+ATLAS Collaboration, The ATLAS Experiment at the CERN Large Hadron Collider,
+JINST 3 (2008) S08003.
+[25]
+ATLAS Collaboration, Performance of the ATLAS trigger system in 2015,
+Eur. Phys. J. C 77 (2017) 317, arXiv: 1611.09661 [hep-ex].
+[26]
+ATLAS Collaboration, The ATLAS Collaboration Software and Firmware,
+ATL-SOFT-PUB-2021-001, 2021, url: https://cds.cern.ch/record/2767187.
+[27]
+ATLAS Collaboration,
+ATLAS data quality operations and performance for 2015–2018 data-taking,
+JINST 15 (2020) P04003, arXiv: 1911.04632 [physics.ins-det].
+[28]
+ATLAS Collaboration, Performance of electron and photon triggers in ATLAS during LHC Run 2,
+Eur. Phys. J. C 80 (2020) 47, arXiv: 1909.00761 [hep-ex].
+[29]
+ATLAS Collaboration, Performance of the ATLAS muon triggers in Run 2,
+JINST 15 (2020) P09015, arXiv: 2004.13447 [hep-ex].
+[30]
+GEANT4 Collaboration, Geant4 – a simulation toolkit, Nucl. Instrum. Meth. A 506 (2003) 250.
+[31]
+ATLAS Collaboration, The ATLAS Simulation Infrastructure, Eur. Phys. J. C 70 (2010) 823,
+arXiv: 1005.4568 [physics.ins-det].
+[32]
+D. J. Lange, The EvtGen particle decay simulation package,
+Nucl. Instrum. Meth. A 462 (2001) 152.
+[33]
+T. Sjöstrand, S. Mrenna, and P. Skands, A brief introduction to PYTHIA 8.1,
+Comput. Phys. Commun. 178 (2008) 852, arXiv: 0710.3820 [hep-ph].
+[34]
+ATLAS Collaboration, The Pythia 8 A3 tune description of ATLAS minimum bias and inelastic
+measurements incorporating the Donnachie–Landshoff diffractive model,
+ATL-PHYS-PUB-2016-017, 2016, url: https://cds.cern.ch/record/2206965.
+[35]
+R. D. Ball et al., Parton distributions with LHC data, Nucl. Phys. B 867 (2013) 244,
+arXiv: 1207.1303 [hep-ph].
+25
+
+[36]
+J. Alwall et al., The automated computation of tree-level and next-to-leading order differential
+cross sections, and their matching to parton shower simulations, JHEP 07 (2014) 079,
+arXiv: 1405.0301 [hep-ph].
+[37]
+R. D. Ball et al., Parton distributions for the LHC run II, JHEP 04 (2015) 040,
+arXiv: 1410.8849 [hep-ph].
+[38]
+ATLAS Collaboration, ATLAS Pythia 8 tunes to 7 TeV data, ATL-PHYS-PUB-2014-021, 2014,
+url: https://cds.cern.ch/record/1966419.
+[39]
+A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks,
+FeynRules 2.0 – A complete toolbox for tree-level phenomenology,
+Comput. Phys. Commun. 185 (2014) 2250, arXiv: 1310.1921 [hep-ph].
+[40]
+C. Degrande, F. Maltoni, J. Wang, and C. Zhang, Automatic computations at next-to-leading order
+in QCD for top-quark flavor-changing neutral processes, Phys. Rev. D 91 (2015) 034024,
+arXiv: 1412.5594 [hep-ph].
+[41]
+M. Bähr et al., Herwig++ physics and manual, Eur. Phys. J. C 58 (2008) 639,
+arXiv: 0803.0883 [hep-ph].
+[42]
+J. Bellm et al., Herwig 7.0/Herwig++ 3.0 release note, Eur. Phys. J. C 76 (2016) 196,
+arXiv: 1512.01178 [hep-ph].
+[43]
+J. Bellm et al., Herwig 7.1 Release Note, (2017), arXiv: 1705.06919 [hep-ph].
+[44]
+L. A. Harland-Lang, A. D. Martin, P. Motylinski, and R. S. Thorne,
+Parton distributions in the LHC era: MMHT 2014 PDFs, Eur. Phys. J. C 75 (2015) 204,
+arXiv: 1412.3989 [hep-ph].
+[45]
+M. Beneke, P. Falgari, S. Klein, and C. Schwinn,
+Hadronic top-quark pair production with NNLL threshold resummation,
+Nucl. Phys. B 855 (2012) 695, arXiv: 1109.1536 [hep-ph].
+[46]
+M. Cacciari, M. Czakon, M. Mangano, A. Mitov, and P. Nason, Top-pair production at hadron
+colliders with next-to-next-to-leading logarithmic soft-gluon resummation,
+Phys. Lett. B 710 (2012) 612, arXiv: 1111.5869 [hep-ph].
+[47]
+P. Bärnreuther, M. Czakon, and A. Mitov, Percent-Level-Precision Physics at the Tevatron:
+Next-to-Next-to-Leading Order QCD Corrections to 𝑞 ¯𝑞 → 𝑡¯𝑡 + 𝑋,
+Phys. Rev. Lett. 109 (2012) 132001, arXiv: 1204.5201 [hep-ph].
+[48]
+M. Czakon and A. Mitov, NNLO corrections to top-pair production at hadron colliders: the
+all-fermionic scattering channels, JHEP 12 (2012) 054, arXiv: 1207.0236 [hep-ph].
+[49]
+M. Czakon and A. Mitov,
+NNLO corrections to top pair production at hadron colliders: the quark-gluon reaction,
+JHEP 01 (2013) 080, arXiv: 1210.6832 [hep-ph].
+[50]
+M. Czakon, P. Fiedler, and A. Mitov,
+Total Top-Quark Pair-Production Cross Section at Hadron Colliders Through 𝑂(𝛼4
+𝑆),
+Phys. Rev. Lett. 110 (2013) 252004, arXiv: 1303.6254 [hep-ph].
+[51]
+M. Czakon and A. Mitov,
+Top++: A program for the calculation of the top-pair cross-section at hadron colliders,
+Comput. Phys. Commun. 185 (2014) 2930, arXiv: 1112.5675 [hep-ph].
+26
+
+[52]
+S. Frixione, G. Ridolfi, and P. Nason,
+A positive-weight next-to-leading-order Monte Carlo for heavy flavour hadroproduction,
+JHEP 09 (2007) 126, arXiv: 0707.3088 [hep-ph].
+[53]
+P. Nason, A new method for combining NLO QCD with shower Monte Carlo algorithms,
+JHEP 11 (2004) 040, arXiv: hep-ph/0409146.
+[54]
+S. Frixione, P. Nason, and C. Oleari,
+Matching NLO QCD computations with parton shower simulations: the POWHEG method,
+JHEP 11 (2007) 070, arXiv: 0709.2092 [hep-ph].
+[55]
+S. Alioli, P. Nason, C. Oleari, and E. Re, A general framework for implementing NLO calculations
+in shower Monte Carlo programs: the POWHEG BOX, JHEP 06 (2010) 043,
+arXiv: 1002.2581 [hep-ph].
+[56]
+H. B. Hartanto, B. Jäger, L. Reina, and D. Wackeroth,
+Higgs boson production in association with top quarks in the POWHEG BOX,
+Phys. Rev. D 91 (2015) 094003, arXiv: 1501.04498 [hep-ph].
+[57]
+ATLAS Collaboration, Studies on top-quark Monte Carlo modelling for Top2016,
+ATL-PHYS-PUB-2016-020, 2016, url: https://cds.cern.ch/record/2216168.
+[58]
+T. Sjöstrand et al., An introduction to PYTHIA 8.2, Comput. Phys. Commun. 191 (2015) 159,
+arXiv: 1410.3012 [hep-ph].
+[59]
+ATLAS Collaboration,
+Improvements in 𝑡¯𝑡 modelling using NLO+PS Monte Carlo generators for Run 2,
+ATL-PHYS-PUB-2018-009, 2018, url: https://cds.cern.ch/record/2630327.
+[60]
+E. Re,
+Single-top 𝑊𝑡-channel production matched with parton showers using the POWHEG method,
+Eur. Phys. J. C 71 (2011) 1547, arXiv: 1009.2450 [hep-ph].
+[61]
+S. Frixione, E. Laenen, P. Motylinski, C. White, and B. R. Webber,
+Single-top hadroproduction in association with a 𝑊 boson, JHEP 07 (2008) 029,
+arXiv: 0805.3067 [hep-ph].
+[62]
+F. Demartin, B. Maier, F. Maltoni, K. Mawatari, and M. Zaro,
+tWH associated production at the LHC, Eur. Phys. J. C 77 (2017) 34,
+arXiv: 1607.05862 [hep-ph].
+[63]
+S. Alioli, P. Nason, C. Oleari, and E. Re,
+NLO vector-boson production matched with shower in POWHEG, JHEP 07 (2008) 060,
+arXiv: 0805.4802 [hep-ph].
+[64]
+ATLAS Collaboration, Measurement of the 𝑍/𝛾∗ boson transverse momentum distribution in 𝑝𝑝
+collisions at √𝑠 = 7 TeV with the ATLAS detector, JHEP 09 (2014) 145,
+arXiv: 1406.3660 [hep-ex].
+[65]
+H.-L. Lai et al., New parton distributions for collider physics, Phys. Rev. D 82 (2010) 074024,
+arXiv: 1007.2241 [hep-ph].
+[66]
+J. Pumplin et al.,
+New Generation of Parton Distributions with Uncertainties from Global QCD Analysis,
+JHEP 07 (2002) 012, arXiv: hep-ph/0201195.
+27
+
+[67]
+P. Golonka and Z. Was,
+PHOTOS Monte Carlo: a precision tool for QED corrections in 𝑍 and 𝑊 decays,
+Eur. Phys. J. C 45 (2006) 97, arXiv: hep-ph/0506026.
+[68]
+N. Davidson, T. Przedzinski, and Z. Was,
+PHOTOS Interface in C++: Technical and physics documentation,
+Comput. Phys. Commun. 199 (2016) 86, arXiv: 1011.0937 [hep-ph].
+[69]
+E. Bothmann et al., Event generation with Sherpa 2.2, SciPost Phys. 7 (2019) 034,
+arXiv: 1905.09127 [hep-ph].
+[70]
+T. Gleisberg and S. Höche, Comix, a new matrix element generator, JHEP 12 (2008) 039,
+arXiv: 0808.3674 [hep-ph].
+[71]
+S. Schumann and F. Krauss,
+A parton shower algorithm based on Catani–Seymour dipole factorisation, JHEP 03 (2008) 038,
+arXiv: 0709.1027 [hep-ph].
+[72]
+S. Höche, F. Krauss, M. Schönherr, and F. Siegert,
+A critical appraisal of NLO+PS matching methods, JHEP 09 (2012) 049,
+arXiv: 1111.1220 [hep-ph].
+[73]
+S. Höche, F. Krauss, M. Schönherr, and F. Siegert,
+QCD matrix elements + parton showers. The NLO case, JHEP 04 (2013) 027,
+arXiv: 1207.5030 [hep-ph].
+[74]
+S. Catani, F. Krauss, B. R. Webber, and R. Kuhn, QCD Matrix Elements + Parton Showers,
+JHEP 11 (2001) 063, arXiv: hep-ph/0109231.
+[75]
+S. Höche, F. Krauss, S. Schumann, and F. Siegert, QCD matrix elements and truncated showers,
+JHEP 05 (2009) 053, arXiv: 0903.1219 [hep-ph].
+[76]
+F. Buccioni et al., OpenLoops 2, Eur. Phys. J. C 79 (2019) 866, arXiv: 1907.13071 [hep-ph].
+[77]
+F. Cascioli, P. Maierhöfer, and S. Pozzorini, Scattering Amplitudes with Open Loops,
+Phys. Rev. Lett. 108 (2012) 111601, arXiv: 1111.5206 [hep-ph].
+[78]
+A. Denner, S. Dittmaier, and L. Hofer,
+Collier: A fortran-based complex one-loop library in extended regularizations,
+Comput. Phys. Commun. 212 (2017) 220, arXiv: 1604.06792 [hep-ph].
+[79]
+P. Nason and G. Zanderighi, 𝑊+𝑊−, 𝑊𝑍 and 𝑍𝑍 production in the POWHEG-BOX-V2,
+Eur. Phys. J. C 74 (2014) 2702, arXiv: 1311.1365 [hep-ph].
+[80]
+ATLAS Collaboration, Vertex Reconstruction Performance of the ATLAS Detector at √𝑠 = 13 TeV,
+ATL-PHYS-PUB-2015-026, 2015, url: https://cds.cern.ch/record/2037717.
+[81]
+ATLAS Collaboration, Electron and photon performance measurements with the ATLAS detector
+using the 2015–2017 LHC proton–proton collision data, JINST 14 (2019) P12006,
+arXiv: 1908.00005 [hep-ex].
+[82]
+ATLAS Collaboration, Evidence for the associated production of the Higgs boson and a top quark
+pair with the ATLAS detector, Phys. Rev. D 97 (2018) 072003, arXiv: 1712.08891 [hep-ex].
+[83]
+ATLAS Collaboration, Muon reconstruction and identification efficiency in ATLAS using the full
+Run 2 𝑝𝑝 collision data set at √𝑠 = 13 TeV, Eur. Phys. J. C 81 (2021) 578,
+arXiv: 2012.00578 [hep-ex].
+28
+
+[84]
+ATLAS Collaboration,
+Jet reconstruction and performance using particle flow with the ATLAS Detector,
+Eur. Phys. J. C 77 (2017) 466, arXiv: 1703.10485 [hep-ex].
+[85]
+M. Cacciari, G. P. Salam, and G. Soyez, The anti-𝑘𝑡 jet clustering algorithm, JHEP 04 (2008) 063,
+arXiv: 0802.1189 [hep-ph].
+[86]
+M. Cacciari, G. P. Salam, and G. Soyez, FastJet user manual, Eur. Phys. J. C 72 (2012) 1896,
+arXiv: 1111.6097 [hep-ph].
+[87]
+ATLAS Collaboration, Jet energy scale and resolution measured in proton–proton collisions at
+√𝑠 = 13 TeV with the ATLAS detector, Eur. Phys. J. C 81 (2020) 689,
+arXiv: 2007.02645 [hep-ex].
+[88]
+ATLAS Collaboration, Performance of pile-up mitigation techniques for jets in 𝑝𝑝 collisions at
+√𝑠 = 8 TeV using the ATLAS detector, Eur. Phys. J. C 76 (2016) 581,
+arXiv: 1510.03823 [hep-ex].
+[89]
+ATLAS Collaboration, ATLAS flavour-tagging algorithms for the LHC Run 2 𝑝𝑝 collision dataset,
+(2022), arXiv: 2211.16345 [physics.data-an].
+[90]
+ATLAS Collaboration, Performance of missing transverse momentum reconstruction with the
+ATLAS detector using proton–proton collisions at √𝑠 = 13 TeV, Eur. Phys. J. C 78 (2018) 903,
+arXiv: 1802.08168 [hep-ex].
+[91]
+ATLAS Collaboration,
+𝐸miss
+T
+performance in the ATLAS detector using 2015–2016 LHC 𝑝𝑝 collisions,
+ATLAS-CONF-2018-023, 2018, url: https://cds.cern.ch/record/2625233.
+[92]
+ATLAS Collaboration, Measurement of the top-quark mass using a leptonic invariant mass in 𝑝𝑝
+collisions at √𝑠 = 13 TeV with the ATLAS detector, (2022), arXiv: 2209.00583 [hep-ex].
+[93]
+A. D. Bukin, Fitting function for asymmetric peaks, 2007,
+arXiv: 0711.4449 [physics.data-an].
+[94]
+A. Kulesza, L. Motyka, T. Stebel, and V. Theeuwes,
+Soft gluon resummation for associated 𝑡¯𝑡𝐻 production at the LHC, JHEP 03 (2016) 065,
+arXiv: 1509.02780 [hep-ph].
+[95]
+J. H. Friedman, Stochastic gradient boosting, Comput. Stat. Data Anal. 38 (2002) 367.
+[96]
+A. Hoecker et al., TMVA - Toolkit for Multivariate Data Analysis, 2007,
+arXiv: physics/0703039 [physics.data-an].
+[97]
+L. Breiman, J. Friedman, C. Stone, and R. Olshen, Classification and Regression Trees,
+Chapman & Hall, 1984.
+[98]
+ATLAS Collaboration, Jet energy scale measurements and their systematic uncertainties in
+proton–proton collisions at √𝑠 = 13 TeV with the ATLAS detector, Phys. Rev. D 96 (2017) 072002,
+arXiv: 1703.09665 [hep-ex].
+[99]
+ATLAS Collaboration, ATLAS 𝑏-jet identification performance and efficiency measurement with 𝑡¯𝑡
+events in 𝑝𝑝 collisions at √𝑠 = 13 TeV, Eur. Phys. J. C 79 (2019) 970,
+arXiv: 1907.05120 [hep-ex].
+[100]
+ATLAS Collaboration, Measurement of 𝑏-tagging efficiency of 𝑐-jets in 𝑡¯𝑡 events using a
+likelihood approach with the ATLAS detector, ATLAS-CONF-2018-001, 2018,
+url: https://cds.cern.ch/record/2306649.
+29
+
+[101]
+ATLAS Collaboration, Calibration of light-flavour 𝑏-jet mistagging rates using ATLAS
+proton–proton collision data at √𝑠 = 13 TeV, ATLAS-CONF-2018-006, 2018,
+url: https://cds.cern.ch/record/2314418.
+[102]
+J. Butterworth et al., PDF4LHC recommendations for LHC Run II, J. Phys. G 43 (2016) 023001,
+arXiv: 1510.03865 [hep-ph].
+[103]
+A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, Parton distributions for the LHC,
+Eur. Phys. J. C 63 (2009) 189, arXiv: 0901.0002 [hep-ph].
+[104]
+A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt,
+Uncertainties on 𝛼𝑆 in global PDF analyses and implications for predicted hadronic cross sections,
+Eur. Phys. J. C 64 (2009) 653, arXiv: 0905.3531 [hep-ph].
+[105]
+J. Gao et al., CT10 next-to-next-to-leading order global analysis of QCD,
+Phys. Rev. D 89 (2014) 033009, arXiv: 1302.6246 [hep-ph].
+[106]
+LHC Higgs Cross Section Working Group,
+Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector,
+CERN-2017-002-M (CERN, Geneva, 2016), arXiv: 1610.07922 [hep-ph].
+[107]
+ATLAS Collaboration, Evidence for 𝑡¯𝑡𝑡¯𝑡 production in the multilepton final state in proton–proton
+collisions at √𝑠 = 13TeV with the ATLAS detector, Eur. Phys. J. C 80 (2020) 1085,
+arXiv: 2007.14858 [hep-ex].
+[108]
+ATLAS Collaboration, Observation of the associated production of a top quark and a 𝑍 boson in
+𝑝𝑝 collisions at √𝑠 = 13 TeV with the ATLAS detector, JHEP 07 (2020) 124,
+arXiv: 2002.07546 [hep-ex].
+[109]
+CMS Collaboration, Measurement of the associated production of a single top quark and a 𝑍
+boson in 𝑝𝑝 collisions at √𝑠 = 13 TeV, Phys. Lett. B 779 (2018) 358,
+arXiv: 1712.02825 [hep-ex].
+[110]
+ATLAS Collaboration, Measurement of 𝑊±𝑍 production cross sections and gauge boson
+polarisation in 𝑝𝑝 collisions at √𝑠 = 13 TeV with the ATLAS detector,
+Eur. Phys. J. C 79 (2019) 535, arXiv: 1902.05759 [hep-ex].
+[111]
+ATLAS Collaboration, Evidence for the 𝐻 → 𝑏 ¯𝑏 decay with the ATLAS detector,
+JHEP 12 (2017) 024, arXiv: 1708.03299 [hep-ex].
+[112]
+ATLAS Collaboration,
+Luminosity determination in 𝑝𝑝 collisions at √𝑠 = 13 TeV using the ATLAS detector at the LHC,
+ATLAS-CONF-2019-021, 2019, url: https://cds.cern.ch/record/2677054.
+[113]
+G. Avoni et al., The new LUCID-2 detector for luminosity measurement and monitoring in ATLAS,
+JINST 13 (2018) P07017.
+[114]
+A. L. Read, Presentation of search results: the 𝐶𝐿𝑆 technique, J. Phys. G 28 (2002) 2693.
+[115]
+G. Cowan, K. Cranmer, E. Gross, and O. Vitells,
+Asymptotic formulae for likelihood-based tests of new physics, Eur. Phys. J. C 71 (2011) 1554,
+arXiv: 1007.1727 [physics.data-an], Erratum: Eur. Phys. J. C 73 (2013) 2501.
+30
+
diff --git a/_tFJT4oBgHgl3EQfqizo/content/tmp_files/load_file.txt b/_tFJT4oBgHgl3EQfqizo/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..663f805098c1f5e775ce5782c8d8dcae3e3853c3
--- /dev/null
+++ b/_tFJT4oBgHgl3EQfqizo/content/tmp_files/load_file.txt
@@ -0,0 +1,1613 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf,len=1612
+page_content='EUROPEAN ORGANISATION FOR NUCLEAR RESEARCH (CERN)  Submitted to: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D  CERN-EP-2022-044  January 30, 2023  Search for flavor-changing neutral-current  couplings between the top quark and the 𝒁 boson  with LHC Run 2 proton–proton collisions at  √𝒔 = 13 TeV with the ATLAS detector  The ATLAS Collaboration  A search for flavor-changing neutral-current couplings between a top quark, an up or charm  quark and a 𝑍 boson is presented, using proton–proton collision data at √𝑠 = 13 TeV collected  by the ATLAS detector at the Large Hadron Collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The analyzed dataset corresponds to an  integrated luminosity of 139 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The search targets both single-top-quark events produced  as 𝑔𝑞 → 𝑡𝑍 (with 𝑞 = 𝑢, 𝑐) and top-quark-pair events, with one top quark decaying through  the 𝑡 → 𝑍𝑞 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The analysis considers events with three leptons (electrons or muons),  a 𝑏-tagged jet, possible additional jets, and missing transverse momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The data are  found to be consistent with the background-only hypothesis and 95% confidence-level limits  on the 𝑡 → 𝑍𝑞 branching ratios are set, assuming only tensor operators of the Standard  Model effective field theory framework contribute to the 𝑡𝑍𝑞 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' These are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 × 10−5  (13 × 10−5) for 𝑡 → 𝑍𝑢 (𝑡 → 𝑍𝑐) for a left-handed 𝑡𝑍𝑞 coupling, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6 × 10−5 (12 × 10−5)  in the case of a right-handed coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' These results are interpreted as 95% CL upper limits  on the strength of corresponding couplings, yielding limits for |𝐶 (13)∗  𝑢𝑊 | and |𝐶 (13)∗  𝑢𝐵  | (|𝐶 (31)  𝑢𝑊 |  and |𝐶 (31)  𝑢𝐵 |) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='15 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='16), and limits for |𝐶 (23)∗  𝑢𝑊 | and |𝐶 (23)∗  𝑢𝐵  | (|𝐶 (32)  𝑢𝑊 | and |𝐶 (32)  𝑢𝐵 |) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='22  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='21), assuming a new-physics energy scale ΛNP of 1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  © 2023 CERN for the benefit of the ATLAS Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='11605v1  [hep-ex]  27 Jan 2023  CERNContents  1  Introduction  2  2  ATLAS detector  3  3  Data and samples of simulated events  4  4  Object reconstruction  8  5  Event reconstruction and selection  9  6  Background estimation and separation from signal  13  7  Systematic uncertainties  15  8  Results  17  9  Conclusions  23  1 Introduction  The top quark is the heaviest elementary particle known and it decays almost exclusively into Wb [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In  the Standard Model of particle physics (SM), flavor-changing neutral-current (FCNC) processes involving  a top quark, an up-type quark and a Z boson are forbidden at tree level and are strongly suppressed by the  GIM mechanism [2] at higher orders, leading to branching ratios for top-quark decays via FCNC processes  of the order of 10−14 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' However, several SM extensions predict such branching ratios to be between 10−4  and 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Examples of SM extensions are the quark-singlet model [4], the two-Higgs-doublet model [5],  the Minimal Supersymmetric Standard Model (MSSM) [6], the MSSM with R-parity violation [7], models  with warped extra dimensions [8], and extended mirror fermion models [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  FCNC couplings can be described by an effective field theory (EFT) [10, 11] that extends the SM Lagrangian  LSM with higher-dimensional operators suppressed by the scale of new physics, ΛNP, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  At order Λ−2  NP the strength of the anomalous couplings is given by the Wilson coefficients 𝐶𝑘 that multiply  dimension-six operators O𝑘,  Leff = LSM +  1  Λ2  NP  ∑︁  𝑘  𝐶𝑘O𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  (1)  The relevant operators for an FCNC process with a top quark and a Z boson, following the notation in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' [12], are the operators O(𝑖 𝑗)  𝑢𝐵 and O(𝑖 𝑗)  𝑢𝑊 with 𝑖 ≠ 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The indices 𝑖 and 𝑗 of the operators refer to the  flavor indices of the quark generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' One index is always equal to 3 as a top quark must be involved,  while the other one is either 1 or 2, corresponding to an up or charm quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The FCNC tZq interactions  can be introduced by vector and tensor couplings, but only the latter are considered in this analysis because  they would produce most of the “FCNC-in-single-top-production” signal [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The FCNC operators can be  left-handed (LH) or right-handed (RH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The order of the indices 𝑖 and 𝑗 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' (1) defines the chirality  2  of the FCNC operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A linear combination of the 𝐶 (13)  𝑢𝐵 and 𝐶 (13)  𝑢𝑊 coefficients corresponds to the tZu  LH coupling while a linear combination of the 𝐶 (31)  𝑢𝐵 and 𝐶 (31)  𝑢𝑊 coefficients defines the tZu RH coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Similarly, the tZc couplings are defined by the 𝐶 (23)  𝑢𝐵 and 𝐶 (23)  𝑢𝑊 coefficients for the LH case, while the 𝐶 (32)  𝑢𝐵  and 𝐶 (32)  𝑢𝑊 coefficients describe the RH case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For each linear combination, the two coefficients assume the  same value with an opposite sign [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Experimental limits on the branching ratio of FCNC t → Zq decays were previously established by  experiments at the Large Electron–Positron Collider (LEP) [13–16], the Hadron–Electron Ring Accelerator  (HERA) [17], the Tevatron [18, 19] and the Large Hadron Collider (LHC) [20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The most stringent  observed limits, B(t → Zu) < 17 × 10−5 and B(t → Zc) < 24 × 10−5 [21], were set by ATLAS in a  search for FCNC processes in tt decays only, using 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1 fb−1 of 𝑝𝑝 collision data at √𝑠 = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The  quoted limits apply to both the left- and right-handed couplings, as the analysis is not sensitive to the  chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  This paper presents a search for FCNC tZq couplings, using 𝑝𝑝 collision data at √𝑠 = 13 TeV collected  by the ATLAS experiment at the LHC and corresponding to an integrated luminosity of 139 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The  search is performed by analyzing the top-quark decays in tt events as well as the production of single top  quarks, as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In the former channel, one of the top quarks decays through an FCNC  process (t → Zq) and the other through the dominant mode (t → Wb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In contrast, in the latter channel the  production of a single top quark proceeds through an FCNC process (gq → tZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Single top production  with FCNC decay contributes negligibly and is not considered in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' While single top-quark  production gives the analysis more sensitivity to the FCNC tZu coupling, the tt decay mode provides  almost equal sensitivity to the FCNC tZu and tZc couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Since the FCNC production and decay  processes are induced by the same couplings, the production cross-section and decay branching ratio are  connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Therefore, the FCNC single-top production cross-section can be interpreted as the branching  ratio of the corresponding FCNC decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Thus, the analysis results for the numbers of production and decay  signal events are translated into branching ratios for t → Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For both of the considered channels, only  the trileptonic final state is selected, in which the Z boson decays into charged leptons and the W boson  from the top quark decays leptonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The final states where either the Z boson or the W boson decays  hadronically are not considered because of the larger backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Therefore, the analysis selects events  with three leptons (electrons or muons), a b-tagged jet, possible additional jets, and missing transverse  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' After the selection, the main background sources are diboson, ttZ and tZ production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' To  improve the separation of signal from background events, a multivariate technique is used, which was not  employed in the previous analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The statistical analysis uses a binned profile likelihood fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  2 ATLAS detector  The ATLAS experiment [24] at the LHC is a multipurpose particle detector with a forward–backward sym-  metric cylindrical geometry and a near 4𝜋 coverage in solid angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1 It consists of an inner tracking detector  surrounded by a thin superconducting solenoid providing a 2 T axial magnetic field, electromagnetic and  1 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the center of the detector  and the 𝑧-axis along the beam pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The 𝑥-axis points from the IP to the center of the LHC ring, and the 𝑦-axis points  upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Cylindrical coordinates (𝑟, 𝜙) are used in the transverse plane, 𝜙 being the azimuthal angle around the 𝑧-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The  pseudorapidity is defined in terms of the polar angle 𝜃 as 𝜂 = − ln tan(𝜃/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Distances in the 𝜂–𝜙 plane are measured in units of  Δ𝑅 ≡  √︃  (Δ𝜂)2 + (Δ𝜙)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  3  g  g  W  b  Z  u / c  g  t  t  (a)  u / c  g  Z  b  W  u / c  t  (b)  Figure 1: Examples of the lowest-order Feynman diagrams for (a) tt production, with one top quark decaying through  the dominant mode in the SM and the other via an FCNC process and for (b) single top-quark production via an  FCNC process in the 𝑠-channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  hadron calorimeters, and a muon spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The inner tracking detector (ID) covers the pseudorapidity  range |𝜂| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' It consists of silicon pixel, silicon microstrip, and transition radiation tracking detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Lead/liquid-argon (LAr) sampling calorimeters provide electromagnetic (EM) energy measurements  with high granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A steel/scintillator-tile hadron calorimeter covers the central pseudorapidity range  (|𝜂| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The endcap and forward regions are instrumented with LAr calorimeters for both the EM and  hadronic energy measurements up to |𝜂| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The muon spectrometer surrounds the calorimeters and is  based on three large superconducting air-core toroidal magnets with eight coils each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The field integral of  the toroids ranges between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 T m across most of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The muon spectrometer includes a  system of precision tracking chambers and fast detectors for triggering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A two-level trigger system [25] is  used to select events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The first-level trigger is implemented in hardware and uses a subset of the detector  information to accept events at a rate below 100 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' This is followed by a software-based trigger that  reduces the accepted event rate to 1 kHz on average depending on the data-taking conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' An extensive  software suite [26] is used in data simulation, in the reconstruction and analysis of real and simulated data,  in detector operations, and in the trigger and data acquisition systems of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  3 Data and samples of simulated events  The data sample used in this analysis corresponds to 139 fb−1 of pp collisions at √𝑠 = 13 TeV collected by  the ATLAS detector during 2015–2018, after requiring stable LHC beams and that all detector subsystems  were operational [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Candidate events were required to satisfy one of the single-electron triggers or one of the single-muon  triggers [25, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Single-lepton triggers with low transverse momentum (𝑝T) thresholds and isolation  requirements were combined in a logical OR with higher-threshold triggers that had a looser identification  criterion and did not have any isolation requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The lowest 𝑝T threshold used for electrons was 24 GeV  (26 GeV) in 2015 (2016–2018), while for muons the corresponding threshold was 20 GeV (26 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  To evaluate the effects of the detector resolution and acceptance on the signal and background, and to  estimate the SM backgrounds, simulated event samples were produced using a Geant4-based Monte Carlo  (MC) detector simulation [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Some of the samples used for evaluating systematic uncertainties did  not use the full Geant4 simulation but instead relied on parameterized showers in the calorimeter [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  4  The top-quark mass in the event generators described below was set to 𝑚𝑡 = 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In all samples, the  decays of bottom and charm hadrons were performed by EvtGen 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 [32], unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The simulated data must account for the fact that significantly more than one inelastic pp collision occurs  per bunch crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The average number of collisions per bunch crossing ranged from 13 to 38 for the  2015–2018 data-taking periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Inelastic collisions were simulated using Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='186 [33] with the A3  set of tuned parameters [34] and the NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3lo [35] set of parton distribution functions (PDFs), and  overlaid on the signal and background MC samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' These simulated events were reweighted to match  the conditions of the collision data, specifically the number of additional pp interactions in the same and  neighboring bunch crossings (pileup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Several MC signal event samples were generated at next-to-leading order (NLO) in QCD with  MadGraph5_aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 [36], using the NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0nlo [37] PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Parton showering and  hadronization were modeled with Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='302 with the NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3lo PDF set and the A14 set of  tuned parameters [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Only events with leptonic decays (including 𝜏-leptons) of the W and Z bosons  were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The TopFCNC Universal FeynRules Output (UFO) model [11, 39, 40] was used for the  computation of top-quark FCNC production and decay processes at NLO in QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Since FCNC processes  in both production and decay are considered in this analysis, separate samples for each mode and for tZu  and tZc couplings were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In order to study the chirality of these couplings, separate samples with  LH and RH couplings were produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Additional signal samples generated with the same version of MadGraph5_aMC@NLO were interfaced  to Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6 [41, 42] instead of Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='302 to assess the uncertainty related to the choice of  parton-shower model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1 default set of tuned parameters [42, 43] was used together with the  MMHT2014lo PDF set [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The decays of bottom and charm hadrons were performed by EvtGen 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  For the normalization, the branching ratios are set to the best observed limits reported in Section 1,  constraining B(𝑡  →  𝑞′𝑊) = 1 − B(𝑡  →  𝑢𝑍/𝑐𝑍), with 𝑞′ = 𝑑, 𝑠, 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The FCNC tt decay signal  is normalized using the tt cross-section prediction at next-to-next-to-leading order (NNLO) in QCD  including the resummation of next-to-next-to-leading logarithmic (NNLL) soft-gluon terms calculated  using Top++ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 [45–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The FCNC single top-quark production signal normalization cross-section is  calculated at NLO using the TopFCNC model as implemented in MadGraph5_aMC@NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The background is estimated using simulated samples that contain at least two leptons and at least two jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  These samples include the production of tt, ttH, ttZ, ttW, tZ, tW, tWZ, Z + jets, diboson, triboson, ttt, tttt,  ttWW, ZH and WH events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The production of tt and ttH events was modeled using the Powheg Box v2 [52–56] generator at NLO with  the NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0nlo PDF set and the ℎdamp parameter2 set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 𝑚𝑡 for tt [57] and to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='75 × (2 𝑚𝑡 + 𝑚H)  for ttH, with 𝑚𝐻 = 125 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The events were interfaced to Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='230 [58] to model the parton  shower, hadronization, and underlying event, with parameters set according to the A14 tune and using the  NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3lo set of PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The decays of bottom and charm hadrons were performed by EvtGen 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Additional tt simulated samples are used to assess modeling uncertainties [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The impact of using a  different parton shower and hadronization model is evaluated by comparing the nominal “Powheg+Pythia  ” tt sample with another event sample produced with the Powheg Box v2 generator, but interfaced with  Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3, which used the Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1 default set of tuned parameters and the MMHT2014lo PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  2 The ℎdamp parameter is a resummation damping factor and one of the parameters that controls the matching of Powheg matrix  elements to the parton shower and thus effectively regulates the high-𝑝T radiation against which the tt system recoils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  5  To estimate the systematic uncertainty in the choice of the ℎdamp parameter, a sample generated in the same  way as the nominal one but with the ℎdamp parameter set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 𝑚𝑡 was produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The production of ttZ and ttW events was modeled using the MadGraph5_aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3 generator at  NLO with the NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0nlo PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The events were interfaced to Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='210, which used the A14  tune and the NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3lo PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Additional ttZ simulated samples are used to assess modeling uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The impact of using a different  parton shower and hadronization model is evaluated by comparing the nominal ttZ sample with an event  sample produced with the MadGraph5_aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 generator interfaced with Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4, which  used the Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 default set of tuned parameters and the MMHT2014lo PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The decays of bottom  and charm hadrons were performed by EvtGen 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The uncertainty due to initial-state radiation (ISR) is  estimated by comparing the nominal event sample with two samples where the Var3c [38] up and down  variations of the A14 tune were employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The SM production of a single top quark in association with a Z boson (tZ) was modeled using the  MadGraph5_aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3 generator at NLO with the NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0nlo PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The events were  interfaced with Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='230, which used the A14 tune and the NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3lo PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Similarly to ttZ, additional tZ simulated samples are used to assess modeling uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The impact  of using a different parton shower and hadronization model is evaluated by comparing the nominal tZ  sample with an event sample produced with the MadGraph5_aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1 generator interfaced with  Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1, which used the Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1 default set of tuned parameters and the MMHT2014lo PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The decays of bottom and charm hadrons were performed by EvtGen 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The uncertainty due to ISR is  estimated by comparing the nominal tZ sample with two additional samples, which had the same settings  as the nominal one, but employed the Var3c up and down variations of the A14 tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The associated production of a single top quark with a W boson (tW) was modeled by the Powheg Box v2 [60]  generator at NLO in QCD using the five-flavor scheme and the NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0nlo set of PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The diagram  removal (DR) scheme [61] was used to remove interference and overlap with tt production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The events  were interfaced to Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='230, which used the A14 tune and the NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3lo set of PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The production of tWZ events was modeled using the MadGraph5_aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3 generator at NLO  with the NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0nlo PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The events were interfaced with Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='212, which used the A14 tune  and the NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3lo PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The DR scheme was employed to handle the interference between the tWZ  and ttZ processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A sample with an alternative scheme described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' [62] was produced to assess the  associated systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The Powheg Box v1 MC generator [63] was used to simulate at NLO accuracy the hard-scattering processes  of Z boson production and decay in the electron, muon, and 𝜏-lepton channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' It was interfaced to  Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='186 for the modeling of the parton shower, hadronization, and underlying event, with parameters  set according to the AZNLO tune [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The CT10nlo [65] PDF set was used for the hard-scattering  processes, whereas the CTEQ6L1 [66] PDF set was used for the parton shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The effect of QED  final-state radiation was simulated with Photos++ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='52 [67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Samples of diboson final states (𝑉𝑉, with 𝑉 = W, Z) were simulated with the Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 [69]  generator depending on the process, including off-shell effects and Higgs boson contributions where  appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Fully leptonic final states and semileptonic final states, where one boson decays leptonically  and the other hadronically, were generated using matrix elements at NLO accuracy in QCD for up to  one additional parton and at LO accuracy for up to three additional parton emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Samples for the  loop-induced processes 𝑔𝑔 → 𝑉𝑉 were generated using LO-accurate matrix elements for up to one  6  additional parton emission for both the cases of fully leptonic and semileptonic final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The matrix  element calculations were matched and merged with the Sherpa parton shower based on Catani–Seymour  dipole factorization [70, 71] using the MEPS@NLO prescription [72–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The virtual QCD corrections  were provided by the OpenLoops library [76–78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0nnlo set of PDFs was used, along  with the dedicated set of tuned parton-shower parameters developed by the Sherpa authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Electroweak  production of a diboson in association with two jets (𝑉𝑉 𝑗 𝑗) was simulated with the Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The LO-accurate matrix elements were matched to a parton shower based on Catani–Seymour dipole  factorization using the MEPS@LO prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Samples were generated using the NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0nnlo PDF  set, along with the dedicated set of tuned parton-shower parameters developed by the Sherpa authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The  decays of bottom and charm hadrons are performed with built-in Sherpa features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' An invariant mass of  𝑚ℓℓ > 4 GeV was required at matrix-element level for any pair of same-flavor charged leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  To assess the uncertainty that the generator contributes to the simulation of diboson final states, alternative  samples are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For these, the Powheg Box v2 [79] generator was used instead of Sherpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The effect  of singly resonant amplitudes and interference effects due to 𝑍/𝛾∗ and same-flavor lepton combinations in  the final state were included where appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Interference effects between 𝑊𝑊 and 𝑍𝑍 for same-flavor  charged leptons and neutrinos were ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Events were interfaced to Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='186 for the modeling of  the parton shower, hadronization, and underlying event, with parameters set according to the AZNLO tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The CT10 PDF set was used for the hard-scattering processes, whereas the CTEQ6L1 PDF set was used  for the parton shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The factorization and renormalization scales were set to the invariant mass of the  boson pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The same invariant mass selection as for the Sherpa samples was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The production of triboson (𝑉𝑉𝑉, with 𝑉 = W, Z) events was simulated with the Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Matrix elements, accurate to NLO for the inclusive process and to LO for up to two additional parton  emissions, were matched and merged with the Sherpa parton shower based on Catani–Seymour dipole  factorization using the MEPS@NLO prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The virtual QCD corrections for matrix elements at NLO  accuracy were provided by the OpenLoops library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Samples were generated using the NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0nnlo  PDF set, along with the dedicated set of tuned parton-shower parameters developed by the Sherpa authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The decays of bottom and charm hadrons are performed with built-in Sherpa features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The production of tttt events was modeled using the MadGraph5_aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 generator at NLO  with the NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1nlo [37] PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The events were interfaced with Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='230, which used the A14  tune and the NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3lo PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The decays of bottom and charm hadrons were simulated using the  EvtGen 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Other rare top-quark processes, namely the production of ttWW and ttt events, were modeled using the  MadGraph5_aMC@NLO generator at LO interfaced with Pythia 8, which used the A14 tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The  associated production of a Higgs boson with a W or Z boson, 𝑉H, was modeled using Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='186 with  the A14 tune and the NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3lo PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Throughout the paper the various MC samples are merged or split as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The ttZ and tWZ backgrounds  are combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The diboson contribution is split according to the origin of the associated jets using  generator-level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Their origin is determined by matching, within a cone of size Δ𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3, jets  to hadrons with 𝑝T > 5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' If one of the jets contains a b- or c-hadron, then it is classified as diboson  + heavy flavor (𝑉𝑉 + HF), otherwise the event is classified as diboson + light flavor (𝑉𝑉 + LF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The tt,  tW, Z + jets, 𝑉𝑉 and tt𝑉 processes with two prompt3 leptons and one nonprompt or fake lepton (a jet  3 Prompt leptons are leptons from the decay of W or Z bosons, either directly or through an intermediate 𝜏 → ℓ𝜈𝜈 decay, or from  the semileptonic decay of top quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  7  misidentified as a lepton) are shown together and called “Fakes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The other minor backgrounds, namely  ttW, ttH, 𝑉H, ttWW, triboson, ttt and tttt, are merged and called “Other bkg.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  4 Object reconstruction  The reconstruction of the basic objects used in the analysis is described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The primary  vertex [80] is selected as the pp vertex candidate with the highest sum of the squared transverse momenta  of all associated tracks with 𝑝T > 500 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Electron candidates are reconstructed from energy clusters in the EM calorimeter that match a reconstructed  track [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The clusters are required to be within the range |𝜂| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='47, excluding the transition  region between the barrel and endcap calorimeters at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='37 < |𝜂| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Each electron candidate’s  transverse impact parameter relative to the beam axis, 𝑑0, divided by its estimated uncertainty must satisfy  |𝑑0|/𝜎(𝑑0) < 5, while the longitudinal distance 𝑧0 from the reconstructed primary vertex to the point  where 𝑑0 is measured must satisfy |𝑧0 sin(𝜃)| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Electron candidates must also satisfy a transverse  momentum requirement of 𝑝T > 15 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A likelihood-based discriminant is constructed from a set of  variables that enhance the electron selection, while rejecting photon conversions and hadrons misidentified  as electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' An 𝜂- and 𝑝T-dependent selection on the likelihood discriminant is applied, and the “Medium”  identification [81] is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Electrons are also required to be isolated using criteria based on ID tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Nonprompt leptons are rejected using a boosted decision tree (BDT) discriminant based on isolation and  b-tagging variables, referred to as the nonprompt-lepton BDT [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The efficiency at the chosen working  point for electrons satisfying the isolation criteria is about 70% for a 𝑝T of 20 GeV and reaches a plateau of  95% at a 𝑝T of 100 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The corresponding rejection factor for leptons from the decay of b-hadrons is  about 50, estimated from a simulated tt sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Correction factors are applied to simulated electrons to  take into account the small differences in trigger, reconstruction, identification and isolation efficiencies  between data and MC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Muon candidates are reconstructed by combining a reconstructed track from the inner detector with  one from the muon spectrometer, and are required to have 𝑝T > 15 GeV and |𝜂| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 and to meet the  “Medium” identification [83] criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Similarly to electrons, muon candidates must have |𝑑0|/𝜎(𝑑0) < 3  and |𝑧0 sin(𝜃)| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' To reject misidentified muon candidates, several quality requirements are  imposed on the muon candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' An isolation requirement based on ID tracks is imposed, and a threshold is  set for the nonprompt-lepton BDT output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The efficiency at the chosen working point for muons satisfying  the isolation criteria is about 80% for a 𝑝T of 20 GeV and reaches a plateau of 99% at a 𝑝T of 100 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The corresponding rejection factor for leptons from the decay of b-hadrons is about 20, estimated from a  simulated tt sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Like for electrons, correction factors are applied to simulated muons to account for  the small differences between data and simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Jets are reconstructed from the particle-flow objects [84] using the anti-𝑘𝑡 algorithm [85, 86] with the  radius parameter set to 𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Their calibration follows the methodology described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Jets are  required to have 𝑝T > 25 GeV and |𝜂| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' To suppress jets arising from pileup, a discriminant called the  “jet vertex tagger” (JVT) is constructed using a two-dimensional likelihood method [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The jet energy  scale and resolution are corrected with 𝜂- and 𝑝T-dependent scale factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  To identify jets containing a b-hadron (b-jets), the “DL1r” multivariate algorithm is employed [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' It uses  impact parameter and secondary and tertiary vertex information from tracks contained in the jet as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Operating points are defined by a threshold value for the 𝑏-tagging discriminant output and are chosen  8  to provide a specific b-jet efficiency in an inclusive tt sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Candidate b-jets must have a 𝑏-tagging  discriminant value that exceeds a threshold corresponding to a 70% b-jet selection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' With this  criterion, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='25% of light-jets, containing neither a b- nor a c-hadron, are misidentified as b-jets, as are 10%  of jets initiated by c-quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Correction factors are derived and applied to correct for differences in b-jet  selection efficiency and the mistagging rates between data and MC simulation [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The missing transverse momentum, with magnitude 𝐸miss  T  , is calculated as the negative of the vector sum of  the transverse momenta of all reconstructed objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' To account for soft hadronic activity, a term including  tracks associated with the primary vertex but not with any of the reconstructed objects is added to the 𝐸miss  T  calculation [90, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  To avoid cases where the detector response to a single physical object is reconstructed as two separate  final-state objects, an overlap removal procedure is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' If electron and muon candidates share a track, the  electron candidate is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' After that, if the Δ𝑅𝑦,𝜙 distance4 between a jet and an electron candidate is  less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2, the jet is discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' If multiple jets satisfy this requirement, only the closest jet is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  For jet–electron distances between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4, the electron candidate is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' If the distance between a  jet and a muon candidate is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2, and the jet has less than three associated tracks, the jet is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Any muon subsequently found at a distance of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4 from a jet is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  5 Event reconstruction and selection  The analysis searches for effects of FCNC tZq couplings both in tt decay and in single-top-quark production  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In the first process, one of the top quarks decays through the dominant mode into a W boson  and a b-quark (hereafter called the “SM top quark”, denoted by 𝑡SM), while the other top quark (hereafter  called the “FCNC top quark”, denoted by 𝑡FCNC) decays into a Z boson and a 𝑢- or 𝑐-quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In the second  process, the production of a single top quark proceeds through an FCNC interaction in association with a  Z boson, while its decay is through the dominant mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In each channel, only the trilepton final state is  targeted, in which the Z and W bosons decay leptonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Therefore, the final state of the FCNC process in  tt decays is characterized by the presence of three leptons, at least two jets, one of which is a b-jet, and  missing transverse momentum from the escaping neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The final state of the FCNC process in single  top-quark production is instead characterized by the presence of three leptons, a b-jet, up to one additional  jet, and missing transverse momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Due to the different final states, two separate signal regions (SRs)  are defined, targeting the two processes: SR1 targets FCNC processes in tt decays while SR2 targets FCNC  processes in single top-quark production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The SRs share common selections for the leptons and they differ  in their top-quark reconstruction and jet multiplicity requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  In both SRs, exactly three leptons (electrons or muons) that do not all have the same charge are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  One of the leptons must have 𝑝T > 27 GeV, because of the trigger thresholds, and must be matched, with  Δ𝑅 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='15, to the lepton reconstructed by the trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Events with a fourth reconstructed lepton are  vetoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' At least one opposite-sign same-flavor lepton pair (OSSF) with an invariant mass in the range  |𝑚ℓℓ − 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 GeV| < 15 GeV is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In the 𝜇ee and e𝜇𝜇 channels the pair is uniquely identified,  whereas in the eee and 𝜇𝜇𝜇 channels both of the possible combinations are considered and the pair with  the invariant mass closer to the Z boson mass is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The lepton not used to reconstruct the Z boson is  4 Δ𝑅𝑦,𝜙 is the Lorentz-invariant distance in the rapidity–azimuthal-angle plane, defined as Δ𝑅𝑦,𝜙 =  √︃  (Δ𝑦)2 + (Δ𝜙)2, where 𝑦  is the rapidity, defined as 𝑦 = (1/2) ln [(𝐸 + 𝑝𝑧)/(𝐸 − 𝑝𝑧)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  9  assumed to be the one coming from the W boson, ℓW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In SR2, to help reject background sources with a  third nonprompt lepton, events are required to have 𝑚T(ℓW, 𝜈) > 40 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  In SR1 the selected events have at least two jets, with exactly one b-tagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In SR2 the selected events have  one or two jets, with exactly one b-tagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For events with exactly two jets, orthogonality between SR1  and SR2 is ensured by using an invariant mass cut on reconstructed top-quark candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' An additional  SR targeting the FCNC tZc coupling in tt decay, based on the presence of a c-jet, was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The  c-tagging was done using the soft-muon tagging technique employed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' With the current dataset,  this SR was found to bring only marginal improvements to the final limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  In the events having at least two jets with one of them being b-tagged, the reconstruction of FCNC and  SM top-quark candidates is based on the “FCNC-in-tt-decay” signal hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The kinematics of the  top-quark candidates are reconstructed from the corresponding decay particles by minimizing the following  expression:  𝜒2  tt =  �  𝑚reco  𝑗𝑎ℓℓ − 𝑚𝑡FCNC  �2  𝜎2  tFCNC  +  �  𝑚reco  𝑗𝑏ℓW 𝜈 − 𝑚𝑡SM  �2  𝜎2  tSM  +  �  𝑚reco  ℓW 𝜈 − 𝑚W  �2  𝜎2  W  ,  (2)  where 𝑚reco  𝑗𝑎ℓℓ, 𝑚reco  𝑗𝑏ℓW 𝜈, and 𝑚reco  ℓW 𝜈 are the reconstructed masses of the Zq, Wb, and ℓW𝜈 systems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The minimization has two independent parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The first is the jet permutation, where any non-b-tagged jet  can be assigned to 𝑗𝑎, while 𝑗𝑏 must correspond to a b-tagged jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The second is the minimization of the 𝜒2  tt  for each permutation by varying the longitudinal component of the neutrino momentum, 𝑝𝜈  𝑧, to determine  the most probable value while its transverse component is set to the missing transverse momentum in the  event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  This procedure assigns a reconstructed jet to the q-quark from the decay of the FCNC top quark and  determines the 𝑝𝜈  𝑧 value to reconstruct the four-momenta of the two top-quark candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' (2), the central values (𝑚𝑡FCNC, 𝑚𝑡SM and 𝑚W) and the widths (𝜎tFCNC, 𝜎tSM and 𝜎W) of the distributions  of the reconstructed masses of the top quark and W boson candidates are taken from reconstructed simulated  FCNC-in-tt-decay signal events that undergo the common object selection procedure just described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' This  is done by matching the true q- and b-quarks in the simulated events to the reconstructed jets, setting  the longitudinal momentum of the neutrino to the 𝑝𝑧 of the true generated neutrino, and the transverse  component to the missing transverse momentum in the event, and then performing a likelihood fit with a  Bukin function6 [93] to the masses of the reconstructed top quarks and W boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The mass values for  the LH coupling are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Compatible mass values are obtained for the RH coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The  fraction of reconstructed top-quark candidates that are matched to the true simulated particles within a  cone of size Δ𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4 is 𝜖𝑡FCNC = 75% for the FCNC top-quark candidates and 𝜖𝑡SM = 54% for the SM  top-quark candidates, where the difference comes from the fact that for the SM top-quark decay the match  of the missing transverse momentum with the generated neutrino is less efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  5 The transverse mass is calculated using the momentum of the lepton associated with the W boson, the 𝐸miss  T  and the azimuthal  angle, 𝜙, between them: 𝑚T(ℓW, 𝜈)=  √︃  2𝑝ℓ  T𝐸miss  T  (1 − cos Δ𝜙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  6 These fits use a generalization of the Gaussian function to allow for asymmetric tails in the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The overall normalization  is fixed to the yield and the shape of the function is determined by five parameters: the peak position, the width of the core, the  asymmetry, the size of the lower tail, and the size of the higher tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' From these parameters, only the peak position and the  width enter the 𝜒2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  10  Table 1: Summary of the mean values and standard deviations of the invariant mass distributions for the top-quark  candidates and the W boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' These values are obtained from the Bukin fits using the FCNC-in-tt-decay signal  samples with the LH coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The two FCNC tZu and tZc coupling samples are combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  FCNC top quark  SM top quark  W boson  𝑚𝑡FCNC [GeV]  𝜎tFCNC [GeV]  𝑚𝑡SM [GeV]  𝜎tSM [GeV]  𝑚W [GeV]  𝜎W [GeV]  FCNC in tt decay (LH)  171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1  166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  Under the FCNC-in-single-top-quark-production signal hypothesis, the SM top-quark candidate is instead  reconstructed in events having one or two jets, with exactly one 𝑏-tagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The missing transverse  momentum is assumed to be the transverse component of the neutrino momentum, while the most probable  value of 𝑝𝜈  𝑧 is determined by minimizing the following expression:  𝜒2  tZ =  �  𝑚reco  𝑗𝑏ℓW 𝜈 − 𝑚𝑡SM  �2  𝜎2  tSM  +  �  𝑚reco  ℓW 𝜈 − 𝑚W  �2  𝜎2  W  ,  (3)  where 𝑚reco  𝑗𝑏ℓW 𝜈 and 𝑚reco  ℓW 𝜈 are the reconstructed masses of the Wb and ℓW𝜈 systems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' (3),  the central values for the masses and widths of the top quark and W boson are taken from reconstructed  simulated FCNC-in-tt-decay signal events, as is done in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7 Therefore, in the events with two  jets, the four-momentum of the SM top-quark candidate reconstructed under the FCNC-in-single-top-  quark-production signal hypothesis is the same as that reconstructed under the FCNC-in-tt-decay signal  hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In this case, the fraction of reconstructed top-quark candidates that are matched to the true  simulated particles within a cone of size Δ𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4 is 𝜖𝑡SM = 71%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  In SR1, the mass of the FCNC top-quark candidate, 𝑚reco  𝑗𝑎ℓℓ, is required to be within 2𝜎tFCNC of 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 GeV,  while no requirement is placed on the mass of the SM top-quark candidate, 𝑚reco  𝑗𝑏ℓW 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In SR2, the mass of the  SM top-quark candidate is required to be within 2𝜎tSM of 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In addition, to ensure orthogonality  with SR1, for events with exactly two jets the mass of the FCNC top-quark candidate is required to be  more than 2𝜎tFCNC from 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Table 2 summarizes the selection criteria applied to the signal regions  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' With these criteria, 496 data events are selected in SR1 and 460 are selected in SR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Figure 2 shows the distributions of the masses of the two top-quark candidates in SR1, and the mass of the  top-quark candidate and the 𝑝T of the reconstructed Z boson in SR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' These kinematic distributions are  some of the key features that distinguish signal events from the backgrounds and they are utilized in the  multivariate analysis described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In SR1, the dominant signal is the FCNC-in-tt-decay events  (shown with solid lines in Figure 2 separately for the tZu and tZc couplings), while the FCNC-in-single-  top-quark-production contribution (shown with dashed lines) is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In contrast, SR2 is more sensitive  to the tZu FCNC-in-single-top-quark-production signal, with similar smaller contributions from the other  three signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' After the event selection the main background sources are ttZ, tZ and diboson production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  7 Using the central values for the masses and widths extracted from the FCNC single-top production signal sample does not have  a significant effect on the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  11  100  150  200  250  300  350  400  450  [GeV]  reco  ν  W  l  bj  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='75  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='25  Data / Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  0  20  40  60  80  100  120  140  160  Events / 10 GeV  ATLAS  1  = 13 TeV, 139 fb  s  SR1, Pre-Fit  Data  Z+tWZ  tt  VV+LF  VV+HF  tZ  Fake lep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Other bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' uncertainty  5  ×  FCNC (u)tZ  5  ×  (uZ)  t  FCNC t  5  ×  FCNC (c)tZ  5  ×  (cZ)  t  FCNC t  (a)  155  160  165 170  175  180  185  190  [GeV]  reco  ll  aj  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='75  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='25  Data / Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  0  20  40  60  80  100  120  140  Events / 5 GeV  ATLAS  1  = 13 TeV, 139 fb  s  SR1, Pre-Fit  Data  Z+tWZ  tt  VV+LF  VV+HF  tZ  Fake lep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Other bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' uncertainty  5  ×  FCNC (u)tZ  5  ×  (uZ)  t  FCNC t  5  ×  FCNC (c)tZ  5  ×  (cZ)  t  FCNC t  (b)  130 140 150 160 170 180 190 200 210  [GeV]  reco  ν  W  l  bj  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='75  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='25  Data / Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  0  20  40  60  80  100  120  Events / 5 GeV  ATLAS  1  = 13 TeV, 139 fb  s  SR2, Pre-Fit  Data  Z+tWZ  tt  VV+LF  VV+HF  tZ  Fake lep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Other bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' uncertainty  5  ×  FCNC (u)tZ  5  ×  (uZ)  t  FCNC t  5  ×  FCNC (c)tZ  5  ×  (cZ)  t  FCNC t  (c)  0  50  100 150 200 250 300 350 400 450  [GeV]  T  p  boson  Z  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='75  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='25  Data / Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  0  20  40  60  80  100  120  140  160  Events / 20 GeV  ATLAS  1  = 13 TeV, 139 fb  s  SR2, Pre-Fit  Data  Z+tWZ  tt  VV+LF  VV+HF  tZ  Fake lep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Other bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' uncertainty  5  ×  FCNC (u)tZ  5  ×  (uZ)  t  FCNC t  5  ×  FCNC (c)tZ  5  ×  (cZ)  t  FCNC t  (d)  Figure 2: Comparison between data and background prediction before the fit (“Pre-Fit”) for some kinematic  distributions in the SRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The distributions are: (a) the mass of the SM top-quark candidate in SR1, (b) the mass of  the FCNC top-quark candidate in SR1, (c) the mass of the SM top-quark candidate in SR2 and (d) the transverse  momentum of the Z boson candidate in SR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The uncertainty band includes both the statistical and systematic  uncertainties in the background prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The four FCNC LH signals are also shown separately, normalized to five  times the cross-section corresponding to the most stringent observed branching ratio limits [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The first (last) bin  in all distributions includes the underflow (overflow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The lower panels show the ratios of the data (“Data”) to the  background prediction (“Bkg.”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  12  Table 2: Overview of the requirements applied to select the events in the signal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' OSSF is an opposite-sign  same-flavor lepton pair, 𝑚Z = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 GeV and 𝑚t = 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Common selections  Exactly 3 leptons with 𝑝T(ℓ1) > 27 GeV  ≥ 1 OSSF pair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' with |𝑚ℓℓ − 𝑚Z| < 15 GeV  SR1  SR2  ≥ 2 jets  1 jet  2 jets  1 b-jet  1 b-jet  1 b-jet  –  𝑚T(ℓW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 𝜈)> 40 GeV  𝑚T(ℓW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 𝜈)> 40 GeV  |𝑚reco  𝑗𝑎ℓℓ − 𝑚t| < 2𝜎tFCNC  –  |𝑚reco  𝑗𝑎ℓℓ − 𝑚t| > 2𝜎tFCNC  –  |𝑚reco  𝑗𝑏ℓ𝑊 𝜈 − 𝑚t| < 2𝜎tSM  |𝑚reco  𝑗𝑏ℓ𝑊 𝜈 − 𝑚t| < 2𝜎tSM  6 Background estimation and separation from signal  Two classes of backgrounds are considered: processes in which three or more prompt leptons are produced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  such as diboson production or the associated production of top quarks (ttZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' tWZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' tZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' ttW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' ttH) and  processes with two prompt leptons in the final state along with one additional nonprompt or fake lepton that  satisfies the selection criteria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' such as tt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' tW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' and Z + jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Such nonprompt or fake leptons can originate  from decays of bottom or charm hadrons, jets misidentified as electrons, leptons from kaon or pion decays,  or electrons from photon conversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  All background contributions are estimated by using MC samples that are normalized to their respective  SM predicted cross-sections calculated at NLO in QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The cross-section of the ttH background includes  NLO+NLL soft-gluon resummation [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the tt + tW nonprompt lepton backgrounds the normalization  is extracted from data, as described later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  After applying the event selection requirements, diboson, ttZ and tZ production constitute the largest  backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For SR1, the dominant backgrounds are ttZ and 𝑉𝑉 + HF production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Monte Carlo  simulation indicates that these represent more than 65% of the total number of selected background events  in this region, with the two processes contributing equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For SR2, 𝑉𝑉 + HF and tZ are the dominant  backgrounds, giving 70% of background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The processes with nonprompt leptons constitute a minor  background, with their contribution being at most 10% of the total selected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Four control regions (CRs) are defined and used in the fit that is described in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The CRs are used  to adjust the normalization and to reduce the associated systematic uncertainties in the main backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The selections applied to define the CRs are summarized in Table 3 and described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  A tt CR is designed to control the tt background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The tt CR is constructed by requiring the presence of  three leptons, with one of the possible pairs having opposite charge, as in the SRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' To veto the presence  of a Z boson, the opposite-sign lepton pair is also required to consist of different flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Events with at  least one jet, with exactly one b-tagged, are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' This region is dominated by tt events with 40%  contamination from other backgrounds, mainly ttW and ttH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A total of 157 data events are selected for the  tt CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  To control the ttZ background, a ttZ CR is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The requirements on the leptons are the same as for  the SRs, while at least four jets, with exactly two b-tagged, are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' This region is dominated by ttZ  13  events with 25% contamination from other backgrounds, mainly tZ and 𝑉𝑉 + HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A total of 286 data  events are selected for the ttZ CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Two mass sideband CRs are also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' These CRs are designed to contain a mixture of the main  background sources (ttZ and diboson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The mass sideband CR1 is defined with almost the same event  selection as SR1, with the differences being that the mass of the FCNC top-quark candidate must be more  than 2𝜎tFCNC from 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 GeV, and the mass of the SM top-quark candidate must also be more than 2𝜎tSM  from 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The mass sideband CR2 is defined with almost the same event selection as SR2, with  the differences being that only events with one jet are considered and that the mass of the SM top-quark  candidate must be more than 2𝜎tSM from 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Totals of 343 and 104 data events are selected for the  mass sidebands CR1 and CR2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Table 3: Overview of the requirements applied to select the events in the control regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' OSSF is an opposite-sign  same-flavor lepton pair, 𝑚Z = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 GeV and 𝑚t = 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Common selections  Exactly 3 leptons with 𝑝T(ℓ1) > 27 GeV  tt CR  ttZ CR  Sideband CR1  Sideband CR2  ≥ 1 OS pair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' no OSSF  ≥ 1 OSSF pair  ≥ 1 OSSF pair  ≥ 1 OSSF pair  with |𝑚ℓℓ − 𝑚Z| < 15 GeV  with |𝑚ℓℓ − 𝑚Z| < 15 GeV  with |𝑚ℓℓ − 𝑚Z| < 15 GeV  –  –  –  𝑚T(ℓW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 𝜈) > 40 GeV  ≥ 1 jet  ≥ 4 jets  ≥ 2 jets  1 jet  1 b-jet  2 b-jets  1 b-jet  1 b-jet  –  –  |𝑚reco  𝑗𝑎ℓℓ − 𝑚t| > 2𝜎tFCNC  –  –  –  |𝑚reco  𝑗𝑏ℓ𝑊 𝜈 − 𝑚t| > 2𝜎tSM  |𝑚reco  𝑗𝑏ℓ𝑊 𝜈 − 𝑚t| > 2𝜎tSM  To better separate the signal from the backgrounds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' a multivariate analysis (MVA) technique is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The  chosen MVA is the gradient boosted decision tree (GBDT) method implemented with TMVA [95, 96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Decision trees [97] recursively partition the parameter space into regions where signal or background  purities are enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Gradient boosting is a method which improves the performance and stability of  decision trees and involves the combination of many trees into a single final discriminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' After boosting,  the final score undergoes a transformation to map the scores onto the interval −1 to +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The most signal-like  events have scores near +1 while the most background-like events have scores near −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A 𝑘-fold cross  validation is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The GBDT training is done separately for the LH and RH samples and in each SR as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In SR1, for  both the FCNC tZu and tZc coupling searches, the expected contribution from FCNC processes in tt decay  is significantly higher than the one from single-top-quark production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Therefore, the GBDT is trained with  only the FCNC-in-tt-decay signal against all backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Since the kinematics of FCNC-in-tt-decay  events for tZu and tZc couplings are similar, the FCNC-in-tt-decay signal samples with the two couplings  are combined to train the GBDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Therefore, in SR1 a single MVA discriminant, 𝐷1, is built for both the  FCNC tZu and tZc coupling searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In contrast, SR2 is particularly sensitive to the FCNC tZu coupling  in single-top-production events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Thus, the corresponding MVA discriminant, 𝐷𝑢  2, is built by training the  GBDT with the the tZu-coupling FCNC-in-single-top-production sample against all backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Despite  the lower sensitivity to the FCNC tZc coupling in SR2, this region is used in combination with SR1 in  the search for a FCNC tZc coupling signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In the total expected FCNC tZc signal yield, the contribution  from the FCNC processes in tt decay events is comparable to the one from the single-top-quark-production  14  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Therefore, in SR2 the MVA discriminant for the search for a FCNC tZc coupling signal, 𝐷𝑐  2, is built  using both the FCNC-in-tt-decay and FCNC-in-single-top-production samples against all backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  For the training of each of the three discriminants, a total of six variables is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' These variables are  chosen from a larger set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Only variables that provide good separation and are well modeled are used in  the final training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the 𝐷1 discriminant the six variables are: the reconstructed masses of the SM and  FCNC top-quark candidates, the Δ𝑅 separation between them, the Δ𝑅 separation between the lepton from  the SM top-quark decay and the reconstructed Z boson, the number of jets, and the transverse momentum  of the jets associated with the u/c-quark from the FCNC top-quark candidate’s decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For both the 𝐷𝑢  2 and  𝐷𝑐  2 discriminants the following six variables are used: the 𝑝T of the Z boson and of the b-tagged jet, the  Δ𝑅 separation between them, the SM top-quark candidate’s mass, the Δ𝑅 separation between the lepton  from the SM top-quark candidate decay and the reconstructed Z boson, and the 𝜒2 from the kinematic fit  under the signal hypothesis of an FCNC process in single-top-quark production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  7 Systematic uncertainties  Systematic uncertainties in the signal acceptance and in the normalization of the individual backgrounds,  as well as uncertainties in the shape of the fitted distributions, are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' These are treated as  being correlated between the different regions, unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The uncertainties are classified  into the following categories:  Reconstruction efficiency and calibration uncertainties:  Systematic uncertainties affecting the recon-  struction efficiency and energy calibration of electrons, muons, jets and b-jets are propagated through the  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The differences between the electron (muon) trigger, reconstruction, selection and isolation efficiencies  in data and those in MC simulation are corrected for by scale factors derived from dedicated Z → e+e−  ( Z → 𝜇+𝜇− ) enriched control samples using a tag-and-probe method [81, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Uncertainties in these  scale factors are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Moreover, uncertainties are included for the electron (muon) energy  (momentum) scale and resolution [81, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  For the jets, an uncertainty for the JVT requirement is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The jet energy scale was derived using  information from test-beam data, LHC collision data and simulation, as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The impact  of the uncertainty in the jet energy resolution is also evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The 𝑏-tagging efficiencies and mistagging rates are measured in data using the same methods as described  in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' [99–101], with the systematic uncertainties due to 𝑏-tagging efficiency and the mistagging rates  calculated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The impact of the uncertainties on the 𝑏-tagging calibration is evaluated separately  for b-, c- and light-jets in the MC samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The uncertainty in 𝐸miss  T  due to a possible miscalibration of the soft-track component of the 𝐸miss  T  is derived  from data–MC comparisons of the 𝑝T balance between the hard and soft 𝐸miss  T  components [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The  uncertainty associated with the leptons and jets is propagated from the corresponding uncertainties in the  energy/momentum scales and resolutions, and is classified together with the uncertainty associated with  the corresponding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  15  Signal and background modeling:  The systematic uncertainties due to MC modeling of the signal and  the main backgrounds are estimated by comparing samples from different MC generators and PDF sets  and by varying the parameters associated with the renormalization and factorization scales, and additional  radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For some processes, some of these uncertainties are found to be negligible and therefore they are  not mentioned in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  For the signal, the effects of the systematic uncertainty in the renormalization and factorization scales, 𝜇r  and 𝜇f, are taken into account by varying these parameters by factors of 2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 with respect to their  default values and comparing the results of these variations with the nominal prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The uncertainty in  the modeling of the parton shower is estimated by comparing the nominal signal sample with one generated  with Herwig 7 instead of Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' PDF uncertainties are found to be negligible and are not included for  the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  For the ttZ and tZ backgrounds, the following uncertainties are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The effect of changing the parton  shower is considered as an uncertainty, following the same strategy used for the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The uncertainty  due to ISR is estimated by comparing the nominal event sample with two samples where the Var3c up and  down variations of the A14 tune were employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Uncertainties from the variation of 𝜇r and 𝜇f are also  included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  For the tWZ background, the effect of changing the modeling of the interference with ttZ is included by  comparing two different diagram removal predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The effect of changing the MC generator for the modeling of the diboson background is considered as an  uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' It is evaluated by comparing the nominal Sherpa sample with one generated with Powheg Box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  This uncertainty is split into the two light- and heavy-flavor components and evaluated separately for each  jet multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Uncertainties in the 𝜇r and 𝜇f scales, as well as in the PDF and in 𝛼s are also included for  the diboson background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  For the tt background, several sources of uncertainty are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The effect of changing the  parton shower is included as an uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The Var3c A14 tune variations, as well as variations of 𝜇r and  𝜇f are also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Additionally, the uncertainty associated with the ℎdamp parameter is evaluated by  using the alternative sample with the ℎdamp value increased to 3 𝑚𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0lo replicas are used to  evaluate the PDF uncertainties for the nominal PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Finally, an uncertainty is added to take into account  the differences in tt background composition between the SRs and the tt CR, which is used to control the tt  background in the fit to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In particular, the fractions of nonprompt leptons originating from each source  are computed, separately for photon conversions and b-hadron decays, in the SRs and in the tt CR, for each  jet multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Then the maximum variation of the fractions between the control region and the signal  regions is taken as an uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Signal and background rate uncertainty:  The tt cross-section uncertainties due to the PDF and 𝛼s are  calculated using the PDF4LHC15 prescription [102] with the MSTW2008nnlo [103, 104], CT10nnlo [65,  105] and NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3lo PDF sets, and are added in quadrature to the effect of the scale uncertainty, resulting  in a total uncertainty of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5% that is assigned to the FCNC-in-tt-decay signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  For the ttZ background, a 12% rate uncertainty is included [106], and for the ttH process the normalization  uncertainty is 15% [106], while for ttW a more conservative 50% is used [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the tZ process,  an uncertainty of 15% in the normalization is applied [108, 109], while for the tWZ process a more  conservative 30% is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For 𝑉𝑉 + LF production, the normalization uncertainty is taken to be 20% [110]  and for 𝑉𝑉 + HF production it is 30% [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Concerning the Z + jets process, a rate uncertainty of 100% is  16  applied, due to the presence of a nonprompt lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A conservative overall normalization uncertainty of  50% is applied to the remaining minor backgrounds (ttt, tttt, 𝑉𝑉𝑉, 𝑉H and ttWW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' These background  components are typically well below 1% in the SRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Luminosity:  The uncertainty in the combined 2015–2018 integrated luminosity is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7% [112], obtained  using the LUCID-2 detector [113] for the primary luminosity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Uncertainty in pileup modeling:  The uncertainty in pileup modeling is accounted for by varying the  reweighting of the MC samples to the data pileup conditions, using the uncertainty in the average number  of interactions per bunch crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  8 Results  A simultaneous binned profile likelihood fit to the data in the SRs and the CRs is performed using MC  distributions of both the signal and background predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Four separate fits are performed to extract LH  and RH results for the FCNC tZu and tZc couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Only the relevant signal templates are used in each fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The templates are binned distributions of the 𝐷1 discriminant in SR1 and in the mass sideband CR1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' the  𝐷𝑢  2 discriminant in SR2 and in the mass sideband CR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' and the total event yields in the tt CR and the ttZ  CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' When fitting to extract limits on the FCNC tZc coupling, the 𝐷𝑐  2 discriminant is used instead of 𝐷𝑢  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The fitted SRs are defined from the SRs described in Section 5 after removing events that constitute  two validation regions (VRs) that are not included in the fit, but the fit results are propagated to those  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The VRs are used to check the stability of the fit and they are obtained by applying a selection  on the GBDT discriminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' VR1 is defined by selecting events with 𝐷1 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6 from the SR1, while  VR2 contains events from SR2 with 𝐷𝑢  2 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7 and 𝐷𝑐  2 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' With the given normalization of  the signal samples, the fraction of signal events that is selected from the SRs to enter the VRs ranges  from 2% to 5%, depending on the SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The signal contamination in the VRs is at most 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The signal  selection efficiency for the FCNC-in-tt-decay signal in SR1 ranges between 4% and 5%, while that for the  FCNC-in-single-top-production signal in SR2 ranges between 3% and 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' In contrast, the signal selection  efficiency for the FCNC-in-tt-decay signal in SR2 and the FCNC-in-single-top-production signal in SR1 is  around 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The statistical analysis used to extract the signal is based on a binned likelihood function L(𝜇, �𝜃) constructed  as a product of Poisson probability terms over all bins in each considered distribution, and Gaussian  constraint terms for �𝜃, a set of nuisance parameters that parameterize effects of MC statistical and systematic  uncertainties in the signal and background expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The signal strength parameter 𝜇 is a multiplicative  factor applied to the number of signal events normalized to a reference branching ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For that, the  most stringent limits mentioned in Section 1 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The nuisance parameters are allowed to vary in  the combined fit to adjust the expectations for signal and background according to the corresponding  systematic uncertainties, and their final values are the adjustment that best fits the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The normalization  of the tt + tW backgrounds is unconstrained in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  17  A test statistic, ˜𝑞𝜇, is constructed according to the profile likelihood ratio:  ˜𝑞𝜇 =  �����������  �����������  −2 ln ��  �  L  �  𝜇, ˆˆ�𝜃 (𝜇)  �  L  �  0, ˆˆ�𝜃 (0)  � ��  �  if ˆ𝜇 < 0,  −2 ln ��  �  L  �  𝜇, ˆˆ�𝜃 (𝜇)  �  L  �  ˆ𝜇, ˆ�𝜃  � ��  �  if 0 ≤ ˆ𝜇 ≤ 𝜇,  0  if ˆ𝜇 > 𝜇,  (4)  where ˆ𝜇 and ˆ�𝜃 are the parameters that maximize the likelihood, and ˆˆ�𝜃 are the nuisance parameter values  that maximize the likelihood for a given 𝜇 hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' This test statistic is used to determine the probability  for accepting the background-only hypothesis for the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Table 4 shows the pre- and post-fit predictions for the signal and background event yields along with the  observed numbers of events in the VRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The post-fit yields refer to the fit for the FCNC tZu LH coupling  extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The data and background expectation are in better agreement after the fit, with an increase  of the 𝑉𝑉 + HF background normalization within its pre-fit uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The post-fit level of agreement  between data and the background prediction in the VRs shows no significant mismodeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Table 4: Predicted and observed yields in the two VRs considered in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The signal and background predictions  are shown before (“Pre-fit”) and after the fit to data for the FCNC tZu LH coupling extraction (“Post-fit”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The quoted  uncertainties include the statistical and systematic uncertainties of the yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the post-fit predictions, they are  computed taking into account correlations among nuisance parameters and among processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the backgrounds  with a nonprompt or fake lepton, the contribution from tt + tW is shown separately from “Other fakes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the  minor backgrounds, the contribution from ttW and ttH are shown separately from “Other bkg.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Pre-fit  Post-fit  VR1  VR2  VR1  VR2  ttZ + tWZ  70  ± 10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2  ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6  70  ± 7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6  𝑉𝑉 + LF  10  ± 5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8  ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  10  ± 5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7  ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0  𝑉𝑉 + HF  56  ± 28  36  ± 14  60  ± 14  47  ± 8  tZ  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7  ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6  tt + tW fakes  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  Other fakes  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1  ttW  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='26  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='25  ttH  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='101 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='032  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='108 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='033  Other bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6  Total background  154  ± 31  69  ± 15  158  ± 13  79  ± 7  Data  151  80  151  80  Data / Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='22  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='15  Tables 5 and 6 show the observed number of events in data and the post-fit predictions for the signal and  background event yields in the SRs and CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The yields refer to the fit for the FCNC tZu LH coupling  extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Good agreement between data and the SM expectation is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The normalization factor  for the tt + tW backgrounds, which is an unconstrained fit parameter, agrees with unity within uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The variations of the post-fit background normalizations are within pre-fit uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' All post-fit values  of the nuisance parameters are less than one standard deviation from the pre-fit values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The statistical  component is the dominant contribution in the total uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The same conclusions are obtained from  the fits for the other FCNC couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  18  Table 5: Predicted and observed yields in the two SRs considered in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The signal and background predictions are  shown after the fit to data for the FCNC tZu LH coupling extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The quoted uncertainties include the statistical  and systematic uncertainties of the yields, computed taking into account correlations among nuisance parameters and  among processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the backgrounds with a nonprompt or fake lepton, the contribution from tt + tW is shown  separately from “Other fakes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the minor backgrounds, the contribution from ttW and ttH are shown separately  from “Other bkg.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  SR1  SR2  (𝐷1 > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6)  (𝐷𝑢  2 > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7 or 𝐷𝑐  2 > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4)  ttZ + tWZ  137  ± 12  36  ±  6  𝑉𝑉 + LF  18  ± 7  24  ±  8  𝑉𝑉 + HF  114  ± 19  162  ± 26  tZ  46  ± 7  108  ± 18  tt + tW fakes  14  ± 4  27  ±  8  Other fakes  7  ± 8  5  ±  6  ttW  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1 ±  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6  ttH  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='89 ±  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='17  Other bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 ±  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='9  FCNC (𝑢)𝑡𝑍  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7  4  ±  8  FCNC tt(𝑢𝑍)  5  ± 9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8 ±  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  Total background  348  ± 15  369  ± 21  Data  345  380  Table 6: Predicted and observed yields in the four CRs considered in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The signal and background predictions  are shown after the fit to data for the FCNC tZu LH coupling extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The quoted uncertainties include the  statistical and systematic uncertainties of the yields, computed taking into account correlations among nuisance  parameters and among processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the backgrounds with a nonprompt or fake lepton, the contribution from  tt + tW is shown separately from “Other fakes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the minor backgrounds, the contribution from ttW and ttH are  shown separately from “Other bkg.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Sideband CR1  Sideband CR2  ttZ CR  tt CR  ttZ + tWZ  102  ± 14  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  230  ± 18  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  𝑉𝑉 + LF  27  ± 11  12  ± 4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='25  𝑉𝑉 + HF  166  ± 25  64  ± 9  17  ± 8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='9  ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  tZ  22  ± 4  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  21  ± 5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='19  tt + tW fakes  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3  93  ± 19  Other fakes  2  ± 4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='09  ttW  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  27  ± 13  ttH  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='07  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1  ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2  Other bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5  FCNC (𝑢)𝑡𝑍  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='10  FCNC tt(𝑢𝑍)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='018 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='035  Total background  338  ± 18  104  ± 8  284  ± 16  157  ± 13  Data  343  104  286  157  19  The 𝜇 parameters are shown in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Table 7: Summary of the signal strength 𝜇 parameters obtained from the fits to extract LH and RH results for the  FCNC tZu and tZc couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the reference branching ratio, the most stringent limits are used [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Vertex  Coupling  𝜇  tZu  LH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='12 (stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=') ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='08 (syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=')  tZu  RH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='12 (stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=') ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='08 (syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=')  tZc  LH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='17 (stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=') ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='14 (syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=')  tZc  RH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='16 (stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=') ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='13 (syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=')  Figure 3 shows the distributions of the fitted variables in the CRs and SRs after the fit for the FCNC tZu  LH coupling extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the FCNC tZc LH coupling extraction, the fitted distributions are presented in  Figure 4, where 𝐷𝑐  2 is used in SR2 and in the mass sideband CR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the tt and ttZ CRs, only the event  yields are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The data and background prediction agree within the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Limits on each FCNC t → Zq branching ratio are computed with the CLs method [114] using the asymptotic  properties of 𝑞𝜇 [115] and assuming that only the corresponding FCNC coupling contributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The observed  and expected 95% confidence-level (CL) limits on the branching ratios are shown in Table 8, where the  limits on the relevant Wilson coefficients are also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The expected limits on the branching ratios  calculated without systematic uncertainties are lower by 20% and 25% for the tZu and tZc couplings,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The leading systematic uncertainties include the uncertainty in the SM tZ background  normalization and the diboson modeling uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Table 8 also shows limits on the FCNC tZu LH and RH couplings obtained when considering only one  SR, either SR1 or SR2, and all CRs in the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The results show that SR2, targeting the FCNC  single-top-production signal, contributes more strongly than SR1 to the combined limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Separate results  for the FCNC tZc coupling are not shown, since the limits are dominated by the FCNC-tt-in-decay signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  20  1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+page_content='25  Data / Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  0  50  100  150  200  250  Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1  ATLAS  1  = 13 TeV, 139 fb  s  Sideband CR1  Post-Fit  Data  Z+tWZ  tt  VV+LF  VV+HF  tZ  Fake lep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+page_content='  Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' uncertainty  500  ×  FCNC (u)tZ  500  ×  (uZ)  t  FCNC t  (a)  1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+page_content='2  ATLAS  1  = 13 TeV, 139 fb  s  Sideband CR2  Post-Fit  Data  Z+tWZ  tt  VV+LF  VV+HF  tZ  Fake lep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Other bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' uncertainty  500  ×  FCNC (u)tZ  500  ×  (uZ)  t  FCNC t  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+page_content='18  ATLAS  1  = 13 TeV, 139 fb  s  SR1  > -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6  1  D  Post-Fit  Data  Z+tWZ  tt  VV+LF  VV+HF  tZ  Fake lep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Other bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+page_content='  Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' uncertainty  50  ×  FCNC (u)tZ  50  ×  (uZ)  t  FCNC t  (d)  Figure 3: Comparison between data and background prediction after the fit to data (“Post-Fit”) for the FCNC tZu LH  coupling extraction for the fitted distributions in the CRs and SRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+page_content='1  ATLAS  1  = 13 TeV, 139 fb  s  Sideband CR1  Post-Fit  Data  Z+tWZ  tt  VV+LF  VV+HF  tZ  Fake lep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+page_content='25  Data / Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  0  10  20  30  40  50  60  Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+page_content='  Other bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' uncertainty  500  ×  FCNC (c)tZ  500  ×  (cZ)  t  FCNC t  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+page_content='8  1  1  D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+page_content='25  Data / Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  0  20  40  60  80  100  120  140  Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='18  ATLAS  1  = 13 TeV, 139 fb  s  SR1  > -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6  1  D  Post-Fit  Data  Z+tWZ  tt  VV+LF  VV+HF  tZ  Fake lep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Other bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' uncertainty  50  ×  FCNC (c)tZ  50  ×  (cZ)  t  FCNC t  (c)  1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+page_content='8  1  c  2  D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+page_content='25  Data / Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  0  20  40  60  80  100  120  140  160  Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='12  ATLAS  1  = 13 TeV, 139 fb  s  SR2  > -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  c  2  > -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7 or D  u  2  D  Post-Fit  Data  Z+tWZ  tt  VV+LF  VV+HF  tZ  Fake lep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Other bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Bkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' uncertainty  50  ×  FCNC (c)tZ  50  ×  (cZ)  t  FCNC t  (d)  Figure 4: Comparison between data and background prediction after the fit to data (“Post-Fit”) for the FCNC tZc LH  coupling extraction for the fitted distributions in the CRs and SRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The distributions are: (a) the 𝐷1 discriminant in  the mass sideband CR1, (b) the 𝐷𝑐  2 discriminant in the mass sideband CR2, (c) the 𝐷1 discriminant in SR1 and (d)  the 𝐷𝑐  2 discriminant in SR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The uncertainty band includes both the statistical and systematic uncertainties in the  background prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The FCNC tZc LH signals are also separately shown, normalized to 500 or 50 times the best  fit of the signal yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The lower panels show the ratios of the data (“Data”) to the background prediction (“Bkg.”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  22  Table 8: Observed and expected 95% CL limits on the FCNC t → Zq branching ratios and the effective coupling  strengths for different vertices and couplings (top eight rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' For the latter, the energy scale is assumed to be  ΛNP = 1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The bottom rows show, for the case of the FCNC t → Zu branching ratio, the observed and expected  95% CL limits when only one of the two SRs, either SR1 or SR2, and all CRs are included in the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Observable  Vertex  Coupling  Observed  Expected  SRs+CRs  B(t → Zq)  tZu  LH  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2×10−5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='9 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4 × 10−5  B(t → Zq)  tZu  RH  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6×10−5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4 × 10−5  B(t → Zq)  tZc  LH  13×10−5  11 +5  −3 × 10−5  B(t → Zq)  tZc  RH  12×10−5  10 +4  −3 × 10−5  |𝐶 (13)∗  𝑢𝑊 | and |𝐶 (13)∗  𝑢𝐵  |  tZu  LH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='13 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='02  |𝐶 (31)  𝑢𝑊 | and |𝐶 (31)  𝑢𝐵 |  tZu  RH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='14 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='02  |𝐶 (23)∗  𝑢𝑊 | and |𝐶 (23)∗  𝑢𝐵  |  tZc  LH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='20 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='04  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='03  |𝐶 (32)  𝑢𝑊 | and |𝐶 (32)  𝑢𝐵 |  tZc  RH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='19 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='04  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='03  SR1+CRs  B(t → Zq)  tZu  LH  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7×10−5  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4 × 10−5  B(t → Zq)  tZu  RH  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5×10−5  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3 × 10−5  SR2+CRs  B(t → Zq)  tZu  LH  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8×10−5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7 × 10−5  B(t → Zq)  tZu  RH  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0×10−5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='9  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8 × 10−5  9 Conclusions  A search for FCNC processes involving a top quark, an up-type quark and a Z boson is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' FCNC  tZq couplings are searched for both in tt decay events, where one top quark decays according to the SM  and the other one decays as t → Zq, and in single top-quark production through the gq → tZ FCNC  process, followed by SM top-quark decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The analysis uses 139 fb−1 of pp collision data collected by the  ATLAS experiment at the LHC between 2015 and 2018 at a center-of-mass energy of 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Events  with three leptons, a b-tagged jet, possible additional jets and missing transverse momentum are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  Multivariate discriminants are used to distinguish signal events from background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  The data are in good agreement with the SM expectations, and no evidence of a signal is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Limits at  95% CL are placed on the t → Zq branching ratios for both the tZu and tZc vertices and for both the RH  and LH couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Assuming a LH coupling, the observed limits on the branching ratios are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2 × 10−5 for  t → Zu and 13 × 10−5 for t → Zc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' These results for t → Zu (t → Zc) improve on the previous observed  limits from ATLAS by a factor of 3 (2), and on the previous expected limits by a factor of 5 (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' These are  the most stringent limits to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The improvement relative to the previous results comes from the inclusion  of the FCNC-in-single-top-quark-production signal and the usage of a multivariate analysis in addition to  the higher integrated luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' These results also constrain the values of Wilson coefficients for effective  field theory operators contributing to the t → Zu and t → Zc FCNC decays of the top quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  23  References  [1]  Particle Data Group, Review of Particle Physics, PTEP 2020 (2020) 083C01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [2]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Glashow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Iliopoulos, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Maiani, Weak Interactions with Lepton-Hadron Symmetry,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 2 (1970) 1285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [3]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Aguilar-Saavedra,  Top flavour-changing neutral interactions: theoretical expectations and experimental detection,  Acta Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Polon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 35 (2004) 2695, arXiv: hep-ph/0409342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [4]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Aguilar-Saavedra, Effects of mixing with quark singlets, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 67 (2003) 035003,  arXiv: hep-ph/0210112, [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 69 (2004) 099901].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [5]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Atwood, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Reina, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Soni,  Phenomenology of two Higgs doublet models with flavor-changing neutral currents,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 55 (1997) 3156, arXiv: hep-ph/9609279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [6]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', Supersymmetry-induced flavor-changing neutral-current top-quark processes at the  CERN Large Hadron Collider, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 75 (2007) 075021, arXiv: hep-ph/0702264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [7]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Yang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Young, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Zhang,  Flavor-changing top quark decays in R-parity-violating supersymmetric models,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 58 (1998) 055001, arXiv: hep-ph/9705341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [8]  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Agashe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Perez, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Soni,  Collider signals of top quark flavor violation from a warped extra dimension,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 75 (2007) 015002, arXiv: hep-ph/0606293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [9]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Hung, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Nugroho, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Yuan,  Top quark rare decays via loop-induced FCNC interactions in extended mirror fermion model,  Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 927 (2018) 166, arXiv: 1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='01690 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [10]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Aguilar-Saavedra, A minimal set of top anomalous couplings, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 812 (2009) 181,  arXiv: 0811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3842 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [11]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Durieux, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Maltoni, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Zhang, Global approach to top-quark flavor-changing interactions,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 91 (2015) 074017, arXiv: 1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='7166 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [12]  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Grzadkowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Iskrzynski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Misiak, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rosiek,  Dimension-six terms in the Standard Model Lagrangian, JHEP 10 (2010) 085,  arXiv: 1008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4884 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [13]  OPAL Collaboration, Search for single top quark production at LEP2,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 521 (2001) 181, arXiv: hep-ex/0110009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [14]  ALEPH Collaboration, Search for single top production in 𝑒+𝑒− collisions at √𝑠 up to 209 GeV,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 543 (2002) 173, arXiv: hep-ex/0206070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [15]  L3 Collaboration, Search for single top production at LEP, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 549 (2002) 290,  arXiv: hep-ex/0210041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [16]  DELPHI Collaboration, Search for single top production via FCNC at LEP at √𝑠 = 189-208 GeV,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 590 (2004) 21, arXiv: hep-ex/0404014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [17]  ZEUS Collaboration, Search for single-top production in 𝑒𝑝 collisions at HERA,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 708 (2012) 27, arXiv: 1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3901 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  24  [18]  CDF Collaboration,  Search for the Flavor-Changing Neutral-Current Decay 𝑡 → 𝑍𝑞 in 𝑝 ¯𝑝 Collisions at √𝑠 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='96 TeV,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 101 (2008) 192002, arXiv: 0805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2109 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [19]  D0 Collaboration, Search for flavor changing neutral currents in decays of top quarks,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 701 (2011) 313, arXiv: 1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4574 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [20]  ATLAS Collaboration, Search for flavour-changing neutral current top-quark decays to 𝑞𝑍 in 𝑝𝑝  collision data collected with the ATLAS detector at √𝑠 = 8 TeV, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 76 (2016) 12,  arXiv: 1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='05796 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [21]  ATLAS Collaboration, Search for flavour-changing neutral current top-quark decays 𝑡 → 𝑞𝑍 in  proton–proton collisions at √𝑠 = 13 TeV with the ATLAS detector, JHEP 07 (2018) 176,  arXiv: 1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='09923 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [22]  CMS Collaboration, Search Search for Flavor-Changing Neutral Currents in Top-Quark Decays  𝑡 → 𝑍𝑞 in 𝑝𝑝 Collisions at √𝑠 = 8 TeV, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 112 (2014) 171802,  arXiv: 1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4194 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [23]  CMS Collaboration, Search for associated production of a 𝑍 boson with a single top quark and for  𝑡𝑍 flavour-changing interactions in 𝑝𝑝 collisions at √𝑠 = 8 TeV, JHEP 07 (2017) 003,  arXiv: 1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='01404 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [24]  ATLAS Collaboration, The ATLAS Experiment at the CERN Large Hadron Collider,  JINST 3 (2008) S08003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [25]  ATLAS Collaboration, Performance of the ATLAS trigger system in 2015,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 77 (2017) 317, arXiv: 1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='09661 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [26]  ATLAS Collaboration, The ATLAS Collaboration Software and Firmware,  ATL-SOFT-PUB-2021-001, 2021, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='ch/record/2767187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [27]  ATLAS Collaboration,  ATLAS data quality operations and performance for 2015–2018 data-taking,  JINST 15 (2020) P04003, arXiv: 1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='04632 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [28]  ATLAS Collaboration, Performance of electron and photon triggers in ATLAS during LHC Run 2,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 80 (2020) 47, arXiv: 1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='00761 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [29]  ATLAS Collaboration, Performance of the ATLAS muon triggers in Run 2,  JINST 15 (2020) P09015, arXiv: 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='13447 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [30]  GEANT4 Collaboration, Geant4 – a simulation toolkit, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A 506 (2003) 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [31]  ATLAS Collaboration, The ATLAS Simulation Infrastructure, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 70 (2010) 823,  arXiv: 1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4568 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [32]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lange, The EvtGen particle decay simulation package,  Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A 462 (2001) 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [33]  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Sjöstrand, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Mrenna, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Skands, A brief introduction to PYTHIA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1,  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 178 (2008) 852, arXiv: 0710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3820 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [34]  ATLAS Collaboration, The Pythia 8 A3 tune description of ATLAS minimum bias and inelastic  measurements incorporating the Donnachie–Landshoff diffractive model,  ATL-PHYS-PUB-2016-017, 2016, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='ch/record/2206965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [35]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', Parton distributions with LHC data, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 867 (2013) 244,  arXiv: 1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1303 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  25  [36]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Alwall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', The automated computation of tree-level and next-to-leading order differential  cross sections, and their matching to parton shower simulations, JHEP 07 (2014) 079,  arXiv: 1405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0301 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [37]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', Parton distributions for the LHC run II, JHEP 04 (2015) 040,  arXiv: 1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='8849 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [38]  ATLAS Collaboration, ATLAS Pythia 8 tunes to 7 TeV data, ATL-PHYS-PUB-2014-021, 2014,  url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='ch/record/1966419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [39]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Alloul, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Christensen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Degrande, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Duhr, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Fuks,  FeynRules 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 – A complete toolbox for tree-level phenomenology,  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 185 (2014) 2250, arXiv: 1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1921 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [40]  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Degrande, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Maltoni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Wang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Zhang, Automatic computations at next-to-leading order  in QCD for top-quark flavor-changing neutral processes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 91 (2015) 034024,  arXiv: 1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5594 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [41]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Bähr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', Herwig++ physics and manual, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 58 (2008) 639,  arXiv: 0803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0883 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [42]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Bellm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0/Herwig++ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0 release note, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 76 (2016) 196,  arXiv: 1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='01178 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [43]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Bellm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1 Release Note, (2017), arXiv: 1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='06919 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [44]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Harland-Lang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Martin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Motylinski, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Thorne,  Parton distributions in the LHC era: MMHT 2014 PDFs, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 75 (2015) 204,  arXiv: 1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3989 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [45]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Beneke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Falgari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Klein, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Schwinn,  Hadronic top-quark pair production with NNLL threshold resummation,  Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 855 (2012) 695, arXiv: 1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1536 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [46]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Cacciari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Czakon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Mangano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Mitov, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Nason, Top-pair production at hadron  colliders with next-to-next-to-leading logarithmic soft-gluon resummation,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 710 (2012) 612, arXiv: 1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5869 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [47]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Bärnreuther, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Czakon, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Mitov, Percent-Level-Precision Physics at the Tevatron:  Next-to-Next-to-Leading Order QCD Corrections to 𝑞 ¯𝑞 → 𝑡¯𝑡 + 𝑋,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 109 (2012) 132001, arXiv: 1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5201 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [48]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Czakon and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Mitov, NNLO corrections to top-pair production at hadron colliders: the  all-fermionic scattering channels, JHEP 12 (2012) 054, arXiv: 1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0236 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [49]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Czakon and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Mitov,  NNLO corrections to top pair production at hadron colliders: the quark-gluon reaction,  JHEP 01 (2013) 080, arXiv: 1210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6832 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [50]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Czakon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Fiedler, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Mitov,  Total Top-Quark Pair-Production Cross Section at Hadron Colliders Through 𝑂(𝛼4  𝑆),  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 110 (2013) 252004, arXiv: 1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6254 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [51]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Czakon and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Mitov,  Top++: A program for the calculation of the top-pair cross-section at hadron colliders,  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 185 (2014) 2930, arXiv: 1112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5675 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  26  [52]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Frixione, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Ridolfi, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Nason,  A positive-weight next-to-leading-order Monte Carlo for heavy flavour hadroproduction,  JHEP 09 (2007) 126, arXiv: 0707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3088 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [53]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Nason, A new method for combining NLO QCD with shower Monte Carlo algorithms,  JHEP 11 (2004) 040, arXiv: hep-ph/0409146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [54]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Frixione, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Nason, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Oleari,  Matching NLO QCD computations with parton shower simulations: the POWHEG method,  JHEP 11 (2007) 070, arXiv: 0709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2092 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [55]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Alioli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Nason, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Oleari, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Re, A general framework for implementing NLO calculations  in shower Monte Carlo programs: the POWHEG BOX, JHEP 06 (2010) 043,  arXiv: 1002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2581 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [56]  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Hartanto, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Jäger, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Reina, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Wackeroth,  Higgs boson production in association with top quarks in the POWHEG BOX,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 91 (2015) 094003, arXiv: 1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='04498 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [57]  ATLAS Collaboration, Studies on top-quark Monte Carlo modelling for Top2016,  ATL-PHYS-PUB-2016-020, 2016, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='ch/record/2216168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [58]  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Sjöstrand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', An introduction to PYTHIA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 191 (2015) 159,  arXiv: 1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3012 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [59]  ATLAS Collaboration,  Improvements in 𝑡¯𝑡 modelling using NLO+PS Monte Carlo generators for Run 2,  ATL-PHYS-PUB-2018-009, 2018, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='ch/record/2630327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [60]  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Re,  Single-top 𝑊𝑡-channel production matched with parton showers using the POWHEG method,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 71 (2011) 1547, arXiv: 1009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2450 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [61]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Frixione, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Laenen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Motylinski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' White, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Webber,  Single-top hadroproduction in association with a 𝑊 boson, JHEP 07 (2008) 029,  arXiv: 0805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3067 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [62]  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Demartin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Maier, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Maltoni, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Mawatari, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Zaro,  tWH associated production at the LHC, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 77 (2017) 34,  arXiv: 1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='05862 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [63]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Alioli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Nason, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Oleari, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Re,  NLO vector-boson production matched with shower in POWHEG, JHEP 07 (2008) 060,  arXiv: 0805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4802 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [64]  ATLAS Collaboration, Measurement of the 𝑍/𝛾∗ boson transverse momentum distribution in 𝑝𝑝  collisions at √𝑠 = 7 TeV with the ATLAS detector, JHEP 09 (2014) 145,  arXiv: 1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3660 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [65]  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', New parton distributions for collider physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 82 (2010) 074024,  arXiv: 1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2241 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [66]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Pumplin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=',  New Generation of Parton Distributions with Uncertainties from Global QCD Analysis,  JHEP 07 (2002) 012, arXiv: hep-ph/0201195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  27  [67]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Golonka and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Was,  PHOTOS Monte Carlo: a precision tool for QED corrections in 𝑍 and 𝑊 decays,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 45 (2006) 97, arXiv: hep-ph/0506026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [68]  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Davidson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Przedzinski, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Was,  PHOTOS Interface in C++: Technical and physics documentation,  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 199 (2016) 86, arXiv: 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0937 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [69]  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Bothmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', Event generation with Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='2, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 7 (2019) 034,  arXiv: 1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='09127 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [70]  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Gleisberg and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Höche, Comix, a new matrix element generator, JHEP 12 (2008) 039,  arXiv: 0808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3674 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [71]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Schumann and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Krauss,  A parton shower algorithm based on Catani–Seymour dipole factorisation, JHEP 03 (2008) 038,  arXiv: 0709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1027 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [72]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Höche, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Krauss, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Schönherr, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Siegert,  A critical appraisal of NLO+PS matching methods, JHEP 09 (2012) 049,  arXiv: 1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1220 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [73]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Höche, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Krauss, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Schönherr, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Siegert,  QCD matrix elements + parton showers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' The NLO case, JHEP 04 (2013) 027,  arXiv: 1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5030 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [74]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Catani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Krauss, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Webber, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Kuhn, QCD Matrix Elements + Parton Showers,  JHEP 11 (2001) 063, arXiv: hep-ph/0109231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [75]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Höche, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Krauss, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Schumann, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Siegert, QCD matrix elements and truncated showers,  JHEP 05 (2009) 053, arXiv: 0903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1219 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [76]  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Buccioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', OpenLoops 2, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 79 (2019) 866, arXiv: 1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='13071 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [77]  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Cascioli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Maierhöfer, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Pozzorini, Scattering Amplitudes with Open Loops,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 108 (2012) 111601, arXiv: 1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='5206 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [78]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Denner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Dittmaier, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Hofer,  Collier: A fortran-based complex one-loop library in extended regularizations,  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 212 (2017) 220, arXiv: 1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='06792 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [79]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Nason and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Zanderighi, 𝑊+𝑊−, 𝑊𝑍 and 𝑍𝑍 production in the POWHEG-BOX-V2,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 74 (2014) 2702, arXiv: 1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1365 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [80]  ATLAS Collaboration, Vertex Reconstruction Performance of the ATLAS Detector at √𝑠 = 13 TeV,  ATL-PHYS-PUB-2015-026, 2015, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='ch/record/2037717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [81]  ATLAS Collaboration, Electron and photon performance measurements with the ATLAS detector  using the 2015–2017 LHC proton–proton collision data, JINST 14 (2019) P12006,  arXiv: 1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='00005 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [82]  ATLAS Collaboration, Evidence for the associated production of the Higgs boson and a top quark  pair with the ATLAS detector, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 97 (2018) 072003, arXiv: 1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='08891 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [83]  ATLAS Collaboration, Muon reconstruction and identification efficiency in ATLAS using the full  Run 2 𝑝𝑝 collision data set at √𝑠 = 13 TeV, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 81 (2021) 578,  arXiv: 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='00578 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  28  [84]  ATLAS Collaboration,  Jet reconstruction and performance using particle flow with the ATLAS Detector,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 77 (2017) 466, arXiv: 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='10485 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [85]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Cacciari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Salam, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Soyez, The anti-𝑘𝑡 jet clustering algorithm, JHEP 04 (2008) 063,  arXiv: 0802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1189 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [86]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Cacciari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Salam, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Soyez, FastJet user manual, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 72 (2012) 1896,  arXiv: 1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6097 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [87]  ATLAS Collaboration, Jet energy scale and resolution measured in proton–proton collisions at  √𝑠 = 13 TeV with the ATLAS detector, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 81 (2020) 689,  arXiv: 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='02645 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [88]  ATLAS Collaboration, Performance of pile-up mitigation techniques for jets in 𝑝𝑝 collisions at  √𝑠 = 8 TeV using the ATLAS detector, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 76 (2016) 581,  arXiv: 1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='03823 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [89]  ATLAS Collaboration, ATLAS flavour-tagging algorithms for the LHC Run 2 𝑝𝑝 collision dataset,  (2022), arXiv: 2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='16345 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='data-an].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [90]  ATLAS Collaboration, Performance of missing transverse momentum reconstruction with the  ATLAS detector using proton–proton collisions at √𝑠 = 13 TeV, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 78 (2018) 903,  arXiv: 1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='08168 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [91]  ATLAS Collaboration,  𝐸miss  T  performance in the ATLAS detector using 2015–2016 LHC 𝑝𝑝 collisions,  ATLAS-CONF-2018-023, 2018, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='ch/record/2625233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [92]  ATLAS Collaboration, Measurement of the top-quark mass using a leptonic invariant mass in 𝑝𝑝  collisions at √𝑠 = 13 TeV with the ATLAS detector, (2022), arXiv: 2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='00583 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [93]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Bukin, Fitting function for asymmetric peaks, 2007,  arXiv: 0711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='4449 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='data-an].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [94]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Kulesza, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Motyka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Stebel, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Theeuwes,  Soft gluon resummation for associated 𝑡¯𝑡𝐻 production at the LHC, JHEP 03 (2016) 065,  arXiv: 1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='02780 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [95]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Friedman, Stochastic gradient boosting, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Data Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' 38 (2002) 367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [96]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Hoecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', TMVA - Toolkit for Multivariate Data Analysis, 2007,  arXiv: physics/0703039 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='data-an].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [97]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Breiman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Friedman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Stone, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Olshen, Classification and Regression Trees,  Chapman & Hall, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [98]  ATLAS Collaboration, Jet energy scale measurements and their systematic uncertainties in  proton–proton collisions at √𝑠 = 13 TeV with the ATLAS detector, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 96 (2017) 072002,  arXiv: 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='09665 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [99]  ATLAS Collaboration, ATLAS 𝑏-jet identification performance and efficiency measurement with 𝑡¯𝑡  events in 𝑝𝑝 collisions at √𝑠 = 13 TeV, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 79 (2019) 970,  arXiv: 1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='05120 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [100]  ATLAS Collaboration, Measurement of 𝑏-tagging efficiency of 𝑐-jets in 𝑡¯𝑡 events using a  likelihood approach with the ATLAS detector, ATLAS-CONF-2018-001, 2018,  url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='ch/record/2306649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  29  [101]  ATLAS Collaboration, Calibration of light-flavour 𝑏-jet mistagging rates using ATLAS  proton–proton collision data at √𝑠 = 13 TeV, ATLAS-CONF-2018-006, 2018,  url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='ch/record/2314418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [102]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Butterworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', PDF4LHC recommendations for LHC Run II, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' G 43 (2016) 023001,  arXiv: 1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='03865 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [103]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Martin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Stirling, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Thorne, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Watt, Parton distributions for the LHC,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 63 (2009) 189, arXiv: 0901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='0002 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [104]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Martin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Stirling, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Thorne, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Watt,  Uncertainties on 𝛼𝑆 in global PDF analyses and implications for predicted hadronic cross sections,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 64 (2009) 653, arXiv: 0905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='3531 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [105]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', CT10 next-to-next-to-leading order global analysis of QCD,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' D 89 (2014) 033009, arXiv: 1302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='6246 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [106]  LHC Higgs Cross Section Working Group,  Handbook of LHC Higgs Cross Sections: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Deciphering the Nature of the Higgs Sector,  CERN-2017-002-M (CERN, Geneva, 2016), arXiv: 1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='07922 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [107]  ATLAS Collaboration, Evidence for 𝑡¯𝑡𝑡¯𝑡 production in the multilepton final state in proton–proton  collisions at √𝑠 = 13TeV with the ATLAS detector, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 80 (2020) 1085,  arXiv: 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='14858 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [108]  ATLAS Collaboration, Observation of the associated production of a top quark and a 𝑍 boson in  𝑝𝑝 collisions at √𝑠 = 13 TeV with the ATLAS detector, JHEP 07 (2020) 124,  arXiv: 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='07546 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [109]  CMS Collaboration, Measurement of the associated production of a single top quark and a 𝑍  boson in 𝑝𝑝 collisions at √𝑠 = 13 TeV, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' B 779 (2018) 358,  arXiv: 1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='02825 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [110]  ATLAS Collaboration, Measurement of 𝑊±𝑍 production cross sections and gauge boson  polarisation in 𝑝𝑝 collisions at √𝑠 = 13 TeV with the ATLAS detector,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 79 (2019) 535, arXiv: 1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='05759 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [111]  ATLAS Collaboration, Evidence for the 𝐻 → 𝑏 ¯𝑏 decay with the ATLAS detector,  JHEP 12 (2017) 024, arXiv: 1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='03299 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [112]  ATLAS Collaboration,  Luminosity determination in 𝑝𝑝 collisions at √𝑠 = 13 TeV using the ATLAS detector at the LHC,  ATLAS-CONF-2019-021, 2019, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='ch/record/2677054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [113]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Avoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=', The new LUCID-2 detector for luminosity measurement and monitoring in ATLAS,  JINST 13 (2018) P07017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [114]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Read, Presentation of search results: the 𝐶𝐿𝑆 technique, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' G 28 (2002) 2693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  [115]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Cowan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Cranmer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Gross, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Vitells,  Asymptotic formulae for likelihood-based tests of new physics, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 71 (2011) 1554,  arXiv: 1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='1727 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='data-an], Erratum: Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content=' C 73 (2013) 2501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
+page_content='  30' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFJT4oBgHgl3EQfqizo/content/2301.11605v1.pdf'}
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+Chiral Anomalies in 3D Spin-Orbit Coupled Metals: Electrical, Thermal, and
+Gravitational anomaly
+Sunit Das,∗ Kamal Das,† and Amit Agarwal‡
+Department of Physics, Indian Institute of Technology Kanpur, Kanpur-208016, India
+The discovery of a chiral anomaly in Weyl semimetals, the non-conservation of chiral charge and
+energy across two opposite chirality Weyl nodes, has sparked immense interest in understanding
+its impact on various physical phenomena. Here, we demonstrate the existence of electrical, ther-
+mal, and gravitational quantum chiral anomalies in 3D spin-orbit coupled systems. Notably, these
+anomalies involve chiral charge transfer across two Fermi surfaces linked to a single Weyl-like point,
+rather than across opposite chirality Weyl nodes as in Weyl semimetals. Our findings reveal that the
+Berry curvature flux piercing the Fermi surface plays a critical role in distinguishing the ‘chirality’ of
+the carriers and the corresponding chiral charge and energy transfer. Importantly, we demonstrate
+that these quantum chiral anomalies lead to interesting thermal spin transport such as the spin
+Nernst effect. Our results suggest that 3D spin-orbit coupled metals offer a promising platform
+for investigating the interplay between quantum chiral anomalies and charge and spin transport in
+non-relativistic systems.
+I.
+INTRODUCTION
+Chiral anomaly refers to the non-conservation of chi-
+ral charges in the presence of collinear electric and mag-
+netic fields. It was first introduced in the context of the
+relativistic field theory of chiral fermions [1–3].
+Later
+it was shown to be achievable in low gap semiconduc-
+tors [4], with signatures in magnetoconductance exper-
+iments.
+Following the discovery of Weyl semimetals
+(WSMs) in recent years, the physics of chiral anomaly
+has been widely studied in condensed matter systems,
+resulting in a variety of non-trivial transport [5–18] and
+optical [19–26] effects.
+Intriguingly, the presence of a
+temperature gradient in Weyl systems can also result in
+an anomaly similar to the axial-gravitational anomaly in
+flat-space time [27–30]. This leads to a range of interest-
+ing magneto-thermal transport phenomena [12, 31–37].
+Central to the physics of chiral anomaly is the con-
+tinuity equation for the chiral charge.
+The continuity
+equation for the chiral charges and energy can be de-
+rived using semiclassical dynamics in crystalline materi-
+als and shows that the Berry curvature monopoles govern
+the chiral anomaly in Weyl metals [38–40]. The concept
+of chiral anomaly has also been extended to other free
+fermionic excitations with no high-energy analog, such as
+multi-Weyl semimetals [41–45], which exhibit two band
+crossings similar to WSMs but with nonlinear momentum
+dispersion along a particular direction, and semimetals
+with a higher number of band crossings near the Weyl
+node [46]. These systems, while possessing a higher chi-
+ral charge, are otherwise similar to Weyl systems in that
+a theory of chiral anomaly requires the presence of two
+opposite chirality Weyl nodes.
+In this paper, we delve into the connection between
+∗ sunitd@iitk.ac.in
+† kamaldas@iitk.ac.in
+‡ amitag@iitk.ac.in
+a)
+b)
+Weyl metals
+3D Spin orbit coupled metals
+a)
+b)
+9
+Here, we have used the fact that the band velocity v�
+does not contribute to the equilibrium current (due to
+angular integration being zero). Now, we use the identity
+rk · (✏�⌦�) = rk✏� · ⌦� + ✏�rk · ⌦� to write the above
+equation as follows,
+j�
+e,eq
+= �e2B
+~2
+Z
+[dk] [rk · (✏�⌦�) � ✏�rk · ⌦�] f �
+� ,(C3)
+= �e2B
+~2
+Z
+[dk]rk · (✏�⌦�)f �
+�
+(C4)
+= e2
+~2 B
+Z
+[dk]✏�⌦� · ˆk@f �
+�
+@k ,
+(C5)
+= �eB
+Z
+[dk] (µ + ✏� � µ) e
+~(v� · ⌦�)
+✓
+�@f �
+�
+@✏�
+◆
+,
+= �e (µC�
+0 + kBTC�
+1 ) B.
+(C6)
+To evaluate Eq. (C4), we have used the fact that rk ·
+⌦� = ±2⇡�3(k), for a system with band touching point,
+which makes the last integral of Eq. (C3) to be zero. Also,
+we have used property of partial integrations to obtain
+Eq. (C5) from the Eq. (C4). Here, we have defined C�
+⌫ as
+follows
+C�
+⌫ =
+Z
+[dk]
+✓✏ � µ
+kBT
+◆⌫ ✓
+�@f �
+�
+@✏�
+◆
+,
+(C7)
+which can also be rewritten in terms of C� given in
+Eq. (8).
+The j�
+✏,eq can be evaluated following similar
+calculations. One remark is in order, although the num-
+ber density of carriers n�=�1 contains the Fermi func-
+tion for both the bands, the equilibrium (also the non-
+equilibrium) currents for � = �1 only come from the
+� = +1 (� = �1) for µ > 0 (µ < 0). This is because the
+currents we obtain are the Fermi surface property.
+(other important aspects to be put in the main text)
+Appendix D: Details of spin current calculations
+To calculate the spin current proportional to the E ·
+B (or rT · B), we consider the band velocity term of
+Eq. (4) and calculate the spin current operator. The band
+velocity operator along the i-direction is given by ˆvi =
+~ki
+m �0 + ↵
+~ �i. Without loss of generality, here we show
+the calculation of spin current in the x-direction. Using
+the expressions of the eigenstates and the spin current
+operator given in the main text, we obtain hu�| ˆJsx
+x |u�i =
+(↵/~ + �~kx sin ✓k cos �k/m). Now, the chiral anomaly
+induced spin current is given by
+jsx
+x
+= ⌧v
+X
+�
+Z
+[dk]
+✓↵
+~ + �~kx
+m sin ✓k cos �k
+◆ ✓
+�µ� + ✏� � µ
+T
+�T �
+◆ ✓
+�@f �
+�
+@✏�
+◆
+.
+(D1)
+In the �µ ! 1 limit, writing the expressions of �µ� and �T � explicitly, we have
+jsx
+x =
+⌧v
+X
+�
+ D�
+1 C�
+0
+D�
+2 D�
+0
+L1 � C�
+0
+D�
+0
+L0
+�
+eE · B �
+ D�
+1 C�
+2
+D�
+2 D�
+0
+L0 � C�
+2
+D�
+2
+L1
+�
+kBrT · B.
+(D2)
+The definition of L⌫ is given in the main text. We eval-
+uate the L⌫ using the Sommerfeld approximation in the
+µ > kBT limit. We obtain the following expressions
+L0
+= �� m2↵2
+6⇡2~5
+[˜µ � ˜µ2 + 2(1 + ˜µ)]
+1 + ˜µ
+,
+(D3)
+L1
+= kBT
+9~3
+(�� + p1 + ˜µ)[�(2 + ˜µ) + p1 + ˜µ]
+(1 + ˜µ)3/2
+.(D4)
+Using these expression along with C�
+⌫ and D�
+⌫ in Eq. (D2),
+we obtain the spin conductivities of Eqs. (32) and (33).
+Following a similar procedure, we can calculate other spin
+currents.
+Now, we show that due to rotational symmetry jsj
+i
+= 0
+for i 6= j. Without loss of generality, we will explicitly
+show the calculation for jsz
+x . The expectation value of
+the spin current operator ˆJsz
+x
+is given by hu�| ˆJsz
+x |u�i =
+� p
+2m sin 2✓k cos �k. Now, as the distribution function is
+independent of ✓k and �k, so the angular integration over
+�k of the hu�| ˆJsz
+x |u�i yields jsz
+x
+= 0. Similarly, all the
+spin currents with spin polarization perpendicular to the
+propagation velocity can be easily shown to be zero due
+to the vanishing angular integration over �k.
+E · B 6= 0 or rT · B 6= 0
+[1] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase e↵ects on
+electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
+[2] Y. Gao, Semiclassical dynamics and nonlinear charge cur-
+a)
+b)
+a)
+b)
+a)
+b)
+FIG. 1.
+Depiction of the quantum chiral anomalies in (a)
+Weyl semi-metals and (b) 3D spin-orbit coupled metals or
+Kramers-Weyl metals. Both systems experience chiral charge
+and energy pumping, manifesting as electrical, thermal, and
+gravitational anomalies, when subjected to a magnetic field
+and collinear electric field (E · B ̸= 0) or a temperature gra-
+dient (∇T · B ̸= 0).
+In contrast to Weyl semimetals, the
+chiral charge pumping in 3D spin-orbit coupled metals occurs
+between two different Fermi surfaces associated with a single
+‘Kramers-Weyl’ node, but with opposite Berry curvature flux
+passing through them.
+chiral anomalies and the Berry curvature flux passing
+through the Fermi surface (FS) [5]. This connection was
+recently explored in Ref. [47, 48].
+Motivated by this,
+we generalize the theory of quantum chiral anomalies
+to Hamiltonians with non-relativistic terms, specifically
+H = hk · σ + σ0k2. Here, the σ represents the real spin
+of the system, σ0 is the identity matrix, and hk is an odd
+function of k. The quadratic kinetic energy-like term in
+arXiv:2301.02965v1  [cond-mat.mes-hall]  8 Jan 2023
+
+2
+the Hamiltonian makes the chiral anomaly in this spin-
+orbit coupled (SOC) metals to be distinctly different from
+that in WSM [see Fig. 1]. These types of systems can be
+found in Kramers-Weyl metals with quadratic corrections
+to their k·p Hamiltonians [49–58] or in systems support-
+ing 3D electron gas with SOC. While some aspects of
+the charge, heat, and spin transport in SOC metals have
+been explored earlier [47, 59, 60], the physics of quantum
+chiral anomalies in these systems is largely unexplored
+and merits further investigation.
+In this paper, we demonstrate that Kramers-Weyl and
+spin-orbit coupled metals can exhibit all three types of
+quantum chiral anomalies—electrical, thermal, and grav-
+itational. We investigate the impact of electric field and
+temperature gradient-induced quantum chiral anomalies
+on charge, heat, and spin transport phenomena. Similar
+to the behavior observed in Weyl semi-metals [17], we
+find that chiral anomalies in 3D SOC systems also result
+in negative longitudinal magneto-resistance and positive
+thermal magneto-resistance. However, a distinct feature
+of 3D SOC systems, as compared to WSMs, is that their
+low-energy Hamiltonian involves real spins.
+We show
+that quantum chiral anomalies in these systems also lead
+to interesting electrical and thermal spin transport in-
+cluding the spin-Nernst effect.
+The structure of the rest of this paper is as follows. In
+Section II, we discuss the origins of chiral anomalies in
+three-dimensional (3D) metals with SOC and Kramers-
+Weyl metals. In Section III, we present a mathematical
+derivation of the continuity equations to demonstrate the
+existence of these anomalies. The effects of these anoma-
+lies on charge and spin transport are examined in Sec-
+tions IV and V, respectively. Finally, we summarize our
+findings in Section VI.
+II.
+ORIGIN OF CHIRAL ANOMALIES IN
+SPIN-ORBIT COUPLED METALS
+To understand the chiral anomaly in 3D spin-orbit cou-
+pled metallic systems (or Kramers-Weyl metals), we first
+revisit the WSM. Specifically, we review the physics of
+chiral anomaly in WSM from the perspective of semi-
+classical dynamics. In WSM, the Hamiltonian for a par-
+ticular Weyl node near the band crossing point can be
+approximated as HWSM = �
+a=x,y,z ℏ(va · k)σa, where
+k is measured from the Weyl node.
+The ‘chirality’ of
+Weyl node is defined as C = sign[vx · vy × vz] [61]. In
+the semiclassical dynamics picture, the existence of chi-
+ral anomaly can be understood by calculating the equi-
+librium current in the presence of an external magnetic
+field but no electric field.
+The equilibrium charge current for each Weyl node (or
+the chiral current) arises from the chiral magnetic ve-
+locity (see Sec. III A with explicit derivation shown in
+Appendix C). The chiral current for WSM can be ex-
+pressed in terms of the Berry curvature flux quantum
+passing through the FS for the WSM [17]. This is consis-
+tent with the intuitive picture of the Weyl nodes acting as
+sinks and sources of the Berry curvature. For the pair of
+Weyl nodes of opposite chirality, their FSs are separated
+in the momentum space (at least for small energies). In
+the presence of an external electric field aligned along
+the magnetic field, the chiral charge carriers are pumped
+across the FSs with distinct Weyl chirality. This flow is
+stabilized by inter-node scattering. This results in dif-
+ferent chiral charge densities on the two Weyl nodes [as
+shown in Fig. 1(a)], and it manifests in several inter-
+esting transport phenomena in WSMs [5, 17, 36].
+We
+emphasize two things here: i) A minimum of a pair of
+Weyl nodes of opposite chirality are needed to produce
+chiral anomaly in WSM, and ii) the chiral anomaly can
+be interpreted as an FS phenomenon, where the chiral
+charges are ‘pumped’ across two FSs enclosing opposite
+quantum of the Berry curvature flux. These two points
+will be crucial in investigating the chiral anomalies in
+Kramers-Weyl metals or 3D SOC metals.
+3D SOC metals or Kramers-Weyl metals are struc-
+turally chiral crystals with broken inversion symme-
+try. They host ‘Weyl’-like nodal points at all the time-
+reversal-invariant momentum (TRIM) points in their
+Brillouin zone. While the form of the SOC can be dif-
+ferent, a common feature of all such materials is that
+they have two FSs for each band crossing point (or the
+Kramers-Weyl node). This is aided by the kinetic en-
+ergy term of the form ℏ2k2/(2m) in their dispersion,
+which is missing in conventional WSM. We have tabu-
+lated all crystalline point groups that support Kramers-
+Weyl points, along with their low energy Hamiltonian in
+the vicinity of the Kramers-Weyl point in Appendix A.
+While our discussion applies to all classes of single crys-
+talline systems of 3D SOC metals or Kramers-Weyl met-
+als listed in Table I, for specific calculations, we consider
+the Hamiltonian [51, 62, 63],
+H = ℏ2k2
+2m σ0 + αk · σ .
+(1)
+Here, m is the effective electron mass, α is the SOC pa-
+rameter, σ = (σx, σy, σz) denotes the vector of the Pauli
+matrices in spin space and k is the Bloch wave vector.
+We note that in contrast to conventional WSM, the Pauli
+matrices here denote the physical spins of the itinerant
+electrons. The energy dispersion for the Hamiltonian in
+Eq. (1) is,
+ϵλ = ℏ2k2
+2m + λαk .
+(2)
+Here, λ = ±1 is the spin-split band index which co-
+incides with the eigenvalues of the operator ˆO = ˆk ·
+σ, and k = |k|.
+The corresponding eigenstates are
+given by |u⟩T
++ = [cos(θk/2), eiφk sin(θk/2)] and |u⟩T
+− =
+[sin(θk/2), −eiφk cos(θk/2)], with cos θk
+≡
+kz/k and
+tan φk ≡ ky/kx. In Fig. 1b, the λ = +1 (λ = −1) band
+is represented by the solid (dashed) line. The two bands
+of the dispersion relation (2) have a band-touching point
+
+3
+(BTP) at ϵ = 0. The λ = +1 band has a minimum at
+ϵ = 0 and increases monotonically as k increases. The
+λ = −1 band is non-monotonic, and it has a minimum
+energy located at ϵmin = −ϵα, with ϵα = mα2/2ℏ2. The
+minimum energy point lies on a circular contour specified
+by |k|2 = k2
+α, where kα = mα/ℏ2.
+Clearly, there are two different types of FSs for any
+value of the Fermi energy greater than the energy of the
+Kramers-Weyl node.
+The inner FS resulting from the
+λ = +1 band has an electronic character. In contrast, the
+outer FS can be interpreted to have the hole character.
+The Berry curvature flux quantum through each of the
+Fermi surfaces is defined as
+Cλ = 1
+2π
+�
+FS
+dS · Ωλ .
+(3)
+Here, dS is the elemental surface area of the FS, and
+Ωλ is the Berry curvature. More interestingly, the flux
+quantum associated with the FSs is equal and opposite.
+We explicitly calculate Cλ = −λ. See Appendix B for de-
+tails. Hence, the Berry curvature flux quantum piercing
+the outer (inner) FS is +1 (−1). We emphasize that this
+scenario is distinctively different from the usual WSM
+with chiral symmetry. In WSM, the pair of FS with the
+opposite sign of the Berry curvature flux quantum cor-
+responds to two distinct Weyl crossing points separated
+by momentum or energy. However, in this case, a single
+Weyl crossing is linked to the two FSs with opposite flux
+quantum. The opposite sign of the Berry curvature flux
+quantum on the two FSs can be used to define charged
+fermions of different ‘flavors’ (akin to chirality in the case
+of WSM) in the two FSs.
+The non-zero flux associated with the two FSs in SOC
+metals gives rise to chiral anomalies. This is captured
+by the non-conservation of the total flavor charge (N λ)
+and energy (Eλ) in presence of a magnetic field (B) and
+an electric field (E) or temperature gradient (∇T). In a
+clean system of 3D SOC metal, we can obtain
+∂N λ
+∂t
+∝ −Cλ
+0 E · B
+and
+∂N λ
+∂t
+∝ −Cλ
+1 ∇T · B . (4)
+A similar calculation for the total energy of each flavour
+of fermions yields,
+∂Eλ
+∂t ∝
+�
+−(µ Cλ
+0 + kBT Cλ
+1 ) E · B
+−(µ Cλ
+1 + kBT Cλ
+2 )∇T · B
+.
+(5)
+Here, µ is the chemical potential, and kBT is the energy
+scale of the temperature. The coefficients Cλ
+ν [Eq. (9)]
+for ν = {0, 1, 2} are the coefficients of the electrical, ther-
+mal, and gravitational chiral anomalies, respectively. See
+Sec. III and Eqs. (14)-(15) for more details. More impor-
+tantly, these are finite only when the Berry curvature
+flux quantum Cλ is finite. Thus, the Berry curvature flux
+quantum plays an important role in defining the par-
+ticles’ flavor (or chirality) and the associated quantum
+flavor anomalies (or chiral anomalies). We highlight the
+chiral charge transfer across the two Fermi surfaces in
+WSM and in 3D SOC metals, with opposite Berry cur-
+vature flux in Fig 1.
+In the next section, we explicitly demonstrate the three
+chiral anomalies in 3D SOC (or Kramers-Weyl) metals
+using the idea of equilibrium and non-equilibrium chiral
+charge and energy currents. We specifically focus on the
+case when the chemical potential is higher in energy than
+the Kramers-Weyl point (µ > 0). The regime when the
+chemical potential is below the energy of the Kramers-
+Weyl point is a bit tricky. We find that in this regime
+there is only one FS. The Fermi surface is associated
+with the λ = −1 band, and the total flux through the
+FS is identically zero. Since the chiral anomaly requires
+two FSs with opposite Berry curvature flux, there is no
+chiral anomaly for µ < 0. However, an interesting Bril-
+louin zone partitioning scheme been proposed in Ref. [47]
+to divide the single FS into two parts having opposite
+Berry curvature flux. We show that such BZ partition-
+ing within a single FS is not physical, and it can lead
+to chiral anomaly-like physics even in a free electron gas
+in absence of a magnetic field and Berry curvature. We
+discuss these subtle issues in detail in Appendix B.
+III.
+CHIRAL CURRENTS AND THE CHIRAL
+ANOMALIES
+In this section, we first show that the existence of equi-
+librium currents in the presence of a magnetic field hints
+at the possible existence of chiral anomalies in the sys-
+tem. Next, we explicitly calculate the continuity equation
+for the chiral charges and energy current in the presence
+of a magnetic field and either a collinear electric field or
+a collinear temperature gradient.
+A.
+Equilibrium chiral current induced by magnetic
+field
+The equations of motion of charge carriers in the pres-
+ence of Berry curvature are described by the following
+semiclassical equation of motion [64, 65]
+˙rλ = Dλ
+�
+vλ + e
+ℏE × Ωλ + e
+ℏ(vλ · Ωλ)B
+�
+,
+(6a)
+ℏ ˙kλ = Dλ
+�
+−eE − evλ × B − e2
+ℏ (E · B)Ωλ
+�
+. (6b)
+Here, ‘−e’ is the electronic charge, vλ is the band ve-
+locity, and Ωλ is the Berry curvature.
+In Eq. (6a),
+Dλ ≡ 1/(1 + e
+ℏΩλ · B) is the phase-space factor, which
+modifies the invariant phase-space volume according to
+[dk] → [dk]D−1
+λ
+[66]. The term e
+ℏ(vλ · Ωλ)B in Eq. (6a)
+is known as the chiral magnetic velocity and as will see
+it plays an important role in anomaly related transport.
+For a given FS, the equilibrium chiral charge and en-
+
+4
+ergy currents are calculated to be [13]
+{jλ
+e,eq, jλ
+ϵ,eq} =
+�
+BZλ
+[dk]{−e, ϵλ} e
+ℏ (vλ · Ωλ) fλ .
+(7)
+In Eq. (7), fλ is the equilibrium Fermi distribution func-
+tion corresponding to the FS λ. We emphasize that the
+chiral magnetic velocity solely determines the chiral cur-
+rents, and the band gradient velocity does not contribute
+to it. Evaluating Eq. (7) for our model Hamiltonian, we
+obtain general relations for the charge and the energy
+current [17, 30, 67],
+jλ
+e,eq= −e
+�
+µCλ
+0 + kBTCλ
+1
+�
+B ,
+(8a)
+jλ
+ϵ,eq=
+�µ
+2 Cλ
+0 + µkBTCλ
+1 + k2
+BT 2
+2
+Cλ
+2
+�
+B .
+(8b)
+Here, we note that all the anomaly coefficients appear
+in the equilibrium current. In Eqs. (8a)-(8b), the coeffi-
+cients are specified by,
+Cλ
+ν =
+e
+4π2ℏ2
+�
+dϵ
+�ϵ − µ
+kBT
+�ν �
+−∂fλ
+∂ϵλ
+�
+Cλ .
+(9)
+It is evident from Eq. (9) that for any quantum system
+with finite Cλ, all the chiral anomaly coefficients are non-
+zero. We mention here that in defining the anomaly co-
+efficients in Eq. (9), we have converted the Fermi sea
+integration of Eq. (7) into Fermi surface integration us-
+ing the rule of partial derivative. We provide the details
+of the calculations in Appendix C.
+The importance of the equilibrium currents given in
+Eqs. (8a)-(8b) is multifold. First of all, the presence of
+finite chiral charges and energy currents in equilibrium is
+an indication of the existence of chiral anomalies. This
+is because, for both chiral anomaly and non-zero chiral
+equilibrium current, non-zero Berry curvature flux is a
+prerequisite. Second, the chiral charge (j+
+e,eq − j−
+e,eq) and
+energy (j+
+ϵ,eq − j−
+ϵ,eq) currents are non-zero. This high-
+lights that in systems hosting a pair of fermions with
+opposite Berry curvature flux quantum, the chiral mag-
+netic velocity induces a dissipationless chiral charge and
+energy current along B [12, 68–71]. Finally, we can ex-
+pect a finite anomaly-induced current in non-equilibrium.
+In equilibrium, the total charge (j+
+e,eq +j−
+e,eq) and energy
+(j+
+ϵ,eq+j−
+ϵ,eq) currents from the two opposite chirality FSs
+will add up to zero due to same chemical potential and
+temperature. However, in the presence of chiral chemical
+potential (µ+ ̸= µ−) and chiral temperature (T+ ̸= T−)
+imbalance induced by the quantum anomalies, these ex-
+pressions will result in finite charge and energy current.
+Note that the general expressions of equilibrium charge
+and energy currents, jλ
+e,eq and jλ
+ϵ,eq, are valid for any 3D
+systems with band touching point. These currents origi-
+nate from the chiral magnetic velocity, e/ℏ(vλ·Ωλ)B. As
+a result, the equilibrium currents are identically zero for
+any two-dimensional system, for which vλ · Ωλ = 0. The
+absence of chiral magnetic velocity in 2D systems for-
+bids the existence of quantum chiral anomalies in two-
+dimensional systems.
+For three-dimensional systems,
+vλ · Ωλ is generally non-zero, which gives rise to finite
+equilibrium currents. However, to have quantum chiral
+anomalies in the system, there should be a pair of FS with
+opposite Berry curvature flux quantum passing through
+them so that jλ
+e/ϵ,eq = −j−λ
+e/ϵ,eq.
+Having discussed the general expressions for the equi-
+librium charge and energy currents, we now calculate all
+the anomaly coefficients for a 3D spin-orbit coupled sys-
+tem. For the Hamiltonian in Eq. (1), the Berry curvature
+is given by Ωλ = −λk/2k3. The chiral anomaly coeffi-
+cients are obtained to be
+{Cλ
+0 , Cλ
+1 , Cλ
+2 } = −λ
+e
+4π2ℏ2 {F0, F1, F2} .
+(10)
+We note that the equilibrium currents of Eqs. (8a) and
+(8b), along with the chiral anomaly coefficients of the
+above equations, do not get affected by the orbital mag-
+netic moment. Here, Fν’s are the dimensionless functions
+of i) x = β(ϵα + µ) for λ = −1 band, and ii) x = βµ for
+λ = +1 band with β = 1/kBT being the inverse temper-
+ature. Their functional form is given by
+F0(x) ≡ 1/(1 + e−x) ,
+F1(x) ≡ x/(1 + ex) + ln[1 + e−x] ,
+(11)
+F2(x) ≡ π2
+3 − x
+�
+x
+1 + ex + 2ln[1 + e−x]
+�
++ 2Li2[−e−x].
+Here, Li2 is the polylogarithmic function of order two.
+With the replacement of (ϵα + µ) → µ, Eqs. (10) and
+(11) become identical to that in the WSMs [17].
+The
+temperature dependence of all three chiral anomaly co-
+efficients is similar to Fig. (6) in Ref. [17]. In the zero
+temperature limit, F0 → 1, and F2 → π2/3. It is worth
+noting that for T = 0, the thermal chiral anomaly coeffi-
+cient Cλ
+1 ∝ F1 → 0 becomes finite only for finite T.
+B.
+Steady state in the presence of chiral anomaly
+The presence of external perturbations, such as an elec-
+tric field E, or a temperature gradient ∇T, drives the
+system out of equilibrium. In the non-equilibrium steady-
+state, the distribution function (gλ) corresponding to the
+FS λ satisfies the following Boltzmann transport equa-
+tion
+∂gλ
+∂t + ˙rλ · ∇r gλ + ˙kλ · ∇k gλ = Icoll{gλ} .
+(12)
+Here, Icoll{gλ} is the collision integral and gλ is the non-
+equilibrium distribution function for each Fermi function.
+Similar to that in WSM, the charge and energy pump-
+ing between the two FSs dictates that the collision inte-
+gral should incorporate both the intra- and inter-Fermi
+surface scattering processes [17, 36, 72]. Within the re-
+laxation time approximation, both the scattering process
+can be captured by the following form of the collision in-
+tegral [13, 73],
+Iλ
+coll = −gλ − ¯gλ
+τ
+− ¯gλ − fλ
+τv
+.
+(13)
+
+5
+Here, ¯gλ represents the ‘local’ steady-state distribution
+function for each FS with a local chemical potential
+µλ ≡ µ + δµλ, and local temperature Tλ ≡ T + δTλ [72],
+and fλ specifies the global equilibrium function.
+The
+first term in the right-hand side of Eq. (13) represents
+the intra-Fermi surface scattering (with scattering rate
+1/τ), which establishes the local equilibrium. The inter-
+Fermi surface scattering has been represented by the sec-
+ond term in Eq. (13) with scattering rate 1/τv. The ra-
+tio of inter- and intra- Fermi surface scattering time for
+Hamiltonian (1) considering screened Coulomb impurity
+potential has been calculated in Ref. [47]. In the small µ
+limit, it is given by τv/τ ∼ (2mα2/ℏ2)2/µ2 [47]. Hence,
+for small µ, similar to the WSM [74, 75], we have τv > τ.
+Now, we construct the continuity equation for the
+particle number and the energy density.
+Substituting
+Eq. (13) in Eq. (12), and then integrating over all the
+momentum states for the FS λ, we obtain
+∂N λ
+∂t
++ eE · BCλ
+0 + ∇r · Jλ = −N λ − N λ
+0
+τv
+.
+(14)
+Here, ∇r ·Jλ = kBCλ
+1 ∇T ·B is the divergence of particle
+current. The quantities {N λ
+0 , N λ} =
+�
+[dk]D−1
+λ {fλ, gλ}
+represents the total particle number density in each
+FS before and after applying the perturbing fields. In
+Eq. (14), the terms E ·BCλ
+0 , and kBCλ
+1 ∇T ·B represents
+the chiral anomaly induced flow of the charge carriers.
+Similarly, the continuity equation for the energy density,
+which we construct by multiplying the energy dispersion
+ϵλ in Eq. (12) and integrating over all the momentum
+states, is obtained to be
+∂Eλ
+∂t +(µCλ
+0 +kBTCλ
+1 ) eE·B+∇r·Jλ
+E = −Eλ − Eλ
+0
+τv
+. (15)
+The second term on the left hand side is −E · jλ
+e,eq that
+represents the work performed by the electric field and
+∇r · Jλ
+E = (µkBCλ
+1 + k2
+BTCλ
+2 ) ∇T · B represents the
+divergence of energy current in presence of ∇T.
+The
+quantities {Eλ
+0 , Eλ} =
+�
+[dk]D−1
+λ ϵλ{fλ, gλ} is the total
+energy density in each FS before and after applying
+external fields, respectively.
+Here, µCλ
+0 and µCλ
+1 spec-
+ify the energy carried out by the chiral charge transfer,
+whereas TCλ
+2 represents the energy pumped out by the
+term ∇T · B [17]. In constructing Eq. (14) and (15), we
+have used the fact that the intra-Fermi surface scatter-
+ing does not change the number of particles and energy
+in each FS.
+IV.
+CHIRAL ANOMALY AND CARRIER
+TRANSPORT
+To calculate the chiral anomaly-induced charge, heat,
+and spin currents, we first calculate the non-equilibrium
+distribution function to linear order in an applied electric
+field. In the linear response regime, we can safely assume
+that the change in chiral chemical potential and temper-
+ature is small, i.e., δµλ < µ, and δTλ < T [13, 17, 72].
+Then, to the lowest order in δµλ and δTλ, the non-
+equilibrium distribution function can be calculated to be
+gλ = fλ +
+�
+−∂fλ
+∂ϵλ
+� � �
+1 − τ
+τv
+� �
+δµλ + ϵλ − µ
+T
+δTλ
+�
+−τDλ
+�
+vλ + e
+ℏ (vλ · Ωλ) B
+�
+·
+�
+eE + (ϵλ − µ) ∇T
+T
+� �
+.
+(16)
+Here, the chiral chemical potential δµλ and δTλ are given
+by [13]
+δµλ = −
+τv
+(Dλ
+2 Dλ
+0 − Dλ
+1
+2)
+��
+Dλ
+2 Cλ
+0 − Dλ
+1 Cλ
+1
+�
+eE · B
++
+�
+Dλ
+2 Cλ
+1 − Dλ
+1 Cλ
+2
+�
+kB∇T · B
+�
+,
+(17)
+kBδTλ = −
+τv
+(Dλ
+2 Dλ
+0 − Dλ
+1
+2)
+��
+Dλ
+0 Cλ
+1 − Dλ
+1 Cλ
+0
+�
+eE · B
++
+�
+Dλ
+0 Cλ
+2 − Dλ
+1 Cλ
+1
+�
+kB∇T · B
+�
+.
+(18)
+In the above equation, we have defined the magnetic field-
+dependent generalized density of states at finite temper-
+ature as
+Dλ
+ν =
+�
+dϵ
+�ϵλ − µ
+kBT
+�ν �
+−∂fλ
+∂ϵλ
+�
+Dλ .
+(19)
+Here, ν = {0, 1, 2}, and Dλ =
+�
+[dk](1 + e/ℏΩλ · B)δ(µ −
+ϵλ) being the density of states corresponding to the FS
+of the band λ. It is evident that both the electric field
+and the temperature gradient components parallel to B
+contribute to generating the system’s chiral chemical po-
+tential and chiral temperature imbalance.
+Having obtained the non-equilibrium distribution func-
+tion, we now calculate the charge and heat current in
+each FS, which are defined as {jλ
+e , jλ
+Q} =
+�
+[dk]{−e, (ϵλ−
+µ)}˙rλgλ. Focusing only on the anomaly induced contri-
+bution ∝ τv, we obtain [13]
+�jλ
+e
+jλ
+Q
+�
+= τvB
+�
+�
+1
+Dλ
+0 (eCλ
+0)2
+ekB
+Dλ
+1
+Dλ
+0 Dλ
+2 Cλ
+0 Cλ
+2
+ekBT
+Dλ
+1
+Dλ
+0 Dλ
+2 Cλ
+0 Cλ
+2
+T
+1
+Dλ
+2 (kBCλ
+2 )2
+�
+�
+×
+�
+E · B
+−∇T · B
+�
+.
+(20)
+In deriving the above equation, we used the fact that in
+the µ ≫ kBT limit (or βµ ≫ 1) limit, Cλ
+1 → 0, and
+Dλ
+0 , Dλ
+2 > Dλ
+1 . Now, the transport coefficients can be ob-
+tained by comparing the total currents
+�
+je,Q = �
+λ jλ
+e,Q
+�
+from Eq. (20) and the phenomenological linear response
+relations [76]: je,a = �
+b[σab Eb − αab ∇bT] and jQ,a =
+�
+b[¯αab Eb − ¯κab ∇bT].
+Here, σ, α, ¯α, and ¯κ de-
+note the electrical, thermo-electric, electro-thermal, and
+constant voltage thermal conductivity matrix, respec-
+tively. Note that the thermo-power matrix is defined as
+Sab = [σ−1α]ab, and the open circuit thermal conductiv-
+ity matrix is expressed as κab = [¯κ − ¯ασ−1α]ab. From
+
+6
+Eq. (20), we see that both the charge and energy cur-
+rents flow along the direction of the magnetic field. This
+is consistent with the fact that these originate from the
+chiral magnetic velocity.
+We calculate the generalized energy density using the
+Sommerfeld approximation in the limit µ ≫ kBT. Re-
+taining only the leading order term in the Sommerfeld
+expansion, we obtain
+Dλ
+ν ≈ m3/2√ϵα
+√
+2π2ℏ3
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(1+λ√1+˜µ)
+2
+√1+˜µ
+F0
+ν = 0,
+˜µ
+2βϵα(1+˜µ)3/2 F2
+ν = 1,
+(1+λ√1+˜µ)
+2
+√1+˜µ
+F2
+ν = 2 .
+(21)
+Here, we have defined the scaled chemical potential,
+˜µ = µ/ϵα. In calculating the above-generalized energy
+densities, we have neglected the magnetic field correc-
+tions, which are very small. Note that i) Dλ
+0 becomes the
+exact density of states in the zero temperature limit for
+the corresponding bands [77], and ii) Dλ
+1 is independent
+of λ i.e., it is identical for both the FSs.
+The chiral anomaly induced transport coefficients (σ,
+α, ¯α, and ¯κ) is obtained from Eq. (20) using the expres-
+sions of Cλ
+ν , and Dλ
+ν . In the µ ≫ kBT limit, for arbitrary
+orientation of the magnetic field, the anomalies induced
+transport coefficients are
+�
+σab αab
+¯αab ¯κab
+�
+=
+τve3B2
+4π2m2α˜µ2 Aab(θ, φ)
+(22)
+×
+�
+� e√1 + ˜µ(2 + ˜µ)
+π2kB
+6βϵα
+(˜µ2+8(1+˜µ))
+˜µ√1+˜µ
+π2
+6β2ϵα
+(˜µ2+8(1+˜µ))
+˜µ√1+˜µ
+π2kB
+3eβ
+√1 + ˜µ(2 + ˜µ)
+�
+� .
+Here, A(θ, φ) is a 3×3 matrix, which captures the angular
+dependence of all the transport coefficients, with (θ, φ)
+denoting the polar, and azimuthal angle of the spheri-
+cal polar coordinate for the magnetic field. The A(θ, φ)
+matrix is obtained to be
+A(θ, φ) =
+�
+�
+sin2 θ cos2 φ
+1
+2 sin2 θ sin 2φ
+1
+2 sin 2θ cos φ
+1
+2 sin2 θ sin 2φ
+sin2 θ sin2 φ
+1
+2 sin 2θ sin φ
+1
+2 sin 2θ cos φ
+1
+2 sin 2θ sin φ
+cos2 θ
+�
+� .
+(23)
+As a consistency check, we note that the longitudinal
+electrical conductivity (σaa) derived above matches with
+that obtained recently in Ref. [47]. The conductivity ma-
+trix of Eq. (22) is valid for the arbitrary direction of the
+applied magnetic field. So, in the planar configuration of
+the magnetic field (θ = π/2), the xy-component of the
+transport coefficients represents various planar Hall ef-
+fects. For instance, the σxy, αxy, ¯αxy, and ¯κxy represents
+the usual planar Hall response, planar Nernst effect, pla-
+nar Ettinghausen effect, and planar Righi-Leduc effects,
+respectively [76]. Hence, our work generalizes the chi-
+ral anomalies induced transport to the thermo-electric,
+and thermal conductivity matrices for spin-orbit coupled
+systems. We emphasize that the chiral anomaly induced
+responses of Eq. (22) become zero for ϵα = 0. This is
+FIG. 2.
+Variation of the chiral anomaly induced electrical
+conductivity with the chemical potential and the spin-orbit
+coupling energy strength. The electrical conductivity is ex-
+pressed in units of σ0 =
+τve4B2
+4
+√
+2π2m3/2ℏ. The anomaly-induced
+response is larger for larger SOC strength and smaller chem-
+ical potential.
+expected because the system’s inversion symmetry is re-
+stored as α → 0, causing the ‘Weyl’ point, related Berry
+curvature, and chiral magnetic velocity to vanish.
+We present the variation of chiral anomaly-induced
+electrical conductivity with µ and ϵα in Fig. 2.
+We
+find that the other conductivity components of Eq. (22)
+also follow a similar qualitative trend in µ and α. The
+anomaly-induced response decreases as µ increases. This
+is consistent with the fact that the chiral anomalies orig-
+inate from the Berry curvature, which peaks in the vicin-
+ity of the band touching points.
+To investigate the impact of the chiral anomaly on
+various longitudinal transport phenomena, we define
+the following generalized magneto-resistance, MRR ≡
+R(B)/R(B = 0)−1. Here, R denotes the different trans-
+port contributions in Eq. (22). In the µ ≫ kBT limit, we
+calculate the Drude conductivities to be
+σD = eτmα
+3ℏ4
+× 2eϵα
+π2 (2 + ˜µ)
+�
+1 + ˜µ ,
+αD = −eτmα
+3ℏ4
+× kB
+3β
+(3˜µ + 4)
+√1 + ˜µ ,
+¯κD = eτmα
+3ℏ4
+× 2ϵα
+eπ2
+π2kB
+3β (2 + ˜µ)
+�
+1 + ˜µ .
+(24)
+In this limit, the longitudinal MR in resistivity is ob-
+tained to be
+MRρ = −
+3τvγ2
+3τvγ2 + 4τ .
+(25)
+Here we have defined, γ =
+eℏ3B
+m2α2 ˜µ.
+The ‘magneto-
+resistance’ in the Seebeck coefficient can be calculated
+
+1.5
+-0.45
+0.30
+1.0-
+0.15
+0.5 -
+30
+10
+20
+407
+to be
+MRS = MRρ
+4(˜µ2 + 3˜µ + 2)
+˜µ(3˜µ + 4)
+.
+(26)
+We note that both of these, MRρ and MRS, show nega-
+tive ‘magneto-resistance’, similar to the band-inversion
+WSM [17].
+However, unlike the case of conventional
+WSM, the relation MRρ/MRS = 1/2 is not satisfied in
+spin-orbit coupled systems.
+In the case of thermo-electric and constant voltage
+thermal conductivity, we find
+MR¯κ = 3τv
+4τ γ2 ,
+(27)
+MRα = −MR¯κ
+˜µ2 + 8(1 + ˜µ)
+˜µ√1 + ˜µ(3˜µ + 4) .
+(28)
+Clearly, MRα is negative while MR¯κ is positive. This is
+similar to the results obtained for WSM in Refs. [17, 78].
+V.
+CHIRAL ANOMALY AND SPIN
+TRANSPORT
+Unlike WSM, where the Pauli matrices in the Hamilto-
+nian represent pseudo-spins, the Pauli matrices in SOC
+systems described by Eq. (1) represent physical spins.
+Consequently, the two bands in SOC systems are spin
+momentum locked with opposite spin orientations on the
+inner and outer FSs [79]. Thus, it is natural to expect
+that chiral anomalies can also influence spin transport
+along with charge transport. Motivated by this, we ex-
+plore the chiral anomalies induced linear spin transport
+(∝ E · B or ∇T · B) in this section. Spin transport in a
+3D SOC system was recently explored in Ref. [79] with-
+out considering the effect of chiral anomaly. In Ref. [47],
+the authors studied electrical chiral anomaly induced lin-
+ear electrical spin current in 3D SOC systems. Here, we
+include the temperature gradient induced spin currents
+and study the chiral anomaly induced spin-Nernst effect,
+in addition to other effects.
+The spin current operator is defined via the anticom-
+mutator relation, ˆJsb
+a
+=
+1
+2{ˆva, ˆsb}, where ˆva is the ve-
+locity operator, ˆsb is the spin operator and a, b denote
+the Cartesian coordinates [80].
+Now, the spin current
+can be calculated as the expectation value of the spin
+current operator weighted by the non-equilibrium distri-
+bution function,
+jsb
+a =
+�
+λ
+�
+[dk]D−1
+λ ⟨uλ(k)| ˆJsb
+a |uλ(k)⟩ gλ .
+(29)
+The matrix of spin transport coefficients is related to the
+spin current via the relation jsb
+a = σsb
+acEc−αsb
+ac∇cT. Here,
+σsb
+ac is the electrical spin conductivity matrix, and αsb
+ac
+is the thermo-electric spin conductivity matrix. These
+tensors represent response coefficients for the spin current
+flowing along the a-direction for spin polarization along
+FIG. 3. The variation of the longitudinal thermoelectric spin
+conductivity with the chemical potential µ and the spin-orbit
+coupling energy strength ϵα. The conductivity αsx
+xk is scaled
+by
+τvekBB
+9
+√
+2ℏ2β√m.
+Similar to the chiral anomaly induced elec-
+trical response, the chiral anomaly induced spin response is
+also larger for larger spin-orbit coupling and smaller chemical
+potential.
+the b-direction, while the electric field or the temperature
+gradient is applied along the c-direction.
+The spin current operator for Hamiltonian (1) is given
+by
+ˆJsb
+a = ℏka
+m σ0 + δab
+α
+ℏ σb ,
+(30)
+where δab = 0 or 1 depending on a ̸= b or a = b, re-
+spectively. Using the eigenstates of Hamiltonian (1), we
+evaluate the expectation value of the above equation to
+be
+⟨uλ| ˆJsb
+a |uλ⟩ = α
+ℏ Iab + λℏk
+m Aab(θk, φk) .
+(31)
+Here, I denotes the 3 × 3 identity matrix, and A(θk, φk)
+is a 3 × 3 matrix defined in Eq. (23). Following the sym-
+metric energy dispersion, the distribution function gλ [see
+Eq. (16)] is independent of θk and φk. As a consequence,
+the angular integration over φk makes all the off-diagonal
+elements of ⟨uλ| ˆJsb
+a |uλ⟩ to be zero, and jsb
+a = 0 for a ̸= b.
+Thus, the spin current is finite only when the spins are
+aligned along the direction of the velocity of the carriers.
+Hence, the chiral anomaly induced spin currents are
+finite only when the spins are polarized along the respec-
+tive directions of current, and we have jsx
+x = jsy
+y = jsz
+z =
+js
+CA. We calculate the spin current induced by the chiral
+anomalies to be [see Appendix D for details]
+js
+CA =τv
+�
+λ
+Cλ
+0
+Dλ
+0
+�Dλ
+1
+Dλ
+2
+L1 − L0
+�
+eE · B
+− Cλ
+2
+Dλ
+2
+�Dλ
+1
+Dλ
+0
+L0 − L1
+�
+kB∇T · B .
+(32)
+
+1.5
+0.170
+1.0-
+-0.125
+0.5
+0.080
+10
+20
+30
+408
+Here, we have defined
+Lν =
+�
+[dk]
+�α
+ℏ + λ ℏ
+mka · ˆk
+� �ϵλ − µ
+kBT
+�ν �
+−∂fλ
+∂ϵλ
+�
+,
+(33)
+with ka = kaˆa being a vector along a-direction with mag-
+nitude equal to the component of k along the a-direction,
+and ˆk = sin θk cos φk ˆx+sin θk sin φk ˆy +cos θk ˆz. We now
+have jsa
+a
+∝ E · B for any arbitrary direction of the ap-
+plied electric field along the k-direction.
+We calculate
+the corresponding chiral anomaly induced electrical spin
+conductivity to be,
+σsx
+xc = σs
+0
+��
+1 + ˜µ −
+π2
+6β2ϵ2α
+(˜µ2 + 9˜µ − 20)
+˜µ2(1 + ˜µ)2
+�
+ˆc· ˆ
+B , (34)
+where we have defined σs
+0 = τve2Bα
+6π2ℏ3 . The second term
+on the right-hand side of Eq. (34) is the finite tempera-
+ture correction to the electrical spin conductivity, which
+vanishes in the T → 0 limit.
+For the thermoelectric part of the spin conductivity,
+we find that it behaves like the electric spin conductiv-
+ity. All the thermoelectric spin currents, where the spin
+is not aligned along the current direction, vanish. We
+obtain, jsb
+a = 0 for b ̸= a, and jsa
+a ∝ ∇T · B. Our calcu-
+lations show that only the conductivity components, αsa
+ac
+are finite, and αsx
+xc = αsy
+yc = αsz
+zc. We calculate the ther-
+moelectric spin conductivity for the temperature gradient
+applied along the c-direction to be,
+αsx
+xc = αs
+0
+� 2
+˜µ2 + ˜µ2 + 3˜µ − 2
+˜µ2√1 + ˜µ − ˜µ2 + 7˜µ + 6
+2˜µ(1 + ˜µ)3/2
+�
+ˆc · ˆ
+B ,
+(35)
+where αs
+0 = τvekBαB
+18ℏ3βϵα . The above expression represents
+the chiral anomaly induced spin-Seebeck (for c = x) or
+the spin-Nernst coefficient (for c ̸= x), with the spins po-
+larized along the x-direction. The variation of αsx
+xk with
+µ and ϵα is presented in Fig. 3. The electrical spin con-
+ductivity also follows similar trends in µ and ϵα. The
+anomaly induced effects in general decrease with increas-
+ing µ and increase with increasing α which is a proxy for
+the degree of inversion symmetry breaking.
+VI.
+CONCLUSION
+In summary, we have provided evidence that quantum
+chiral anomalies can be understood as a feature of FSs.
+Specifically, the chirality of charge carriers can be deter-
+mined by the sign of the Berry curvature quantum pass-
+ing through the associated Fermi surface. This has signif-
+icant implications for 3D SOC metals or Kramers-Weyl
+metals, where chiral charge pumping can occur across
+the two Fermi surfaces associated with a single Kramers-
+Weyl node. To the best of our knowledge, this kind of
+chiral anomaly has no analog in relativistic field theories
+of chiral fermions. We have also demonstrated the exis-
+tence of three distinct types of quantum chiral anomalies
+– electrical, thermal, and gravitational – in 3D SOC met-
+als and Kramers-Weyl metals.
+The effect of these quantum chiral anomalies can be ob-
+served in electrical and thermo-electric charge and spin
+transport in 3D SOC metals and Kramers-Weyl metals.
+While the electrical transport signatures of chiral anoma-
+lies in 3D spin-orbit coupled metals are similar to those in
+Weyl semimetals, the signatures in electrical and thermo-
+electric spin transport are unique to 3D SOC metals. We
+have shown that spin conductivities are finite only when
+spins are polarized along the direction of carrier flow. we
+found that the chiral anomaly-induced spin conductivi-
+ties are proportional to the strength of the magnetic field,
+unlike charge conductivities which scale with the square
+of the magnetic field. Our findings contribute to the un-
+derstanding of chiral anomaly induced charge, heat, and
+spin transport in 3D SOC metals and Kramers-Weyl sys-
+tems.
+ACKNOWLEDGEMENTS
+We acknowledge the Science and Engineering Research
+Board (SERB, via project MTR/2019/001520) for finan-
+cial support. S. D. thanks the MHRD, India for fund-
+ing through the Prime Minister’s Research Fellowship
+(PMRF). We sincerely thank Atasi Chakraborty for the
+useful discussions.
+Appendix A: 3D non-centrosymmetric SOC metals
+and Kramers-Weyl metals
+In this appendix, we discuss the SOC-induced chiral
+anomaly in other 3D systems with different forms of the
+SOC, compared to Eq. (1). Comparing the list of single
+crystalline point groups which support 3D spin-orbit cou-
+pled metals [81] with the list of Kramers Weyl metals [50],
+we find that these are identical. However, 3D electron
+gas with SOC can also arise in some heterostructures of
+two different single crystals. Both of these systems have
+doubly degenerate band touching points, which we refer
+to as ‘Kramers-Weyl’ points. Kramers-Weyl metals are
+realized in structurally chiral crystals that lack mirror,
+inversion, or roto-inversion symmetry [50]. There are 65
+Sohncke chiral space groups corresponding to 11 chiral
+point groups which characterize the structurally chiral
+crystals [53].
+The bands of non-magnetic chiral crystals are at least
+doubly degenerate at the time-reversal-invariant mo-
+menta (TRIM) points due to Kramers theorem [50].
+However, the SOC lifts the Kramer’s degeneracy at all
+other points in the momentum space, leaving behind
+‘Weyl’-like Kramers-Weyl nodes at the TRIM points.
+All these band degenerate points are topologically non-
+trivial, carrying finite Chern numbers [50]. In general,
+the chiral crystals can host multiple band crossings at
+
+9
+the TRIM points in the Brillouin zone along with multi-
+fold band degeneracy [49–51, 53, 57].
+In this paper, we focus on Kramers-Weyl metals
+that have a two-fold degenerate Kramers-Weyl point at
+TRIM. In Table I, we summarize the chiral space groups
+and point groups which support Kramers-Weyl fermions,
+along with some material examples [47, 50, 81, 82]. The
+generic Kramers-Weyl system will have a low energy
+Hamiltonian of the form: H = �
+ab ℏ2kakb/(2mab) +
+hk · σ, in the vicinity of the Kramers-Weyl point for
+which |hk| = 0.
+Here, a, b = x, y, z, mab is the effec-
+tive mass tensor, and k is the momentum with respect
+to the Kramers-Weyl point. The specific form of sym-
+metry allowed hk, for each of the chiral point groups is
+also summarized in Table I. Each of these Kramers-Weyl
+points has a chiral charge with value ±1. For example,
+the Hamiltonian (1) with isotropic SOC term αk · σ can
+be realized in point groups T and O in K2Sn2O3, β-RhSi,
+CoSi crystals [50, 51, 55–58].
+TABLE I. The space groups and the point groups for topologically non-trivial chiral crystals hosting Kramers-Weyl Fermions
+with chiral charge ±1. Some material examples, along with the form of the symmetry-allowed SOC terms in the vicinity of the
+Kramers-Weyl points for each space group are also presented.
+Space group Point group (Laue class)
+Material
+SOC term
+1
+C1(1)
+Li6CuB4O10
+(α1kx + α2ky + α3kz)σx + (α4kx + α5ky + α6kz)σy
++(α7kx + α8ky + α9kz)σz
+3-5
+C2(2)
+Pb3GeO5
+(α1kx + α2ky)σx + (α3kx + α4ky)σy + α5kzσz
+16-24
+D2(222)
+AlPS4
+α1kxσx + α2kyσy + α3kzσz
+143-146
+C3(3)
+β-Ag3IS
+75-80
+C4(4)
+BaCu2Te2O6Cl2
+(α1kx + α2ky)σx + (α1ky − α2kx)σy + α3kzσz
+168-173
+C6(6)
+α-In2Se3
+149-155
+D3(32)
+Ag3BO3
+89-98
+D4(422)
+CdAs2
+α1(kxσx + kyσy) + α2kzσz
+177-182
+D6(622)
+NbGe2
+195-199
+T(23)
+K2Sn2O3, β-RhSi
+α1(kxσx + kyσy + kzσz)
+207-214
+O(432)
+BaSi2, SrSi2
+Appendix B: Berry curvature flux quantum and
+chiral anomaly for negative chemical potential
+In this Appendix, we calculate the Berry curvature
+flux quantum for each Fermi surface and discuss the chi-
+ral anomaly for Fermi energies below the Kramers-Weyl
+node, i.e., µ < 0. We start by calculating the Berry cur-
+vature flux quantum for the FSs. The Berry curvature
+flux through any FS is defined as Cλ =
+1
+2π
+�
+FS dS · Ωλ,
+where dS is the elemental surface area of the FS. Using
+the divergence theorem, and capturing the Fermi surface
+via the Heaviside step function [Θ(µ − ϵλ)], we have
+Cλ= 1
+2π
+�
+dk∇k · ΩλΘ(µ − ϵλ)
+= − 1
+2π
+�
+dk Ωλ · ∇kΘ(µ − ϵλ)
+= ℏ
+2π
+�
+dk Ωλ · vλδ(µ − ϵλ) .
+(B1)
+Note that in the zero-temperature limit, the above ex-
+pression reduces to the electrical chiral anomaly coeffi-
+cient defined in Eq. (10). Below, we explicitly calculate
+the Cλ.
+Case I (µ > 0):— For µ > 0, there are two Fermi
+wave vectors kF
+λ = −λkα +
+�
+k2α + 2mµ/ℏ2 with λ = ±,
+corresponding to two FSs of the two bands.
+The kF
++
+(kF
+−) corresponds to the inner (outer) FS. Now, using
+the expressions of vλ, Ωλ, and the δ-function property,
+Cλ for each band λ becomes
+Cλ= ℏ
+2π
+�
+dk −λ
+2k2
+�ℏk
+m + λα
+ℏ
+�
+δ(µ − ϵλ) ,
+= −λ
+�
+dk
+�ℏ2k
+m + λα
+� δ(kF
+λ − k)
+|ϵ′
+λ|
+.
+(B2)
+Here, ϵ′
+λ is the first derivative of ϵλ with respect to k.
+Evaluating this integral yields Cλ = −λ.
+Case II (µ < 0):— For µ < 0, there is only one car-
+rier pocket, which looks like an annular sphere. It has
+two surfaces, the inner and the outer surfaces. Also, the
+energy dispersion is non-monotonic (see Fig. 4). Conse-
+quently, in the region near the Kramers-Weyl node, the
+band velocity is negative, while in other regions, the band
+velocity is positive. As a result, we have regions within
+the same pocket that have opposite signs of the chiral
+magnetic velocity (∝ vλ · Ωλ).
+This ensures that the
+Berry curvature flux through the entire FS calculated us-
+ing Eq. (B1) is zero. Hence, we expect that there should
+
+10
+not be any chiral anomaly for µ < 0.
+However, in Ref. [47], the authors discussed the chiral
+anomaly for µ < 0 with the idea of partitioning the FS
+into two regions based on the sign of the chiral mag-
+netic velocity.
+Below, we discuss this in detail.
+We
+show the partitioning of the FS in Fig 4, with the blue
+and red regions representing the two different partitions.
+Here, the χ is used as the index for denoting the inner
+(outer) region of the FS, with χ = −1 for the blue region
+(χ = +1 for the red region). To calculate the Berry cur-
+vature flux using Eq. (B1), we first compute the Fermi
+wave vectors corresponding to the two different regions
+of the Fermi pocket of the λ = −1 band.
+The Fermi
+wave vectors corresponding to the inner (χ = −1) and
+outer (χ = −1) boundary of the Fermi pocket is given by
+kF
+χ = kα + χ
+�
+k2α + 2mµ/ℏ2. Recall that kα = mα/ℏ2,
+which corresponds to the minima in the energy of the
+λ = −1 band. The χ = − (+) region of the Fermi pocket
+correspond the kF
+− < k < kα (kα < k < kF
++). These re-
+gions are represented by blue and red colors, respectively,
+in Fig. 4. For λ = −1 band, the Cλ is given by
+Cλ = ℏ
+2π
+�
+dk 1
+2k2
+�ℏk
+m − α
+ℏ
+�
+δ(µ − ϵ−) .
+(B3)
+Now, for either of the two regions, the above equation
+reduces to
+Cχ
+λ =
+�
+dk
+�ℏ2k
+m − α
+� δ(kF
+χ − k)
+|ϵ′
+−|
+.
+(B4)
+As the band velocity, ϵ′
+− = ℏ2k/m − α is negative (pos-
+itive) for the region with kF
+− < k < kα (kα < k < kF
++),
+Eq. (B4) yields Cχ
+λ = χ. We note again that the sign of
+Cχ
+λ is essentially tied to the sign of the chiral magnetic ve-
+locity proportional to the (Ωλ · vλ) term. We emphasize
+that without partition of the FS, Eq. (B3) itself yields
+zero due to the two different roots of the δ-function (kF
++
+and kF
+−). This partitioning of the Brillouin zone, as per
+the sign of the chiral magnetic velocity, allows one to de-
+fine two regions of FS with opposite Berry curvature flux
+quantum. This had been used in Ref. [47] to discuss the
+continuity equation and the associated electrical chiral
+anomaly for µ < 0, on the same footing as we have dis-
+cussed for µ > 0 [47] in the main text. While partitioning
+a single electron pocket to define carriers of different ‘fla-
+vors’ is mathematically appealing, we believe that this
+way of defining the chiral anomaly is superfluous and
+not physical.
+Here, we present a counter-example to establish the
+above claim. Consider a 3D electron gas (without any
+SOC, without any magnetic field), with an electric field
+applied along the x-direction. We can divide the Fermi
+sphere of the system into two halves with positive and
+negative velocities along the x-axis and treat the parti-
+cles with opposite velocities as having different flavors
+(s = ±).
+In the bottom panels (c) and (d) of Fig. 4,
+we have schematically shown this partitioning of the
+FS. The particles in the blue (red) region with s = +1
+a)
+b)
+d)
+c)
+FIG. 4. a) The band dispersion and the Brillouin zone par-
+titioning for the λ = −1 band of a 3D SOC system. For the
+blue-shaded region with negative band velocity, the Berry cur-
+vature flux is −1, while the Berry curvature flux is +1 for the
+red-shaded region with positive band velocity. b) The corre-
+sponding cross-section of the Fermi surface for µ < 0 for the
+λ = −1 band, highlighting the two partitions of the Fermi
+pocket. c) The band dispersion of 3D electron gas without
+SOC. This trivial system can also be partitioned into red and
+blue regions depending on the sign of the x component of the
+band velocity. d) The cross-section of the spherical FS for a
+3D electron gas in the kx − ky plane.
+(s = −1), have positive (negative) velocity. In the pres-
+ence of only an electric field along the x-direction, the
+collisionless Boltzmann equation [Eq. (12) with λ → s
+and Icoll{gλ} = 0], upto first order in the electric field
+strength, becomes
+∂gs
+∂t − eEvs
+x
+∂fs
+∂ϵ = 0 .
+(B5)
+Here, gs (fs) is the non-equilibrium (equilibrium Fermi
+Dirac) distribution function for the s region of the Bril-
+louin zone. Integrating the above equation over all the
+momentum states within the respective partition of the
+FS, we obtain
+∂Ns
+∂t + s eE
+2π3
+�2mµ
+ℏ2
+�3/2
+= 0 .
+(B6)
+The above equation resembles Eq. (4), indicating the pos-
+sibility of a “chiral anomaly” in a normal 3D electron gas.
+However, this cannot be physical and is very unlikely to
+be correct. Due to this, we believe that the partitioning
+of the BZ to divide one electron/hole pocket into mul-
+tiple partitions is not physically acceptable.
+However,
+the partitioning of the BZ to include full electron/hole
+pockets is acceptable, and this forms the basis of valley
+physics in 2D and chiral anomaly related physics in 3D
+systems.
+Having discussed the chiral anomaly for µ < 0, we
+conclude this Appendix with a small discussion on the
+
+11
+chirality of ‘Weyl’-type nodes and the Berry curvature
+flux quantum. For the WSM, the Berry curvature flux
+through the FS of a node represents the ‘chirality’ of that
+node, irrespective of the conduction or the valence band.
+This is easily seen because, in the m → ∞ limit, the
+Hamiltonian in Eq. (1) reduces to the Hamiltonian for
+a single Weyl node HWSM.
+In contrast to the bands
+of Hamiltonian in Eq. (1), both bands of HWSM are
+monotonous (around the nodal point) and only one FS
+exists at any particular energy. Then a straightforward
+calculation following Eq. (B2) yields, Cλ = −sign(α) for
+both the conduction and valence band of HWSM. Because
+the Cλ depends on the sign of α, the Berry curvature flux
+quantum becomes opposite for opposite chirality nodes
+where α has the opposite sign.
+This establishes that
+for WSM, the chirality of each Weyl node can be rep-
+resented as the Berry curvature flux quantum through
+the node [5, 17, 61, 83]. However, for the Kramers-Weyl
+nodes, the Berry curvature flux quantum and the chiral-
+ity of the node are not identical.
+The chirality of the
+Kramers-Weyl nodes depends on the sign of α for Hamil-
+tonian (1), which is specific to a given TRIM point of the
+material [50].
+Appendix C: Calculation of equilibrium currents
+In this Appendix, we derive the expressions of the equi-
+librium currents obtained in Eqs. (8a) and (8b). In the
+presence of only a magnetic field, the velocity of the cen-
+ter of mass of the wave packets for the carriers in each
+band is given by ˙rλ = Dλ
+�
+vλ + e
+ℏ(vλ · Ωλ)B
+�
+. The equi-
+librium charge current for the FS λ (corresponding to
+each band) is given by
+jλ
+e,eq = −e
+�
+[dk]˙rfλ = −eB
+�
+[dk] e
+ℏ(vλ · Ωλ)fλ . (C1)
+Here, we have used the fact that the band velocity vλ
+does not contribute to the equilibrium current (due to
+angular integration being zero). Now, we use the identity
+∇k ·(ϵλΩλ) = ∇kϵλ·Ωλ+ϵλ∇k ·Ωλ to express the above
+equation as,
+jλ
+e,eq= −e2B
+ℏ2
+�
+[dk] [∇k · (ϵλΩλ) − ϵλ∇k · Ωλ] fλ (C2)
+= −e2B
+ℏ2
+�
+[dk]∇k · (ϵλΩλ)fλ
+(C3)
+= e2
+ℏ2 B
+�
+[dk]ϵλΩλ · ˆk∂fλ
+∂k ,
+(C4)
+= −eB
+�
+[dk] (µ + ϵλ − µ) e
+ℏ(vλ · Ωλ)
+�
+−∂fλ
+∂ϵλ
+�
+,
+= −e
+�
+µCλ
+0 + kBTCλ
+1
+�
+B.
+(C5)
+To evaluate Eq. (C3), we have used the fact that ∇k ·
+Ωλ = ±2πδ3(k), for a system with doubly degenerate
+band touching point with linear dispersion. This makes
+the last integral of Eq. (C2) to be zero.
+To obtain
+Eq. (C4) from Eq. (C3), we have used integrations by
+parts. Here, we have defined Cλ
+ν as,
+Cλ
+ν =
+�
+[dk] e
+ℏvλ · Ωλ
+�ϵ − µ
+kBT
+�ν �
+−∂fλ
+∂ϵλ
+�
+.
+(C6)
+These can also be rewritten in terms of Cλ given in
+Eq. (10).
+The energy current, jλ
+ϵ,eq, can be evaluated
+in a similar manner.
+Appendix D: Details of spin current calculations
+To calculate the spin current proportional to the E ·B
+(or ∇T · B), we consider the band velocity term of
+Eq. (6a) and calculate the spin current operator. The
+band velocity operator along the i-direction is given by
+ˆvi = ℏki
+m σ0 + α
+ℏ σi. Without loss of generality, here we
+show the calculation of spin current in the x-direction.
+Using the expressions of the eigenstates and the spin
+current operator given in the main text, we obtain
+⟨uλ| ˆJsx
+x |uλ⟩ = (α/ℏ + λℏkx sin θk cos φk/m). Now, the
+chiral anomaly induced spin current is given by
+jsx
+x = τv
+�
+λ
+�
+[dk]
+�α
+ℏ + λℏkx
+m sin θk cos φk
+� �
+δµλ + ϵλ − µ
+T
+δTλ
+� �
+−∂fλ
+∂ϵλ
+�
+.
+(D1)
+In the βµ → ∞ limit, writing the expressions of δµλ and δTλ explicitly, we have
+jsx
+x =τv
+�
+λ
+� Dλ
+1 Cλ
+0
+Dλ
+2 Dλ
+0
+L1 − Cλ
+0
+Dλ
+0
+L0
+�
+eE · B −
+� Dλ
+1 Cλ
+2
+Dλ
+2 Dλ
+0
+L0 − Cλ
+2
+Dλ
+2
+L1
+�
+kB∇T · B.
+(D2)
+The definition of Lν is given in the main text. We eval-
+uate the Lν using the Sommerfeld approximation in the
+µ ≫ kBT limit. We obtain the following expressions
+L0= −λ m2α2
+6π2ℏ5
+[˜µ − ˜µ2 + 2(1 + ˜µ)]
+1 + ˜µ
+,
+(D3)
+L1= kBT
+9ℏ3
+(−λ + √1 + ˜µ)[λ(2 + ˜µ) + √1 + ˜µ]
+(1 + ˜µ)3/2
+. (D4)
+
+12
+Using these expression along with Cλ
+ν and Dλ
+ν in Eq. (D2),
+we obtain the spin conductivities of Eqs. (34) and (35).
+Following a similar procedure, we can calculate other spin
+currents.
+We show that due to rotational symmetry jsj
+i
+= 0
+for i ̸= j. Without loss of generality, we will explicitly
+show the calculation for jsz
+x . The expectation value of
+the spin current operator ˆJsz
+x
+is given by ⟨uλ| ˆJsz
+x |uλ⟩ =
+λ p
+2m sin 2θk cos φk. Now, as the distribution function is
+independent of θk and φk, so the angular integration over
+φk of the ⟨uλ| ˆJsz
+x |uλ⟩ yields jsz
+x
+= 0. Similarly, all the
+spin currents with spin polarization perpendicular to the
+propagation velocity can be easily shown to be zero due
+to the vanishing angular integration over φk.
+[1] S. L. Adler, Axial-vector vertex in spinor electrodynam-
+ics, Phys. Rev. 177, 2426 (1969).
+[2] H. Nielsen and M. Ninomiya, Absence of neutrinos on a
+lattice: (i). proof by homotopy theory, Nuclear Physics
+B 185, 20 (1981).
+[3] H. Nielsen and M. Ninomiya, Absence of neutrinos on a
+lattice: (ii). intuitive topological proof, Nuclear Physics
+B 193, 173 (1981).
+[4] H. Nielsen and M. Ninomiya, The adler-bell-jackiw
+anomaly and weyl fermions in a crystal, Physics Letters
+B 130, 389 (1983).
+[5] D. T. Son and B. Z. Spivak, Chiral anomaly and classical
+negative magnetoresistance of weyl metals, Phys. Rev. B
+88, 104412 (2013).
+[6] J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan,
+M. Hirschberger, W. Wang, R. J. Cava, and N. P. Ong,
+Evidence for the chiral anomaly in the dirac semimetal
+na3bi, Science 350, 413 (2015).
+[7] A. A. Burkov, Giant planar hall effect in topological met-
+als, Phys. Rev. B 96, 041110 (2017).
+[8] C.-L. Zhang, S.-Y. Xu, I. Belopolski, Z. Yuan, Z. Lin,
+B. Tong, G. Bian, N. Alidoust, C.-C. Lee, S.-M. Huang,
+T.-R. Chang, G. Chang, C.-H. Hsu, H.-T. Jeng, M. Ne-
+upane, D. S. Sanchez, H. Zheng, J. Wang, H. Lin,
+C. Zhang, H.-Z. Lu, S.-Q. Shen, T. Neupert, M. Za-
+hid Hasan, and S. Jia, Signatures of the adler–bell–jackiw
+chiral anomaly in a weyl fermion semimetal, Nature Com-
+munications 7, 10735 (2016).
+[9] Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosi´c,
+A. V. Fedorov, R. D. Zhong, J. A. Schneeloch, G. D.
+Gu, and T. Valla, Chiral magnetic effect in zrte5, Nature
+Physics 12, 550 (2016).
+[10] S. Nandy, G. Sharma, A. Taraphder, and S. Tewari, Chi-
+ral anomaly as the origin of the planar hall effect in weyl
+semimetals, Phys. Rev. Lett. 119, 176804 (2017).
+[11] N. Kumar, S. N. Guin, C. Felser, and C. Shekhar, Planar
+hall effect in the weyl semimetal gdptbi, Phys. Rev. B
+98, 041103 (2018).
+[12] K.-S. Kim, H.-J. Kim, and M. Sasaki, Boltzmann equa-
+tion approach to anomalous transport in a weyl metal,
+Phys. Rev. B 89, 195137 (2014).
+[13] K. Das, S. K. Singh, and A. Agarwal, Chiral anomalies
+induced transport in weyl metals in quantizing magnetic
+field, Phys. Rev. Research 2, 033511 (2020).
+[14] K. Das and A. Agarwal, Linear magnetochiral transport
+in tilted type-i and type-ii weyl semimetals, Phys. Rev.
+B 99, 085405 (2019).
+[15] K. Das and A. Agarwal, Berry curvature induced ther-
+mopower in type-i and type-ii weyl semimetals, Phys.
+Rev. B 100, 085406 (2019).
+[16] K. Das and A. Agarwal, Intrinsic hall conductivities in-
+duced by the orbital magnetic moment, Phys. Rev. B
+103, 125432 (2021).
+[17] K. Das and A. Agarwal, Thermal and gravitational chiral
+anomaly induced magneto-transport in weyl semimetals,
+Phys. Rev. Research 2, 013088 (2020).
+[18] D. Mandal, K. Das, and A. Agarwal, Chiral anomaly and
+nonlinear magnetotransport in time reversal symmetric
+weyl semimetals, Phys. Rev. B 106, 035423 (2022).
+[19] F. H¨utt, D. Kamenskyi, D. Neubauer, C. Shekhar,
+C. Felser, M. Dressel, and A. V. Pronin, Terahertz trans-
+mission through taas single crystals in simultaneously ap-
+plied magnetic and electric fields: Possible optical signa-
+tures of the chiral anomaly in a weyl semimetal, Results
+in Physics 15, 102630 (2019).
+[20] P. Hosur and X.-L. Qi, Tunable circular dichroism due
+to the chiral anomaly in weyl semimetals, Phys. Rev. B
+91, 081106 (2015).
+[21] P. E. C. Ashby and J. P. Carbotte, Chiral anomaly and
+optical absorption in weyl semimetals, Phys. Rev. B 89,
+245121 (2014).
+[22] J. Ma and D. A. Pesin, Chiral magnetic effect and natural
+optical activity in metals with or without weyl points,
+Phys. Rev. B 92, 235205 (2015).
+[23] T. Morimoto and N. Nagaosa, Chiral anomaly and giant
+magnetochiral anisotropy in noncentrosymmetric weyl
+semimetals, Phys. Rev. Lett. 117, 146603 (2016).
+[24] M. M. Jadidi, M. Kargarian, M. Mittendorff, Y. Aytac,
+B. Shen, J. C. K¨onig-Otto, S. Winnerl, N. Ni, A. L.
+Gaeta, T. E. Murphy, and H. D. Drew, Nonlinear op-
+tical control of chiral charge pumping in a topological
+weyl semimetal, Phys. Rev. B 102, 245123 (2020).
+[25] A. Thakur, K. Sadhukhan, and A. Agarwal, Dynamic
+current-current susceptibility in three-dimensional dirac
+and weyl semimetals, Phys. Rev. B 97, 035403 (2018).
+[26] K. Sonowal, A. Singh, and A. Agarwal, Giant optical ac-
+tivity and kerr effect in type-i and type-ii weyl semimet-
+als, Phys. Rev. B 100, 085436 (2019).
+[27] K. Landsteiner, E. Meg´ıas, and F. Pena-Benitez, Gravi-
+tational anomaly and transport phenomena, Phys. Rev.
+Lett. 107, 021601 (2011).
+[28] A. Lucas, R. A. Davison, and S. Sachdev, Hydrodynamic
+theory of thermoelectric transport and negative magne-
+toresistance in weyl semimetals, Proceedings of the Na-
+tional Academy of Sciences 113, 9463 (2016).
+[29] J. Gooth, A. C. Niemann, T. Meng, A. G. Grushin,
+K. Landsteiner, B. Gotsmann, F. Menges, M. Schmidt,
+C. Shekhar, V. S¨uß, R. H¨uhne, B. Rellinghaus, C. Felser,
+B. Yan, and K. Nielsch, Experimental signatures of the
+mixed axial–gravitational anomaly in the weyl semimetal
+nbp, Nature 547, 324 (2017).
+[30] M. Stone and J. Kim, Mixed anomalies: Chiral vortical
+effect and the sommerfeld expansion, Phys. Rev. D 98,
+025012 (2018).
+
+13
+[31] R. Lundgren, P. Laurell, and G. A. Fiete, Thermoelectric
+properties of weyl and dirac semimetals, Phys. Rev. B 90,
+165115 (2014).
+[32] M. Hirschberger, S. Kushwaha, Z. Wang, Q. Gibson,
+S. Liang, C. Belvin, B. A. Bernevig, R. J. Cava, and
+N. P. Ong, The chiral anomaly and thermopower of weyl
+fermions in the half-heusler gdptbi, Nature Materials 15,
+1161 (2016).
+[33] Z. Jia, C. Li, X. Li, J. Shi, Z. Liao, D. Yu, and X. Wu,
+Thermoelectric signature of the chiral anomaly in cd3as2,
+Nature Communications 7, 13013 (2016).
+[34] B. Z. Spivak and A. V. Andreev, Magnetotransport phe-
+nomena related to the chiral anomaly in weyl semimetals,
+Phys. Rev. B 93, 085107 (2016).
+[35] U. Stockert, R. D. dos Reis, M. O. Ajeesh, S. J. Watz-
+man, M. Schmidt, C. Shekhar, J. P. Heremans, C. Felser,
+M. Baenitz, and M. Nicklas, Thermopower and ther-
+mal conductivity in the weyl semimetal nbp, Journal of
+Physics: Condensed Matter 29, 325701 (2017).
+[36] V. A. Zyuzin, Magnetotransport of weyl semimetals due
+to the chiral anomaly, Phys. Rev. B 95, 245128 (2017).
+[37] D. Vu, W. Zhang, C. Sahin, M. E. Flatte, N. Trivedi, and
+J. P. Heremans, Thermal chiral anomaly in the magnetic-
+field-induced ideal weyl phase of bi1-xsbx, Nature Mate-
+rials 20, 1525 (2021).
+[38] D. T. Son and N. Yamamoto, Berry curvature, triangle
+anomalies, and the chiral magnetic effect in fermi liquids,
+Phys. Rev. Lett. 109, 181602 (2012).
+[39] D. T. Son and N. Yamamoto, Kinetic theory with berry
+curvature from quantum field theories, Phys. Rev. D 87,
+085016 (2013).
+[40] M. A. Stephanov and Y. Yin, Chiral kinetic theory, Phys.
+Rev. Lett. 109, 162001 (2012).
+[41] C. Fang, M. J. Gilbert, X. Dai, and B. A. Bernevig,
+Multi-weyl topological semimetals stabilized by point
+group symmetry, Phys. Rev. Lett. 108, 266802 (2012).
+[42] X. Li, B. Roy, and S. Das Sarma, Weyl fermions with
+arbitrary monopoles in magnetic fields: Landau levels,
+longitudinal magnetotransport, and density-wave order-
+ing, Phys. Rev. B 94, 195144 (2016).
+[43] Z.-M. Huang, J. Zhou, and S.-Q. Shen, Topological re-
+sponses from chiral anomaly in multi-weyl semimetals,
+Phys. Rev. B 96, 085201 (2017).
+[44] R.
+M.
+A.
+Dantas,
+F.
+Pe˜na-Benitez,
+B.
+Roy,
+and
+P. Sur´owka, Magnetotransport in multi-weyl semimet-
+als: a kinetic theory approach, Journal of High Energy
+Physics 2018, 69 (2018).
+[45] S. Das, K. Das, and A. Agarwal, Nonlinear magnetocon-
+ductivity in weyl and multi-weyl semimetals in quantiz-
+ing magnetic field, Phys. Rev. B 105, 235408 (2022).
+[46] L. Lepori,
+M. Burrello, and E. Guadagnini, Axial
+anomaly in multi-weyl and triple-point semimetals, Jour-
+nal of High Energy Physics 2018, 110 (2018).
+[47] S. Cheon, G. Y. Cho, K.-S. Kim, and H.-W. Lee, Chi-
+ral anomaly in noncentrosymmetric systems induced by
+spin-orbit coupling, Phys. Rev. B 105, L180303 (2022).
+[48] L.-L. Gao and X.-G. Huang, Chiral anomaly in non-
+relativistic systems: Berry curvature and chiral kinetic
+theory, Chinese Physics Letters 39, 021101 (2022).
+[49] B. Bradlyn,
+J. Cano,
+Z. Wang,
+M. G. Vergniory,
+C. Felser, R. J. Cava, and B. A. Bernevig, Beyond dirac
+and weyl fermions: Unconventional quasiparticles in con-
+ventional crystals, Science 353, aaf5037 (2016).
+[50] G. Chang, B. J. Wieder, F. Schindler, D. S. Sanchez,
+I. Belopolski, S.-M. Huang, B. Singh, D. Wu, T.-R.
+Chang, T. Neupert, S.-Y. Xu, H. Lin, and M. Z. Hasan,
+Topological quantum properties of chiral crystals, Nature
+Materials 17, 978 (2018).
+[51] W.-Y. He, X. Y. Xu, and K. T. Law, Kramers weyl
+semimetals as quantum solenoids and their applications
+in spin-orbit torque devices, Communications Physics 4,
+66 (2021).
+[52] C.-L. Zhang, F. Schindler, H. Liu, T.-R. Chang, S.-Y. Xu,
+G. Chang, W. Hua, H. Jiang, Z. Yuan, J. Sun, H.-T. Jeng,
+H.-Z. Lu, H. Lin, M. Z. Hasan, X. C. Xie, T. Neupert, and
+S. Jia, Ultraquantum magnetoresistance in the kramers-
+weyl semimetal candidate β−ag2Se, Phys. Rev. B 96,
+165148 (2017).
+[53] N. B. M. Schr¨oter, D. Pei, M. G. Vergniory, Y. Sun,
+K.
+Manna,
+F.
+de
+Juan,
+J.
+A.
+Krieger,
+V.
+S¨uss,
+M.
+Schmidt,
+P.
+Dudin,
+B.
+Bradlyn,
+T.
+K.
+Kim,
+T. Schmitt, C. Cacho, C. Felser, V. N. Strocov, and
+Y. Chen, Chiral topological semimetal with multifold
+band crossings and long fermi arcs, Nature Physics 15,
+759 (2019).
+[54] W. Tan, X. Jiang, Y. Li, X. Wu, J. Wang, and B. Huang,
+A unified understanding of diverse spin textures of
+kramers–weyl fermions in nonmagnetic chiral crystals,
+Advanced Functional Materials 32, 2208023 (2022).
+[55] Z. Rao, H. Li, T. Zhang, S. Tian, C. Li, B. Fu, C. Tang,
+L. Wang, Z. Li, W. Fan, J. Li, Y. Huang, Z. Liu, Y. Long,
+C. Fang, H. Weng, Y. Shi, H. Lei, Y. Sun, T. Qian, and
+H. Ding, Observation of unconventional chiral fermions
+with long fermi arcs in cosi, Nature 567, 496 (2019).
+[56] D. S. Sanchez, I. Belopolski, T. A. Cochran, X. Xu, J.-
+X. Yin, G. Chang, W. Xie, K. Manna, V. S¨uß, C.-Y.
+Huang, N. Alidoust, D. Multer, S. S. Zhang, N. Shumiya,
+X. Wang, G.-Q. Wang, T.-R. Chang, C. Felser, S.-Y.
+Xu, S. Jia, H. Lin, and M. Z. Hasan, Topological chiral
+crystals with helicoid-arc quantum states, Nature 567,
+500 (2019).
+[57] D. Dutta, B. Ghosh, B. Singh, H. Lin, A. Politano,
+A. Bansil, and A. Agarwal, Collective plasmonic modes
+in the chiral multifold fermionic material cosi, Phys. Rev.
+B 105, 165104 (2022).
+[58] D. Takane, Z. Wang, S. Souma, K. Nakayama, T. Naka-
+mura, H. Oinuma, Y. Nakata, H. Iwasawa, C. Cacho,
+T. Kim, K. Horiba, H. Kumigashira, T. Takahashi,
+Y. Ando, and T. Sato, Observation of chiral fermions
+with a large topological charge and associated fermi-
+arc surface states in cosi, Phys. Rev. Lett. 122, 076402
+(2019).
+[59] S. Verma, T. Biswas, and T. K. Ghosh, Thermoelectric
+and optical probes for a fermi surface topology change in
+noncentrosymmetric metals, Phys. Rev. B 100, 045201
+(2019).
+[60] O. Pal, B. Dey, and T. K. Ghosh, Berry curvature in-
+duced magnetotransport in 3d noncentrosymmetric met-
+als, Journal of Physics: Condensed Matter 34, 025702
+(2021).
+[61] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl
+and dirac semimetals in three-dimensional solids, Rev.
+Mod. Phys. 90, 015001 (2018).
+[62] J. Kang and J. Zang, Transport theory of metallic b20
+helimagnets, Phys. Rev. B 91, 134401 (2015).
+[63] K. V. Samokhin, Effects of impurities on the upper crit-
+ical field Hc2 in superconductors without inversion sym-
+metry, Phys. Rev. B 78, 144511 (2008).
+
+14
+[64] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on
+electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
+[65] T. Morimoto, S. Zhong, J. Orenstein, and J. E. Moore,
+Semiclassical theory of nonlinear magneto-optical re-
+sponses
+with
+applications
+to
+topological
+dirac/weyl
+semimetals, Phys. Rev. B 94, 245121 (2016).
+[66] D. Xiao, J. Shi, and Q. Niu, Berry phase correction to
+electron density of states in solids, Phys. Rev. Lett. 95,
+137204 (2005).
+[67] J. Ma and D. A. Pesin, Chiral magnetic effect and natural
+optical activity in metals with or without weyl points,
+Phys. Rev. B 92, 235205 (2015).
+[68] K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Chi-
+ral magnetic effect, Phys. Rev. D 78, 074033 (2008).
+[69] Q. Li and D. E. Kharzeev, Chiral magnetic effect in
+condensed matter systems, Nuclear Physics A 956, 107
+(2016).
+[70] D. E. Kharzeev, The chiral magnetic effect and anomaly-
+induced transport, Progress in Particle and Nuclear
+Physics 75, 133 (2014).
+[71] D. E. Kharzeev, Y. Kikuchi, and R. Meyer, Chiral mag-
+netic effect without chirality source in asymmetric weyl
+semimetals, The European Physical Journal B 91, 83
+(2018).
+[72] S. K. Yip, Kinetic equation and magneto-conductance for
+weyl metal in the clean limit (2015).
+[73] M.-X. Deng, G. Y. Qi, R. Ma, R. Shen, R.-Q. Wang,
+L. Sheng, and D. Y. Xing, Quantum oscillations of the
+positive longitudinal magnetoconductivity: A fingerprint
+for identifying weyl semimetals, Phys. Rev. Lett. 122,
+036601 (2019).
+[74] A. A. Burkov, Negative longitudinal magnetoresistance
+in dirac and weyl metals, Phys. Rev. B 91, 245157 (2015).
+[75] J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan,
+M. Hirschberger, W. Wang, R. J. Cava, and N. P. Ong,
+Evidence for the chiral anomaly in the dirac semimetal
+na3bi, Science 350, 413 (2015).
+[76] N. Ashcroft and N. Mermin, Solid State Physics, HRW
+international editions (Holt,
+Rinehart and Winston,
+1976).
+[77] S. Verma, A. Kundu, and T. K. Ghosh, Dynamical po-
+larization and plasmons in noncentrosymmetric metals,
+Phys. Rev. B 102, 195208 (2020).
+[78] C. Schindler, S. Galeski, W. Schnelle, R. Wawrzy´nczak,
+W. Abdel-Haq, S. N. Guin, J. Kroder, N. Kumar, C. Fu,
+H. Borrmann, C. Shekhar, C. Felser, T. Meng, A. G.
+Grushin, Y. Zhang, Y. Sun, and J. Gooth, Anisotropic
+electrical and thermal magnetotransport in the magnetic
+semimetal gdptbi, Phys. Rev. B 101, 125119 (2020).
+[79] P. Kapri, B. Dey, and T. K. Ghosh, Role of berry curva-
+ture in the generation of spin currents in rashba systems,
+Phys. Rev. B 103, 165401 (2021).
+[80] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,
+and T. Jungwirth, Spin hall effects, Rev. Mod. Phys. 87,
+1213 (2015).
+[81] K. Samokhin, Spin–orbit coupling and semiclassical elec-
+tron dynamics in noncentrosymmetric metals, Annals of
+Physics 324, 2385 (2009).
+[82] G. H. Fecher, J. K¨ubler, and C. Felser, Chirality in the
+solid state: Chiral crystal structures in chiral and achiral
+space groups, Materials 15, 5812 (2022).
+[83] M. J. Park, S. Cheon, and H.-W. Lee, Nondivergent chiral
+charge pumping in weyl semimetals, Phys. Rev. B 106,
+075140 (2022).
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf,len=1320
+page_content='Chiral Anomalies in 3D Spin-Orbit Coupled Metals: Electrical, Thermal, and  Gravitational anomaly  Sunit Das,∗ Kamal Das,† and Amit Agarwal‡  Department of Physics, Indian Institute of Technology Kanpur, Kanpur-208016, India  The discovery of a chiral anomaly in Weyl semimetals, the non-conservation of chiral charge and  energy across two opposite chirality Weyl nodes, has sparked immense interest in understanding  its impact on various physical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Here, we demonstrate the existence of electrical, ther-  mal, and gravitational quantum chiral anomalies in 3D spin-orbit coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Notably, these  anomalies involve chiral charge transfer across two Fermi surfaces linked to a single Weyl-like point,  rather than across opposite chirality Weyl nodes as in Weyl semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Our findings reveal that the  Berry curvature flux piercing the Fermi surface plays a critical role in distinguishing the ‘chirality’ of  the carriers and the corresponding chiral charge and energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Importantly, we demonstrate  that these quantum chiral anomalies lead to interesting thermal spin transport such as the spin  Nernst effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Our results suggest that 3D spin-orbit coupled metals offer a promising platform  for investigating the interplay between quantum chiral anomalies and charge and spin transport in  non-relativistic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  INTRODUCTION  Chiral anomaly refers to the non-conservation of chi-  ral charges in the presence of collinear electric and mag-  netic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' It was first introduced in the context of the  relativistic field theory of chiral fermions [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Later  it was shown to be achievable in low gap semiconduc-  tors [4], with signatures in magnetoconductance exper-  iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Following the discovery of Weyl semimetals  (WSMs) in recent years, the physics of chiral anomaly  has been widely studied in condensed matter systems,  resulting in a variety of non-trivial transport [5–18] and  optical [19–26] effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Intriguingly, the presence of a  temperature gradient in Weyl systems can also result in  an anomaly similar to the axial-gravitational anomaly in  flat-space time [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This leads to a range of interest-  ing magneto-thermal transport phenomena [12, 31–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Central to the physics of chiral anomaly is the con-  tinuity equation for the chiral charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The continuity  equation for the chiral charges and energy can be de-  rived using semiclassical dynamics in crystalline materi-  als and shows that the Berry curvature monopoles govern  the chiral anomaly in Weyl metals [38–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The concept  of chiral anomaly has also been extended to other free  fermionic excitations with no high-energy analog, such as  multi-Weyl semimetals [41–45], which exhibit two band  crossings similar to WSMs but with nonlinear momentum  dispersion along a particular direction, and semimetals  with a higher number of band crossings near the Weyl  node [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' These systems, while possessing a higher chi-  ral charge, are otherwise similar to Weyl systems in that  a theory of chiral anomaly requires the presence of two  opposite chirality Weyl nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  In this paper, we delve into the connection between  ∗ sunitd@iitk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='in  † kamaldas@iitk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='in  ‡ amitag@iitk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='in  a)  b)  Weyl metals  3D Spin orbit coupled metals  a)  b)  9  Here, we have used the fact that the band velocity v�  does not contribute to the equilibrium current (due to  angular integration being zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Now, we use the identity  rk · (✏�⌦�) = rk✏� · ⌦� + ✏�rk · ⌦� to write the above  equation as follows,  j�  e,eq  = �e2B  ~2  Z  [dk] [rk · (✏�⌦�) � ✏�rk · ⌦�] f �  � ,(C3)  = �e2B  ~2  Z  [dk]rk · (✏�⌦�)f �  �  (C4)  = e2  ~2 B  Z  [dk]✏�⌦� · ˆk@f �  �  @k ,  (C5)  = �eB  Z  [dk] (µ + ✏� � µ) e  ~(v� · ⌦�)  ✓  �@f �  �  @✏�  ◆  ,  = �e (µC�  0 + kBTC�  1 ) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (C6)  To evaluate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (C4), we have used the fact that rk ·  ⌦� = ±2⇡�3(k), for a system with band touching point,  which makes the last integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (C3) to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Also,  we have used property of partial integrations to obtain  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (C5) from the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Here, we have defined C�  ⌫ as  follows  C�  ⌫ =  Z  [dk]  ✓✏ � µ  kBT  ◆⌫ ✓  �@f �  �  @✏�  ◆  ,  (C7)  which can also be rewritten in terms of C� given in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The j�  ✏,eq can be evaluated following similar  calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' One remark is in order, although the num-  ber density of carriers n�=�1 contains the Fermi func-  tion for both the bands, the equilibrium (also the non-  equilibrium) currents for � = �1 only come from the  � = +1 (� = �1) for µ > 0 (µ < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This is because the  currents we obtain are the Fermi surface property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (other important aspects to be put in the main text)  Appendix D: Details of spin current calculations  To calculate the spin current proportional to the E ·  B (or rT · B), we consider the band velocity term of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (4) and calculate the spin current operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The band  velocity operator along the i-direction is given by ˆvi =  ~ki  m �0 + ↵  ~ �i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Without loss of generality, here we show  the calculation of spin current in the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Using  the expressions of the eigenstates and the spin current  operator given in the main text, we obtain hu�| ˆJsx  x |u�i =  (↵/~ + �~kx sin ✓k cos �k/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Now, the chiral anomaly  induced spin current is given by  jsx  x  = ⌧v  X  �  Z  [dk]  ✓↵  ~ + �~kx  m sin ✓k cos �k  ◆ ✓  �µ� + ✏� � µ  T  �T �  ◆ ✓  �@f �  �  @✏�  ◆  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (D1)  In the �µ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 1 limit, writing the expressions of �µ� and �T � explicitly, we have  jsx  x =  ⌧v  X  �  \uf8ff D�  1 C�  0  D�  2 D�  0  L1 � C�  0  D�  0  L0  �  eE · B �  \uf8ff D�  1 C�  2  D�  2 D�  0  L0 � C�  2  D�  2  L1  �  kBrT · B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (D2)  The definition of L⌫ is given in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We eval-  uate the L⌫ using the Sommerfeld approximation in the  µ > kBT limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We obtain the following expressions  L0  = �� m2↵2  6⇡2~5  [˜µ � ˜µ2 + 2(1 + ˜µ)]  1 + ˜µ  ,  (D3)  L1  = kBT  9~3  (�� + p1 + ˜µ)[�(2 + ˜µ) + p1 + ˜µ]  (1 + ˜µ)3/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (D4)  Using these expression along with C�  ⌫ and D�  ⌫ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (D2),  we obtain the spin conductivities of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (32) and (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Following a similar procedure, we can calculate other spin  currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Now, we show that due to rotational symmetry jsj  i  = 0  for i 6= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Without loss of generality, we will explicitly  show the calculation for jsz  x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The expectation value of  the spin current operator ˆJsz  x  is given by hu�| ˆJsz  x |u�i =  � p  2m sin 2✓k cos �k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Now, as the distribution function is  independent of ✓k and �k, so the angular integration over  �k of the hu�| ˆJsz  x |u�i yields jsz  x  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Similarly, all the  spin currents with spin polarization perpendicular to the  propagation velocity can be easily shown to be zero due  to the vanishing angular integration over �k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  E · B 6= 0 or rT · B 6= 0  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xiao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Chang, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Niu, Berry phase e↵ects on  electronic properties, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 82, 1959 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Gao, Semiclassical dynamics and nonlinear charge cur-  a)  b)  a)  b)  a)  b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Depiction of the quantum chiral anomalies in (a)  Weyl semi-metals and (b) 3D spin-orbit coupled metals or  Kramers-Weyl metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Both systems experience chiral charge  and energy pumping, manifesting as electrical, thermal, and  gravitational anomalies, when subjected to a magnetic field  and collinear electric field (E · B ̸= 0) or a temperature gra-  dient (∇T · B ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  In contrast to Weyl semimetals, the  chiral charge pumping in 3D spin-orbit coupled metals occurs  between two different Fermi surfaces associated with a single  ‘Kramers-Weyl’ node, but with opposite Berry curvature flux  passing through them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  chiral anomalies and the Berry curvature flux passing  through the Fermi surface (FS) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This connection was  recently explored in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Motivated by this,  we generalize the theory of quantum chiral anomalies  to Hamiltonians with non-relativistic terms, specifically  H = hk · σ + σ0k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Here, the σ represents the real spin  of the system, σ0 is the identity matrix, and hk is an odd  function of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The quadratic kinetic energy-like term in  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='02965v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='mes-hall]  8 Jan 2023  2  the Hamiltonian makes the chiral anomaly in this spin-  orbit coupled (SOC) metals to be distinctly different from  that in WSM [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' These types of systems can be  found in Kramers-Weyl metals with quadratic corrections  to their k·p Hamiltonians [49–58] or in systems support-  ing 3D electron gas with SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' While some aspects of  the charge, heat, and spin transport in SOC metals have  been explored earlier [47, 59, 60], the physics of quantum  chiral anomalies in these systems is largely unexplored  and merits further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  In this paper, we demonstrate that Kramers-Weyl and  spin-orbit coupled metals can exhibit all three types of  quantum chiral anomalies—electrical, thermal, and grav-  itational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We investigate the impact of electric field and  temperature gradient-induced quantum chiral anomalies  on charge, heat, and spin transport phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Similar  to the behavior observed in Weyl semi-metals [17], we  find that chiral anomalies in 3D SOC systems also result  in negative longitudinal magneto-resistance and positive  thermal magneto-resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' However, a distinct feature  of 3D SOC systems, as compared to WSMs, is that their  low-energy Hamiltonian involves real spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  We show  that quantum chiral anomalies in these systems also lead  to interesting electrical and thermal spin transport in-  cluding the spin-Nernst effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The structure of the rest of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In  Section II, we discuss the origins of chiral anomalies in  three-dimensional (3D) metals with SOC and Kramers-  Weyl metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In Section III, we present a mathematical  derivation of the continuity equations to demonstrate the  existence of these anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The effects of these anoma-  lies on charge and spin transport are examined in Sec-  tions IV and V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Finally, we summarize our  findings in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  ORIGIN OF CHIRAL ANOMALIES IN  SPIN-ORBIT COUPLED METALS  To understand the chiral anomaly in 3D spin-orbit cou-  pled metallic systems (or Kramers-Weyl metals), we first  revisit the WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Specifically, we review the physics of  chiral anomaly in WSM from the perspective of semi-  classical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In WSM, the Hamiltonian for a par-  ticular Weyl node near the band crossing point can be  approximated as HWSM = �  a=x,y,z ℏ(va · k)σa, where  k is measured from the Weyl node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The ‘chirality’ of  Weyl node is defined as C = sign[vx · vy × vz] [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In  the semiclassical dynamics picture, the existence of chi-  ral anomaly can be understood by calculating the equi-  librium current in the presence of an external magnetic  field but no electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The equilibrium charge current for each Weyl node (or  the chiral current) arises from the chiral magnetic ve-  locity (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' III A with explicit derivation shown in  Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The chiral current for WSM can be ex-  pressed in terms of the Berry curvature flux quantum  passing through the FS for the WSM [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This is consis-  tent with the intuitive picture of the Weyl nodes acting as  sinks and sources of the Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' For the pair of  Weyl nodes of opposite chirality, their FSs are separated  in the momentum space (at least for small energies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In  the presence of an external electric field aligned along  the magnetic field, the chiral charge carriers are pumped  across the FSs with distinct Weyl chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This flow is  stabilized by inter-node scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This results in dif-  ferent chiral charge densities on the two Weyl nodes [as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 1(a)], and it manifests in several inter-  esting transport phenomena in WSMs [5, 17, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  We  emphasize two things here: i) A minimum of a pair of  Weyl nodes of opposite chirality are needed to produce  chiral anomaly in WSM, and ii) the chiral anomaly can  be interpreted as an FS phenomenon, where the chiral  charges are ‘pumped’ across two FSs enclosing opposite  quantum of the Berry curvature flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' These two points  will be crucial in investigating the chiral anomalies in  Kramers-Weyl metals or 3D SOC metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  3D SOC metals or Kramers-Weyl metals are struc-  turally chiral crystals with broken inversion symme-  try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' They host ‘Weyl’-like nodal points at all the time-  reversal-invariant momentum (TRIM) points in their  Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' While the form of the SOC can be dif-  ferent, a common feature of all such materials is that  they have two FSs for each band crossing point (or the  Kramers-Weyl node).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This is aided by the kinetic en-  ergy term of the form ℏ2k2/(2m) in their dispersion,  which is missing in conventional WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We have tabu-  lated all crystalline point groups that support Kramers-  Weyl points, along with their low energy Hamiltonian in  the vicinity of the Kramers-Weyl point in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  While our discussion applies to all classes of single crys-  talline systems of 3D SOC metals or Kramers-Weyl met-  als listed in Table I, for specific calculations, we consider  the Hamiltonian [51, 62, 63],  H = ℏ2k2  2m σ0 + αk · σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (1)  Here, m is the effective electron mass, α is the SOC pa-  rameter, σ = (σx, σy, σz) denotes the vector of the Pauli  matrices in spin space and k is the Bloch wave vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  We note that in contrast to conventional WSM, the Pauli  matrices here denote the physical spins of the itinerant  electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The energy dispersion for the Hamiltonian in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (1) is,  ϵλ = ℏ2k2  2m + λαk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (2)  Here, λ = ±1 is the spin-split band index which co-  incides with the eigenvalues of the operator ˆO = ˆk ·  σ, and k = |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The corresponding eigenstates are  given by |u⟩T  + = [cos(θk/2), eiφk sin(θk/2)] and |u⟩T  − =  [sin(θk/2), −eiφk cos(θk/2)], with cos θk  ≡  kz/k and  tan φk ≡ ky/kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 1b, the λ = +1 (λ = −1) band  is represented by the solid (dashed) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The two bands  of the dispersion relation (2) have a band-touching point  3  (BTP) at ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The λ = +1 band has a minimum at  ϵ = 0 and increases monotonically as k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The  λ = −1 band is non-monotonic, and it has a minimum  energy located at ϵmin = −ϵα, with ϵα = mα2/2ℏ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The  minimum energy point lies on a circular contour specified  by |k|2 = k2  α, where kα = mα/ℏ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Clearly, there are two different types of FSs for any  value of the Fermi energy greater than the energy of the  Kramers-Weyl node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The inner FS resulting from the  λ = +1 band has an electronic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In contrast, the  outer FS can be interpreted to have the hole character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The Berry curvature flux quantum through each of the  Fermi surfaces is defined as  Cλ = 1  2π  �  FS  dS · Ωλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (3)  Here, dS is the elemental surface area of the FS, and  Ωλ is the Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' More interestingly, the flux  quantum associated with the FSs is equal and opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  We explicitly calculate Cλ = −λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' See Appendix B for de-  tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hence, the Berry curvature flux quantum piercing  the outer (inner) FS is +1 (−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We emphasize that this  scenario is distinctively different from the usual WSM  with chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In WSM, the pair of FS with the  opposite sign of the Berry curvature flux quantum cor-  responds to two distinct Weyl crossing points separated  by momentum or energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' However, in this case, a single  Weyl crossing is linked to the two FSs with opposite flux  quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The opposite sign of the Berry curvature flux  quantum on the two FSs can be used to define charged  fermions of different ‘flavors’ (akin to chirality in the case  of WSM) in the two FSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The non-zero flux associated with the two FSs in SOC  metals gives rise to chiral anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This is captured  by the non-conservation of the total flavor charge (N λ)  and energy (Eλ) in presence of a magnetic field (B) and  an electric field (E) or temperature gradient (∇T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In a  clean system of 3D SOC metal, we can obtain  ∂N λ  ∂t  ∝ −Cλ  0 E · B  and  ∂N λ  ∂t  ∝ −Cλ  1 ∇T · B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (4)  A similar calculation for the total energy of each flavour  of fermions yields,  ∂Eλ  ∂t ∝  �  −(µ Cλ  0 + kBT Cλ  1 ) E · B  −(µ Cλ  1 + kBT Cλ  2 )∇T · B  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (5)  Here, µ is the chemical potential, and kBT is the energy  scale of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The coefficients Cλ  ν [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (9)]  for ν = {0, 1, 2} are the coefficients of the electrical, ther-  mal, and gravitational chiral anomalies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' See  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' III and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (14)-(15) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' More impor-  tantly, these are finite only when the Berry curvature  flux quantum Cλ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Thus, the Berry curvature flux  quantum plays an important role in defining the par-  ticles’ flavor (or chirality) and the associated quantum  flavor anomalies (or chiral anomalies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We highlight the  chiral charge transfer across the two Fermi surfaces in  WSM and in 3D SOC metals, with opposite Berry cur-  vature flux in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  In the next section, we explicitly demonstrate the three  chiral anomalies in 3D SOC (or Kramers-Weyl) metals  using the idea of equilibrium and non-equilibrium chiral  charge and energy currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We specifically focus on the  case when the chemical potential is higher in energy than  the Kramers-Weyl point (µ > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The regime when the  chemical potential is below the energy of the Kramers-  Weyl point is a bit tricky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We find that in this regime  there is only one FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The Fermi surface is associated  with the λ = −1 band, and the total flux through the  FS is identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Since the chiral anomaly requires  two FSs with opposite Berry curvature flux, there is no  chiral anomaly for µ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' However, an interesting Bril-  louin zone partitioning scheme been proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' [47]  to divide the single FS into two parts having opposite  Berry curvature flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We show that such BZ partition-  ing within a single FS is not physical, and it can lead  to chiral anomaly-like physics even in a free electron gas  in absence of a magnetic field and Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We  discuss these subtle issues in detail in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  CHIRAL CURRENTS AND THE CHIRAL  ANOMALIES  In this section, we first show that the existence of equi-  librium currents in the presence of a magnetic field hints  at the possible existence of chiral anomalies in the sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Next, we explicitly calculate the continuity equation  for the chiral charges and energy current in the presence  of a magnetic field and either a collinear electric field or  a collinear temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Equilibrium chiral current induced by magnetic  field  The equations of motion of charge carriers in the pres-  ence of Berry curvature are described by the following  semiclassical equation of motion [64, 65]  ˙rλ = Dλ  �  vλ + e  ℏE × Ωλ + e  ℏ(vλ · Ωλ)B  �  ,  (6a)  ℏ ˙kλ = Dλ  �  −eE − evλ × B − e2  ℏ (E · B)Ωλ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (6b)  Here, ‘−e’ is the electronic charge, vλ is the band ve-  locity, and Ωλ is the Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (6a),  Dλ ≡ 1/(1 + e  ℏΩλ · B) is the phase-space factor, which  modifies the invariant phase-space volume according to  [dk] → [dk]D−1  λ  [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The term e  ℏ(vλ · Ωλ)B in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (6a)  is known as the chiral magnetic velocity and as will see  it plays an important role in anomaly related transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  For a given FS, the equilibrium chiral charge and en-  4  ergy currents are calculated to be [13]  {jλ  e,eq, jλ  ϵ,eq} =  �  BZλ  [dk]{−e, ϵλ} e  ℏ (vλ · Ωλ) fλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (7)  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (7), fλ is the equilibrium Fermi distribution func-  tion corresponding to the FS λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We emphasize that the  chiral magnetic velocity solely determines the chiral cur-  rents, and the band gradient velocity does not contribute  to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Evaluating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (7) for our model Hamiltonian, we  obtain general relations for the charge and the energy  current [17, 30, 67],  jλ  e,eq= −e  �  µCλ  0 + kBTCλ  1  �  B ,  (8a)  jλ  ϵ,eq=  �µ  2 Cλ  0 + µkBTCλ  1 + k2  BT 2  2  Cλ  2  �  B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (8b)  Here, we note that all the anomaly coefficients appear  in the equilibrium current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (8a)-(8b), the coeffi-  cients are specified by,  Cλ  ν =  e  4π2ℏ2  �  dϵ  �ϵ − µ  kBT  �ν �  −∂fλ  ∂ϵλ  �  Cλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (9)  It is evident from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (9) that for any quantum system  with finite Cλ, all the chiral anomaly coefficients are non-  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We mention here that in defining the anomaly co-  efficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (9), we have converted the Fermi sea  integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (7) into Fermi surface integration us-  ing the rule of partial derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We provide the details  of the calculations in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The importance of the equilibrium currents given in  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (8a)-(8b) is multifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' First of all, the presence of  finite chiral charges and energy currents in equilibrium is  an indication of the existence of chiral anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This  is because, for both chiral anomaly and non-zero chiral  equilibrium current, non-zero Berry curvature flux is a  prerequisite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Second, the chiral charge (j+  e,eq − j−  e,eq) and  energy (j+  ϵ,eq − j−  ϵ,eq) currents are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This high-  lights that in systems hosting a pair of fermions with  opposite Berry curvature flux quantum, the chiral mag-  netic velocity induces a dissipationless chiral charge and  energy current along B [12, 68–71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Finally, we can ex-  pect a finite anomaly-induced current in non-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  In equilibrium, the total charge (j+  e,eq +j−  e,eq) and energy  (j+  ϵ,eq+j−  ϵ,eq) currents from the two opposite chirality FSs  will add up to zero due to same chemical potential and  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' However, in the presence of chiral chemical  potential (µ+ ̸= µ−) and chiral temperature (T+ ̸= T−)  imbalance induced by the quantum anomalies, these ex-  pressions will result in finite charge and energy current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Note that the general expressions of equilibrium charge  and energy currents, jλ  e,eq and jλ  ϵ,eq, are valid for any 3D  systems with band touching point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' These currents origi-  nate from the chiral magnetic velocity, e/ℏ(vλ·Ωλ)B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' As  a result, the equilibrium currents are identically zero for  any two-dimensional system, for which vλ · Ωλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The  absence of chiral magnetic velocity in 2D systems for-  bids the existence of quantum chiral anomalies in two-  dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  For three-dimensional systems,  vλ · Ωλ is generally non-zero, which gives rise to finite  equilibrium currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' However, to have quantum chiral  anomalies in the system, there should be a pair of FS with  opposite Berry curvature flux quantum passing through  them so that jλ  e/ϵ,eq = −j−λ  e/ϵ,eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Having discussed the general expressions for the equi-  librium charge and energy currents, we now calculate all  the anomaly coefficients for a 3D spin-orbit coupled sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' For the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (1), the Berry curvature  is given by Ωλ = −λk/2k3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The chiral anomaly coeffi-  cients are obtained to be  {Cλ  0 , Cλ  1 , Cλ  2 } = −λ  e  4π2ℏ2 {F0, F1, F2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (10)  We note that the equilibrium currents of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (8a) and  (8b), along with the chiral anomaly coefficients of the  above equations, do not get affected by the orbital mag-  netic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Here, Fν’s are the dimensionless functions  of i) x = β(ϵα + µ) for λ = −1 band, and ii) x = βµ for  λ = +1 band with β = 1/kBT being the inverse temper-  ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Their functional form is given by  F0(x) ≡ 1/(1 + e−x) ,  F1(x) ≡ x/(1 + ex) + ln[1 + e−x] ,  (11)  F2(x) ≡ π2  3 − x  �  x  1 + ex + 2ln[1 + e−x]  �  + 2Li2[−e−x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Here, Li2 is the polylogarithmic function of order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  With the replacement of (ϵα + µ) → µ, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (10) and  (11) become identical to that in the WSMs [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The  temperature dependence of all three chiral anomaly co-  efficients is similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (6) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In the zero  temperature limit, F0 → 1, and F2 → π2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' It is worth  noting that for T = 0, the thermal chiral anomaly coeffi-  cient Cλ  1 ∝ F1 → 0 becomes finite only for finite T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Steady state in the presence of chiral anomaly  The presence of external perturbations, such as an elec-  tric field E, or a temperature gradient ∇T, drives the  system out of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In the non-equilibrium steady-  state, the distribution function (gλ) corresponding to the  FS λ satisfies the following Boltzmann transport equa-  tion  ∂gλ  ∂t + ˙rλ · ∇r gλ + ˙kλ · ∇k gλ = Icoll{gλ} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (12)  Here, Icoll{gλ} is the collision integral and gλ is the non-  equilibrium distribution function for each Fermi function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Similar to that in WSM, the charge and energy pump-  ing between the two FSs dictates that the collision inte-  gral should incorporate both the intra- and inter-Fermi  surface scattering processes [17, 36, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Within the re-  laxation time approximation, both the scattering process  can be captured by the following form of the collision in-  tegral [13, 73],  Iλ  coll = −gλ − ¯gλ  τ  − ¯gλ − fλ  τv  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (13)  5  Here, ¯gλ represents the ‘local’ steady-state distribution  function for each FS with a local chemical potential  µλ ≡ µ + δµλ, and local temperature Tλ ≡ T + δTλ [72],  and fλ specifies the global equilibrium function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The  first term in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (13) represents  the intra-Fermi surface scattering (with scattering rate  1/τ), which establishes the local equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The inter-  Fermi surface scattering has been represented by the sec-  ond term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (13) with scattering rate 1/τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The ra-  tio of inter- and intra- Fermi surface scattering time for  Hamiltonian (1) considering screened Coulomb impurity  potential has been calculated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In the small µ  limit, it is given by τv/τ ∼ (2mα2/ℏ2)2/µ2 [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hence,  for small µ, similar to the WSM [74, 75], we have τv > τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Now, we construct the continuity equation for the  particle number and the energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Substituting  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (13) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (12), and then integrating over all the  momentum states for the FS λ, we obtain  ∂N λ  ∂t  + eE · BCλ  0 + ∇r · Jλ = −N λ − N λ  0  τv  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (14)  Here, ∇r ·Jλ = kBCλ  1 ∇T ·B is the divergence of particle  current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The quantities {N λ  0 , N λ} =  �  [dk]D−1  λ {fλ, gλ}  represents the total particle number density in each  FS before and after applying the perturbing fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (14), the terms E ·BCλ  0 , and kBCλ  1 ∇T ·B represents  the chiral anomaly induced flow of the charge carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Similarly, the continuity equation for the energy density,  which we construct by multiplying the energy dispersion  ϵλ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (12) and integrating over all the momentum  states, is obtained to be  ∂Eλ  ∂t +(µCλ  0 +kBTCλ  1 ) eE·B+∇r·Jλ  E = −Eλ − Eλ  0  τv  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (15)  The second term on the left hand side is −E · jλ  e,eq that  represents the work performed by the electric field and  ∇r · Jλ  E = (µkBCλ  1 + k2  BTCλ  2 ) ∇T · B represents the  divergence of energy current in presence of ∇T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The  quantities {Eλ  0 , Eλ} =  �  [dk]D−1  λ ϵλ{fλ, gλ} is the total  energy density in each FS before and after applying  external fields, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Here, µCλ  0 and µCλ  1 spec-  ify the energy carried out by the chiral charge transfer,  whereas TCλ  2 represents the energy pumped out by the  term ∇T · B [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In constructing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (14) and (15), we  have used the fact that the intra-Fermi surface scatter-  ing does not change the number of particles and energy  in each FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  CHIRAL ANOMALY AND CARRIER  TRANSPORT  To calculate the chiral anomaly-induced charge, heat,  and spin currents, we first calculate the non-equilibrium  distribution function to linear order in an applied electric  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In the linear response regime, we can safely assume  that the change in chiral chemical potential and temper-  ature is small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=', δµλ < µ, and δTλ < T [13, 17, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Then, to the lowest order in δµλ and δTλ, the non-  equilibrium distribution function can be calculated to be  gλ = fλ +  �  −∂fλ  ∂ϵλ  � � �  1 − τ  τv  � �  δµλ + ϵλ − µ  T  δTλ  �  −τDλ  �  vλ + e  ℏ (vλ · Ωλ) B  � �  eE + (ϵλ − µ) ∇T  T  � �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (16)  Here, the chiral chemical potential δµλ and δTλ are given  by [13]  δµλ = −  τv  (Dλ  2 Dλ  0 − Dλ  1  2)  ��  Dλ  2 Cλ  0 − Dλ  1 Cλ  1  �  eE · B  +  �  Dλ  2 Cλ  1 − Dλ  1 Cλ  2  �  kB∇T · B  �  ,  (17)  kBδTλ = −  τv  (Dλ  2 Dλ  0 − Dλ  1  2)  ��  Dλ  0 Cλ  1 − Dλ  1 Cλ  0  �  eE · B  +  �  Dλ  0 Cλ  2 − Dλ  1 Cλ  1  �  kB∇T · B  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (18)  In the above equation, we have defined the magnetic field-  dependent generalized density of states at finite temper-  ature as  Dλ  ν =  �  dϵ  �ϵλ − µ  kBT  �ν �  −∂fλ  ∂ϵλ  �  Dλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (19)  Here, ν = {0, 1, 2}, and Dλ =  �  [dk](1 + e/ℏΩλ · B)δ(µ −  ϵλ) being the density of states corresponding to the FS  of the band λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' It is evident that both the electric field  and the temperature gradient components parallel to B  contribute to generating the system’s chiral chemical po-  tential and chiral temperature imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Having obtained the non-equilibrium distribution func-  tion, we now calculate the charge and heat current in  each FS, which are defined as {jλ  e , jλ  Q} =  �  [dk]{−e, (ϵλ−  µ)}˙rλgλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Focusing only on the anomaly induced contri-  bution ∝ τv, we obtain [13]  �jλ  e  jλ  Q  �  = τvB  �  �  1  Dλ  0 (eCλ  0)2  ekB  Dλ  1  Dλ  0 Dλ  2 Cλ  0 Cλ  2  ekBT  Dλ  1  Dλ  0 Dλ  2 Cλ  0 Cλ  2  T  1  Dλ  2 (kBCλ  2 )2  �  �  ×  �  E · B  −∇T · B  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (20)  In deriving the above equation, we used the fact that in  the µ ≫ kBT limit (or βµ ≫ 1) limit, Cλ  1 → 0, and  Dλ  0 , Dλ  2 > Dλ  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Now, the transport coefficients can be ob-  tained by comparing the total currents  �  je,Q = �  λ jλ  e,Q  �  from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (20) and the phenomenological linear response  relations [76]: je,a = �  b[σab Eb − αab ∇bT] and jQ,a =  �  b[¯αab Eb − ¯κab ∇bT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Here, σ, α, ¯α, and ¯κ de-  note the electrical, thermo-electric, electro-thermal, and  constant voltage thermal conductivity matrix, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Note that the thermo-power matrix is defined as  Sab = [σ−1α]ab, and the open circuit thermal conductiv-  ity matrix is expressed as κab = [¯κ − ¯ασ−1α]ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' From  6  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (20), we see that both the charge and energy cur-  rents flow along the direction of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This  is consistent with the fact that these originate from the  chiral magnetic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  We calculate the generalized energy density using the  Sommerfeld approximation in the limit µ ≫ kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Re-  taining only the leading order term in the Sommerfeld  expansion, we obtain  Dλ  ν ≈ m3/2√ϵα  √  2π2ℏ3  �  �  �  �  �  �  �  �  �  (1+λ√1+˜µ)  2  √1+˜µ  F0  ν = 0,  ˜µ  2βϵα(1+˜µ)3/2 F2  ν = 1,  (1+λ√1+˜µ)  2  √1+˜µ  F2  ν = 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (21)  Here, we have defined the scaled chemical potential,  ˜µ = µ/ϵα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In calculating the above-generalized energy  densities, we have neglected the magnetic field correc-  tions, which are very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Note that i) Dλ  0 becomes the  exact density of states in the zero temperature limit for  the corresponding bands [77], and ii) Dλ  1 is independent  of λ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=', it is identical for both the FSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The chiral anomaly induced transport coefficients (σ,  α, ¯α, and ¯κ) is obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (20) using the expres-  sions of Cλ  ν , and Dλ  ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In the µ ≫ kBT limit, for arbitrary  orientation of the magnetic field, the anomalies induced  transport coefficients are  �  σab αab  ¯αab ¯κab  �  =  τve3B2  4π2m2α˜µ2 Aab(θ, φ)  (22)  ×  �  � e√1 + ˜µ(2 + ˜µ)  π2kB  6βϵα  (˜µ2+8(1+˜µ))  ˜µ√1+˜µ  π2  6β2ϵα  (˜µ2+8(1+˜µ))  ˜µ√1+˜µ  π2kB  3eβ  √1 + ˜µ(2 + ˜µ)  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Here, A(θ, φ) is a 3×3 matrix, which captures the angular  dependence of all the transport coefficients, with (θ, φ)  denoting the polar, and azimuthal angle of the spheri-  cal polar coordinate for the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The A(θ, φ)  matrix is obtained to be  A(θ, φ) =  �  �  sin2 θ cos2 φ  1  2 sin2 θ sin 2φ  1  2 sin 2θ cos φ  1  2 sin2 θ sin 2φ  sin2 θ sin2 φ  1  2 sin 2θ sin φ  1  2 sin 2θ cos φ  1  2 sin 2θ sin φ  cos2 θ  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (23)  As a consistency check, we note that the longitudinal  electrical conductivity (σaa) derived above matches with  that obtained recently in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The conductivity ma-  trix of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (22) is valid for the arbitrary direction of the  applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' So, in the planar configuration of  the magnetic field (θ = π/2), the xy-component of the  transport coefficients represents various planar Hall ef-  fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' For instance, the σxy, αxy, ¯αxy, and ¯κxy represents  the usual planar Hall response, planar Nernst effect, pla-  nar Ettinghausen effect, and planar Righi-Leduc effects,  respectively [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hence, our work generalizes the chi-  ral anomalies induced transport to the thermo-electric,  and thermal conductivity matrices for spin-orbit coupled  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We emphasize that the chiral anomaly induced  responses of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (22) become zero for ϵα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This is  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Variation of the chiral anomaly induced electrical  conductivity with the chemical potential and the spin-orbit  coupling energy strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The electrical conductivity is ex-  pressed in units of σ0 =  τve4B2  4  √  2π2m3/2ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The anomaly-induced  response is larger for larger SOC strength and smaller chem-  ical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  expected because the system’s inversion symmetry is re-  stored as α → 0, causing the ‘Weyl’ point, related Berry  curvature, and chiral magnetic velocity to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  We present the variation of chiral anomaly-induced  electrical conductivity with µ and ϵα in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  We  find that the other conductivity components of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (22)  also follow a similar qualitative trend in µ and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The  anomaly-induced response decreases as µ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This  is consistent with the fact that the chiral anomalies orig-  inate from the Berry curvature, which peaks in the vicin-  ity of the band touching points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  To investigate the impact of the chiral anomaly on  various longitudinal transport phenomena, we define  the following generalized magneto-resistance, MRR ≡  R(B)/R(B = 0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Here, R denotes the different trans-  port contributions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In the µ ≫ kBT limit, we  calculate the Drude conductivities to be  σD = eτmα  3ℏ4  × 2eϵα  π2 (2 + ˜µ)  �  1 + ˜µ ,  αD = −eτmα  3ℏ4  × kB  3β  (3˜µ + 4)  √1 + ˜µ ,  ¯κD = eτmα  3ℏ4  × 2ϵα  eπ2  π2kB  3β (2 + ˜µ)  �  1 + ˜µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (24)  In this limit, the longitudinal MR in resistivity is ob-  tained to be  MRρ = −  3τvγ2  3τvγ2 + 4τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (25)  Here we have defined, γ =  eℏ3B  m2α2 ˜µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The ‘magneto-  resistance’ in the Seebeck coefficient can be calculated  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='0-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='5 -  30  10  20  407  to be  MRS = MRρ  4(˜µ2 + 3˜µ + 2)  ˜µ(3˜µ + 4)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (26)  We note that both of these, MRρ and MRS, show nega-  tive ‘magneto-resistance’, similar to the band-inversion  WSM [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  However, unlike the case of conventional  WSM, the relation MRρ/MRS = 1/2 is not satisfied in  spin-orbit coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  In the case of thermo-electric and constant voltage  thermal conductivity, we find  MR¯κ = 3τv  4τ γ2 ,  (27)  MRα = −MR¯κ  ˜µ2 + 8(1 + ˜µ)  ˜µ√1 + ˜µ(3˜µ + 4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (28)  Clearly, MRα is negative while MR¯κ is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This is  similar to the results obtained for WSM in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' [17, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  CHIRAL ANOMALY AND SPIN  TRANSPORT  Unlike WSM, where the Pauli matrices in the Hamilto-  nian represent pseudo-spins, the Pauli matrices in SOC  systems described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (1) represent physical spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Consequently, the two bands in SOC systems are spin  momentum locked with opposite spin orientations on the  inner and outer FSs [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Thus, it is natural to expect  that chiral anomalies can also influence spin transport  along with charge transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Motivated by this, we ex-  plore the chiral anomalies induced linear spin transport  (∝ E · B or ∇T · B) in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Spin transport in a  3D SOC system was recently explored in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' [79] with-  out considering the effect of chiral anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' [47],  the authors studied electrical chiral anomaly induced lin-  ear electrical spin current in 3D SOC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Here, we  include the temperature gradient induced spin currents  and study the chiral anomaly induced spin-Nernst effect,  in addition to other effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The spin current operator is defined via the anticom-  mutator relation, ˆJsb  a  =  1  2{ˆva, ˆsb}, where ˆva is the ve-  locity operator, ˆsb is the spin operator and a, b denote  the Cartesian coordinates [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Now, the spin current  can be calculated as the expectation value of the spin  current operator weighted by the non-equilibrium distri-  bution function,  jsb  a =  �  λ  �  [dk]D−1  λ ⟨uλ(k)| ˆJsb  a |uλ(k)⟩ gλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (29)  The matrix of spin transport coefficients is related to the  spin current via the relation jsb  a = σsb  acEc−αsb  ac∇cT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Here,  σsb  ac is the electrical spin conductivity matrix, and αsb  ac  is the thermo-electric spin conductivity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' These  tensors represent response coefficients for the spin current  flowing along the a-direction for spin polarization along  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The variation of the longitudinal thermoelectric spin  conductivity with the chemical potential µ and the spin-orbit  coupling energy strength ϵα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The conductivity αsx  xk is scaled  by  τvekBB  9  √  2ℏ2β√m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Similar to the chiral anomaly induced elec-  trical response, the chiral anomaly induced spin response is  also larger for larger spin-orbit coupling and smaller chemical  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  the b-direction, while the electric field or the temperature  gradient is applied along the c-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The spin current operator for Hamiltonian (1) is given  by  ˆJsb  a = ℏka  m σ0 + δab  α  ℏ σb ,  (30)  where δab = 0 or 1 depending on a ̸= b or a = b, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Using the eigenstates of Hamiltonian (1), we  evaluate the expectation value of the above equation to  be  ⟨uλ| ˆJsb  a |uλ⟩ = α  ℏ Iab + λℏk  m Aab(θk, φk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (31)  Here, I denotes the 3 × 3 identity matrix, and A(θk, φk)  is a 3 × 3 matrix defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Following the sym-  metric energy dispersion, the distribution function gλ [see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (16)] is independent of θk and φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' As a consequence,  the angular integration over φk makes all the off-diagonal  elements of ⟨uλ| ˆJsb  a |uλ⟩ to be zero, and jsb  a = 0 for a ̸= b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Thus, the spin current is finite only when the spins are  aligned along the direction of the velocity of the carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Hence, the chiral anomaly induced spin currents are  finite only when the spins are polarized along the respec-  tive directions of current, and we have jsx  x = jsy  y = jsz  z =  js  CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We calculate the spin current induced by the chiral  anomalies to be [see Appendix D for details]  js  CA =τv  �  λ  Cλ  0  Dλ  0  �Dλ  1  Dλ  2  L1 − L0  �  eE · B  − Cλ  2  Dλ  2  �Dλ  1  Dλ  0  L0 − L1  �  kB∇T · B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (32)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='170  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='0-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='080  10  20  30  408  Here, we have defined  Lν =  �  [dk]  �α  ℏ + λ ℏ  mka · ˆk  � �ϵλ − µ  kBT  �ν �  −∂fλ  ∂ϵλ  �  ,  (33)  with ka = kaˆa being a vector along a-direction with mag-  nitude equal to the component of k along the a-direction,  and ˆk = sin θk cos φk ˆx+sin θk sin φk ˆy +cos θk ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We now  have jsa  a  ∝ E · B for any arbitrary direction of the ap-  plied electric field along the k-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  We calculate  the corresponding chiral anomaly induced electrical spin  conductivity to be,  σsx  xc = σs  0  ��  1 + ˜µ −  π2  6β2ϵ2α  (˜µ2 + 9˜µ − 20)  ˜µ2(1 + ˜µ)2  �  ˆc· ˆ  B , (34)  where we have defined σs  0 = τve2Bα  6π2ℏ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The second term  on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (34) is the finite tempera-  ture correction to the electrical spin conductivity, which  vanishes in the T → 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  For the thermoelectric part of the spin conductivity,  we find that it behaves like the electric spin conductiv-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' All the thermoelectric spin currents, where the spin  is not aligned along the current direction, vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We  obtain, jsb  a = 0 for b ̸= a, and jsa  a ∝ ∇T · B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Our calcu-  lations show that only the conductivity components, αsa  ac  are finite, and αsx  xc = αsy  yc = αsz  zc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We calculate the ther-  moelectric spin conductivity for the temperature gradient  applied along the c-direction to be,  αsx  xc = αs  0  � 2  ˜µ2 + ˜µ2 + 3˜µ − 2  ˜µ2√1 + ˜µ − ˜µ2 + 7˜µ + 6  2˜µ(1 + ˜µ)3/2  �  ˆc · ˆ  B ,  (35)  where αs  0 = τvekBαB  18ℏ3βϵα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The above expression represents  the chiral anomaly induced spin-Seebeck (for c = x) or  the spin-Nernst coefficient (for c ̸= x), with the spins po-  larized along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The variation of αsx  xk with  µ and ϵα is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The electrical spin con-  ductivity also follows similar trends in µ and ϵα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The  anomaly induced effects in general decrease with increas-  ing µ and increase with increasing α which is a proxy for  the degree of inversion symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  CONCLUSION  In summary, we have provided evidence that quantum  chiral anomalies can be understood as a feature of FSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Specifically, the chirality of charge carriers can be deter-  mined by the sign of the Berry curvature quantum pass-  ing through the associated Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This has signif-  icant implications for 3D SOC metals or Kramers-Weyl  metals, where chiral charge pumping can occur across  the two Fermi surfaces associated with a single Kramers-  Weyl node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' To the best of our knowledge, this kind of  chiral anomaly has no analog in relativistic field theories  of chiral fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We have also demonstrated the exis-  tence of three distinct types of quantum chiral anomalies  – electrical, thermal, and gravitational – in 3D SOC met-  als and Kramers-Weyl metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The effect of these quantum chiral anomalies can be ob-  served in electrical and thermo-electric charge and spin  transport in 3D SOC metals and Kramers-Weyl metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  While the electrical transport signatures of chiral anoma-  lies in 3D spin-orbit coupled metals are similar to those in  Weyl semimetals, the signatures in electrical and thermo-  electric spin transport are unique to 3D SOC metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We  have shown that spin conductivities are finite only when  spins are polarized along the direction of carrier flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' we  found that the chiral anomaly-induced spin conductivi-  ties are proportional to the strength of the magnetic field,  unlike charge conductivities which scale with the square  of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Our findings contribute to the un-  derstanding of chiral anomaly induced charge, heat, and  spin transport in 3D SOC metals and Kramers-Weyl sys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  We acknowledge the Science and Engineering Research  Board (SERB, via project MTR/2019/001520) for finan-  cial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' thanks the MHRD, India for fund-  ing through the Prime Minister’s Research Fellowship  (PMRF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We sincerely thank Atasi Chakraborty for the  useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Appendix A: 3D non-centrosymmetric SOC metals  and Kramers-Weyl metals  In this appendix, we discuss the SOC-induced chiral  anomaly in other 3D systems with different forms of the  SOC, compared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Comparing the list of single  crystalline point groups which support 3D spin-orbit cou-  pled metals [81] with the list of Kramers Weyl metals [50],  we find that these are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' However, 3D electron  gas with SOC can also arise in some heterostructures of  two different single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Both of these systems have  doubly degenerate band touching points, which we refer  to as ‘Kramers-Weyl’ points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kramers-Weyl metals are  realized in structurally chiral crystals that lack mirror,  inversion, or roto-inversion symmetry [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' There are 65  Sohncke chiral space groups corresponding to 11 chiral  point groups which characterize the structurally chiral  crystals [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The bands of non-magnetic chiral crystals are at least  doubly degenerate at the time-reversal-invariant mo-  menta (TRIM) points due to Kramers theorem [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  However, the SOC lifts the Kramer’s degeneracy at all  other points in the momentum space, leaving behind  ‘Weyl’-like Kramers-Weyl nodes at the TRIM points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  All these band degenerate points are topologically non-  trivial, carrying finite Chern numbers [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In general,  the chiral crystals can host multiple band crossings at  9  the TRIM points in the Brillouin zone along with multi-  fold band degeneracy [49–51, 53, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  In this paper, we focus on Kramers-Weyl metals  that have a two-fold degenerate Kramers-Weyl point at  TRIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In Table I, we summarize the chiral space groups  and point groups which support Kramers-Weyl fermions,  along with some material examples [47, 50, 81, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The  generic Kramers-Weyl system will have a low energy  Hamiltonian of the form: H = �  ab ℏ2kakb/(2mab) +  hk · σ, in the vicinity of the Kramers-Weyl point for  which |hk| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Here, a, b = x, y, z, mab is the effec-  tive mass tensor, and k is the momentum with respect  to the Kramers-Weyl point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The specific form of sym-  metry allowed hk, for each of the chiral point groups is  also summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Each of these Kramers-Weyl  points has a chiral charge with value ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' For example,  the Hamiltonian (1) with isotropic SOC term αk · σ can  be realized in point groups T and O in K2Sn2O3, β-RhSi,  CoSi crystals [50, 51, 55–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The space groups and the point groups for topologically non-trivial chiral crystals hosting Kramers-Weyl Fermions  with chiral charge ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Some material examples, along with the form of the symmetry-allowed SOC terms in the vicinity of the  Kramers-Weyl points for each space group are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='Space group Point group (Laue class)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='Material  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='SOC term  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='C1(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='Li6CuB4O10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='(α1kx + α2ky + α3kz)σx + (α4kx + α5ky + α6kz)σy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='+(α7kx + α8ky + α9kz)σz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='3-5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='C2(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='Pb3GeO5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='(α1kx + α2ky)σx + (α3kx + α4ky)σy + α5kzσz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='16-24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='D2(222)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='AlPS4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='α1kxσx + α2kyσy + α3kzσz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='143-146  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='C3(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='β-Ag3IS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='75-80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='C4(4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='BaCu2Te2O6Cl2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='(α1kx + α2ky)σx + (α1ky − α2kx)σy + α3kzσz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='168-173  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='C6(6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='α-In2Se3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='149-155  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='D3(32)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='Ag3BO3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='89-98  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='D4(422)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='CdAs2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='α1(kxσx + kyσy) + α2kzσz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='177-182  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='D6(622)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='NbGe2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='195-199  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='T(23)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='K2Sn2O3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' β-RhSi  α1(kxσx + kyσy + kzσz)  207-214  O(432)  BaSi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' SrSi2  Appendix B: Berry curvature flux quantum and  chiral anomaly for negative chemical potential  In this Appendix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' we calculate the Berry curvature  flux quantum for each Fermi surface and discuss the chi-  ral anomaly for Fermi energies below the Kramers-Weyl  node,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=', µ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We start by calculating the Berry cur-  vature flux quantum for the FSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The Berry curvature  flux through any FS is defined as Cλ =  1  2π  �  FS dS · Ωλ,  where dS is the elemental surface area of the FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Using  the divergence theorem, and capturing the Fermi surface  via the Heaviside step function [Θ(µ − ϵλ)], we have  Cλ= 1  2π  �  dk∇k · ΩλΘ(µ − ϵλ)  = − 1  2π  �  dk Ωλ · ∇kΘ(µ − ϵλ)  = ℏ  2π  �  dk Ωλ · vλδ(µ − ϵλ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (B1)  Note that in the zero-temperature limit, the above ex-  pression reduces to the electrical chiral anomaly coeffi-  cient defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Below, we explicitly calculate  the Cλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Case I (µ > 0):— For µ > 0, there are two Fermi  wave vectors kF  λ = −λkα +  �  k2α + 2mµ/ℏ2 with λ = ±,  corresponding to two FSs of the two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The kF  +  (kF  −) corresponds to the inner (outer) FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Now, using  the expressions of vλ, Ωλ, and the δ-function property,  Cλ for each band λ becomes  Cλ= ℏ  2π  �  dk −λ  2k2  �ℏk  m + λα  ℏ  �  δ(µ − ϵλ) ,  = −λ  �  dk  �ℏ2k  m + λα  � δ(kF  λ − k)  |ϵ′  λ|  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (B2)  Here, ϵ′  λ is the first derivative of ϵλ with respect to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Evaluating this integral yields Cλ = −λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Case II (µ < 0):— For µ < 0, there is only one car-  rier pocket, which looks like an annular sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' It has  two surfaces, the inner and the outer surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Also, the  energy dispersion is non-monotonic (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Conse-  quently, in the region near the Kramers-Weyl node, the  band velocity is negative, while in other regions, the band  velocity is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' As a result, we have regions within  the same pocket that have opposite signs of the chiral  magnetic velocity (∝ vλ · Ωλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  This ensures that the  Berry curvature flux through the entire FS calculated us-  ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (B1) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hence, we expect that there should  10  not be any chiral anomaly for µ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  However, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' [47], the authors discussed the chiral  anomaly for µ < 0 with the idea of partitioning the FS  into two regions based on the sign of the chiral mag-  netic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Below, we discuss this in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  We  show the partitioning of the FS in Fig 4, with the blue  and red regions representing the two different partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Here, the χ is used as the index for denoting the inner  (outer) region of the FS, with χ = −1 for the blue region  (χ = +1 for the red region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' To calculate the Berry cur-  vature flux using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (B1), we first compute the Fermi  wave vectors corresponding to the two different regions  of the Fermi pocket of the λ = −1 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The Fermi  wave vectors corresponding to the inner (χ = −1) and  outer (χ = −1) boundary of the Fermi pocket is given by  kF  χ = kα + χ  �  k2α + 2mµ/ℏ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Recall that kα = mα/ℏ2,  which corresponds to the minima in the energy of the  λ = −1 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The χ = − (+) region of the Fermi pocket  correspond the kF  − < k < kα (kα < k < kF  +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' These re-  gions are represented by blue and red colors, respectively,  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' For λ = −1 band, the Cλ is given by  Cλ = ℏ  2π  �  dk 1  2k2  �ℏk  m − α  ℏ  �  δ(µ − ϵ−) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (B3)  Now, for either of the two regions, the above equation  reduces to  Cχ  λ =  �  dk  �ℏ2k  m − α  � δ(kF  χ − k)  |ϵ′  −|  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (B4)  As the band velocity, ϵ′  − = ℏ2k/m − α is negative (pos-  itive) for the region with kF  − < k < kα (kα < k < kF  +),  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (B4) yields Cχ  λ = χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We note again that the sign of  Cχ  λ is essentially tied to the sign of the chiral magnetic ve-  locity proportional to the (Ωλ · vλ) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We emphasize  that without partition of the FS, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (B3) itself yields  zero due to the two different roots of the δ-function (kF  +  and kF  −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This partitioning of the Brillouin zone, as per  the sign of the chiral magnetic velocity, allows one to de-  fine two regions of FS with opposite Berry curvature flux  quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This had been used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' [47] to discuss the  continuity equation and the associated electrical chiral  anomaly for µ < 0, on the same footing as we have dis-  cussed for µ > 0 [47] in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' While partitioning  a single electron pocket to define carriers of different ‘fla-  vors’ is mathematically appealing, we believe that this  way of defining the chiral anomaly is superfluous and  not physical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Here, we present a counter-example to establish the  above claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Consider a 3D electron gas (without any  SOC, without any magnetic field), with an electric field  applied along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We can divide the Fermi  sphere of the system into two halves with positive and  negative velocities along the x-axis and treat the parti-  cles with opposite velocities as having different flavors  (s = ±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  In the bottom panels (c) and (d) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 4,  we have schematically shown this partitioning of the  FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The particles in the blue (red) region with s = +1  a)  b)  d)  c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' a) The band dispersion and the Brillouin zone par-  titioning for the λ = −1 band of a 3D SOC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' For the  blue-shaded region with negative band velocity, the Berry cur-  vature flux is −1, while the Berry curvature flux is +1 for the  red-shaded region with positive band velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' b) The corre-  sponding cross-section of the Fermi surface for µ < 0 for the  λ = −1 band, highlighting the two partitions of the Fermi  pocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' c) The band dispersion of 3D electron gas without  SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This trivial system can also be partitioned into red and  blue regions depending on the sign of the x component of the  band velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' d) The cross-section of the spherical FS for a  3D electron gas in the kx − ky plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (s = −1), have positive (negative) velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In the pres-  ence of only an electric field along the x-direction, the  collisionless Boltzmann equation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (12) with λ → s  and Icoll{gλ} = 0], upto first order in the electric field  strength, becomes  ∂gs  ∂t − eEvs  x  ∂fs  ∂ϵ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (B5)  Here, gs (fs) is the non-equilibrium (equilibrium Fermi  Dirac) distribution function for the s region of the Bril-  louin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Integrating the above equation over all the  momentum states within the respective partition of the  FS, we obtain  ∂Ns  ∂t + s eE  2π3  �2mµ  ℏ2  �3/2  = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (B6)  The above equation resembles Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (4), indicating the pos-  sibility of a “chiral anomaly” in a normal 3D electron gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  However, this cannot be physical and is very unlikely to  be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Due to this, we believe that the partitioning  of the BZ to divide one electron/hole pocket into mul-  tiple partitions is not physically acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  However,  the partitioning of the BZ to include full electron/hole  pockets is acceptable, and this forms the basis of valley  physics in 2D and chiral anomaly related physics in 3D  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Having discussed the chiral anomaly for µ < 0, we  conclude this Appendix with a small discussion on the  11  chirality of ‘Weyl’-type nodes and the Berry curvature  flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' For the WSM, the Berry curvature flux  through the FS of a node represents the ‘chirality’ of that  node, irrespective of the conduction or the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  This is easily seen because, in the m → ∞ limit, the  Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (1) reduces to the Hamiltonian for  a single Weyl node HWSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  In contrast to the bands  of Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (1), both bands of HWSM are  monotonous (around the nodal point) and only one FS  exists at any particular energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Then a straightforward  calculation following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (B2) yields, Cλ = −sign(α) for  both the conduction and valence band of HWSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Because  the Cλ depends on the sign of α, the Berry curvature flux  quantum becomes opposite for opposite chirality nodes  where α has the opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  This establishes that  for WSM, the chirality of each Weyl node can be rep-  resented as the Berry curvature flux quantum through  the node [5, 17, 61, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' However, for the Kramers-Weyl  nodes, the Berry curvature flux quantum and the chiral-  ity of the node are not identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The chirality of the  Kramers-Weyl nodes depends on the sign of α for Hamil-  tonian (1), which is specific to a given TRIM point of the  material [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Appendix C: Calculation of equilibrium currents  In this Appendix, we derive the expressions of the equi-  librium currents obtained in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (8a) and (8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' In the  presence of only a magnetic field, the velocity of the cen-  ter of mass of the wave packets for the carriers in each  band is given by ˙rλ = Dλ  �  vλ + e  ℏ(vλ · Ωλ)B  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The equi-  librium charge current for the FS λ (corresponding to  each band) is given by  jλ  e,eq = −e  �  [dk]˙rfλ = −eB  �  [dk] e  ℏ(vλ · Ωλ)fλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (C1)  Here, we have used the fact that the band velocity vλ  does not contribute to the equilibrium current (due to  angular integration being zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Now, we use the identity  ∇k ·(ϵλΩλ) = ∇kϵλ·Ωλ+ϵλ∇k ·Ωλ to express the above  equation as,  jλ  e,eq= −e2B  ℏ2  �  [dk] [∇k · (ϵλΩλ) − ϵλ∇k · Ωλ] fλ (C2)  = −e2B  ℏ2  �  [dk]∇k · (ϵλΩλ)fλ  (C3)  = e2  ℏ2 B  �  [dk]ϵλΩλ · ˆk∂fλ  ∂k ,  (C4)  = −eB  �  [dk] (µ + ϵλ − µ) e  ℏ(vλ · Ωλ)  �  −∂fλ  ∂ϵλ  �  ,  = −e  �  µCλ  0 + kBTCλ  1  �  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (C5)  To evaluate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (C3), we have used the fact that ∇k ·  Ωλ = ±2πδ3(k), for a system with doubly degenerate  band touching point with linear dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' This makes  the last integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (C2) to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  To obtain  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (C4) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (C3), we have used integrations by  parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Here, we have defined Cλ  ν as,  Cλ  ν =  �  [dk] e  ℏvλ · Ωλ  �ϵ − µ  kBT  �ν �  −∂fλ  ∂ϵλ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (C6)  These can also be rewritten in terms of Cλ given in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  The energy current, jλ  ϵ,eq, can be evaluated  in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Appendix D: Details of spin current calculations  To calculate the spin current proportional to the E ·B  (or ∇T · B), we consider the band velocity term of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (6a) and calculate the spin current operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The  band velocity operator along the i-direction is given by  ˆvi = ℏki  m σ0 + α  ℏ σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Without loss of generality, here we  show the calculation of spin current in the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Using the expressions of the eigenstates and the spin  current operator given in the main text, we obtain  ⟨uλ| ˆJsx  x |uλ⟩ = (α/ℏ + λℏkx sin θk cos φk/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Now, the  chiral anomaly induced spin current is given by  jsx  x = τv  �  λ  �  [dk]  �α  ℏ + λℏkx  m sin θk cos φk  � �  δµλ + ϵλ − µ  T  δTλ  � �  −∂fλ  ∂ϵλ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (D1)  In the βµ → ∞ limit, writing the expressions of δµλ and δTλ explicitly, we have  jsx  x =τv  �  λ  � Dλ  1 Cλ  0  Dλ  2 Dλ  0  L1 − Cλ  0  Dλ  0  L0  �  eE · B −  � Dλ  1 Cλ  2  Dλ  2 Dλ  0  L0 − Cλ  2  Dλ  2  L1  �  kB∇T · B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  (D2)  The definition of Lν is given in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We eval-  uate the Lν using the Sommerfeld approximation in the  µ ≫ kBT limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' We obtain the following expressions  L0= −λ m2α2  6π2ℏ5  [˜µ − ˜µ2 + 2(1 + ˜µ)]  1 + ˜µ  ,  (D3)  L1= kBT  9ℏ3  (−λ + √1 + ˜µ)[λ(2 + ˜µ) + √1 + ˜µ]  (1 + ˜µ)3/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (D4)  12  Using these expression along with Cλ  ν and Dλ  ν in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (D2),  we obtain the spin conductivities of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' (34) and (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Following a similar procedure, we can calculate other spin  currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  We show that due to rotational symmetry jsj  i  = 0  for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Without loss of generality, we will explicitly  show the calculation for jsz  x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' The expectation value of  the spin current operator ˆJsz  x  is given by ⟨uλ| ˆJsz  x |uλ⟩ =  λ p  2m sin 2θk cos φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Now, as the distribution function is  independent of θk and φk, so the angular integration over  φk of the ⟨uλ| ˆJsz  x |uλ⟩ yields jsz  x  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Similarly, all the  spin currents with spin polarization perpendicular to the  propagation velocity can be easily shown to be zero due  to the vanishing angular integration over φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Adler, Axial-vector vertex in spinor electrodynam-  ics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 177, 2426 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Nielsen and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ninomiya, Absence of neutrinos on a  lattice: (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' proof by homotopy theory, Nuclear Physics  B 185, 20 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Nielsen and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ninomiya, Absence of neutrinos on a  lattice: (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' intuitive topological proof, Nuclear Physics  B 193, 173 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [4] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Nielsen and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ninomiya, The adler-bell-jackiw  anomaly and weyl fermions in a crystal, Physics Letters  B 130, 389 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Son and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Spivak, Chiral anomaly and classical  negative magnetoresistance of weyl metals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B  88, 104412 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xiong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kushwaha, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Liang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Krizan,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hirschberger, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Cava, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ong,  Evidence for the chiral anomaly in the dirac semimetal  na3bi, Science 350, 413 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Burkov, Giant planar hall effect in topological met-  als, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 96, 041110 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Belopolski, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Yuan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lin,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Tong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Bian, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Alidoust, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Huang,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Chang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Chang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hsu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Jeng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ne-  upane, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sanchez, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lin,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Neupert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Za-  hid Hasan, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Jia, Signatures of the adler–bell–jackiw  chiral anomaly in a weyl fermion semimetal, Nature Com-  munications 7, 10735 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [9] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kharzeev, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Huang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Pletikosi´c,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Fedorov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zhong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Schneeloch, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Gu, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Valla, Chiral magnetic effect in zrte5, Nature  Physics 12, 550 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Nandy, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sharma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Taraphder, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Tewari, Chi-  ral anomaly as the origin of the planar hall effect in weyl  semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 119, 176804 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [11] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Guin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Felser, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shekhar, Planar  hall effect in the weyl semimetal gdptbi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B  98, 041103 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [12] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kim, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sasaki, Boltzmann equa-  tion approach to anomalous transport in a weyl metal,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 89, 195137 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [13] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Das, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Singh, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Agarwal, Chiral anomalies  induced transport in weyl metals in quantizing magnetic  field, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Research 2, 033511 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [14] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Das and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Agarwal, Linear magnetochiral transport  in tilted type-i and type-ii weyl semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  B 99, 085405 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [15] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Das and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Agarwal, Berry curvature induced ther-  mopower in type-i and type-ii weyl semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 100, 085406 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [16] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Das and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Agarwal, Intrinsic hall conductivities in-  duced by the orbital magnetic moment, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B  103, 125432 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Das and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Agarwal, Thermal and gravitational chiral  anomaly induced magneto-transport in weyl semimetals,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Research 2, 013088 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Mandal, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Das, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Agarwal, Chiral anomaly and  nonlinear magnetotransport in time reversal symmetric  weyl semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 106, 035423 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [19] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' H¨utt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kamenskyi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Neubauer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shekhar,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Felser, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Dressel, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Pronin, Terahertz trans-  mission through taas single crystals in simultaneously ap-  plied magnetic and electric fields: Possible optical signa-  tures of the chiral anomaly in a weyl semimetal, Results  in Physics 15, 102630 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hosur and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Qi, Tunable circular dichroism due  to the chiral anomaly in weyl semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B  91, 081106 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [21] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ashby and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Carbotte, Chiral anomaly and  optical absorption in weyl semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 89,  245121 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ma and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Pesin, Chiral magnetic effect and natural  optical activity in metals with or without weyl points,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 92, 235205 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [23] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Morimoto and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Nagaosa, Chiral anomaly and giant  magnetochiral anisotropy in noncentrosymmetric weyl  semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 117, 146603 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Jadidi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kargarian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Mittendorff, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Aytac,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' K¨onig-Otto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Winnerl, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Gaeta, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Murphy, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Drew, Nonlinear op-  tical control of chiral charge pumping in a topological  weyl semimetal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 102, 245123 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Thakur, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sadhukhan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Agarwal, Dynamic  current-current susceptibility in three-dimensional dirac  and weyl semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 97, 035403 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [26] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sonowal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Singh, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Agarwal, Giant optical ac-  tivity and kerr effect in type-i and type-ii weyl semimet-  als, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 100, 085436 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [27] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Landsteiner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Meg´ıas, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Pena-Benitez, Gravi-  tational anomaly and transport phenomena, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 107, 021601 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lucas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Davison, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sachdev, Hydrodynamic  theory of thermoelectric transport and negative magne-  toresistance in weyl semimetals, Proceedings of the Na-  tional Academy of Sciences 113, 9463 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Gooth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Niemann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Meng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Grushin,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Landsteiner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Gotsmann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Menges, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Schmidt,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shekhar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' S¨uß, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' H¨uhne, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rellinghaus, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Felser,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Yan, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Nielsch, Experimental signatures of the  mixed axial–gravitational anomaly in the weyl semimetal  nbp, Nature 547, 324 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Stone and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kim, Mixed anomalies: Chiral vortical  effect and the sommerfeld expansion, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' D 98,  025012 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  13  [31] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lundgren, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Laurell, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Fiete, Thermoelectric  properties of weyl and dirac semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 90,  165115 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hirschberger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kushwaha, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Gibson,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Liang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Belvin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Bernevig, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Cava, and  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ong, The chiral anomaly and thermopower of weyl  fermions in the half-heusler gdptbi, Nature Materials 15,  1161 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [33] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Jia, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Liao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Yu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wu,  Thermoelectric signature of the chiral anomaly in cd3as2,  Nature Communications 7, 13013 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [34] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Spivak and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Andreev, Magnetotransport phe-  nomena related to the chiral anomaly in weyl semimetals,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 93, 085107 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [35] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Stockert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' dos Reis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ajeesh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Watz-  man, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Schmidt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shekhar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Heremans, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Felser,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Baenitz, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Nicklas, Thermopower and ther-  mal conductivity in the weyl semimetal nbp, Journal of  Physics: Condensed Matter 29, 325701 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [36] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zyuzin, Magnetotransport of weyl semimetals due  to the chiral anomaly, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 95, 245128 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Vu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sahin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Flatte, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Trivedi, and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Heremans, Thermal chiral anomaly in the magnetic-  field-induced ideal weyl phase of bi1-xsbx, Nature Mate-  rials 20, 1525 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [38] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Son and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Yamamoto, Berry curvature, triangle  anomalies, and the chiral magnetic effect in fermi liquids,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 109, 181602 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Son and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Yamamoto, Kinetic theory with berry  curvature from quantum field theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' D 87,  085016 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Stephanov and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Yin, Chiral kinetic theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 109, 162001 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [41] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Fang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Gilbert, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Dai, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Bernevig,  Multi-weyl topological semimetals stabilized by point  group symmetry, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 108, 266802 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [42] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Roy, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Das Sarma, Weyl fermions with  arbitrary monopoles in magnetic fields: Landau levels,  longitudinal magnetotransport, and density-wave order-  ing, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 94, 195144 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [43] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zhou, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shen, Topological re-  sponses from chiral anomaly in multi-weyl semimetals,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 96, 085201 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [44] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Dantas,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Pe˜na-Benitez,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Roy,  and  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sur´owka, Magnetotransport in multi-weyl semimet-  als: a kinetic theory approach, Journal of High Energy  Physics 2018, 69 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [45] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Das, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Das, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Agarwal, Nonlinear magnetocon-  ductivity in weyl and multi-weyl semimetals in quantiz-  ing magnetic field, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 105, 235408 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [46] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lepori,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Burrello, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Guadagnini, Axial  anomaly in multi-weyl and triple-point semimetals, Jour-  nal of High Energy Physics 2018, 110 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [47] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Cheon, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Cho, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kim, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lee, Chi-  ral anomaly in noncentrosymmetric systems induced by  spin-orbit coupling, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 105, L180303 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [48] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Gao and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Huang, Chiral anomaly in non-  relativistic systems: Berry curvature and chiral kinetic  theory, Chinese Physics Letters 39, 021101 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [49] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Bradlyn,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Cano,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wang,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Vergniory,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Felser, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Cava, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Bernevig, Beyond dirac  and weyl fermions: Unconventional quasiparticles in con-  ventional crystals, Science 353, aaf5037 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [50] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Chang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wieder, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Schindler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sanchez,  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Belopolski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Huang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Singh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Chang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Neupert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hasan,  Topological quantum properties of chiral crystals, Nature  Materials 17, 978 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [51] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Law, Kramers weyl  semimetals as quantum solenoids and their applications  in spin-orbit torque devices, Communications Physics 4,  66 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [52] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Schindler, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Chang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xu,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Chang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hua, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Jiang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Jeng,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hasan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xie, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Neupert, and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Jia, Ultraquantum magnetoresistance in the kramers-  weyl semimetal candidate β−ag2Se, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 96,  165148 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [53] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Schr¨oter, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Pei, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Vergniory, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sun,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Manna,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  de  Juan,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Krieger,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  S¨uss,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Schmidt,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Dudin,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Bradlyn,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Kim,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Schmitt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Cacho, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Felser, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Strocov, and  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Chen, Chiral topological semimetal with multifold  band crossings and long fermi arcs, Nature Physics 15,  759 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [54] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Tan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Huang,  A unified understanding of diverse spin textures of  kramers–weyl fermions in nonmagnetic chiral crystals,  Advanced Functional Materials 32, 2208023 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [55] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Tian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Fu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Tang,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Long,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Fang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Weng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Qian, and  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ding, Observation of unconventional chiral fermions  with long fermi arcs in cosi, Nature 567, 496 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [56] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sanchez, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Belopolski, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Cochran, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Yin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Chang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xie, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Manna, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' S¨uß, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Huang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Alidoust, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Multer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zhang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shumiya,  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Chang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Felser, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Jia, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hasan, Topological chiral  crystals with helicoid-arc quantum states, Nature 567,  500 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [57] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Dutta, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ghosh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Singh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Politano,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Bansil, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Agarwal, Collective plasmonic modes  in the chiral multifold fermionic material cosi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  B 105, 165104 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [58] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Takane, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Souma, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Nakayama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Naka-  mura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Oinuma, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Nakata, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Iwasawa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Cacho,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Horiba, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kumigashira, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Takahashi,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ando, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sato, Observation of chiral fermions  with a large topological charge and associated fermi-  arc surface states in cosi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 122, 076402  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [59] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Verma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Biswas, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ghosh, Thermoelectric  and optical probes for a fermi surface topology change in  noncentrosymmetric metals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 100, 045201  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [60] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Pal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Dey, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ghosh, Berry curvature in-  duced magnetotransport in 3d noncentrosymmetric met-  als, Journal of Physics: Condensed Matter 34, 025702  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [61] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Armitage, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Mele, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Vishwanath, Weyl  and dirac semimetals in three-dimensional solids, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 90, 015001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [62] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zang, Transport theory of metallic b20  helimagnets, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 91, 134401 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [63] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Samokhin, Effects of impurities on the upper crit-  ical field Hc2 in superconductors without inversion sym-  metry, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 78, 144511 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  14  [64] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xiao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Chang, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Niu, Berry phase effects on  electronic properties, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 82, 1959 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [65] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Morimoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zhong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Orenstein, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Moore,  Semiclassical theory of nonlinear magneto-optical re-  sponses  with  applications  to  topological  dirac/weyl  semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 94, 245121 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [66] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xiao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shi, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Niu, Berry phase correction to  electron density of states in solids, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 95,  137204 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [67] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ma and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Pesin, Chiral magnetic effect and natural  optical activity in metals with or without weyl points,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 92, 235205 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [68] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Fukushima, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kharzeev, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Warringa, Chi-  ral magnetic effect, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' D 78, 074033 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [69] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Li and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kharzeev, Chiral magnetic effect in  condensed matter systems, Nuclear Physics A 956, 107  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [70] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kharzeev, The chiral magnetic effect and anomaly-  induced transport, Progress in Particle and Nuclear  Physics 75, 133 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [71] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kharzeev, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kikuchi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Meyer, Chiral mag-  netic effect without chirality source in asymmetric weyl  semimetals, The European Physical Journal B 91, 83  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [72] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Yip, Kinetic equation and magneto-conductance for  weyl metal in the clean limit (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [73] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Deng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Qi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wang,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sheng, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xing, Quantum oscillations of the  positive longitudinal magnetoconductivity: A fingerprint  for identifying weyl semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 122,  036601 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [74] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Burkov, Negative longitudinal magnetoresistance  in dirac and weyl metals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 91, 245157 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [75] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Xiong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kushwaha, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Liang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Krizan,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Hirschberger, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Cava, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ong,  Evidence for the chiral anomaly in the dirac semimetal  na3bi, Science 350, 413 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [76] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ashcroft and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Mermin, Solid State Physics, HRW  international editions (Holt,  Rinehart and Winston,  1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [77] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Verma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kundu, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ghosh, Dynamical po-  larization and plasmons in noncentrosymmetric metals,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 102, 195208 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [78] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Schindler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Galeski, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Schnelle, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wawrzy´nczak,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Abdel-Haq, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Guin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kroder, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kumar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Fu,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Borrmann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Shekhar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Felser, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Meng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  Grushin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sun, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Gooth, Anisotropic  electrical and thermal magnetotransport in the magnetic  semimetal gdptbi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 101, 125119 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [79] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Kapri, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Dey, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Ghosh, Role of berry curva-  ture in the generation of spin currents in rashba systems,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 103, 165401 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [80] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Sinova, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Valenzuela, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Wunderlich, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Back,  and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Jungwirth, Spin hall effects, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' 87,  1213 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [81] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Samokhin, Spin–orbit coupling and semiclassical elec-  tron dynamics in noncentrosymmetric metals, Annals of  Physics 324, 2385 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [82] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Fecher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' K¨ubler, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Felser, Chirality in the  solid state: Chiral crystal structures in chiral and achiral  space groups, Materials 15, 5812 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='  [83] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Park, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Cheon, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Lee, Nondivergent chiral  charge pumping in weyl semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
+page_content=' B 106,  075140 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfKgN7/content/2301.02965v1.pdf'}
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+arXiv:2301.03869v1  [hep-ph]  10 Jan 2023
+Effective-Lagrangian study of ψ′(J/ψ) → V P and the insights
+into ρπ puzzle
+Lin-Wan Yana, Yun-Hua Chenb, Chun-Gui Duana, Zhi-Hui Guoa
+a Department of Physics and Hebei Key Laboratory of Photophysics Research and Application,
+Hebei Normal University, Shijiazhuang 050024, China
+b School of Mathematics and Physics,
+University of Science and Technology Beijing, Beijing 100083, China
+Abstract
+Within the effective Lagrangian approach, we carry out a unified study of the J/ψ(ψ′) →
+V P, J/ψ → Pγ and relevant radiative decays of light-flavor hadrons. Large amount of
+experimental data, including the various decay widths and electromagnetic form factors, are
+fitted to constrain the numerous hadron couplings. Relative strengths between the strong
+and electromagnetic interactions are revealed in the J/ψ → V P and ψ′ → V P processes.
+The effect from the strong interaction is found to dominate in the J/ψ → ρπ decay, while
+the electromagnetic interaction turns out to be the dominant effect in ψ′ → ρπ decay,
+which provides an explanation to the ρπ puzzle. For the J/ψ → K∗ ¯K + ¯K∗K and ψ′ →
+K∗ ¯K + ¯K∗K, the former process is dominated by the strong interactions, and the effects
+from the electromagnetic parts are found to be comparable with those of strong interactions
+in the latter process. Different SU(3) breaking effects from the electromagnetic parts appear
+in the charged and neutral channels for the ψ′ → K∗ ¯K + ¯K∗K processes explain the rather
+different ratios between B(ψ′ → K∗+K− + K∗−K+)/B(J/ψ → K∗+K− + K∗−K+) and
+B(ψ′ → K∗0 ¯K0 + ¯K∗0K0)/B(J/ψ → K∗0 ¯K0 + ¯K∗0K0).
+1
+Introduction
+The strong suppression of the branching ratios of the ψ′ → ρπ and ψ′ → K∗ ¯K + c.c.
+processes, relative to those of the corresponding decay channels of the J/ψ, has been a long-
+standing puzzle in charmonium physics. The annihilation of the c¯c into three gluons is usually
+assumed to be the dominant mechanism that rules the decays of the J/ψ and ψ′ to the light-
+flavor hadrons [1–3]. The annihilation amplitudes of the latter processes and also the decays to
+the lepton pairs are proportional to the wave functions of the S-wave charmonium states J/ψ
+and ψ′ at the origin. As a result, the branching ratios with the light-flavor-hadron (h) decays
+of the ψ′ and J/ψ can be predicted by their leptonic decay widths [4], i.e.,
+Qh ≡
+B(ψ′ → h)
+B(J/ψ → h) =
+B(ψ′ → e+e−)
+B(J/ψ → e+e−) = (13.3 ± 0.4)% .
+(1)
+However, according to the recent PDG averages [4], the ratio of Qρπ,
+B(ψ′ → ρπ)
+B(J/ψ → ρπ) = (0.2 ± 0.1)% ,
+(2)
+1
+
+and the various ratios of QK∗ ¯
+K,
+B(ψ′ → K∗+K− + c.c.)
+B(J/ψ → K∗+K− + c.c.)
+=
+(0.5+0.2
+−0.1)% ,
+B(ψ′ → K∗0 ¯K0 + c.c.)
+B(J/ψ → K∗0 ¯K0 + c.c.)
+=
+(2.6+0.8
+−0.7)% ,
+B(ψ′ → K∗ ¯K + c.c.)
+B(J/ψ → K∗ ¯K + c.c.)
+=
+(1.4+0.5
+−0.4)% ,
+(3)
+are drastically different from the prediction in Eq. (1).
+These contradictions are generally
+referred as the ρπ puzzle, which was first established in Ref. [5] four decades ago. Tremendous
+efforts have been made to address this problem, including the proposal of a vector glueball near
+the J/ψ mass [6], higher Fock components in the charmonium states [7], the intrinsic charm
+portions in the light-flavor vector ρ [8], the nodes in the wave functions [9], the meson mixing
+mechanisms [10], the final-state interactions [11–14]. Another important issue in those decays
+is the sizable SU(3) breaking effects in the charged and neutral K∗ ¯K decays of the J/ψ and ψ′.
+I.e., the strict SU(3) flavor symmetry would give the prediction QK∗+K−+c.c. = QK∗0 ¯
+K0+c.c.,
+which is however severely violated according to the experimental measurements in Eq. (3).
+In this work, we aim at a unified description of the processes J/ψ(ψ′) → V P and γ(∗)P,
+with V and P the light-flavor vector and pseudoscalar mesons in order. In such kinds of de-
+cay processes, one needs to simultaneously take into account of the single-Okubo-Zweig-Iizuka
+(OZI) or even the doubly suppressed OZI strong interaction effects, the electromagnetic contri-
+butions and the SU(3) flavor symmetry breaking terms. The effective Lagrangian approach can
+provide an excellent framework to properly include all the aforementioned effects. Regarding
+the well-celebrated OZI rule, a quantitative way to understand such suppression mechanism
+is the large NC QCD [15].
+In the effective field theory (EFT) approach, the NC counting
+order can be directly related to the number of traces in the flavor space [16,17]. Typically one
+additional flavor trace will introduce one more 1/NC suppression order to the EFT operator.
+The single OZI/double OZI effects can be systematically incorporated via the EFT operators
+with the proper numbers of flavor traces. Furthermore, the chiral EFT is constructed accord-
+ing to the spontaneous and explicit chiral symmetry breaking patterns of QCD. The SU(3)
+flavor symmetry breaking effects can be then introduced through the basic building tensors of
+the EFT involving the small but nonvanishing light-flavor quark masses. Although the chiral
+power counting scheme based on the momentum expansion is not valid in the massive J/ψ
+or ψ′ decays, the basic building blocks and methodology of the EFT Lagrangians are useful
+to conveniently take into account all the relevant ingredients describing the J/ψ(ψ′) → V P
+and J/ψ → γ(∗)P processes, including the OZI strong interaction parts, the electromagnetic
+contributions and the SU(3) flavor symmetry breaking effects. This formalism has been suc-
+cessfully applied to the light-flavor decay processes of V → Pγ(∗), e+e− → K∗ ¯K + c.c. and
+J/ψ → V P, Pγ(∗) in a series of works in Refs. [18–20]. In this work we push forward the study
+along the line of this research to address the mysterious ρπ puzzle by including similar decay
+processes of ψ′. In addition, we also perform the global analyses of the large amount of updated
+branching ratios of various decay processes from the PDG [4] and the newly measured different
+decay widths from the BESIII collaboration [21].
+This paper is organized as follows. In Sec. 2, we introduce the relevant effective Lagrangians
+and elaborate the calculations of the decay amplitudes. The global fit to the various experi-
+mental data and the phenomenological consequences are analyzed in detail in Sec. 3. We give
+the short summary and conclusions in Sec. 4.
+2
+
+2
+Effective Lagrangian and calculations of transition ampli-
+tudes
+The primary aim of this work is to study the various decay processes of the J/ψ and ψ′
+into a light-flavor vector and a light pseudoscalar meson, and the light-flavor meson radiative
+decays and relevant form factors. Therefore we need to include not only the transition operators
+between the charmonia and the light-flavor mesons, but also the EFT operators describing the
+interactions among the light-meson themselves. To tightly constrain the free couplings, we
+simultaneously take into account the experimental data from both the decay processes with
+only light-flavor mesons and also the processes involving the J/ψ and ψ′.
+Resonance chiral theory (RχT) [22] provides a reliable framework to study the interactions
+of the light-flavor resonances and the light pseudoscalar mesons (π, K, η), the latter of which
+are treated as the pseudo-Nambu-Goldstone bosons (pNGBs) resulting from the spontaneous
+symmetry breaking of QCD. As an extension of the chiral perturbation theory (χPT), RχT
+explicitly introduces the heavier degrees of freedom of QCD, i.e., the light-flavor resonances,
+such as the vectors ρ, K∗, ω, φ, the axial vectors, scalars, etc, into the chiral Lagrangians,
+together with the pNGBs and external source fields, like the photons. The RχT operators are
+constructed in a chiral covariant way, therefore the physical amplitudes calculated in the RχT
+automatically fulfill the requirements of chiral symmetry of QCD in the low energy region. On
+the other hand, the large NC expansion of QCD [23] has been widely used as another useful
+guide to arrange the operators and amplitudes of the RχT [24]. In addition, from the large NC
+point of view, the QCD UA(1) anomaly effect, which is considered to be the most responsible
+factor for the large mass of the physical state η′, is however 1/NC suppressed. As a result,
+the η′ state would become the ninth pNGB both in large NC and chiral limits. Based on this
+argument, the nonet of the pNGBs (π, K, η, η′) can be systematically included in the effective
+Lagrangian [25]. We closely follow this guideline to include the singlet η0 state in the RχT and
+adopt the general two-mixing-angle formalism to study the physical processes with the η and
+η′ mesons. Next we briefly introduce the relevant RχT Lagrangians.
+In the present work, only the light-flavor vector resonances will be relevant to our study
+and the minimal interaction operators with the vectors in even-intrinsic-parity sector of the
+RχT is given by [22]
+L (2)
+V
+=
+FV
+2
+√
+2
+⟨Vµνf µν
++ ⟩ + iGV
+√
+2
+⟨Vµνuµuν⟩ ,
+(4)
+where the nonent of the vector resonances is incorporated via the 3 × 3 matrix
+Vµν =
+
+
+
+
+
+1
+√
+2ρ0 +
+1
+√
+6ω8 +
+1
+√
+3ω0
+ρ+
+K∗+
+ρ−
+− 1
+√
+2ρ0 +
+1
+√
+6ω8 +
+1
+√
+3ω0
+K∗0
+K∗−
+K
+∗0
+− 2
+√
+6ω8 +
+1
+√
+3ω0
+
+
+
+
+
+µν
+,
+(5)
+the basic chiral tensors with the pNGBs and the external source fields are defined as
+U = u2 = ei
+√
+2Φ
+F
+,
+uµ = i
+�
+u†(∂µ − irµ)u − u(∂µ − uℓµ)u†�
+,
+f µν
+± = uF µν
+L u† ± u†F µν
+R u ,
+F µν
+L(R) = ∂µl(r)ν − ∂νl(r)µ ,
+χ± = u†χu† ± uχ†u ,
+χ = 2B0(s + ip)
+(6)
+3
+
+and the flavor contents of the nonet pNGB matrix read
+Φ =
+
+
+
+
+
+1
+√
+2π0 +
+1
+√
+6η8 +
+1
+√
+3η0
+π+
+K+
+π−
+− 1
+√
+2π0 +
+1
+√
+6η8 +
+1
+√
+3η0
+K0
+K−
+K
+0
+− 2
+√
+6η8 +
+1
+√
+3η0
+
+
+
+
+ .
+(7)
+The quark-mass terms are introduced by taking the scalar external source filed s in Eq. (6) as
+s = diag{mu, md, ms}. In this work, we will take mu = md = ˆm throughout, i.e. neglecting
+the isospin breaking effects from the strong interaction parts. The physical vectors ω and φ
+can be well described by assuming the ideal mixing of the octet ω8 and the singlet ω0 [4]
+ω0
+=
+�
+2
+3ω −
+�
+1
+3φ,
+ω8
+=
+�
+2
+3φ +
+�
+1
+3ω.
+(8)
+In contrast, the mixing pattern of the η8 and η0 is more involved. The modern chiral prescrip-
+tion introduces the sophisticated two-mixing-angle scheme [26, 27] to address the η-η′ mixing
+system
+
+ η
+η′
+
+ = 1
+F
+
+ F8 cos θ8
+−F0 sin θ0
+F8 sin θ8
+F0 cos θ0
+
+
+
+ η8
+η0
+
+ ,
+(9)
+where F0 and F8 are the weak decay constants of the singlet and octet axial-vector currents,
+respectively. The conventional mixing formula with a single mixing angle can be naturally
+recovered by taking F8 = F0 = F and θ0 = θ8 in Eq.(9). Equivalently one can also use the
+quark-flavor basis to describe the two-mixing-angle formalism
+
+ η
+η′
+
+ = 1
+F
+
+ Fq cos θq
+−Fs sin θs
+Fq sin θq
+Fs cos θs
+
+
+
+ ηq
+ηs
+
+ ,
+(10)
+where the quark-flavor contents of the states ηq = (η8 +
+√
+2η0)/
+√
+3 and ηs = (−
+√
+2η8 + η0)/
+√
+3
+are (¯uu + ¯dd)/
+√
+2 and ¯ss, respectively.
+The vector resonances in Eq. (4) are expressed in terms of the anti-symmetric tensors,
+instead of the commonly used Proca fields.
+It is demonstrated in Refs. [22, 28] that it is
+convenient to use the anti-symmetric tensors to describe the vector resonances in RχT, since
+the high energy behaviors of the resulting amplitudes and form factors automatically match
+the QCD constraints without requiring the inclusion of extra local chiral counter terms in
+the anti-symmetric tensor formalism. The RχT Lagrangians in the odd-intrinsic-parity sector
+comprise two different classes, namely the V V P and V JP types, with J the external sources.
+Those RχT operators written in terms of the anti-symmetric tensor fields that are relevant
+to the O(p4) chiral low energy constants, are worked out in Ref. [29], and the relevant RχT
+Lagrangians and discussions on the V V P Green functions by explicitly including the dynamical
+singlet η0 state are given in Ref. [18]. A more complete basis of the odd-intrinsic-parity RχT
+operators that contribute to the O(p6) chiral low energy constants, is given in Ref. [30]. A
+proliferation of the unknown resonance couplings arise in the more complete RχT Lagrangians,
+4
+
+as expected. This can hinder one from giving the definite conclusions on the phenomenological
+discussions [31]. From the practical point of view, we will work with the RχT operator basis
+from Refs. [18, 29] and we believe that the higher order effects from the extra operators in
+Ref. [30] can be accounted for by the uncertainties of the resonance couplings in the former
+two references. For the sake of completeness, we give the explicit expressions of the relevant
+RχT Lagrangians [18,29]
+LV V P =
+d1εµνρσ⟨{V µν, V ρα}∇αuσ⟩ + id2εµνρσ⟨{V µν, V ρσ}χ−⟩ + d3εµνρσ⟨{∇αV µν, V ρα}uσ⟩
++d4εµνρσ⟨{∇σV µν, V ρα}uα⟩ − id5M2
+V
+�
+2
+3εµνρσ⟨V µνV ρσ⟩ ln(det u) ,
+(11)
+and
+LV JP =
+c1
+MV
+εµνρσ⟨{V µν, f ρα
++ }∇αuσ⟩ + c2
+MV
+εµνρσ⟨{V µα, f ρσ
++ }∇αuν⟩ + ic3
+MV
+εµνρσ⟨{V µν, f ρσ
++ }χ−⟩
++ ic4
+MV
+εµνρσ⟨V µν[f ρσ
+− , χ+]⟩ + c5
+MV
+εµνρσ⟨{∇αV µν, f ρα
++ }uσ⟩ + c6
+MV
+εµνρσ⟨{∇αV µα, f ρσ
++ }uν⟩
++ c7
+MV
+εµνρσ⟨{∇σV µν, f ρα
++ }uα⟩ − ic8MV
+�
+2
+3εµνρσ⟨V µν ˜f ρσ
++ ⟩ ln(det u) ,
+(12)
+where the covariant derivative acting on the chiral field X is given by
+∇µX = ∂µX + [Γµ, X] ,
+Γµ = 1
+2
+�
+u+(∂µ − irµ)u + u(∂µ − ilµ)u+�
+.
+(13)
+As previously mentioned in the Introduction, both the strong and electromagnetic interac-
+tions can be important in the J/ψ(ψ′) → V P processes. The effects from the strong interactions
+are taken into account by the direct J/ψ(ψ′)V P transition operators [18]
+Lψ(ψ′)V P =
+Mψ(ψ′)h(′)
+1 εµνρσψ(′)µ⟨uνV ρσ⟩ +
+1
+Mψ(ψ′)
+h(′)
+2 εµνρσψ(′)µ⟨{uν, V ρσ}χ+⟩
++Mψ(ψ′)h(′)
+3 εµνρσψ(′)µ⟨uν⟩⟨V ρσ⟩ ,
+(14)
+where the couplings h(′)
+i=1,2,3 corresponding to the J/ψ and ψ′ will be separately fitted to the
+experimental data of the two charmonium states. Two types of EFT operators are introduced
+to account for the electromagnetic effects, which include the direct ψPγ transition operators
+LψP γ = g1εµνρσψµ⟨uνf ρσ
++ ⟩ +
+1
+M2
+ψ
+g2εµνρσψµ⟨{uν, f ρσ
++ }χ+⟩ ,
+(15)
+and the conversion vertex of the charmonium and the photon
+Lψγ = −1
+2
+√
+2
+fψ
+Mψ
+⟨ ˆψµνf µν
++ ⟩ ,
+(16)
+being ˆψµν = ∂µψν − ∂νψµ. The values of the couplings g1, g2 and fψ are different for the J/ψ
+and ψ′ and they will be determined by the relevant experimental data. Different powers of the
+Mψ(ψ′) are introduced in Eqs. (14)-(16), so that the couplings appearing in those Lagrangians
+are dimensionless.
+5
+
+It is found [20,32] that the J/ψ → η(′)γ(∗) amplitudes are dominated by the ηc mediating
+diagrams, i.e., via the J/ψ → ηcγ(∗) → η(′)γ(∗) intermediate processes. The decay amplitude
+of the ψ → η(′)γ(∗) can be written as
+Mmixing
+ψ→η(′)γ∗ = e εµνρσǫµ
+ψǫν
+γ∗qρkσ ληcη(′) gψηcγ∗(s) eiδP ,
+(17)
+being P = η, η′, where the electromagnetic transition form factor between the ψ and ηc takes
+form [33–35]
+gψηcγ∗(s) = gψηcγ∗(0)e
+s
+16β2 .
+(18)
+For the mixing parameters ληcη(′) between the ηc and η(′) states, we take the determinations
+ληcη = −4.6 × 10−3 and ληcη′ = −1.2 × 10−2 from Ref. [32]. The phenomenological phase
+factors δη(′) in front of the ηc mediating diagrams need to be separately fitted to the data of
+the J/ψ .
+V
+P
+=
++
+P
+V
+P
+V ′
+γ(∗)
+γ(∗)
+γ(∗)
+V
+(a)
+(b)
+Figure 1:
+Diagrams relevant to the V → Pγ(∗) processes: (a) direct type and (b) indirect
+type.
+γ∗
+γ∗
+γ∗
+V
+V
+V
++
++
++
+=
+V
+P
+ψ
+V
+P
+P
+P
+η(′)
+ηc
+ψ
+ψ
+ψ
+ψ
+(a)
+(b)
+(c)
+(d)
+Figure 2:
+Feynman diagrams for the processes J/ψ → V P. The notations of the solid square
+in diagram (c) and the open circle in diagram (d) are explained in the text.
+The various Feynman diagrams relevant to our study are illustrated in Figs.1, 2 and 3. To
+be more specific, the diagrams in Fig. 1 contribute to the light-flavor processes V → Pγ(∗) and
+P → V γ(∗). The amplitudes of the J/ψ(ψ′) → V P and J/ψ → Pγ(∗) receive contributions
+from the diagrams in Figs. 2 and 3, respectively. The formulas relevant to the V → Pγ(∗) and
+P → V γ(∗) processes are worked out in Ref. [18], and the expressions of the J/ψ → V P, Pγ(∗)
+amplitudes are calculated in Ref. [20]. The corresponding decay amplitudes of the ψ′ state
+share similar expressions as those involving J/ψ, with obvious replacements of the resonance
+6
+
+P
+ψ
+γ(∗)
+=
+γ(∗)
++
+V
+P
+ψ
++
+ηc
+ψ
+P
+ψ
+γ(∗)
+γ(∗)
+η(′)
+(a)
+(b)
+(c)
+Figure 3:
+Feynman diagrams for the processes J/ψ → Pγ(∗)
+couplings. Nevertheless, for the sake of completeness and to set up the notations, we further
+elaborate the amplitudes of the processes of ψ′ → V P .
+For the ψ′ → V P decay, the first diagram (a) in Fig. 2 denotes the contributions from the
+strong interactions, i.e., from the Lagrangians in Eq. (14). Other diagrams in Fig. 2 correspond
+to the electromagnetic effects. The ψ′ → V P amplitude can be written as
+Mψ′→V P = εµνρσǫµ
+ψ′ǫν
+V qρkσGψ′→V P ,
+(19)
+where the polarization vectors of the ψ′ and V are given by ǫµ
+ψ′ and ǫν
+V , q and k stand for the
+four-momentum of the ψ′ and V , respectively. The effective couplings Gψ′→V P include various
+contributions from the individual diagram of Fig. 2. The explicit expressions of Gψ′→V P for
+the various processes are given in Appendix (A). The decay widths of ψ′ → V P read
+Γ(ψ′ → V P) =
+1
+96πM3
+ψ′
+λ(Mψ′, MV , mP)
+3
+2 ��Gψ′→V P
+��2 ,
+(20)
+with the K¨all´en function λ(x, y, z) = x2 + y2 + z2 − 2xy − 2xz − 2yz.
+Similarly, the corresponding amplitude of the radiative process J/ψ(q) → γ∗(k)P(q − k)
+can be given in terms of one effective coupling as well
+Mψ→P γ∗ = e εµνρσǫµ
+ψǫν
+γ∗qρkσGψ→P γ∗(s) ,
+(21)
+with s = k2. The effective coupling Gψ→P γ∗ can receive contributions from all the diagrams in
+Fig. 3. The explicit expressions are given in Ref. [20]. The formula of the decay width of the
+J/ψ → Pγ process finds it form
+Γψ→P γ = 1
+3α
+�
+M2
+ψ − M2
+P
+2Mψ
+�3
+|Gψ→P γ∗(0)|2 .
+(22)
+The expression of the width for the Dalitz decay process J/ψ → Pγ∗ → Pl+l− is given by
+Γψ→P l+l− =
+� (Mψ−mP )2
+4m2
+l
+α2(2m2
+l + s)
+72M3
+ψπs3
+�
+s(s − 4m2
+l )
+�
+λ(s, Mψ, mP)]
+3
+2 |Gψ→P γ∗(s)|2ds .
+(23)
+3
+Comprehensive fits and phenomenological discussions
+Compared to the previous studies in Refs. [18,20], we incorporate in this work the data from
+the various ψ′ → V P decays, apart from other types of data from the V → Pγ(∗), P → V γ(∗)
+7
+
+and J/ψ → V P, Pγ(∗) processes, to perform a comprehensive fit, so that the ρπ puzzle can be
+addressed. In addition, we update numerous types of data according to the most recent PDG
+averages [4], and timely revise the determinations of the resonance couplings.
+In total, we include 135 data points from several different types of processes in the com-
+prehensive fit. To be more specific, the data from the pure light-flavor processes amount to 70,
+and they consist of both the decay widths, such as those of the ω → ηγ, η′ → ωγ, η → γγ, etc,
+and the form factors of the φ → ηγ∗ and η(′) → γγ∗. For the data related to the J/ψ, they
+include all the available widths of the J/ψ → V P, Pγ and Pe+e− from the PDG [4], and also
+the recent BESIII measurements of the invariant-mass distributions of the lepton pairs in the
+transition of J/ψ → ηγ∗ [21]. Regarding the data of the ψ′, we will include in the joint fit all
+the available widths of the ψ′ → V P processes from PDG [4].
+An efficient way to reduce the number of unknown couplings in the RχT is to impose
+the high energy constraints dictated by QCD to the various form factors and Green functions
+calculated from the RχT Lagrangians in Eqs. (4), (11) and (12). Furthermore, the high energy
+behaviors of the resulting amplitudes after imposing such constraints will mimic the properties
+as predicted by QCD. Following the previous discussions in Refs. [18–20, 29, 36–38], we take
+the following high energy constraints on the various couplings
+4c3 + c1 = 0 ,
+c1 − c2 + c5 = 0 ,
+c5 − c6 = NC
+64π2
+MV
+√
+2FV
+,
+d1 + 8d2 − d3 = F 2
+8F 2
+V
+,
+d3 = − NC
+64π2
+M2
+V
+F 2
+V ,
+c8 = −
+√
+2M2
+0
+√
+3M2
+V
+c1 ,
+(24)
+where the pion weak decay constant takes the normalization F = 92.4 MeV throughout, the
+UA(1) anomaly parameter is set to be M0 = 900 MeV [39, 40], the chiral-limit mass of the
+lowest vector resonance multiplet is fixed at MV = Mρ = 775 MeV and the vector-photon
+transition coupling FV will be fitted. By taking into account the the leptonic widths of the
+J/ψ and ψ′, we can determine the charmonium-photon transition coupling fJ/ψ(ψ′) in Eq. (16),
+whose explicit values are found to be
+fJ/ψ = 293.8 ± 3.5 MeV ,
+fψ′ = 208.1 ± 5.1 MeV .
+(25)
+We are then left with 23 undetermined parameters, including the four η-η′ mixing param-
+eters introduced in Eq. (9), four couplings FV , c3, c4 and d2 that emerge from the light-flavor
+resonance interactions, nine parameters exclusively entering in the J/ψ decays and six pa-
+rameters that are dedicated to the ψ′ processes. The couplings that describe the interactions
+of the light-flavor resonances will also enter in the charmonia decays.
+Therefore the joint
+fits by simultaneously including the relevant data of the light-flavor mesons, the data from
+the J/ψ → V P, Pγ(∗) and the ψ′ ones, will obviously give more stringent constraints on the
+couplings than the situation by including just one of these data sets. Furthermore, such com-
+prehensive studies in a unified framework are also expected to give a further insight into ρπ
+puzzle elaborated in the Introduction.
+The resulting parameters from the joint fit are given in Table. 1. The updated parameters
+related to the light-flavor resonances, the J/ψ decays and the η-η′ mixing are well consistent
+with the previous determinations [18–20] where the data of the ψ′ processes are not included
+in these studies. For the ψ′ → Pγ(∗) processes, which could receive significant contributions
+from the ψ′ → J/ψP transition vertexes, i.e. ψ′ → J/ψη → γη [41], are not considered in
+this work. Therefore we will not discuss such kinds of processes here. The relative phases
+8
+
+F8
+(1.41 ± 0.02)Fπ
+F0
+(1.36 ± 0.03)Fπ
+θ8
+(−24.3 ± 0.4)◦
+θ0
+(−12.8 ± 0.5)◦
+FV
+139.04 ± 1.72
+c3
+0.0046 ± 0.0003
+c4
+−0.0014 ± 0.0001
+d2
+0.100 ± 0.008
+h1
+(−2.35 ± 0.06) × 10−5
+h2
+(-3.08±0.60)×10−5
+h3
+(3.39±0.22)×10−6
+g1
+(-2.40±0.06)×10−5
+g2
+(-2.23±0.48)×10−4
+r1
+0.40±0.04
+h′
+1
+(0.33±0.23)×10−6
+h′
+2
+(-4.01±0.32)×10−5
+h′
+3
+(0.85±0.47)×10−6
+g′
+1
+(-1.70±0.47)×10−4
+g′
+2
+(0.18±0.95)×10−3
+δη
+(117.12 ± 3.81)◦
+δη′
+(50.03 ± 16.01)◦
+β
+512.86 ± 7.36 MeV
+β′
+112.97 ± 0.98 MeV
+F (∗)
+q
+(1.24±0.02)Fπ
+F (∗)
+s
+(1.52±0.02)Fπ
+θ(∗)
+q
+(37.3 ± 0.7)◦
+θ(∗)
+s
+(35.1 ± 0.4)◦
+χ2/d.o.f
+157.25/(135-23)=1.40
+Table 1:
+Parameters from the joint fit. The quantities marked with asterisk are predictions,
+instead of free parameters in the fit.
+δη(′) of Eq. (17) for the ηc mediating effects in the ψ′ → V η(′) decay processes, are found to
+be insensitive to our present studies. As a result, the phases of δη(′) in the ψ′ decays will be
+fixed to zero throughout. The previous study in Ref. [18] pointed out a strong correlation
+between the d2 and d5 parameters, and we find that this correlation still holds in our joint
+fit. The resulting relation turns out to be d5 = 3.57d2 + 0.01. Regarding the four parameters
+F8, F0, θ8 and θ0 related with the η-η′ mixing, our current determinations of the central values
+and uncertainties more or less resemble those in Ref. [20]. In Ref. [18], only the data from
+the light-flavor sector were considered and the resulting η-η′ mixing parameters were found to
+bear large uncertainties. The simultaneous inclusion of the relevant data from the J/ψ and
+ψ′ processes, together with the light-flavor ones, can obviously pin down the uncertainties of
+the η-η′ mixing parameters [20,42–44]. In Table 1, we also give the predictions to the mixing
+parameters in the quark-flavor basis.
+Generally speaking, the numerous types of data are well reproduced in our comprehensive
+fit.
+The comparisons of the various decay widths for the pure light-flavor processes from
+the revised fit and the updated PDG values are shown in Table. 2. Similar comparisons for
+the partial decay widths of the J/ψ and ψ′ are given in Tables. 3 and 4, respectively. The
+resulting curves of the form factors for the η → γγ∗, η′ → γγ∗, φ → ηγ∗ and J/ψ → η′γ∗ are
+shown together with the experimental data in Fig. 4. The fitted results of the recent BESIII
+measurements on the e+e− spectra in the J/ψ → ηe+e− processes are illustrated in Fig. 5. We
+point out a subtlety about the effects of the light-flavor vector resonances in the e+e− spectra.
+In the J/ψ → η′e+e− decays, the light-flavor vectors are removed in the BESIII analysis [45],
+and as a result we have also subtracted the contributions from the intermediate light vector
+exchanges in accord with the experimental setups. This explains the smooth line shapes of
+the electromagnetic J/ψ → η′e+e− transition form factors shown in Fig. 4. Regarding the
+J/ψ → ηe+e− process, we keep the effects of the intermediate light-flavor vector resonances,
+in order to be consistent with the setups of the experimental analyses in Ref. [21]. It should
+be stressed that the prominent peaks of the narrow vectors ω and φ can be diluted due to the
+large bin widths of the experimental energy resolutions. To clearly show the influence of the
+9
+
+Exp
+Fit
+Γω→πγ
+724.78 ± 34.64
+705.65 ± 17.40
+Γρ0→π0γ
+70.08 ± 12.37
+73.23 ± 1.81
+ΓK∗0→K0γ
+116.36 ± 11.27
+108.95 ± 2.69
+Γω→πe−e+
+6.68 ± 0.63
+6.40 ± 0.16
+Γω→πµ−µ+
+1.16 ± 0.18
+0.63 ± 0.02
+Γω→ηγ
+3.91 ± 0.41
+5.30 ± 0.11
+Γρ0→ηγ
+44.73 ± 3.39
+43.93 ± 0.96
+Γφ→ηγ
+55.28 ± 1.23
+55.01 ± 1.00
+Γφ→η′γ
+0.26 ± 0.01
+0.26 ± 0.01
+Γη′→ωγ
+4.74 ± 0.29
+5.05 ± 0.18
+Γη→γγ
+0.52 ± 0.02
+0.50 ± 0.01
+Γη′→γγ
+4.34 ± 0.20
+3.92 ± 0.11
+Γη→γe−e+
+(9.04 ± 0.89) × 10−3
+(8.32 ± 0.23) × 10−3
+Γη→γµ−µ+
+(0.41 ± 0.07) × 10−3
+(0.39 ± 0.01) × 10−3
+Γη′→γµ−µ+
+(2.12 ± 0.61) × 10−2
+(1.47 ± 0.04) × 10−2
+Γφ→ηe−e+
+0.459 ± 0.018
+0.460 ± 0.008
+Table 2:
+The decay widths in units of KeV for the light-flavor hadrons.
+Exp
+Fit
+J/ψ → ρ0π0
+5.6 ± 0.7
+5.5 ± 0.3
+J/ψ → ρπ
+16.9 ± 1.5
+16.2 ± 1.0
+J/ψ → ρ0η
+0.193 ± 0.023
+0.185 ± 0.021
+J/ψ → ρ0η′
+0.081 ± 0.008
+0.080 ± 0.007
+J/ψ → ωπ0
+0.45 ± 0.05
+0.45 ± 0.04
+J/ψ → ωη
+1.74 ± 0.20
+1.65 ± 0.09
+J/ψ → ωη′
+0.189 ± 0.018
+0.189 ± 0.018
+J/ψ → φη
+0.74 ± 0.08
+0.76 ± 0.06
+J/ψ → φη′
+0.46 ± 0.05
+0.45 ± 0.05
+J/ψ → K∗+K− + c.c.
+6.0 ± 1.0
+6.6 ± 0.3
+J/ψ → K∗0 ¯
+K0 + c.c.
+4.2 ± 0.4
+3.8 ± 0.2
+J/ψ → π0γ
+0.0356 ± 0.0017
+0.0341 ± 0.0016
+J/ψ → ηγ
+1.085 ± 0.018
+1.085 ± 0.013
+J/ψ → η′γ
+5.25 ± 0.07
+5.35 ± 0.04
+J/ψ → π0e+e−
+(0.076 ± 0.014) × 10−2
+(0.129 ± 0.004) × 10−2
+J/ψ → ηe+e−
+(1.42 ± 0.08) × 10−2
+(1.35 ± 0.02) × 10−2
+J/ψ → η′e+e−
+(6.59 ± 0.18) × 10−2
+(6.08 ± 0.05) × 10−2
+Table 3:
+Branching fractions(×10−3) of the decay processes for J/ψ .
+10
+
+bin widths, we give the histograms by using the energy bin width at 50 MeV. It is evident that
+the signals of narrow light vector resonances can be obviously enhanced when the energy bin
+width is reduced.
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.1
+1
+-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8
+0.1
+1
+10
+100
+1000
+0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
+-4
+-2
+0
+2
+4
+6
+8
+10
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+pa
+2m3
+h
+/4 F
+h
+(s,0)
+2 [keV]
+s[GeV2]
+h
+gg
+*
+pa
+2m3
+h
+/4 F
+h'
+(s,0)
+2 [keV]
+s[GeV2]
+h
+gg
+*
+|F
+fhg(s)/F
+fhg(0)|
+2
+s1/2 [GeV]
+f
+hg
+*
+|FJ/yh
+|
+2
+M(e+e-) (GeV/c2)
+J/y
+h
+g
+*
+Figure 4:
+The form factors for the η → γγ∗, η′ → γγ∗, φ → ηγ∗ and J/ψ → η′γ∗. The
+red solid lines are obtained by taking the central values of the parameters in Table 1, and the
+shaded areas correspond to the error bands at 1-σ level. The experimental data on the form
+factors for the η → γγ∗ , η′ → γγ∗, φ → ηγ∗ and J/ψ → η′γ∗ are taken from Refs. [46–51],
+Refs. [46–48,51,52], Ref. [49] and Ref. [45],respectively.
+With the fitted parameters in Table 1, it is then interesting to decipher the roles of different
+mechanisms and resonances played in a given process.
+The J/ψ → Pl+l− processes can provide an environment to study the intermediate hadron
+resonances [20,53]. Recently, the J/ψ → ηγ∗(→ e+e−) form factors are reported by the BESIII
+collaboration in Ref. [21], in which the experimental analysis includes only the ρ resonance in
+the e+e− spectra, apart from the QED contributions. However, it is pointed out that the ρ
+contribution should come from an isospin violated intermediate process J/ψ → ηρ → ηe+e−.
+In contrast, the contributions from the ω and φ are expected to be more important, since
+they enter via the isospin conserved intermediate processes J/ψ → ηω and J/ψ → ηφ, whose
+branching ratios are around eight and four times larger than that of the J/ψ → ρη in order.
+As a result, we expect that the effect of the ρ resonance is much suppressed, compared to
+the contributions from ω and φ. Due to the narrow widths of the latter two resonances, they
+manifest themselves as prominent peaks in the e+e− spectra, as shown in Fig. 5. However,
+these narrow peaks can be easily washed out when the energy resolution is low. E.g., we also
+explicitly give the energy distributions of the e+e− in Fig. 5 when taking the energy bin width
+11
+
+Exp
+Fit
+ψ′ → ρπ
+0.032 ± 0.012
+0.037 ± 0.010
+ψ′ → ρ0η
+0.022 ± 0.006
+0.021 ± 0.005
+ψ′ → ρ0η′
+0.019 ± 0.017
+0.028 ± 0.008
+ψ′ → ωπ0
+0.021 ± 0.006
+0.021 ± 0.004
+ψ′ → ωη
+—
+0.005 ± 0.003
+ψ′ → ωη′
+0.032 ± 0.025
+0.033 ± 0.019
+ψ′ → φη
+0.031 ± 0.0031
+0.032 ± 0.003
+ψ′ → φη′
+0.0154 ± 0.0020
+0.016 ± 0.0019
+ψ′ → K∗+K− + c.c.
+0.029 ± 0.004
+0.029 ± 0.004
+ψ′ → K∗0 ¯
+K0 + c.c.
+0.109 ± 0.020
+0.080 ± 0.011
+Table 4:
+Branching fractions(×10−3) of the decay processes for ψ′. The ψ′ → ωη channel is
+not included in the fit, instead the result corresponds to our prediction, which is around two
+times smaller than the upper limit 1.1 × 10−5 reported in PDG [4].
+at 50 MeV and 100 MeV. In the latter case the signals of the narrow ω and φ become faintly
+visible. As pointed out in Refs. [20,41,54], we also confirm the importance of the η(′)−ηc mixing
+mechanism in the J/ψ → η(′)γ(∗) decay processes. A future experimental measurement with
+higher energy resolution will be definitely helpful to discriminate the roles of different hadrons
+in the J/ψ → ηe+e− process. In Table 5, we give our predictions to the branching ratios of
+various J/ψ → Pl+l− processes and also make comparisons with the results in Refs. [20,55,56].
+Exp
+This Work
+Ref. [20]
+Ref. [55]
+Ref. [56]
+ψ → π0e+e−
+0.076 ± 0.014
+0.1294 ± 0.0044
+0.1191 ± 0.0138
+0.0389+0.0037
+−0.0033
+——
+ψ → ηe+e−
+1.42 ± 0.08
+1.35 ± 0.02
+1.16 ± 0.08
+1.21 ± 0.04
+1.38
+ψ → η′e+e−
+6.59 ± 0.18
+6.08 ± 0.05
+5.76 ± 0.16
+5.66 ± 0.16
+6.06
+ψ → π0µ+µ−
+——
+0.0304 ± 0.0010
+0.0280 ± 0.0032
+0.0101+0.0010
+−0.0009
+——
+ψ → ηµ+µ−
+——
+0.40 ± 0.01
+0.32 ± 0.02
+0.30 ± 0.01
+0.46
+ψ → η′µ+µ−
+——
+1.64 ± 0.02
+1.46 ± 0.04
+1.31 ± 0.04
+1.72
+Table 5: Branching ratios (×10−5) for J/ψ → Pl+l− .
+Our study reveals an interesting feature that can shed light on the ρπ puzzle in the
+J/ψ(ψ′) → V P decays. For this purpose, let’s focus on the interplay between the electro-
+magnetic and strong interactions in the J/ψ(ψ′) → V P processes. We separately show the
+contributions from the strong and electromagnetic interactions to the isospin conserved and
+violated decays for J/ψ and ψ′ in Tables 6 and 7, respectively. The contributions from the
+strong interactions are given by the hi=1,2,3 terms in Eq. (14), while the electromagnetic contri-
+butions are obtained by taking hi=1,2,3 = 0. For the J/ψ → V P decays, the contributions from
+strong interactions turn out to play major roles in most of the isospin conserved channels, with
+the exception of the J/ψ → φη′ process, where the strengths of the two types of interactions
+are comparable. While the isospin violated channels can only receive contributions from the
+electromagnetic interactions, since the isospin breaking effects from the strong interaction parts
+are not included in this work. According to the results shown in Table 7, for the ψ′ → V P
+processes, the strong interactions are found to play comparable roles in many of the isospin
+12
+
+ 0
+ 10
+ 20
+ 30
+ 40
+ 0.2 0.4 0.6 0.8  1  1.2 1.4 1.6 1.8  2  2.2 2.4
+|FJ/ψη|2
+me+e- [GeV/c2]
+1E-06
+1E-05
+1E-04
+ 0.2 0.4 0.6 0.8  1  1.2 1.4 1.6 1.8  2  2.2 2.4
+dB(J/ψ→e+e-η/dq (GeV/c2)-1
+me+e- [GeV/c2]
+Figure 5:
+The form factors and differential branching fractions for the J/ψ → ηe+e−. The
+experimental data are from the Ref. [21]. The red solid lines represent the curves with the
+central values of the parameters in Table.1, and the shaded areas stand for the error bands.
+The histograms are obtained by taking different energy bin width at 50 MeV.
+Isospin conserved cases
+Exp
+Strong interaction
+EM interaction
+| GJ/ψ→ρ0π0 |
+2.537 ± 0.154
+2.899 ± 0.075
+0.385 ± 0.006
+| GJ/ψ→ρπ |
+4.408 ± 0.191
+5.022 ± 0.129
+0.709 ± 0.009
+| GJ/ψ→ωη |
+1.497 ± 0.084
+1.586 ± 0.037
+0.132 ± 0.009
+| GJ/ψ→ωη′ |
+0.562 ± 0.026
+0.647 ± 0.028
+0.119 ± 0.008
+| GJ/ψ→φη |
+1.060 ± 0.056
+1.270 ± 0.045
+0.198 ± 0.031
+| GJ/ψ→φη′ |
+0.974 ± 0.052
+1.074 ± 0.049
+2.031 ± 0.044
+| GJ/ψ→K∗+K− |
+2.011 ± 0.161
+2.313 ± 0.048
+0.216 ± 0.036
+| GJ/ψ→K∗0 ¯
+K0 |
+1.686 ± 0.078
+2.308 ± 0.048
+0.715 ± 0.009
+Isospin violated cases
+Exp
+EM interaction
+Strong interaction
+| GJ/ψ→ρ0η |
+0.498 ± 0.029
+0.487 ± 0.028
+−−
+| GJ/ψ→ρ0η′ |
+0.367 ± 0.018
+0.365 ± 0.017
+−−
+| GJ/ψ→ωπ0 |
+0.721 ± 0.039
+0.720 ± 0.035
+−−
+Table 6:
+The effective couplings of Gψ→V P in units of 10−6MeV−1.
+13
+
+Isospin conserved cases
+Exp
+Strong interaction
+EM interaction
+| Gψ′ → ρπ |
+0.255 ± 0.044
+0.029 ± 0.036
+0.255 ± 0.016
+| Gψ′ → ωη |
+—
+0.103 ± 0.038
+0.036 ± 0.003
+| Gψ′ → ωη′ |
+0.288 ± 0.096
+0.212 ± 0.079
+0.079 ± 0.013
+| Gψ′ → φη |
+0.275 ± 0.013
+0.285 ± 0.030
+0.565 ± 0.029
+| Gψ′ → φη′ |
+0.213 ± 0.013
+0.197 ± 0.068
+0.412 ± 0.067
+| Gψ′ → K∗+K− |
+0.181 ± 0.012
+0.263 ± 0.021
+0.093 ± 0.021
+| Gψ′ → K∗0 ¯
+K0 |
+0.352 ± 0.031
+0.267 ± 0.021
+0.568 ± 0.007
+Isospin violated cases
+Exp
+EM interaction
+Strong interaction
+| Gψ′ → ρ0η |
+0.219 ± 0.028
+0.216 ± 0.026
+−−
+| Gψ′ → ρ0η′ |
+0.222 ± 0.083
+0.271 ± 0.040
+−−
+| Gψ′ → ωπ0 |
+0.207 ± 0.028
+0.208 ± 0.019
+−−
+Table 7:
+The effective couplings of Gψ′→V P in units of 10−6MeV−1.
+conserved channels as those from the electromagnetic parts. Especially, the electromagnetic
+interaction turns out to play the dominant role in the ψ′ → ρπ process and the effects from
+the strong interactions are found to be very small. In contrast, the strong interactions domi-
+nate the decay of J/ψ → ρπ process and the electromagnetic effects appear to be small. This
+provides a sensible explanation to the ρπ puzzle.
+For the charged K∗+K− + c.c. and neutral K∗0 ¯K0 + c.c. decay processes of J/ψ or ψ′, the
+SU(3) breaking effects can originate from the strong interactions via the h2 term in Eq. (14),
+which turns out to be the same for both charged and neutral processes, and the electromagnetic
+interactions via the cj terms in Eq. (12), where the c4 operator is found to solely contribute
+to the charged process [19]. The contributions from the electromagnetic parts to the J/ψ →
+K∗ ¯K + c.c. processes are obviously smaller than those from the strong interactions, which
+explains the similar branching ratios between J/ψ → K∗+K−+c.c. and J/ψ → K∗0 ¯K0+c.c.. In
+contrast, our study reveals that the magnitudes of the strong interactions in the ψ′ → K∗ ¯K+c.c.
+can be comparable with those of the electromagnetic parts. While, the SU(3) breaking effects
+in the electromagnetic parts are quite different for the charged and neutral decay processes due
+to the c4 operator [19]. This gives a new insight and also a reasonable explanation to the very
+different branching ratios of the ψ′ → K∗+K− + c.c. and ψ′ → K∗0 ¯K0 + c.c..
+4
+Summary and conclusions
+We use the effective Lagrangian approach to simultaneously investigate the processes of
+J/ψ(ψ′) → V P, J/ψ → Pγ,J/ψ → Pl+l−, the radiative decays of light-flavor hadrons and
+their relevant form factors. High energy constraints on the resonance couplings are used to
+reduce the number of free parameters. The remaining resonance couplings are then determined
+through the joint fit to a large amount of experimental data, including the updated PDG
+averages of the various partial decay widths and the most recent J/ψ → ηγ∗ form factors from
+BESIII.
+Thanks to the use of effective Lagrangian, the different types of contributions from the
+OZI allowed/suppressed strong interactions, SU(3) breaking terms and electromagnetic ef-
+fects can be easily identified in our study. We pay special attention to the relative magni-
+14
+
+tudes from the strong and electromagnetic interactions in the J/ψ → V P and ψ′ → V P
+processes, so as to provide an insight into the ρπ puzzle.
+An anatomy of the J/ψ → ρπ
+and ψ′ → ρπ amplitudes reveals that the strong interaction dominates the former process
+and the electromagnetic interaction prevails the latter one.
+For the obviously distinct ra-
+tios between the charged B(ψ′ → K∗+K− + c.c.)/B(J/ψ → K∗+K− + c.c.) and the neu-
+tral B(ψ′ → K∗0 ¯K0 + c.c.)/B(J/ψ → K∗0 ¯K0 + c.c.) processes, our study uncovers that the
+J/ψ → K∗ ¯K+c.c. processes are mainly ruled by the strong interactions, where the SU(3) break-
+ing effects enter similarly in both the charged and neutral amplitudes, while the ψ′ → K∗ ¯K+c.c.
+decays are found to be importantly affected by the electromagnetic interactions, where the
+SU(3) symmetry breaking terms appear differently in the charged and neutral processes.
+Acknowledgements
+We thank Lu Niu for an early-stage contribution to this work. This work is partially funded
+by the Natural Science Foundation of China under Grant Nos. 11975090, 12150013, 11975028
+and 11974043.
+A
+The expressions of the effective couplings for ψ′ → V P
+The expressions of the effective couplings in the ψ′ → V P processes defined in Eq. (20)
+take the form:
+Gψ′→ρ0π0 = 2
+√
+2
+FπMρ
+h′
+1Mψ′ + 8
+√
+2
+FπMρ
+h′
+2m2
+π
+1
+Mψ′ + 32πα
+FπMρ
+FV g′
+1 + 128πα
+FπMρ
+FV g′
+2
+m2
+π
+M2
+ψ′
++ 8
+√
+2πα
+3
+fψ′
+Mψ′ Fρπγ∗(M2
+ψ′) ,
+(A.1)
+Gψ′→ρ+π− = 2
+√
+2
+FπMρ
+h′
+1Mψ′ + 8
+√
+2
+FπMρ
+h′
+2m2
+π
+1
+Mψ′ + 8
+√
+2πα
+3
+fψ′
+Mψ′ Fρπγ∗(M2
+ψ′) ,
+(A.2)
+Gψ′→ρ0η =32
+√
+2πα
+3FMρ
+FV g′
+1(a1 − a3) + 128
+√
+2πα
+3FMρM2
+ψ′
+FV g′
+2[a1m2
+π − a3(2m2
+K − m2
+π)]
+− 8πα FV
+Mρ
+ληηcgψ′ηcγ∗(M2
+ρ )eiδ′
+η + 8
+√
+2πα
+3
+fψ′
+Mψ′ Fρηγ∗(M2
+ψ′) ,
+(A.3)
+Gψ′→ρ0η′ =32
+√
+2πα
+3FMρ
+FV g′
+1(a2 − a4) + 128
+√
+2πα
+3FMρM2
+ψ′
+FV g′
+2[a2m2
+π − a4(2m2
+K − m2
+π)]
+− 8πα FV
+Mρ
+λη′ηcgψ′ηcγ∗(M2
+ρ )eiδ′
+η′ + 8
+√
+2πα
+3
+fψ′
+Mψ′ Fρη′γ∗(M2
+ψ′) ,
+(A.4)
+Gψ′→ωπ0 = 32πα
+3FπMω
+FV g′
+1 + 128πα
+3FπMω
+FV g′
+2
+m2
+π
+M2
+ψ′
++ 8
+√
+2πα
+3
+fψ′
+Mψ′ Fωπγ∗(M2
+ψ′) ,
+(A.5)
+15
+
+Gψ′→ωη =
+4
+FMω
+a1h′
+1Mψ′ +
+16
+FMω
+a1h′
+2m2
+π
+1
+Mψ′ +
+4
+FMω
+(2a1 + a3)h′
+3Mψ′ + 32
+√
+2πα
+9FMω
+FV g′
+1(a1 − a3)
++ 128
+√
+2πα
+9FMωM2
+ψ′
+FV g′
+2[a1m2
+π − a3(2m2
+K − m2
+π)] − 8
+3πα FV
+Mω
+ληηcgψ′ηcγ∗(M2
+ω)eiδ′
+η
++ 8
+√
+2πα
+3
+fψ′
+Mψ′ Fωηγ∗(M2
+ψ′) ,
+(A.6)
+Gψ′→ωη′ =
+4
+FMω
+a2h′
+1Mψ′ +
+16
+FMω
+a2h′
+2m2
+π
+1
+Mψ′ +
+4
+FMω
+(2a2 + a4)h′
+3Mψ′ + 32
+√
+2πα
+9FMω
+FV g′
+1(a2 − a4)
++ 128
+√
+2πα
+9FMωM2
+ψ′
+FV g′
+2[a2m2
+π − a4(2m2
+K − m2
+π)] − 8
+3πα FV
+Mω
+λη′ηcgψ′ηcγ∗(M2
+ω)eiδ′
+η′
++ 8
+√
+2πα
+3
+fψ′
+Mψ′ Fωη′γ∗(M2
+ψ′) ,
+(A.7)
+Gψ′→φη = − 2
+√
+2
+FMφ
+a3h′
+1Mψ′ − 8
+√
+2
+FMφ
+a3h′
+2(2m2
+K − m2
+π) 1
+Mψ′ − 2
+√
+2
+FMφ
+(2a1 + a3)h′
+3Mψ′
++ 64πα
+9FMφ
+FV g′
+1(a1 − a3) +
+256πα
+9FMφM2
+ψ′
+FV g′
+2[a1m2
+π − a3(2m2
+K − m2
+π)]
+− 8
+√
+2
+3Mφ
+παFV ληηcgψ′ηcγ∗(M2
+φ)eiδ′
+η + 8
+√
+2πα
+3
+fψ′
+Mψ′ Fφηγ∗(M2
+ψ′) ,
+(A.8)
+Gψ′→φη′ = − 2
+√
+2
+FMφ
+a4h′
+1Mψ′ − 8
+√
+2
+FMφ
+a4h′
+2(2m2
+K − m2
+π) 1
+Mψ′ − 2
+√
+2
+FMφ
+(2a2 + a4)h′
+3Mψ′
++ 64πα
+9FMφ
+FV g′
+1(a2 − a4) +
+256πα
+9FMφM2
+ψ′
+FV g′
+2[a2m2
+π − a4(2m2
+K − m2
+π)]
+− 8
+√
+2
+3Mφ
+παFV λη′ηcgψ′ηcγ∗(M2
+φ)eiδ′
+η′ + 8
+√
+2πα
+3
+fψ′
+Mψ′ Fφη′γ∗(M2
+ψ′) ,
+(A.9)
+Gψ′→K∗+K− =
+2
+√
+2
+FKMK∗ h′
+1Mψ′ +
+8
+√
+2
+FKMK∗ h′
+2m2
+K
+1
+Mψ′ + 8
+√
+2πα
+3
+fψ′
+Mψ′ FK∗+K−γ∗(M2
+ψ′) , (A.10)
+Gψ′→K∗0 ¯
+K0 =
+2
+√
+2
+FKMK∗ h′
+1Mψ′ +
+8
+√
+2
+FKMK∗ h′
+2m2
+K
+1
+Mψ′ + 8
+√
+2πα
+3
+fψ′
+Mψ′ FK∗0 ¯
+K0γ∗(M2
+ψ′) ,
+(A.11)
+16
+
+with
+a1 =
+F
+cos(θ0 − θ8)(cos θ0
+√
+6F8
+− sin θ8
+√
+3F0
+) ,
+a2 =
+F
+cos(θ0 − θ8)( sin θ0
+√
+6F8
++ cos θ8
+√
+3F0
+) ,
+a3 =
+F
+cos(θ0 − θ8)(−2 cos θ0
+√
+6F8
+− sin θ8
+√
+3F0
+) ,
+a4 =
+F
+cos(θ0 − θ8)(−2 sin θ0
+√
+6F8
++ cos θ8
+√
+3F0
+) .
+(A.12)
+The expressions FK∗+K−γ∗ and FK∗0 ¯
+K0γ∗ that correspond to the electromagnetic contributions
+to the effective couplings of Gψ′→K∗+K− and Gψ′→K∗0 ¯
+K0 are given by
+FK∗+K−γ∗(s) =
+−2
+√
+2
+3FKMV MK∗ [(c1 + c2 + 8c3 − c5)m2
+K + (c2 + c5 − c1 − 2c6)M2
+K∗ + (c1 − c2 + c5)s
++ 24c4(m2
+K − m2
+π)] +
+2FV
+3FKMK∗ [(d1 + 8d2 − d3)m2
+K + d3(M2
+K∗ + s)]
+× [Dω(s) + 3Dρ(s) − 2Dφ(s)],
+(A.13)
+and
+FK∗0 ¯
+K0γ∗(s) =
+4
+√
+2
+3FKMV MK∗ [(c1 + c2 + 8c3 − c5)m2
+K + (c2 + c5 − c1 − 2c6)M2
+K∗ + (c1 − c2 + c5)s]
++
+2FV
+3FKMK∗ [(d1 + 8d2 − d3)m2
+K + d3(M2
+K∗ + s)][Dω(s) − 3Dρ(s) − 2Dφ(s)] .
+(A.14)
+It is pointed out that SU(3) breaking effect caused by the c4 term only enters in the charged
+FK∗+K−γ∗ amplitude and is absent in the neutral FK∗0 ¯
+K0γ∗ process. For the expressions of
+other amplitudes FV P γ(∗), they are given in Appendix of Ref. [20] and we do not repeat them
+here.
+References
+[1] T. Appelquist, A. De Rujula, H. D. Politzer and S. L. Glashow, Phys. Rev. Lett. 34, 365
+(1975) doi:10.1103/PhysRevLett.34.365
+[2] T.
+Appelquist
+and
+H.
+D.
+Politzer,
+Phys.
+Rev.
+Lett.
+34,
+43
+(1975)
+doi:10.1103/PhysRevLett.34.43
+[3] A.
+De
+Rujula
+and
+S.
+L.
+Glashow,
+Phys.
+Rev.
+Lett.
+34,
+46-49
+(1975)
+doi:10.1103/PhysRevLett.34.46
+[4] R.
+L.
+Workman
+et
+al.
+[Particle
+Data
+Group],
+PTEP
+2022
+(2022),
+083C01
+doi:10.1093/ptep/ptac097
+[5] M.
+E.
+B.
+Franklin
+et
+al.
+Phys.
+Rev.
+Lett.
+51,
+963-966
+(1983)
+doi:10.1103/PhysRevLett.51.963
+17
+
+[6] M.
+Suzuki,
+Phys.
+Rev.
+D
+63
+(2001),
+054021
+doi:10.1103/PhysRevD.63.054021
+[arXiv:hep-ph/0006296 [hep-ph]].
+[7] Y.
+Q.
+Chen
+and
+E.
+Braaten,
+Phys.
+Rev.
+Lett.
+80
+(1998),
+5060-5063
+doi:10.1103/PhysRevLett.80.5060 [arXiv:hep-ph/9801226 [hep-ph]].
+[8] S.
+J.
+Brodsky
+and
+M.
+Karliner,
+Phys.
+Rev.
+Lett.
+78
+(1997),
+4682-4685
+doi:10.1103/PhysRevLett.78.4682 [arXiv:hep-ph/9704379 [hep-ph]].
+[9] S. S. Pinsky, Phys. Lett. B 236 (1990), 479-482 doi:10.1016/0370-2693(90)90387-L
+[10] T.
+Feldmann
+and
+P.
+Kroll,
+Phys.
+Rev.
+D
+62
+(2000),
+074006
+doi:10.1103/PhysRevD.62.074006 [arXiv:hep-ph/0003096 [hep-ph]].
+[11] X.
+Q.
+Li,
+D.
+V.
+Bugg and B.
+S.
+Zou,
+Phys.
+Rev.
+D 55 (1997),
+1421-1424
+doi:10.1103/PhysRevD.55.1421
+[12] Q.
+Zhao,
+Phys.
+Lett.
+B
+697
+(2011),
+52-57
+doi:10.1016/j.physletb.2011.01.043
+[arXiv:1012.1165 [hep-ph]].
+[13] Q.
+Wang,
+G.
+Li
+and
+Q.
+Zhao,
+Phys.
+Rev.
+D
+85
+(2012),
+074015
+doi:10.1103/PhysRevD.85.074015 [arXiv:1201.1681 [hep-ph]].
+[14] J.
+M.
+Gerard
+and
+A.
+Martini,
+Phys.
+Lett.
+B
+730
+(2014),
+264-266
+doi:10.1016/j.physletb.2014.01.068 [arXiv:1312.3081 [hep-ph]].
+[15] G. ’t Hooft, Nucl. Phys. B 72 (1974) 461; 75 (1974) 461; E. Witten, Nucl. Phys. B 160
+(1979) 57.
+[16] J.
+Gasser
+and
+H.
+Leutwyler,
+Annals
+Phys.
+158,
+142
+(1984)
+doi:10.1016/0003-
+4916(84)90242-2
+[17] J. Gasser and H. Leutwyler, Nucl. Phys. B 250, 465-516 (1985) doi:10.1016/0550-
+3213(85)90492-4
+[18] Y. H. Chen,
+Z. H. Guo and H. Q. Zheng,
+Phys. Rev. D 85,
+054018 (2012)
+doi:10.1103/PhysRevD.85.054018 [arXiv:1201.2135 [hep-ph]].
+[19] Y. H. Chen, Z. H. Guo and H. Q. Zheng, Phys. Rev. D 90, no.3, 034013 (2014)
+doi:10.1103/PhysRevD.90.034013 [arXiv:1311.3366 [hep-ph]].
+[20] Y.
+H.
+Chen,
+Z.
+H.
+Guo
+and
+B.
+S.
+Zou,
+Phys.
+Rev.
+D
+91,
+014010
+(2015)
+doi:10.1103/PhysRevD.91.014010 [arXiv:1411.1159 [hep-ph]].
+[21] M. Ablikim et al. [BESIII], Phys. Rev. D 99, no.1, 012006 (2019) [erratum: Phys. Rev. D
+104, no.9, 099901 (2021)] doi:10.1103/PhysRevD.99.012006 [arXiv:1810.03091 [hep-ex]].
+[22] G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B 321, 311-342 (1989)
+doi:10.1016/0550-3213(89)90346-5
+[23] G. ’t Hooft, Nucl. Phys. B 72, 461 (1974) doi:10.1016/0550-3213(74)90154-0;
+G. ’t Hooft, Nucl. Phys. B 75, 461-470 (1974) doi:10.1016/0550-3213(74)90088-1;
+E. Witten, Nucl. Phys. B 160, 57-115 (1979) doi:10.1016/0550-3213(79)90232-3;
+18
+
+[24] V. Cirigliano, G. Ecker, M. Eidemuller, R. Kaiser, A. Pich and J. Portoles, Nucl. Phys.
+B 753, 139-177 (2006) doi:10.1016/j.nuclphysb.2006.07.010 [arXiv:hep-ph/0603205 [hep-
+ph]].
+[25] E. Witten, Nucl. Phys. B 156, 269-283 (1979) doi:10.1016/0550-3213(79)90031-2;
+S.
+R.
+Coleman
+and
+E.
+Witten,
+Phys.
+Rev.
+Lett.
+45,
+100
+(1980)
+doi:10.1103/PhysRevLett.45.100;
+G. Veneziano, Nucl. Phys. B 159, 213-224 (1979) doi:10.1016/0550-3213(79)90332-8
+[26] H. Leutwyler, Nucl. Phys. B Proc. Suppl. 64, 223-231 (1998) doi:10.1016/S0920-
+5632(97)01065-7 [arXiv:hep-ph/9709408 [hep-ph]].
+[27] R. Kaiser and H. Leutwyler, [arXiv:hep-ph/9806336 [hep-ph]].
+[28] G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafael, Phys. Lett. B 223, 425-432
+(1989) doi:10.1016/0370-2693(89)91627-4
+[29] P. D. Ruiz-Femenia, A. Pich and J. Portoles, JHEP 07, 003 (2003) doi:10.1088/1126-
+6708/2003/07/003 [arXiv:hep-ph/0306157 [hep-ph]].
+[30] K.
+Kampf
+and
+J.
+Novotny,
+Phys.
+Rev.
+D
+84,
+014036
+(2011)
+doi:10.1103/PhysRevD.84.014036 [arXiv:1104.3137 [hep-ph]].
+[31] J.
+A.
+Miranda
+and
+P.
+Roig,
+Phys.
+Rev.
+D
+102,
+114017
+(2020)
+doi:10.1103/PhysRevD.102.114017 [arXiv:2007.11019 [hep-ph]].
+[32] K. T. Chao, Nucl. Phys. B 335, 101-114 (1990) doi:10.1016/0550-3213(90)90172-A
+[33] J. J. Dudek, R. G. Edwards and D. G. Richards, Phys. Rev. D 73, 074507 (2006)
+doi:10.1103/PhysRevD.73.074507 [arXiv:hep-ph/0601137 [hep-ph]].
+[34] J. J. Dudek, R. Edwards and C. E. Thomas, Phys. Rev. D 79, 094504 (2009)
+doi:10.1103/PhysRevD.79.094504 [arXiv:0902.2241 [hep-ph]].
+[35] Y. Chen, D. C. Du, B. Z. Guo, N. Li, C. Liu, H. Liu, Y. B. Liu, J. P. Ma, X. F. Meng
+and Z. Y. Niu, et al. Phys. Rev. D 84, 034503 (2011) doi:10.1103/PhysRevD.84.034503
+[arXiv:1104.2655 [hep-lat]].
+[36] Z. H. Guo and P. Roig, Phys. Rev. D 82 (2010), 113016 doi:10.1103/PhysRevD.82.113016
+[arXiv:1009.2542 [hep-ph]].
+[37] P.
+Roig
+and
+J.
+J.
+Sanz
+Cillero,
+Phys.
+Lett.
+B
+733
+(2014),
+158-163
+doi:10.1016/j.physletb.2014.04.034 [arXiv:1312.6206 [hep-ph]].
+[38] C. Chen, C. G. Duan and Z. H. Guo, JHEP 08 (2022), 144 doi:10.1007/JHEP08(2022)144
+[arXiv:2201.12764 [hep-ph]].
+[39] Z.
+H.
+Guo
+and
+J.
+A.
+Oller,
+Phys.
+Rev.
+D
+84,
+034005
+(2011)
+doi:10.1103/PhysRevD.84.034005 [arXiv:1104.2849 [hep-ph]].
+[40] For
+a
+review
+see,
+T.
+Feldmann,
+Int.
+J.
+Mod.
+Phys.
+A
+15,no.2,159
+(2000).
+doi:10.1142/S0217751X00000082 [arXiv:9907491 [hep-ph]].
+19
+
+[41] Q. Zhao, Phys. Lett. B 697, 52 (2011) doi:10.1016/j.physletb.2011.01.043 [arXiv:1012.1165
+[hep-ph]]
+[42] T.
+Feldmann,
+P.
+Kroll
+and
+B.
+Stech,
+Phys.
+Rev.
+D
+58
+(1998),
+114006
+doi:10.1103/PhysRevD.58.114006 [arXiv:hep-ph/9802409 [hep-ph]].
+[43] A. Bramon, R. Escribano and M. D. Scadron, Phys. Lett. B 403 (1997), 339-343
+doi:10.1016/S0370-2693(97)00508-X [arXiv:hep-ph/9703313 [hep-ph]].
+[44] R. Escribano, Eur. Phys. J. C 65 (2010), 467-473 doi:10.1140/epjc/s10052-009-1206-9
+[arXiv:0807.4201 [hep-ph]].
+[45] M.
+Ablikim
+et
+al.
+[BESIII],
+Phys.
+Rev.
+D
+89,
+no.9,
+092008
+(2014)
+doi:10.1103/PhysRevD.89.092008 [arXiv:1403.7042 [hep-ex]].
+[46] R. I. Dzhelyadin et al. Phys. Lett. B 94 (1980), 548 doi:10.1016/0370-2693(80)90937-5
+[47] R. I. Dzhelyadin et al. Phys. Lett. B 88 (1979), 379 doi:10.1016/0370-2693(79)90490-8
+[48] H. J. Behrend et al. [CELLO], Z. Phys. C 49 (1991), 401-410 doi:10.1007/BF01549692
+[49] M. N. Achasov et al. Phys. Lett. B 504 (2001), 275-281 doi:10.1016/S0370-2693(01)00320-
+3
+[50] R.
+Arnaldi
+et
+al.
+[NA60],
+Phys.
+Lett.
+B
+677
+(2009),
+260-266
+doi:10.1016/j.physletb.2009.05.029 [arXiv:0902.2547 [hep-ph]].
+[51] H.
+Aihara
+et
+al.
+[TPC/Two
+Gamma],
+Phys.
+Rev.
+Lett.
+64
+(1990),
+172
+doi:10.1103/PhysRevLett.64.172
+[52] M. Acciarri
+et al. [L3],
+Phys. Lett. B 418 (1998),
+399-410
+doi:10.1016/S0370-
+2693(97)01219-7
+[53] B.
+Kubis
+and
+F.
+Niecknig,
+Phys.
+Rev.
+D
+91
+(2015)
+no.3,
+036004
+doi:10.1103/PhysRevD.91.036004 [arXiv:1412.5385 [hep-ph]].
+[54] Kuang-Ta Chao, Nucl.Phys.B 335 101-114 (1990) doi:10.1016/0550-3213(90)90172-A
+[55] J. Fu, H. B. Li, X. Qin, and M. Z. Yang, Mod. Phys. Lett. A 27, (2012) 1250223
+doi:10.1142/S0217732312502239 [arXiv:1111.4055 [hep-ph]]
+[56] J.
+K.
+He
+and
+C.
+J.
+Fan,
+Phys.
+Rev.
+D
+105
+(2022)
+no.9,
+094034
+doi:10.1103/PhysRevD.105.094034 [arXiv:2005.13568 [hep-ph]].
+20
+
diff --git a/a9E2T4oBgHgl3EQfaAcV/content/tmp_files/load_file.txt b/a9E2T4oBgHgl3EQfaAcV/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf,len=1369
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='03869v1  [hep-ph]  10 Jan 2023  Effective-Lagrangian study of ψ′(J/ψ) → V P and the insights  into ρπ puzzle  Lin-Wan Yana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Yun-Hua Chenb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Chun-Gui Duana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Zhi-Hui Guoa  a Department of Physics and Hebei Key Laboratory of Photophysics Research and Application,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Hebei Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Shijiazhuang 050024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' China  b School of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  University of Science and Technology Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Beijing 100083,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' China  Abstract  Within the effective Lagrangian approach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' we carry out a unified study of the J/ψ(ψ′) →  V P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' J/ψ → Pγ and relevant radiative decays of light-flavor hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Large amount of  experimental data, including the various decay widths and electromagnetic form factors, are  fitted to constrain the numerous hadron couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Relative strengths between the strong  and electromagnetic interactions are revealed in the J/ψ → V P and ψ′ → V P processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  The effect from the strong interaction is found to dominate in the J/ψ → ρπ decay, while  the electromagnetic interaction turns out to be the dominant effect in ψ′ → ρπ decay,  which provides an explanation to the ρπ puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' For the J/ψ → K∗ ¯K + ¯K∗K and ψ′ →  K∗ ¯K + ¯K∗K, the former process is dominated by the strong interactions, and the effects  from the electromagnetic parts are found to be comparable with those of strong interactions  in the latter process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Different SU(3) breaking effects from the electromagnetic parts appear  in the charged and neutral channels for the ψ′ → K∗ ¯K + ¯K∗K processes explain the rather  different ratios between B(ψ′ → K∗+K− + K∗−K+)/B(J/ψ → K∗+K− + K∗−K+) and  B(ψ′ → K∗0 ¯K0 + ¯K∗0K0)/B(J/ψ → K∗0 ¯K0 + ¯K∗0K0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  1  Introduction  The strong suppression of the branching ratios of the ψ′ → ρπ and ψ′ → K∗ ¯K + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  processes, relative to those of the corresponding decay channels of the J/ψ, has been a long-  standing puzzle in charmonium physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The annihilation of the c¯c into three gluons is usually  assumed to be the dominant mechanism that rules the decays of the J/ψ and ψ′ to the light-  flavor hadrons [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The annihilation amplitudes of the latter processes and also the decays to  the lepton pairs are proportional to the wave functions of the S-wave charmonium states J/ψ  and ψ′ at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' As a result, the branching ratios with the light-flavor-hadron (h) decays  of the ψ′ and J/ψ can be predicted by their leptonic decay widths [4], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=',  Qh ≡  B(ψ′ → h)  B(J/ψ → h) =  B(ψ′ → e+e−)  B(J/ψ → e+e−) = (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4)% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (1)  However, according to the recent PDG averages [4], the ratio of Qρπ,  B(ψ′ → ρπ)  B(J/ψ → ρπ) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1)% ,  (2)  1  and the various ratios of QK∗ ¯  K,  B(ψ′ → K∗+K− + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=')  B(J/ψ → K∗+K− + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=')  =  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1)% ,  B(ψ′ → K∗0 ¯K0 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=')  B(J/ψ → K∗0 ¯K0 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=')  =  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='7)% ,  B(ψ′ → K∗ ¯K + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=')  B(J/ψ → K∗ ¯K + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=')  =  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='5  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4)% ,  (3)  are drastically different from the prediction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  These contradictions are generally  referred as the ρπ puzzle, which was first established in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [5] four decades ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Tremendous  efforts have been made to address this problem, including the proposal of a vector glueball near  the J/ψ mass [6], higher Fock components in the charmonium states [7], the intrinsic charm  portions in the light-flavor vector ρ [8], the nodes in the wave functions [9], the meson mixing  mechanisms [10], the final-state interactions [11–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Another important issue in those decays  is the sizable SU(3) breaking effects in the charged and neutral K∗ ¯K decays of the J/ψ and ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=', the strict SU(3) flavor symmetry would give the prediction QK∗+K−+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' = QK∗0 ¯  K0+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=',  which is however severely violated according to the experimental measurements in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  In this work, we aim at a unified description of the processes J/ψ(ψ′) → V P and γ(∗)P,  with V and P the light-flavor vector and pseudoscalar mesons in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' In such kinds of de-  cay processes, one needs to simultaneously take into account of the single-Okubo-Zweig-Iizuka  (OZI) or even the doubly suppressed OZI strong interaction effects, the electromagnetic contri-  butions and the SU(3) flavor symmetry breaking terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The effective Lagrangian approach can  provide an excellent framework to properly include all the aforementioned effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Regarding  the well-celebrated OZI rule, a quantitative way to understand such suppression mechanism  is the large NC QCD [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  In the effective field theory (EFT) approach, the NC counting  order can be directly related to the number of traces in the flavor space [16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Typically one  additional flavor trace will introduce one more 1/NC suppression order to the EFT operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  The single OZI/double OZI effects can be systematically incorporated via the EFT operators  with the proper numbers of flavor traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Furthermore, the chiral EFT is constructed accord-  ing to the spontaneous and explicit chiral symmetry breaking patterns of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The SU(3)  flavor symmetry breaking effects can be then introduced through the basic building tensors of  the EFT involving the small but nonvanishing light-flavor quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Although the chiral  power counting scheme based on the momentum expansion is not valid in the massive J/ψ  or ψ′ decays, the basic building blocks and methodology of the EFT Lagrangians are useful  to conveniently take into account all the relevant ingredients describing the J/ψ(ψ′) → V P  and J/ψ → γ(∗)P processes, including the OZI strong interaction parts, the electromagnetic  contributions and the SU(3) flavor symmetry breaking effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' This formalism has been suc-  cessfully applied to the light-flavor decay processes of V → Pγ(∗), e+e− → K∗ ¯K + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' and  J/ψ → V P, Pγ(∗) in a series of works in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' In this work we push forward the study  along the line of this research to address the mysterious ρπ puzzle by including similar decay  processes of ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' In addition, we also perform the global analyses of the large amount of updated  branching ratios of various decay processes from the PDG [4] and the newly measured different  decay widths from the BESIII collaboration [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 2, we introduce the relevant effective Lagrangians  and elaborate the calculations of the decay amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The global fit to the various experi-  mental data and the phenomenological consequences are analyzed in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' We give  the short summary and conclusions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  2  2  Effective Lagrangian and calculations of transition ampli-  tudes  The primary aim of this work is to study the various decay processes of the J/ψ and ψ′  into a light-flavor vector and a light pseudoscalar meson, and the light-flavor meson radiative  decays and relevant form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Therefore we need to include not only the transition operators  between the charmonia and the light-flavor mesons, but also the EFT operators describing the  interactions among the light-meson themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' To tightly constrain the free couplings, we  simultaneously take into account the experimental data from both the decay processes with  only light-flavor mesons and also the processes involving the J/ψ and ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Resonance chiral theory (RχT) [22] provides a reliable framework to study the interactions  of the light-flavor resonances and the light pseudoscalar mesons (π, K, η), the latter of which  are treated as the pseudo-Nambu-Goldstone bosons (pNGBs) resulting from the spontaneous  symmetry breaking of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' As an extension of the chiral perturbation theory (χPT), RχT  explicitly introduces the heavier degrees of freedom of QCD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=', the light-flavor resonances,  such as the vectors ρ, K∗, ω, φ, the axial vectors, scalars, etc, into the chiral Lagrangians,  together with the pNGBs and external source fields, like the photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The RχT operators are  constructed in a chiral covariant way, therefore the physical amplitudes calculated in the RχT  automatically fulfill the requirements of chiral symmetry of QCD in the low energy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' On  the other hand, the large NC expansion of QCD [23] has been widely used as another useful  guide to arrange the operators and amplitudes of the RχT [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' In addition, from the large NC  point of view, the QCD UA(1) anomaly effect, which is considered to be the most responsible  factor for the large mass of the physical state η′, is however 1/NC suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' As a result,  the η′ state would become the ninth pNGB both in large NC and chiral limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Based on this  argument, the nonet of the pNGBs (π, K, η, η′) can be systematically included in the effective  Lagrangian [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' We closely follow this guideline to include the singlet η0 state in the RχT and  adopt the general two-mixing-angle formalism to study the physical processes with the η and  η′ mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Next we briefly introduce the relevant RχT Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  In the present work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' only the light-flavor vector resonances will be relevant to our study  and the minimal interaction operators with the vectors in even-intrinsic-parity sector of the  RχT is given by [22]  L (2)  V  =  FV  2  √  2  ⟨Vµνf µν  + ⟩ + iGV  √  2  ⟨Vµνuµuν⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (4)  where the nonent of the vector resonances is incorporated via the 3 × 3 matrix  Vµν =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ed  1  √  2ρ0 +  1  √  6ω8 +  1  √  3ω0  ρ+  K∗+  ρ−  − 1  √  2ρ0 +  1  √  6ω8 +  1  √  3ω0  K∗0  K∗−  K  ∗0  − 2  √  6ω8 +  1  √  3ω0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f8  µν  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (5)  the basic chiral tensors with the pNGBs and the external source fields are defined as  U = u2 = ei  √  2Φ  F  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  uµ = i  �  u†(∂µ − irµ)u − u(∂µ − uℓµ)u†�  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  f µν  ± = uF µν  L u† ± u†F µν  R u ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  F µν  L(R) = ∂µl(r)ν − ∂νl(r)µ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  χ± = u†χu† ± uχ†u ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  χ = 2B0(s + ip)  (6)  3  and the flavor contents of the nonet pNGB matrix read  Φ =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ed  1  √  2π0 +  1  √  6η8 +  1  √  3η0  π+  K+  π−  − 1  √  2π0 +  1  √  6η8 +  1  √  3η0  K0  K−  K  0  − 2  √  6η8 +  1  √  3η0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (7)  The quark-mass terms are introduced by taking the scalar external source filed s in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (6) as  s = diag{mu, md, ms}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' In this work, we will take mu = md = ˆm throughout, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' neglecting  the isospin breaking effects from the strong interaction parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The physical vectors ω and φ  can be well described by assuming the ideal mixing of the octet ω8 and the singlet ω0 [4]  ω0  =  �  2  3ω −  �  1  3φ,  ω8  =  �  2  3φ +  �  1  3ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (8)  In contrast, the mixing pattern of the η8 and η0 is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The modern chiral prescrip-  tion introduces the sophisticated two-mixing-angle scheme [26, 27] to address the η-η′ mixing  system  \uf8eb  \uf8ed η  η′  \uf8f6  \uf8f8 = 1  F  \uf8eb  \uf8ed F8 cos θ8  −F0 sin θ0  F8 sin θ8  F0 cos θ0  \uf8f6  \uf8f8  \uf8eb  \uf8ed η8  η0  \uf8f6  \uf8f8 ,  (9)  where F0 and F8 are the weak decay constants of the singlet and octet axial-vector currents,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The conventional mixing formula with a single mixing angle can be naturally  recovered by taking F8 = F0 = F and θ0 = θ8 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Equivalently one can also use the  quark-flavor basis to describe the two-mixing-angle formalism  \uf8eb  \uf8ed η  η′  \uf8f6  \uf8f8 = 1  F  \uf8eb  \uf8ed Fq cos θq  −Fs sin θs  Fq sin θq  Fs cos θs  \uf8f6  \uf8f8  \uf8eb  \uf8ed ηq  ηs  \uf8f6  \uf8f8 ,  (10)  where the quark-flavor contents of the states ηq = (η8 +  √  2η0)/  √  3 and ηs = (−  √  2η8 + η0)/  √  3  are (¯uu + ¯dd)/  √  2 and ¯ss, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  The vector resonances in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (4) are expressed in terms of the anti-symmetric tensors,  instead of the commonly used Proca fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  It is demonstrated in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [22, 28] that it is  convenient to use the anti-symmetric tensors to describe the vector resonances in RχT, since  the high energy behaviors of the resulting amplitudes and form factors automatically match  the QCD constraints without requiring the inclusion of extra local chiral counter terms in  the anti-symmetric tensor formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The RχT Lagrangians in the odd-intrinsic-parity sector  comprise two different classes, namely the V V P and V JP types, with J the external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Those RχT operators written in terms of the anti-symmetric tensor fields that are relevant  to the O(p4) chiral low energy constants, are worked out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [29], and the relevant RχT  Lagrangians and discussions on the V V P Green functions by explicitly including the dynamical  singlet η0 state are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' A more complete basis of the odd-intrinsic-parity RχT  operators that contribute to the O(p6) chiral low energy constants, is given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' A  proliferation of the unknown resonance couplings arise in the more complete RχT Lagrangians,  4  as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' This can hinder one from giving the definite conclusions on the phenomenological  discussions [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' From the practical point of view, we will work with the RχT operator basis  from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [18, 29] and we believe that the higher order effects from the extra operators in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [30] can be accounted for by the uncertainties of the resonance couplings in the former  two references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' For the sake of completeness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' we give the explicit expressions of the relevant  RχT Lagrangians [18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='29]  LV V P =  d1εµνρσ⟨{V µν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' V ρα}∇αuσ⟩ + id2εµνρσ⟨{V µν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' V ρσ}χ−⟩ + d3εµνρσ⟨{∇αV µν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' V ρα}uσ⟩  +d4εµνρσ⟨{∇σV µν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' V ρα}uα⟩ − id5M2  V  �  2  3εµνρσ⟨V µνV ρσ⟩ ln(det u) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (11)  and  LV JP =  c1  MV  εµνρσ⟨{V µν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' f ρα  + }∇αuσ⟩ + c2  MV  εµνρσ⟨{V µα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' f ρσ  + }∇αuν⟩ + ic3  MV  εµνρσ⟨{V µν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' f ρσ  + }χ−⟩  + ic4  MV  εµνρσ⟨V µν[f ρσ  − ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' χ+]⟩ + c5  MV  εµνρσ⟨{∇αV µν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' f ρα  + }uσ⟩ + c6  MV  εµνρσ⟨{∇αV µα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' f ρσ  + }uν⟩  + c7  MV  εµνρσ⟨{∇σV µν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' f ρα  + }uα⟩ − ic8MV  �  2  3εµνρσ⟨V µν ˜f ρσ  + ⟩ ln(det u) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (12)  where the covariant derivative acting on the chiral field X is given by  ∇µX = ∂µX + [Γµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' X] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Γµ = 1  2  �  u+(∂µ − irµ)u + u(∂µ − ilµ)u+�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (13)  As previously mentioned in the Introduction, both the strong and electromagnetic interac-  tions can be important in the J/ψ(ψ′) → V P processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The effects from the strong interactions  are taken into account by the direct J/ψ(ψ′)V P transition operators [18]  Lψ(ψ′)V P =  Mψ(ψ′)h(′)  1 εµνρσψ(′)µ⟨uνV ρσ⟩ +  1  Mψ(ψ′)  h(′)  2 εµνρσψ(′)µ⟨{uν, V ρσ}χ+⟩  +Mψ(ψ′)h(′)  3 εµνρσψ(′)µ⟨uν⟩⟨V ρσ⟩ ,  (14)  where the couplings h(′)  i=1,2,3 corresponding to the J/ψ and ψ′ will be separately fitted to the  experimental data of the two charmonium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Two types of EFT operators are introduced  to account for the electromagnetic effects, which include the direct ψPγ transition operators  LψP γ = g1εµνρσψµ⟨uνf ρσ  + ⟩ +  1  M2  ψ  g2εµνρσψµ⟨{uν, f ρσ  + }χ+⟩ ,  (15)  and the conversion vertex of the charmonium and the photon  Lψγ = −1  2  √  2  fψ  Mψ  ⟨ ˆψµνf µν  + ⟩ ,  (16)  being ˆψµν = ∂µψν − ∂νψµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The values of the couplings g1, g2 and fψ are different for the J/ψ  and ψ′ and they will be determined by the relevant experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Different powers of the  Mψ(ψ′) are introduced in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (14)-(16), so that the couplings appearing in those Lagrangians  are dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  5  It is found [20,32] that the J/ψ → η(′)γ(∗) amplitudes are dominated by the ηc mediating  diagrams, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=', via the J/ψ → ηcγ(∗) → η(′)γ(∗) intermediate processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The decay amplitude  of the ψ → η(′)γ(∗) can be written as  Mmixing  ψ→η(′)γ∗ = e εµνρσǫµ  ψǫν  γ∗qρkσ ληcη(′) gψηcγ∗(s) eiδP ,  (17)  being P = η, η′, where the electromagnetic transition form factor between the ψ and ηc takes  form [33–35]  gψηcγ∗(s) = gψηcγ∗(0)e  s  16β2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (18)  For the mixing parameters ληcη(′) between the ηc and η(′) states, we take the determinations  ληcη = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6 × 10−3 and ληcη′ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 × 10−2 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The phenomenological phase  factors δη(′) in front of the ηc mediating diagrams need to be separately fitted to the data of  the J/ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  V  P  =  +  P  V  P  V ′  γ(∗)  γ(∗)  γ(∗)  V  (a)  (b)  Figure 1:  Diagrams relevant to the V → Pγ(∗) processes: (a) direct type and (b) indirect  type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  γ∗  γ∗  γ∗  V  V  V  +  +  +  =  V  P  ψ  V  P  P  P  η(′)  ηc  ψ  ψ  ψ  ψ  (a)  (b)  (c)  (d)  Figure 2:  Feynman diagrams for the processes J/ψ → V P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The notations of the solid square  in diagram (c) and the open circle in diagram (d) are explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  The various Feynman diagrams relevant to our study are illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1, 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' To  be more specific, the diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 1 contribute to the light-flavor processes V → Pγ(∗) and  P → V γ(∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The amplitudes of the J/ψ(ψ′) → V P and J/ψ → Pγ(∗) receive contributions  from the diagrams in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The formulas relevant to the V → Pγ(∗) and  P → V γ(∗) processes are worked out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [18], and the expressions of the J/ψ → V P, Pγ(∗)  amplitudes are calculated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The corresponding decay amplitudes of the ψ′ state  share similar expressions as those involving J/ψ, with obvious replacements of the resonance  6  P  ψ  γ(∗)  =  γ(∗)  +  V  P  ψ  +  ηc  ψ  P  ψ  γ(∗)  γ(∗)  η(′)  (a)  (b)  (c)  Figure 3:  Feynman diagrams for the processes J/ψ → Pγ(∗)  couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Nevertheless, for the sake of completeness and to set up the notations, we further  elaborate the amplitudes of the processes of ψ′ → V P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  For the ψ′ → V P decay, the first diagram (a) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 2 denotes the contributions from the  strong interactions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=', from the Lagrangians in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Other diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 2 correspond  to the electromagnetic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The ψ′ → V P amplitude can be written as  Mψ′→V P = εµνρσǫµ  ψ′ǫν  V qρkσGψ′→V P ,  (19)  where the polarization vectors of the ψ′ and V are given by ǫµ  ψ′ and ǫν  V , q and k stand for the  four-momentum of the ψ′ and V , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The effective couplings Gψ′→V P include various  contributions from the individual diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The explicit expressions of Gψ′→V P for  the various processes are given in Appendix (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The decay widths of ψ′ → V P read  Γ(ψ′ → V P) =  1  96πM3  ψ′  λ(Mψ′, MV , mP)  3  2 ��Gψ′→V P  ��2 ,  (20)  with the K¨all´en function λ(x, y, z) = x2 + y2 + z2 − 2xy − 2xz − 2yz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Similarly, the corresponding amplitude of the radiative process J/ψ(q) → γ∗(k)P(q − k)  can be given in terms of one effective coupling as well  Mψ→P γ∗ = e εµνρσǫµ  ψǫν  γ∗qρkσGψ→P γ∗(s) ,  (21)  with s = k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The effective coupling Gψ→P γ∗ can receive contributions from all the diagrams in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The explicit expressions are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The formula of the decay width of the  J/ψ → Pγ process finds it form  Γψ→P γ = 1  3α  �  M2  ψ − M2  P  2Mψ  �3  |Gψ→P γ∗(0)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (22)  The expression of the width for the Dalitz decay process J/ψ → Pγ∗ → Pl+l− is given by  Γψ→P l+l− =  � (Mψ−mP )2  4m2  l  α2(2m2  l + s)  72M3  ψπs3  �  s(s − 4m2  l )  �  λ(s, Mψ, mP)]  3  2 |Gψ→P γ∗(s)|2ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (23)  3  Comprehensive fits and phenomenological discussions  Compared to the previous studies in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [18,20], we incorporate in this work the data from  the various ψ′ → V P decays, apart from other types of data from the V → Pγ(∗), P → V γ(∗)  7  and J/ψ → V P, Pγ(∗) processes, to perform a comprehensive fit, so that the ρπ puzzle can be  addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' In addition, we update numerous types of data according to the most recent PDG  averages [4], and timely revise the determinations of the resonance couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  In total, we include 135 data points from several different types of processes in the com-  prehensive fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' To be more specific, the data from the pure light-flavor processes amount to 70,  and they consist of both the decay widths, such as those of the ω → ηγ, η′ → ωγ, η → γγ, etc,  and the form factors of the φ → ηγ∗ and η(′) → γγ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' For the data related to the J/ψ, they  include all the available widths of the J/ψ → V P, Pγ and Pe+e− from the PDG [4], and also  the recent BESIII measurements of the invariant-mass distributions of the lepton pairs in the  transition of J/ψ → ηγ∗ [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Regarding the data of the ψ′, we will include in the joint fit all  the available widths of the ψ′ → V P processes from PDG [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  An efficient way to reduce the number of unknown couplings in the RχT is to impose  the high energy constraints dictated by QCD to the various form factors and Green functions  calculated from the RχT Lagrangians in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (4), (11) and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Furthermore, the high energy  behaviors of the resulting amplitudes after imposing such constraints will mimic the properties  as predicted by QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Following the previous discussions in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [18–20, 29, 36–38], we take  the following high energy constraints on the various couplings  4c3 + c1 = 0 ,  c1 − c2 + c5 = 0 ,  c5 − c6 = NC  64π2  MV  √  2FV  ,  d1 + 8d2 − d3 = F 2  8F 2  V  ,  d3 = − NC  64π2  M2  V  F 2  V ,  c8 = −  √  2M2  0  √  3M2  V  c1 ,  (24)  where the pion weak decay constant takes the normalization F = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4 MeV throughout, the  UA(1) anomaly parameter is set to be M0 = 900 MeV [39, 40], the chiral-limit mass of the  lowest vector resonance multiplet is fixed at MV = Mρ = 775 MeV and the vector-photon  transition coupling FV will be fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' By taking into account the the leptonic widths of the  J/ψ and ψ′, we can determine the charmonium-photon transition coupling fJ/ψ(ψ′) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (16),  whose explicit values are found to be  fJ/ψ = 293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='5 MeV ,  fψ′ = 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1 MeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (25)  We are then left with 23 undetermined parameters, including the four η-η′ mixing param-  eters introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (9), four couplings FV , c3, c4 and d2 that emerge from the light-flavor  resonance interactions, nine parameters exclusively entering in the J/ψ decays and six pa-  rameters that are dedicated to the ψ′ processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The couplings that describe the interactions  of the light-flavor resonances will also enter in the charmonia decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Therefore the joint  fits by simultaneously including the relevant data of the light-flavor mesons, the data from  the J/ψ → V P, Pγ(∗) and the ψ′ ones, will obviously give more stringent constraints on the  couplings than the situation by including just one of these data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Furthermore, such com-  prehensive studies in a unified framework are also expected to give a further insight into ρπ  puzzle elaborated in the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  The resulting parameters from the joint fit are given in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The updated parameters  related to the light-flavor resonances, the J/ψ decays and the η-η′ mixing are well consistent  with the previous determinations [18–20] where the data of the ψ′ processes are not included  in these studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' For the ψ′ → Pγ(∗) processes, which could receive significant contributions  from the ψ′ → J/ψP transition vertexes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' ψ′ → J/ψη → γη [41], are not considered in  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Therefore we will not discuss such kinds of processes here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The relative phases  8  F8  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='02)Fπ  F0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='03)Fπ  θ8  (−24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4)◦  θ0  (−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='5)◦  FV  139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='04 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='72  c3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0046 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0003  c4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0001  d2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='008  h1  (−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='06) × 10−5  h2  (-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='60)×10−5  h3  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='22)×10−6  g1  (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='06)×10−5  g2  (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='48)×10−4  r1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='04  h′  1  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='33±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='23)×10−6  h′  2  (-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='32)×10−5  h′  3  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='47)×10−6  g′  1  (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='70±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='47)×10−4  g′  2  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='95)×10−3  δη  (117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='12 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='81)◦  δη′  (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='03 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01)◦  β  512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='86 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='36 MeV  β′  112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='98 MeV  F (∗)  q  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='02)Fπ  F (∗)  s  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='02)Fπ  θ(∗)  q  (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='7)◦  θ(∗)  s  (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4)◦  χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='f  157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='25/(135-23)=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='40  Table 1:  Parameters from the joint fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The quantities marked with asterisk are predictions,  instead of free parameters in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  δη(′) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (17) for the ηc mediating effects in the ψ′ → V η(′) decay processes, are found to  be insensitive to our present studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' As a result, the phases of δη(′) in the ψ′ decays will be  fixed to zero throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The previous study in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [18] pointed out a strong correlation  between the d2 and d5 parameters, and we find that this correlation still holds in our joint  fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The resulting relation turns out to be d5 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='57d2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Regarding the four parameters  F8, F0, θ8 and θ0 related with the η-η′ mixing, our current determinations of the central values  and uncertainties more or less resemble those in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [18], only the data from  the light-flavor sector were considered and the resulting η-η′ mixing parameters were found to  bear large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The simultaneous inclusion of the relevant data from the J/ψ and  ψ′ processes, together with the light-flavor ones, can obviously pin down the uncertainties of  the η-η′ mixing parameters [20,42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' In Table 1, we also give the predictions to the mixing  parameters in the quark-flavor basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Generally speaking, the numerous types of data are well reproduced in our comprehensive  fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  The comparisons of the various decay widths for the pure light-flavor processes from  the revised fit and the updated PDG values are shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Similar comparisons for  the partial decay widths of the J/ψ and ψ′ are given in Tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The  resulting curves of the form factors for the η → γγ∗, η′ → γγ∗, φ → ηγ∗ and J/ψ → η′γ∗ are  shown together with the experimental data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The fitted results of the recent BESIII  measurements on the e+e− spectra in the J/ψ → ηe+e− processes are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' We  point out a subtlety about the effects of the light-flavor vector resonances in the e+e− spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  In the J/ψ → η′e+e− decays, the light-flavor vectors are removed in the BESIII analysis [45],  and as a result we have also subtracted the contributions from the intermediate light vector  exchanges in accord with the experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' This explains the smooth line shapes of  the electromagnetic J/ψ → η′e+e− transition form factors shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Regarding the  J/ψ → ηe+e− process, we keep the effects of the intermediate light-flavor vector resonances,  in order to be consistent with the setups of the experimental analyses in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' It should  be stressed that the prominent peaks of the narrow vectors ω and φ can be diluted due to the  large bin widths of the experimental energy resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' To clearly show the influence of the  9  Exp  Fit  Γω→πγ  724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='78 ± 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='64  705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='65 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='40  Γρ0→π0γ  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='08 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='37  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='23 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='81  ΓK∗0→K0γ  116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='36 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='27  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='95 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='69  Γω→πe−e+  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='63  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='16  Γω→πµ−µ+  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='02  Γω→ηγ  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='41  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='11  Γρ0→ηγ  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='73 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='39  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='96  Γφ→ηγ  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='28 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='23  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='00  Γφ→η′γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01  Γη′→ωγ  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='18  Γη→γγ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01  Γη′→γγ  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='11  Γη→γe−e+  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='89) × 10−3  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='23) × 10−3  Γη→γµ−µ+  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='07) × 10−3  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01) × 10−3  Γη′→γµ−µ+  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='61) × 10−2  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='04) × 10−2  Γφ→ηe−e+  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='459 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='018  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='460 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='008  Table 2:  The decay widths in units of KeV for the light-flavor hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Exp  Fit  J/ψ → ρ0π0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='3  J/ψ → ρπ  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='5  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0  J/ψ → ρ0η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='193 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='023  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='185 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='021  J/ψ → ρ0η′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='081 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='080 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='007  J/ψ → ωπ0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='04  J/ψ → ωη  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='09  J/ψ → ωη′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='189 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='018  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='189 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='018  J/ψ → φη  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='06  J/ψ → φη′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='05  J/ψ → K∗+K− + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='3  J/ψ → K∗0 ¯  K0 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2  J/ψ → π0γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0356 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0017  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0341 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0016  J/ψ → ηγ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='085 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='018  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='085 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='013  J/ψ → η′γ  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='07  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='04  J/ψ → π0e+e−  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='076 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='014) × 10−2  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='129 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='004) × 10−2  J/ψ → ηe+e−  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='08) × 10−2  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='02) × 10−2  J/ψ → η′e+e−  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='18) × 10−2  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='05) × 10−2  Table 3:  Branching fractions(×10−3) of the decay processes for J/ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  10  bin widths, we give the histograms by using the energy bin width at 50 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' It is evident that  the signals of narrow light vector resonances can be obviously enhanced when the energy bin  width is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1  1  10  100  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='45  4  2  0  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content="0  0  1  2  3  4  5  6  7  8  9  10  pa  2m3  h  /4 F  h  (s,0)  2 [keV]  s[GeV2]  h  gg pa  2m3  h  /4 F  h'  (s,0)  2 [keV]  s[GeV2]  h  gg |F  fhg(s)/F  fhg(0)|  2  s1/2 [GeV]  f  hg |FJ/yh  |  2  M(e+e-) (GeV/c2)  J/y  h  g Figure 4:  The form factors for the η → γγ∗, η′ → γγ∗, φ → ηγ∗ and J/ψ → η′γ∗." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The  red solid lines are obtained by taking the central values of the parameters in Table 1, and the  shaded areas correspond to the error bands at 1-σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The experimental data on the form  factors for the η → γγ∗ , η′ → γγ∗, φ → ηγ∗ and J/ψ → η′γ∗ are taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [46–51],  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [46–48,51,52], Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [49] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [45],respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  With the fitted parameters in Table 1, it is then interesting to decipher the roles of different  mechanisms and resonances played in a given process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  The J/ψ → Pl+l− processes can provide an environment to study the intermediate hadron  resonances [20,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Recently, the J/ψ → ηγ∗(→ e+e−) form factors are reported by the BESIII  collaboration in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [21], in which the experimental analysis includes only the ρ resonance in  the e+e− spectra, apart from the QED contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' However, it is pointed out that the ρ  contribution should come from an isospin violated intermediate process J/ψ → ηρ → ηe+e−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  In contrast, the contributions from the ω and φ are expected to be more important, since  they enter via the isospin conserved intermediate processes J/ψ → ηω and J/ψ → ηφ, whose  branching ratios are around eight and four times larger than that of the J/ψ → ρη in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  As a result, we expect that the effect of the ρ resonance is much suppressed, compared to  the contributions from ω and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Due to the narrow widths of the latter two resonances, they  manifest themselves as prominent peaks in the e+e− spectra, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' However,  these narrow peaks can be easily washed out when the energy resolution is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=', we also  explicitly give the energy distributions of the e+e− in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 5 when taking the energy bin width  11  Exp  Fit  ψ′ → ρπ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='032 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='037 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='010  ψ′ → ρ0η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='022 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='021 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='005  ψ′ → ρ0η′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='019 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='017  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='028 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='008  ψ′ → ωπ0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='021 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='021 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='004  ψ′ → ωη  —  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='005 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='003  ψ′ → ωη′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='032 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='033 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='019  ψ′ → φη  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='031 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0031  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='032 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='003  ψ′ → φη′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0154 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='016 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0019  ψ′ → K∗+K− + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='029 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='029 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='004  ψ′ → K∗0 ¯  K0 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='109 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='080 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='011  Table 4:  Branching fractions(×10−3) of the decay processes for ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The ψ′ → ωη channel is  not included in the fit, instead the result corresponds to our prediction, which is around two  times smaller than the upper limit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1 × 10−5 reported in PDG [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  at 50 MeV and 100 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' In the latter case the signals of the narrow ω and φ become faintly  visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' As pointed out in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [20,41,54], we also confirm the importance of the η(′)−ηc mixing  mechanism in the J/ψ → η(′)γ(∗) decay processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' A future experimental measurement with  higher energy resolution will be definitely helpful to discriminate the roles of different hadrons  in the J/ψ → ηe+e− process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' In Table 5, we give our predictions to the branching ratios of  various J/ψ → Pl+l− processes and also make comparisons with the results in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [20,55,56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Exp  This Work  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [20]  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [55]  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [56]  ψ → π0e+e−  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='076 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1294 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0044  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1191 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0138  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0389+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0037  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0033  ——  ψ → ηe+e−  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='38  ψ → η′e+e−  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='18  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='05  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='16  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='16  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='06  ψ → π0µ+µ−  ——  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0304 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0280 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0032  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0101+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0010  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='0009  ——  ψ → ηµ+µ−  ——  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='46  ψ → η′µ+µ−  ——  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='72  Table 5: Branching ratios (×10−5) for J/ψ → Pl+l− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Our study reveals an interesting feature that can shed light on the ρπ puzzle in the  J/ψ(ψ′) → V P decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' For this purpose, let’s focus on the interplay between the electro-  magnetic and strong interactions in the J/ψ(ψ′) → V P processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' We separately show the  contributions from the strong and electromagnetic interactions to the isospin conserved and  violated decays for J/ψ and ψ′ in Tables 6 and 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The contributions from the  strong interactions are given by the hi=1,2,3 terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (14), while the electromagnetic contri-  butions are obtained by taking hi=1,2,3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' For the J/ψ → V P decays, the contributions from  strong interactions turn out to play major roles in most of the isospin conserved channels, with  the exception of the J/ψ → φη′ process, where the strengths of the two types of interactions  are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' While the isospin violated channels can only receive contributions from the  electromagnetic interactions, since the isospin breaking effects from the strong interaction parts  are not included in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' According to the results shown in Table 7, for the ψ′ → V P  processes, the strong interactions are found to play comparable roles in many of the isospin  12  0  10  20  30  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4  |FJ/ψη|2  me+e- [GeV/c2]  1E-06  1E-05  1E-04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4  dB(J/ψ→e+e-η/dq (GeV/c2)-1  me+e- [GeV/c2]  Figure 5:  The form factors and differential branching fractions for the J/ψ → ηe+e−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The  experimental data are from the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The red solid lines represent the curves with the  central values of the parameters in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1, and the shaded areas stand for the error bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  The histograms are obtained by taking different energy bin width at 50 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Isospin conserved cases  Exp  Strong interaction  EM interaction  | GJ/ψ→ρ0π0 |  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='537 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='154  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='899 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='385 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='006  | GJ/ψ→ρπ |  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='408 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='191  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='022 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='129  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='709 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='009  | GJ/ψ→ωη |  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='497 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='084  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='586 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='037  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='132 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='009  | GJ/ψ→ωη′ |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='562 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='026  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='647 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='119 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='008  | GJ/ψ→φη |  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='060 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='056  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='270 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='198 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='031  | GJ/ψ→φη′ |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='974 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='052  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='074 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='049  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='031 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='044  | GJ/ψ→K∗+K− |  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='011 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='161  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='313 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='048  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='216 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='036  | GJ/ψ→K∗0 ¯  K0 |  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='686 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='078  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='308 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='048  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='715 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='009  Isospin violated cases  Exp  EM interaction  Strong interaction  | GJ/ψ→ρ0η |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='498 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='029  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='487 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='028  −−  | GJ/ψ→ρ0η′ |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='367 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='018  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='365 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='017  −−  | GJ/ψ→ωπ0 |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='721 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='039  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='720 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='035  −−  Table 6:  The effective couplings of Gψ→V P in units of 10−6MeV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  13  Isospin conserved cases  Exp  Strong interaction  EM interaction  | Gψ′ → ρπ |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='255 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='044  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='029 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='036  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='255 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='016  | Gψ′ → ωη |  —  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='103 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='038  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='036 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='003  | Gψ′ → ωη′ |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='288 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='096  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='212 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='079  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='079 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='013  | Gψ′ → φη |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='275 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='013  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='285 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='565 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='029  | Gψ′ → φη′ |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='213 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='013  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='197 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='068  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='412 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='067  | Gψ′ → K∗+K− |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='181 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='263 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='021  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='093 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='021  | Gψ′ → K∗0 ¯  K0 |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='352 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='031  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='267 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='021  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='568 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='007  Isospin violated cases  Exp  EM interaction  Strong interaction  | Gψ′ → ρ0η |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='219 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='216 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='026  −−  | Gψ′ → ρ0η′ |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='222 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='083  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='271 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='040  −−  | Gψ′ → ωπ0 |  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='207 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='208 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='019  −−  Table 7:  The effective couplings of Gψ′→V P in units of 10−6MeV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  conserved channels as those from the electromagnetic parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Especially, the electromagnetic  interaction turns out to play the dominant role in the ψ′ → ρπ process and the effects from  the strong interactions are found to be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' In contrast, the strong interactions domi-  nate the decay of J/ψ → ρπ process and the electromagnetic effects appear to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' This  provides a sensible explanation to the ρπ puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  For the charged K∗+K− + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' and neutral K∗0 ¯K0 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' decay processes of J/ψ or ψ′, the  SU(3) breaking effects can originate from the strong interactions via the h2 term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (14),  which turns out to be the same for both charged and neutral processes, and the electromagnetic  interactions via the cj terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (12), where the c4 operator is found to solely contribute  to the charged process [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The contributions from the electromagnetic parts to the J/ψ →  K∗ ¯K + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' processes are obviously smaller than those from the strong interactions, which  explains the similar branching ratios between J/ψ → K∗+K−+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' and J/ψ → K∗0 ¯K0+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='. In  contrast, our study reveals that the magnitudes of the strong interactions in the ψ′ → K∗ ¯K+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  can be comparable with those of the electromagnetic parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' While, the SU(3) breaking effects  in the electromagnetic parts are quite different for the charged and neutral decay processes due  to the c4 operator [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' This gives a new insight and also a reasonable explanation to the very  different branching ratios of the ψ′ → K∗+K− + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' and ψ′ → K∗0 ¯K0 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='.  4  Summary and conclusions  We use the effective Lagrangian approach to simultaneously investigate the processes of  J/ψ(ψ′) → V P, J/ψ → Pγ,J/ψ → Pl+l−, the radiative decays of light-flavor hadrons and  their relevant form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' High energy constraints on the resonance couplings are used to  reduce the number of free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' The remaining resonance couplings are then determined  through the joint fit to a large amount of experimental data, including the updated PDG  averages of the various partial decay widths and the most recent J/ψ → ηγ∗ form factors from  BESIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Thanks to the use of effective Lagrangian, the different types of contributions from the  OZI allowed/suppressed strong interactions, SU(3) breaking terms and electromagnetic ef-  fects can be easily identified in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' We pay special attention to the relative magni-  14  tudes from the strong and electromagnetic interactions in the J/ψ → V P and ψ′ → V P  processes, so as to provide an insight into the ρπ puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  An anatomy of the J/ψ → ρπ  and ψ′ → ρπ amplitudes reveals that the strong interaction dominates the former process  and the electromagnetic interaction prevails the latter one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  For the obviously distinct ra-  tios between the charged B(ψ′ → K∗+K− + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' )/B(J/ψ → K∗+K− + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=') and the neu-  tral B(ψ′ → K∗0 ¯K0 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' )/B(J/ψ → K∗0 ¯K0 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=') processes, our study uncovers that the  J/ψ → K∗ ¯K+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' processes are mainly ruled by the strong interactions, where the SU(3) break-  ing effects enter similarly in both the charged and neutral amplitudes, while the ψ′ → K∗ ¯K+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  decays are found to be importantly affected by the electromagnetic interactions, where the  SU(3) symmetry breaking terms appear differently in the charged and neutral processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Acknowledgements  We thank Lu Niu for an early-stage contribution to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' This work is partially funded  by the Natural Science Foundation of China under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 11975090, 12150013, 11975028  and 11974043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  A  The expressions of the effective couplings for ψ′ → V P  The expressions of the effective couplings in the ψ′ → V P processes defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' (20)  take the form:  Gψ′→ρ0π0 = 2  √  2  FπMρ  h′  1Mψ′ + 8  √  2  FπMρ  h′  2m2  π  1  Mψ′ + 32πα  FπMρ  FV g′  1 + 128πα  FπMρ  FV g′  2  m2  π  M2  ψ′  + 8  √  2πα  3  fψ′  Mψ′ Fρπγ∗(M2  ψ′) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1)  Gψ′→ρ+π− = 2  √  2  FπMρ  h′  1Mψ′ + 8  √  2  FπMρ  h′  2m2  π  1  Mψ′ + 8  √  2πα  3  fψ′  Mψ′ Fρπγ∗(M2  ψ′) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2)  Gψ′→ρ0η =32  √  2πα  3FMρ  FV g′  1(a1 − a3) + 128  √  2πα  3FMρM2  ψ′  FV g′  2[a1m2  π − a3(2m2  K − m2  π)]  − 8πα FV  Mρ  ληηcgψ′ηcγ∗(M2  ρ )eiδ′  η + 8  √  2πα  3  fψ′  Mψ′ Fρηγ∗(M2  ψ′) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='3)  Gψ′→ρ0η′ =32  √  2πα  3FMρ  FV g′  1(a2 − a4) + 128  √  2πα  3FMρM2  ψ′  FV g′  2[a2m2  π − a4(2m2  K − m2  π)]  − 8πα FV  Mρ  λη′ηcgψ′ηcγ∗(M2  ρ )eiδ′  η′ + 8  √  2πα  3  fψ′  Mψ′ Fρη′γ∗(M2  ψ′) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4)  Gψ′→ωπ0 = 32πα  3FπMω  FV g′  1 + 128πα  3FπMω  FV g′  2  m2  π  M2  ψ′  + 8  √  2πα  3  fψ′  Mψ′ Fωπγ∗(M2  ψ′) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='5)  15  Gψ′→ωη =  4  FMω  a1h′  1Mψ′ +  16  FMω  a1h′  2m2  π  1  Mψ′ +  4  FMω  (2a1 + a3)h′  3Mψ′ + 32  √  2πα  9FMω  FV g′  1(a1 − a3)  + 128  √  2πα  9FMωM2  ψ′  FV g′  2[a1m2  π − a3(2m2  K − m2  π)] − 8  3πα FV  Mω  ληηcgψ′ηcγ∗(M2  ω)eiδ′  η  + 8  √  2πα  3  fψ′  Mψ′ Fωηγ∗(M2  ψ′) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6)  Gψ′→ωη′ =  4  FMω  a2h′  1Mψ′ +  16  FMω  a2h′  2m2  π  1  Mψ′ +  4  FMω  (2a2 + a4)h′  3Mψ′ + 32  √  2πα  9FMω  FV g′  1(a2 − a4)  + 128  √  2πα  9FMωM2  ψ′  FV g′  2[a2m2  π − a4(2m2  K − m2  π)] − 8  3πα FV  Mω  λη′ηcgψ′ηcγ∗(M2  ω)eiδ′  η′  + 8  √  2πα  3  fψ′  Mψ′ Fωη′γ∗(M2  ψ′) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='7)  Gψ′→φη = − 2  √  2  FMφ  a3h′  1Mψ′ − 8  √  2  FMφ  a3h′  2(2m2  K − m2  π) 1  Mψ′ − 2  √  2  FMφ  (2a1 + a3)h′  3Mψ′  + 64πα  9FMφ  FV g′  1(a1 − a3) +  256πα  9FMφM2  ψ′  FV g′  2[a1m2  π − a3(2m2  K − m2  π)]  − 8  √  2  3Mφ  παFV ληηcgψ′ηcγ∗(M2  φ)eiδ′  η + 8  √  2πα  3  fψ′  Mψ′ Fφηγ∗(M2  ψ′) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='8)  Gψ′→φη′ = − 2  √  2  FMφ  a4h′  1Mψ′ − 8  √  2  FMφ  a4h′  2(2m2  K − m2  π) 1  Mψ′ − 2  √  2  FMφ  (2a2 + a4)h′  3Mψ′  + 64πα  9FMφ  FV g′  1(a2 − a4) +  256πα  9FMφM2  ψ′  FV g′  2[a2m2  π − a4(2m2  K − m2  π)]  − 8  √  2  3Mφ  παFV λη′ηcgψ′ηcγ∗(M2  φ)eiδ′  η′ + 8  √  2πα  3  fψ′  Mψ′ Fφη′γ∗(M2  ψ′) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='9)  Gψ′→K∗+K− =  2  √  2  FKMK∗ h′  1Mψ′ +  8  √  2  FKMK∗ h′  2m2  K  1  Mψ′ + 8  √  2πα  3  fψ′  Mψ′ FK∗+K−γ∗(M2  ψ′) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='10)  Gψ′→K∗0 ¯  K0 =  2  √  2  FKMK∗ h′  1Mψ′ +  8  √  2  FKMK∗ h′  2m2  K  1  Mψ′ + 8  √  2πα  3  fψ′  Mψ′ FK∗0 ¯  K0γ∗(M2  ψ′) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='11)  16  with  a1 =  F  cos(θ0 − θ8)(cos θ0  √  6F8  − sin θ8  √  3F0  ) ,  a2 =  F  cos(θ0 − θ8)( sin θ0  √  6F8  + cos θ8  √  3F0  ) ,  a3 =  F  cos(θ0 − θ8)(−2 cos θ0  √  6F8  − sin θ8  √  3F0  ) ,  a4 =  F  cos(θ0 − θ8)(−2 sin θ0  √  6F8  + cos θ8  √  3F0  ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='12)  The expressions FK∗+K−γ∗ and FK∗0 ¯  K0γ∗ that correspond to the electromagnetic contributions  to the effective couplings of Gψ′→K∗+K− and Gψ′→K∗0 ¯  K0 are given by  FK∗+K−γ∗(s) =  −2  √  2  3FKMV MK∗ [(c1 + c2 + 8c3 − c5)m2  K + (c2 + c5 − c1 − 2c6)M2  K∗ + (c1 − c2 + c5)s  + 24c4(m2  K − m2  π)] +  2FV  3FKMK∗ [(d1 + 8d2 − d3)m2  K + d3(M2  K∗ + s)]  × [Dω(s) + 3Dρ(s) − 2Dφ(s)],  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='13)  and  FK∗0 ¯  K0γ∗(s) =  4  √  2  3FKMV MK∗ [(c1 + c2 + 8c3 − c5)m2  K + (c2 + c5 − c1 − 2c6)M2  K∗ + (c1 − c2 + c5)s]  +  2FV  3FKMK∗ [(d1 + 8d2 − d3)m2  K + d3(M2  K∗ + s)][Dω(s) − 3Dρ(s) − 2Dφ(s)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='14)  It is pointed out that SU(3) breaking effect caused by the c4 term only enters in the charged  FK∗+K−γ∗ amplitude and is absent in the neutral FK∗0 ¯  K0γ∗ process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' For the expressions of  other amplitudes FV P γ(∗), they are given in Appendix of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [20] and we do not repeat them  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  References  [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Appelquist, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' De Rujula, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Politzer and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Glashow, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 34, 365  (1975) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='365  [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Appelquist  and  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Politzer,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  34,  43  (1975)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='43  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  De  Rujula  and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Glashow,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  34,  46-49  (1975)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='46  [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Workman  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [Particle  Data  Group],  PTEP  2022  (2022),  083C01  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1093/ptep/ptac097  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Franklin  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  51,  963-966  (1983)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='963  17  [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Suzuki,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  D  63  (2001),  054021  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='054021  [arXiv:hep-ph/0006296 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Chen  and  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Braaten,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  80  (1998),  5060-5063  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='5060 [arXiv:hep-ph/9801226 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Brodsky  and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Karliner,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  78  (1997),  4682-4685  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4682 [arXiv:hep-ph/9704379 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Pinsky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 236 (1990), 479-482 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0370-2693(90)90387-L  [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Feldmann  and  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Kroll,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  D  62  (2000),  074006  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='074006 [arXiv:hep-ph/0003096 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [11] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Li,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Bugg and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Zou,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='  D 55 (1997),  1421-1424  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1421  [12] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Zhao,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  B  697  (2011),  52-57  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='043  [arXiv:1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1165 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [13] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Wang,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Li  and  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Zhao,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  D  85  (2012),  074015  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='074015 [arXiv:1201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1681 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Gerard  and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Martini,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  B  730  (2014),  264-266  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='068 [arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='3081 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [15] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' ’t Hooft, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 72 (1974) 461;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 75 (1974) 461;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Witten, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 160  (1979) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Gasser  and  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Leutwyler,  Annals  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  158,  142  (1984)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0003-  4916(84)90242-2  [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Gasser and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Leutwyler, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 250, 465-516 (1985) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0550-  3213(85)90492-4  [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Chen,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Guo and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Zheng,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' D 85,  054018 (2012)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='054018 [arXiv:1201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2135 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Guo and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Zheng, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' D 90, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='3, 034013 (2014)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='034013 [arXiv:1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='3366 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [20] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Chen,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Guo  and  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Zou,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  D  91,  014010  (2015)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='014010 [arXiv:1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1159 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Ablikim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [BESIII], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' D 99, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1, 012006 (2019) [erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' D  104, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='9, 099901 (2021)] doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='012006 [arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='03091 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [22] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Ecker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Gasser, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Pich and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' de Rafael, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 321, 311-342 (1989)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0550-3213(89)90346-5  [23] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' ’t Hooft, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 72, 461 (1974) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0550-3213(74)90154-0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' ’t Hooft, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 75, 461-470 (1974) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0550-3213(74)90088-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Witten, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 160, 57-115 (1979) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0550-3213(79)90232-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  18  [24] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Cirigliano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Ecker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Eidemuller, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Kaiser, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Pich and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Portoles, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  B 753, 139-177 (2006) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='nuclphysb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='010 [arXiv:hep-ph/0603205 [hep-  ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [25] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Witten, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 156, 269-283 (1979) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0550-3213(79)90031-2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Coleman  and  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Witten,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  45,  100  (1980)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Veneziano, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 159, 213-224 (1979) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0550-3213(79)90332-8  [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Leutwyler, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' 64, 223-231 (1998) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/S0920-  5632(97)01065-7 [arXiv:hep-ph/9709408 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [27] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Kaiser and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Leutwyler, [arXiv:hep-ph/9806336 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [28] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Ecker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Gasser, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Leutwyler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Pich and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' de Rafael, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 223, 425-432  (1989) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0370-2693(89)91627-4  [29] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Ruiz-Femenia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Pich and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Portoles, JHEP 07, 003 (2003) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1088/1126-  6708/2003/07/003 [arXiv:hep-ph/0306157 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [30] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Kampf  and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Novotny,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  D  84,  014036  (2011)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='014036 [arXiv:1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='3137 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Miranda  and  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Roig,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  D  102,  114017  (2020)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='114017 [arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='11019 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [32] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Chao, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 335, 101-114 (1990) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0550-3213(90)90172-A  [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Dudek, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Edwards and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content=' D 73, 074507 (2006)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='  [34] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' D 79, 094504 (2009)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='  [35] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Meng  and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Niu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' D 84, 034503 (2011) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='2655 [hep-lat]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [36] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Guo and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Roig, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' D 82 (2010), 113016 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='  [37] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Roig  and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Sanz  Cillero,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  B  733  (2014),  158-163  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='034 [arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='6206 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [38] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Duan and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Guo, JHEP 08 (2022), 144 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
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+page_content='12764 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [39] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Guo  and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Oller,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  D  84,  034005  (2011)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='034005 [arXiv:1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2849 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [40] For  a  review  see,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Feldmann,  Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  A  15,no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2,159  (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1142/S0217751X00000082 [arXiv:9907491 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  19  [41] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Zhao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 697, 52 (2011) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='043 [arXiv:1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1165  [hep-ph]]  [42] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Feldmann,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Kroll  and  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Stech,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  D  58  (1998),  114006  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='114006 [arXiv:hep-ph/9802409 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Bramon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Escribano and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Scadron, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 403 (1997), 339-343  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/S0370-2693(97)00508-X [arXiv:hep-ph/9703313 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [44] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Escribano, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' C 65 (2010), 467-473 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1140/epjc/s10052-009-1206-9  [arXiv:0807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4201 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [45] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Ablikim  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [BESIII],  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  D  89,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='9,  092008  (2014)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='092008 [arXiv:1403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='7042 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [46] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Dzhelyadin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 94 (1980), 548 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0370-2693(80)90937-5  [47] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Dzhelyadin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 88 (1979), 379 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0370-2693(79)90490-8  [48] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Behrend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [CELLO], Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' C 49 (1991), 401-410 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1007/BF01549692  [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Achasov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 504 (2001), 275-281 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/S0370-2693(01)00320-  3  [50] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Arnaldi  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [NA60],  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  B  677  (2009),  260-266  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='029 [arXiv:0902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='2547 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [51] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Aihara  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [TPC/Two  Gamma],  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  64  (1990),  172  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='172  [52] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Acciarri  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' [L3],  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B 418 (1998),  399-410  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/S0370-  2693(97)01219-7  [53] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Kubis  and  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Niecknig,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  D  91  (2015)  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='3,  036004  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='036004 [arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='5385 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  [54] Kuang-Ta Chao, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='B 335 101-114 (1990) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1016/0550-3213(90)90172-A  [55] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Fu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Qin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Yang, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content=' A 27, (2012) 1250223  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1142/S0217732312502239 [arXiv:1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='4055 [hep-ph]]  [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  He  and  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Fan,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  D  105  (2022)  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='9,  094034  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='094034 [arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='13568 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
+page_content='  20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfaAcV/content/2301.03869v1.pdf'}
diff --git a/bdFJT4oBgHgl3EQf9C1W/content/tmp_files/2301.11686v1.pdf.txt b/bdFJT4oBgHgl3EQf9C1W/content/tmp_files/2301.11686v1.pdf.txt
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+Communications in Mathematics n (2023), no. m, 00–13
+DOI: https://doi.org/10.46298/cm.ABCD
+©2023 Mohammed Y. Abass and Habeeb M. Abood
+This is an open access article licensed under the CC BY-SA 4.0
+1
+Generalized curvature tensor and the hypersurfaces of the
+Hermitian manifold for the class of Kenmotsu type
+Mohammed Y. Abass and Habeeb M. Abood
+Abstract. This paper determined the components of the generalized curvature ten-
+sor for the class of Kenmotsu type and established the mentioned class is η-Einstein
+manifold when the generalized curvature tensor is flat; the converse holds true un-
+der suitable conditions. It also introduced the notion of generalized Φ-holomorphic
+sectional (GΦSH-) curvature tensor and thus found the necessary and sufficient con-
+ditions for the class of Kenmotsu type to be of constant GΦSH-curvature. In addi-
+tion, the notion of Φ-generalized semi-symmetric was introduced and its relationship
+with the class of Kenmotsu type and η-Einstein manifold established. Furthermore,
+this paper generalized the notion of the manifold of constant curvature and deduced
+its relationship with the aforementioned ideas. It finally showed that the class of
+Kenmotsu type exists as a hypersurface of the Hermitian manifold and derived a
+relation between the components of the Riemannian curvature tensors of the almost
+Hermitian manifold and its hypersurfaces.
+1
+Introduction
+The notion of generalized curvature tensor was introduced by Shaikh and Kundu [19]
+to generalize famous curvature tensors such as conformal curvature tensor, concircular
+tensor, and conharmonic tensor. Yildiz and De [22] introduced and studied Φ-projectively
+semisymmetric and Φ-Weyl semisymmetric while Kenmotsu [13] and Kirichenko and Khari-
+tonova [16] discussed Φ-holomorphic sectional curvature tensor. On the other hand, in-
+vestigation of the geometry of the submanifolds of the Riemannian manifold has won the
+MSC 2020: AMS classification 53C25, 53D10, 53D15.
+Keywords: Almost contact metric manifolds; Einstein manifold; Kenmotsu manifold; Φ-holomorphic
+sectional curvature tensor; hypersurfaces on Hermitian manifolds.
+Affiliation:
+Mohammed Yousif Abass – Department of Mathematics, College of Science, University of
+Basrah, Basrah, Iraq.
+E-mail: mohammedyousif42@yahoo.com
+Habeeb Mtashar Abood – Department of Mathematics, College of Education for Pure
+Sciences, University of Basrah, Basrah, Iraq
+E-mail: iraqsafwan2006@gmail.com
+arXiv:2301.11686v1  [math.DG]  27 Jan 2023
+
+=P sciences2
+Mohammed Y. Abass and Habeeb M. Abood
+interest of authors such as Alegre and Carriazo [3], Sular and ¨Ozg¨ur [20] and Chen [6].
+The special subject in the study of the geometry of submanifolds is the hypersurface of the
+Riemannian manifolds, which has been discussed by Goldberg [9]. We concentrated on the
+geometry of the hypersurfaces of the almost Hermitian manifolds that have almost contact
+structures on the associated G-structure space. The last mentioned topic was studied by
+Banaru and Kirichenko [5]. Moreover, Ignatochkina [12], Ignatochkina and Morozov [11],
+and Nikiforova and Ignatochkina [17] studied the transformations and conformal transfor-
+mations on hypersurfaces induced from almost Hermitian manifolds.
+The aim of this article is organized according to the differential geometry of the gen-
+eralized curvature tensor of the almost contact metric manifolds, especially the class of
+Kenmotsu type and the class of Kenmotsu type as a hypersurface of the Hermitian mani-
+fold.
+2
+Preliminaries
+We use the notations M 2n+1, X(M) and ∇ to denote the smooth manifold M of
+dimension 2n + 1, the Lie algebra of smooth vector fields of M, and the Riemannian
+connection respectively.
+Definition 2.1. [14] A smooth manifold M 2n+1 with the quadruple (Φ, ξ, η, g) is called
+an almost contact metric manifold or briefly ACR-manifold, where Φ : X(M) → X(M),
+ξ ∈ X(M), g is the Riemannian metric and η(·) = g(·, ξ), such that
+Φ(ξ) = 0;
+η(ξ) = 1;
+η ◦ Φ = 0;
+Φ2 = −id + η ⊗ ξ;
+g(ΦX, ΦY ) = g(X, Y ) − η(X)η(Y );
+∀X, Y ∈ X(M).
+In the present article, we fix the components of the Riemannian metric g of ACR-
+manifold M 2n+1 as follows:
+g00 = 1;
+ga0 = gab = gˆaˆb = 0;
+gˆab = δa
+b ;
+gij = gji,
+(1)
+where a, b = 1, 2, ..., n, ˆa = a + n and i, j = 0, 1, ..., 2n. Moreover, the components of the
+endomorphism Φ are given by
+Φ0
+0 = Φa
+ˆb = 0;
+Φa
+b =
+√
+−1δa
+b ;
+Φj
+i = −Φ
+ˆi
+ˆj,
+(2)
+where ˆˆi = i. So, for all X, Y ∈ X(M), we have
+X = Xiεi;
+g(X, Y ) = gijXiY j;
+Φ(X) = Φi
+jXjεi,
+where Xi ∈ C∞(M) and (p; ε0 = ξ, ε1, ..., ε2n) is an A-frame over M 2n+1 such that p ∈ M,
+εa =
+1
+√
+2(id − √−1Φ)ea, εˆa =
+1
+√
+2(id + √−1Φ)ea, and {ξ, e1, ..., en, Φe1, ..., Φen} is a basis
+of X(M). The set of all such A-frame that given above is called an associated G-structure
+space (AG-structure space). For more detail, we refer to the citation [14].
+
+Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu
+type
+3
+Definition 2.2. [1] A class of ACR-manifold such that the following identity:
+∇X(Φ)Y − ∇ΦX(Φ)ΦY = −η(Y )ΦX,
+∀ X, Y ∈ X(M)
+hold is called a class of Kenmotsu type.
+Lemma 2.3. [1] On the AG-structure space, the class of Kenmotsu type satisfies the fol-
+lowing relations:
+Aad
+[bc] − Bad
+[cb] − Bah
+[b B
+d
+|h|c] = 0;
+Aacd
+b
+− Ba[cd]
+b + Ba[c
+hB|h|d]
+b = 0;
+Aa
+[bcd] = 0
+A[bc]
+ad + B
+[cb]
+ad
++ B
+[b
+ah
+B|h|c]
+d = 0;
+Ab
+acd + B
+b
+a[cd] − B
+h
+a[c B
+b
+|h|d] = 0;
+A[bcd]
+a
+= 0;
+where [·| · |·] denotes the anti-symmetric operator of the involving indexes except | · | and
+c, d, h = 1, 2, ..., n.
+We denote by R, r, Q the Riemann curvature tensor, Ricci tensor and Ricci operator
+of ACR-manifold respectively.
+Theorem 2.4. [1] The components of R for the class of Kenmotsu type over the AG-
+structure space are given by
+1. Ra
+0c0 = −δa
+c;
+Ra
+�bcd = 2(Bab
+[cd] − δa
+[c δb
+d]);
+Ra
+�bc �d = Babd
+c − Bab
+h Bhd
+c;
+2. Ra
+bcd = 2Aa
+bcd;
+Ra
+bc �d = Aad
+bc − Bah
+c B
+d
+bh
+− δa
+c δd
+b,
+where R(X, Y )Z = Ri
+jklXkY lZjεi, k, l = 0, 1, ..., 2n and the remaining components of R
+are given by the first Bianchi identity or the conjugate (i.e. Ri
+jkl = Rˆi
+ˆjˆkˆl; ˆ0 = 0) to the
+above components or identical to zero.
+Theorem 2.5. [1] The components of r of the class for Kenmotsu type over the AG-
+structure space are as follows:
+1. r00 = −2n;
+rab = −2Ac
+abc + B
+c
+cab − B
+h
+ca
+B
+c
+hb ;
+2. ra0 = 0;
+r�ab = −2(nδa
+b + Bca
+[bc]) + Aac
+cb − Bah
+b B
+c
+ch ,
+where r(X, Y ) = rijXiY j, rij = rji and the remaining components of r are conjugate to
+the above components.
+Definition 2.6. [1] An ACR-manifold (M 2n+1, Φ, ξ, η, g) with Ricci tensor r, is called
+1. Einstein manifold, if rij = λgij, where λ is an Einstein constant.
+2. η-Einstein manifold, if rij = λgij + µηiηj, where λ, µ are scalars.
+3. has Φ-invariant property, if ra0 = rab = 0.
+
+4
+Mohammed Y. Abass and Habeeb M. Abood
+Definition 2.7. [19] The projective, concircular and generalized curvature tensors of type
+(4, 0) on ACR-manifold (M 2n+1, Φ, ξ, η, g) are defined by the following formulas respec-
+tively:
+P(X, Y, Z, W) = R(X, Y, Z, W) − 1
+2n{g(X, Z)r(Y, W) − g(X, W)r(Y, Z)};
+C(X, Y, Z, W) = R(X, Y, Z, W) −
+s
+2n(2n + 1){g(X, Z)g(Y, W) − g(X, W)g(Y, Z)};
+B(X, Y, Z, W) = a0R(X, Y, Z, W) + a1{g(X, Z)r(Y, W) − g(X, W)r(Y, Z) + r(X, Z).
+g(Y, W) − r(X, W)g(Y, Z)} + 2a2s{g(X, Z)g(Y, W) − g(X, W)g(Y, Z)};
+for all X, Y, Z, W ∈ X(M), where s is the scalar curvature, a0, a1, a2 are scalars and for
+any tensor T of type (3, 1), we get T(X, Y, Z, W) = g(T(Z, W)Y, X) a tensor of type (4,
+0).
+We can rewrite the above tensors on AG-structure space as follows:
+Pijkl = Rijkl − 1
+2n{gik rjl − gil rjk};
+(3)
+Cijkl = Rijkl −
+s
+2n(2n + 1){gik gjl − gil gjk};
+(4)
+Bijkl = a0Rijkl + a1{gik rjl − gil rjk + rik gjl − ril gjk} + 2a2s{gik gjl − gil gjk}.
+(5)
+We note that the generalized curvature tensor B satisfies the first Bianchi identity.
+3
+Properties of Generalized Curvature Tensor
+In this section, we shall investigate some properties of the generalized curvature tensor
+on the class of Kenmotsu type.
+Theorem 3.1. On AG-structure space, the components of generalized curvature tensor are
+given by
+1. Ba0b0 = a1 rab;
+2. Bˆa0b0 = −(a0 + 2na1 − 2a2s)δa
+b + a1 rˆab;
+3. Bˆabcd = 2a0 Aa
+bcd + a1{δa
+c rbd − δa
+d rbc};
+4. Bˆabc ˆd = a0(Aad
+bc − Bah
+c B
+d
+bh ) + a1{δa
+c Qd
+b + δd
+b Qa
+c} + (2a2s − a0)δa
+c δd
+b;
+5. Bˆaˆbcd = 2a0 Bab
+[cd] + 4a1 δ[a
+[c Qb]
+d] + 2(2a2s − a0) δ[a
+[c δb]
+d];
+and the remaining components are identical to zero or given by the first Bianchi identity
+or the conjugate to the above components.
+
+Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu
+type
+5
+Proof. Since r(X, Y ) = g(X, QY ), then rij = gikQk
+j. Consquently, regarding the equation
+(1), we have
+rˆab = gˆakQk
+b = gˆa0Q0
+b + gˆacQc
+b + gˆaˆcQˆc
+b = Qa
+b.
+Since B defined on the class of Kenmotsu type, then the substitutions of the values of
+Rijkl = Rˆi
+jkl and gij from the Theorem 2.4 and the equation (1) in the equation (5), we
+get the desired.
+Theorem 3.2. The class of Kenmotsu type (M 2n+1, Φ, ξ, η, g) has flat generalized curvature
+tensor if and only if, M is η-Einstein manifold with λ =
+1
+a1(a0 + 2na1 − 2a2s), Aa
+bcd = 0,
+µ = −(2n + λ), Aad
+bc = Bah
+c B
+d
+bh
++ a1
+a0µ δa
+cδd
+b and Bab
+[cd] =
+a1
+a0µ δa
+[cδb
+d], provided that
+a0, a1 ̸= 0.
+Proof. Suppose that M 2n+1 has flat generalized curvature tensor with a0 ̸= 0 and a1 ̸= 0,
+then Bijkl = 0 and from the Theorem 3.1, we have
+rab = 0;
+rˆab = 1
+a1
+(a0 + 2na1 − 2a2s)δa
+b ;
+Aa
+bcd = 0.
+Then according to the Definition 2.6, we get λ =
+1
+a1(a0 + 2na1 − 2a2s). Since M is the
+class of Kenmotsu type, then from the Theorem 2.5, we have r00 = −2n = λ + µ and this
+gives µ. Again, the Theorem 3.1; item 4 gives Aad
+bc = Bah
+c B
+d
+bh
++ a1
+a0µ δa
+c δd
+b. Moreover,
+the Theorem 3.1; item 5 gives Bab
+[cd] = a1
+a0µ δa
+[c δb
+d]. The converse is also true.
+Now, we introduce the notion of generalized Φ-holomorphic sectional (GΦHS-) curva-
+ture tensor as follows:
+Definition 3.3. A GΦHS-curvature tensor S of ACR-manifold (M 2n+1, Φ, ξ, η, g) is a map
+defined by
+S(X) = B(ΦX, X, X, ΦX)
+(g(X, X))2
+;
+∀ X ∈ ker(η);
+X ̸= 0.
+Moreover, M is called of pointwise constant GΦHS-curvature if S(X) = γ and γ does not
+depend on X.
+Clearly that, GΦHS-curvature tensor is Φ-holomorphic sectional (ΦHS-) curvature
+tensor if and only if, a0 = 1, and a1 = a2 = 0. Therefore, we can drive the necessary and
+sufficient condition of ACR-manifold to be has pointwise constant GΦHS-curvature on
+AG-structure space.
+Theorem 3.4. An ACR-manifold (M 2n+1, Φ, ξ, η, g) has pointwise constant GΦHS- cur-
+vature if and only if, on AG-structure space, the generalized curvature tensor B of M
+satisfies
+B(a d)
+(bc) = γ
+2
+�δad
+bc ,
+where �δad
+bc = δa
+b δd
+c + δa
+cδd
+b and (··) denotes the symmetric operator of the including indexes.
+
+6
+Mohammed Y. Abass and Habeeb M. Abood
+Proof. Since the tensor B has the same properties of Riemannian curvature tensor R, then
+we can follow the same proof was found in [14] or equivalently in [21].
+Theorem 3.5. The class of Kenmotsu type (M 2n+1, Φ, ξ, η, g) has pointwise constant GΦ
+HS-curvature if and only if, on AG-structure space, M satisfies the following equality:
+Aad
+bc = B
+[ad]
+bc
+− B
+a
+hb
+Bdh
+c − 2a1
+a0
+δ(a
+(b Qd)
+c) + γ − 2a2s + a0
+2a0
+�δad
+bc .
+Proof. Suppose that M is the class of Kenmotsu type and has pointwise constant GΦ
+HS-curvature. Regarding the Theorem 3.4 and the Theorem 3.1; item 4, we get
+A(ad)
+(bc) = B(a|h|
+(b B
+d)
+c)h
+− 2a1
+a0
+δ(a
+(b Qd)
+c) + γ − 2a2s + a0
+2a0
+�δad
+bc .
+The above equation can be rewritten as follows:
+A(ad)
+(bc) = −B
+(a
+h(b
+Bd)h
+c) − 2a1
+a0
+δ(a
+(b Qd)
+c) + γ − 2a2s + a0
+2a0
+�δad
+bc .
+Since Aad
+bc = A[ad]
+[bc] + A[ad]
+(bc) + A(ad)
+[bc] + A(ad)
+(bc) , then taking into account the Lemma 2.3 and the
+above result, we attain the requirement.
+Recently, Yıldız and De [22] introduced the notions of Φ-projectively semisymmetric
+and Φ-Weyl semisymmetric. Regarding these ideas, we can introduce the following defini-
+tion:
+Definition 3.6. An ACR-manifold (M 2n+1, Φ, ξ, η, g) is called Φ-generalized semi-symmetric
+if B(Z, W) · Φ = 0, for all Z, W ∈ X(M), or equivalently
+B(X, ΦY, Z, W) + B(ΦX, Y, Z, W) = 0;
+∀ X, Y, Z, W ∈ X(M).
+Lemma 3.7. On AG-structure space, the ACR-manifold (M 2n+1, Φ, ξ, η, g) is Φ-generalized
+semi (ΦGS-) symmetric if and only if,
+Ba0b0 = Bˆa0b0 = Ba0bc = B�a0bc = Ba0�bc = Babcd = Bˆaˆbcd = 0.
+Proof. According to the Definition 3.6, we have M is Φ-generalized semi-symmetric if and
+only if,
+B(X, ΦY, Z, W) + B(ΦX, Y, Z, W) = 0;
+∀ X, Y, Z, W ∈ X(M).
+On the AG-structure space, the above identity equivalent to the following:
+Biqkl Φq
+j + Btjkl Φt
+i = 0;
+q, t = 0, 1, ..., 2n.
+If we take
+(i, j, k, l) = (a, 0, b, 0), (ˆa, 0, b, 0), (a, 0, b, c), (�a, 0, b, c), (a, 0,�b, c), (a, b, c, d), (ˆa,ˆb, c, d),
+and using the equation (2), we obtain the result.
+
+Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu
+type
+7
+It is not hard to conclude the following:
+Corollary 3.8. The ACR-manifold (M 2n+1, Φ, ξ, η, g) of flat generalized curvature tensor is
+usually ΦGS-symmetric.
+Corollary 3.9. The class of Kenmotsu type (M 2n+1, Φ, ξ, η, g) has flat generalized curvature
+tensor if and only if, M is ΦGS-symmetric with Aa
+bcd = 0 and Aad
+bc = Bah
+c B
+d
+bh
++ a1
+a0µ δa
+cδd
+b,
+where µ = − 1
+a1(a0 + 4na1 − 2a2s), provided that a0, a1 ̸= 0.
+Proof. Suppose that M is the class of Kenmotsu type and it has flat generalized curvature
+tensor, then from the Corollary 3.8, M is ΦGS-symmetric and from the Theorem 3.1, we
+get the other conditions.
+Conversely, If M is ΦGS-symmetric with the above conditions then according to the
+Lemma 3.7 and the Theorem 3.1, M has flat generalized curvature tensor.
+Theorem 3.10. The class of Kenmotsu type (M 2n+1, Φ, ξ, η, g) is ΦGS-symmetric if and
+only if, M is η-Einstein manifold with λ =
+1
+a1(a0 + 2na1 − 2a2s), µ = −(2n + λ) and
+Bab
+[cd] = a1
+a0µ δa
+[cδb
+d], provided that a0, a1 ̸= 0.
+Proof. Suppose that M is Φ-generalized semi-symmetric class of Kenmotsu type, then
+from the Lemma 3.7 and Theorem 3.1, we have
+rab = 0;
+rˆab = 1
+a1
+(a0 + 2na1 − 2a2s)δa
+b ;
+Bab
+[cd] = − 1
+a0
+(a0 + 4na1 − 2a2s)δa
+[cδb
+d].
+Regarding the Definition 2.6 and Theorem 2.5, we attain the values of λ and µ.
+The converse is verified directly.
+Corollary 3.11. The class of Kenmotsu type (M 2n+1, Φ, ξ, η, g) is ΦGS-symmetric and has
+GΦHS-curvature if and only if, M is η-Einstein manifold with λ =
+1
+a1(a0 + 2na1 − 2a2s),
+µ = −(2n + λ), Bab
+[cd] = a1
+a0µδa
+[cδb
+d], and Aad
+bc =
+γ
+2a0 �δad
+bc − B
+a
+hb
+Bdh
+c + a1
+a0µδa
+b δd
+c, provided
+that a0, a1 ̸= 0.
+Proof. Suppose that M is the class of Kenmotsu type, then the necessary and sufficient
+conditions of the present corollary are satisfy from the Theorems 3.5 and 3.10.
+4
+Generalized Curvature Tensor Related with Another Tensors
+In this section, we introduce a generalization of the notion of ACR-manifold of constant
+curvature that used by Abood and Al-Hussaini [2]. We shall show this idea in the following
+definition:
+Definition 4.1. An ACR-manifold (M 2n+1, Φ, ξ, η, g) is said to be has constant generalized
+curvature κ if the following identity holds:
+B(X, Y, Z, W) = κ{g(X, Z)g(Y, W) − g(X, W)g(Y, Z)};
+∀ X, Y, Z, W ∈ X(M).
+
+8
+Mohammed Y. Abass and Habeeb M. Abood
+On the AG-structure space, the Definition 4.1 equivalent to the identity below.
+Bijkl = κ{gik gjl − gil gjk}.
+(6)
+Directly, regarding the Definition 4.1, Definition 2.7 and the definition of the conharmonic
+curvature tensor (see [8]), we have the following result:
+Theorem 4.2. Suppose that M 2n+1 is an ACR-manifold of constant generalized curvature
+κ = 2a2s. Then M has flat conharmonic curvature tensor if and only if, a0 = 1 and
+a1 = −
+1
+2n−1.
+Theorem 4.3. An ACR-manifold (M 2n+1, Φ, ξ, η, g) has constant generalized curvature κ
+if and only if, on the AG-structure space, B has the following components:
+1. Bˆa0b0 = κ δa
+b ;
+2. Bˆabc ˆd = κ δa
+cδd
+b;
+3. Bˆaˆbcd = 2κ δa
+[cδb
+d];
+and the remaining components are identical to zero or establishing from the above compo-
+nents by the first Bianchi identity or by taking the conjugate operation.
+Proof. The result follows from the equation (6) by taking
+(i, j, k, l) = (ˆa, 0, b, 0), (ˆa, b, c, ˆd), (ˆa,ˆb, c, d);
+and using the equation (1).
+Theorem 4.4. The ACR-manifold (M 2n+1, Φ, ξ, η, g) is ΦGS-symmetric if and only if, M
+has constant generalized curvature κ = 0.
+Proof. The claim of this theorem is achieving from the Lemma 3.7 and Theorem 4.3.
+Theorem 4.5. If an ACR-manifold (M 2n+1, Φ, ξ, η, g) has constant generalized curvature
+κ, then M has pointwise constant GΦHS-curvature equal to γ = κ.
+Proof. The allegation of the present theorem occurs from the Theorems 3.4 and 4.3.
+Theorem 4.6. The class of Kenmotsu type (M 2n+1, Φ, ξ, η, g) has constant generalized cur-
+vature κ if and only if, M is η-Einstein manifold with λ =
+1
+a1(a0 + 2na1 − 2a2s + κ),
+Aa
+bcd = 0, µ = −(2n + λ), Aad
+bc = Bah
+c B
+d
+bh
++ a1
+a0µ δa
+cδd
+b and Bab
+[cd] = a1
+a0µ δa
+[cδb
+d], provided
+that a0, a1 ̸= 0.
+Proof. The assertion of this theorem can be happen, if we combining the results of the
+Theorems 3.1 and 4.3.
+
+Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu
+type
+9
+Now, we find the geometric properties of ACR-manifold if the generalized curvature
+tensor, the concircular tensor and the projective tensor are related.
+Suppose that (M 2n+1, Φ, ξ, η, g) is an ACR-manifold satisfies the following condition:
+B(X, Y, Z, W) = a0
+3 {P(X, Y, Z, W) − P(Y, X, Z, W) + C(X, Y, Z, W)}.
+(7)
+Regarding the equations (3), (4) and (5), we can write the equation (7) on AG-structure
+space as follows:
+(a1 + a0
+6n){gik rjl −gil rjk +rik gjl −ril gjk}+(2a2 +
+a0
+6n(2n + 1))s{gik gjl −gil gjk} = 0. (8)
+The contracting of the equation (8), that is multiply it by gik ( the components of g−1 on
+AG-structure space), we can deduce that
+rjl = −(α + 2nβ)s
+(2n − 1)α gjl,
+(9)
+where α = a1 + a0
+6n and β = 2a2 +
+a0
+6n(2n+1). Moreover, the contracting of the equation (9)
+gives a0 + 4na1 + 4n(2n + 1)a2 = 0. Then we can state the following theorem:
+Theorem 4.7. Any ACR-manifold (M 2n+1, Φ, ξ, η, g) which satisfies the identity (7) is an
+Einstein manifold with a0 + 4na1 + 4n(2n + 1)a2 = 0, provided that α ̸= 0. Moreover, if
+M is the class of Kenmotsu type then s = 2n(2n−1)α
+α+2nβ , provided that α + 2nβ ̸= 0.
+Proof. The first part of this theorem is obvious from the above discussion. Now, if M is
+the class of Kenmotsu type then from the Theorem 2.5, we have r00 = −2n. Then the
+result is establishing from the equations (1) and (9).
+5
+The Hypersurfaces of the Hermitian Manifold
+Suppose that (M 2n−1, Φ, ξ, η, g) is an ACR-manifold, then there exists an almost com-
+plex structure J on M × R defined by J(X, f d
+dt) = (ΦX − fξ, η(X) d
+dt), where X ∈ X(M),
+t ∈ R and f is a smooth function on R. The Riemannian metric h on M × R is defined by
+h((X, f1
+d
+dt), (Y, f2
+d
+dt)) = g(X, Y ) + f1 f2;
+∀ X, Y ∈ X(M);
+f1, f2 ∈ C∞(R).
+The structure on M × R is Hermitian if and only if, the structure on M is normal (see
+[18]). Since the class of Kenmotsu type is normal because its the class C3 ⊕C4 ⊕C5, where
+C5 is taken here to be α-Kenmotsu manifold with α = 1 (see [7] for more detail about the
+classes C3 and C4). Then the product manifold of the class of Kenmotsu type and the real
+line is Hermitian (i.e. W3 ⊕ W4, see [10]).
+Now, we discuss the opposite problem, that is, if (N 2n, J, h) is the Hermitian manifold,
+then can be find a hypersuface of N which is the class of Kenmotsu type? We depend on
+the citation [5] for the background.
+Suppose that a, b, c = 1, 2, ..., n−1 and σij = σji; i, j = 1, 2, ..., 2n−1 are the components
+of the second quadratic form as mentioned in [5].
+
+10
+Mohammed Y. Abass and Habeeb M. Abood
+Theorem 5.1. [5] An ACR-manifold which a hypersuface of an almost Hermitian manifold
+has the following first family of the Cartan structure equations:
+dωa = ωa
+b ∧ ωb + Bab
+c
+ωc ∧ ωb + Babc ωb ∧ ωc + (
+√
+2Ban
+b
++
+√
+−1σa
+b )ωb ∧ ω
++ (
+√
+−1σab −
+√
+2 �Bnab − 1
+√
+2Bab
+n − 1
+√
+2
+�Babn)ωb ∧ ω;
+dωa = −ωb
+a ∧ ωb + Bc
+ab ωc ∧ ωb + Babc ωb ∧ ωc + (
+√
+2Bb
+an −
+√
+−1σb
+a)ωb ∧ ω
+− (
+√
+−1σab +
+√
+2 �Bnab + 1
+√
+2Bn
+ab + 1
+√
+2
+�Babn)ωb ∧ ω;
+dω =
+√
+2Bnab ωa ∧ ωb +
+√
+2Bnab ωa ∧ ωb + (
+√
+2Bna
+b
+−
+√
+2Ba
+nb − 2
+√
+−1σa
+b )ωb ∧ ωa
++ ( �Bnbn + Bn
+nb +
+√
+−1σnb)ω ∧ ωb + ( �Bnbn + Bnb
+n −
+√
+−1σb
+n)ω ∧ ωb.
+From Banaru [4], we see that the Hermitian manifold N satisfies Bαβγ = Bαβγ = 0,
+where α, β, γ = 1, 2, ..., n, then the Theorem 5.1 reduce to the following form:
+Theorem 5.2. An ACR-manifold which a hypersuface of the Hermitian manifold has the
+following first family of the Cartan structure equations:
+dωa = ωa
+b ∧ ωb + Bab
+c
+ωc ∧ ωb + (
+√
+2Ban
+b
++
+√
+−1σa
+b )ωb ∧ ω + (
+√
+−1σab − 1
+√
+2Bab
+n )ωb ∧ ω;
+dωa = −ωb
+a ∧ ωb + Bc
+ab ωc ∧ ωb + (
+√
+2Bb
+an −
+√
+−1σb
+a)ωb ∧ ω − (
+√
+−1σab + 1
+√
+2Bn
+ab)ωb ∧ ω;
+dω = (
+√
+2Bna
+b
+−
+√
+2Ba
+nb − 2
+√
+−1σa
+b )ωb ∧ ωa + (Bn
+nb +
+√
+−1σnb)ω ∧ ωb
++ (Bnb
+n −
+√
+−1σb
+n)ω ∧ ωb.
+Regarding Abood and Abass [1], we note that the class of Kenmotsu type satisfies the
+following theorem:
+Theorem 5.3. [1] The class of Kenmotsu type M 2n−1 has the following first group of Cartan
+structure equations:
+dωa = ωa
+b ∧ ωb + Bab
+c ωc ∧ ωb − ωa ∧ ω;
+dωa = −ωb
+a ∧ ωb + B
+c
+ab
+ωc ∧ ωb − ωa ∧ ω;
+dω = 0,
+where Bab
+c and B
+c
+ab
+are the components of the first Kirichenko’s tensor as explained in
+[15].
+Now, if the class of Kenmotsu type M 2n−1 is a hypersurface of the Hermitian manifold
+N 2n, then the cartan structure equations that mentioned in the Theorems 5.2 and 5.3 must
+
+Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu
+type
+11
+be equal. Then we get
+Bab
+c = Bab
+c;
+√
+2Ban
+b
++
+√
+−1σa
+b = −δa
+b ;
+√
+−1σab − 1
+√
+2Bab
+n = 0;
+Bc
+ab = B
+c
+ab ;
+√
+2Bb
+an −
+√
+−1σb
+a = −δb
+a;
+√
+−1σab + 1
+√
+2Bn
+ab = 0;
+(10)
+√
+2Bna
+b
+−
+√
+2Ba
+nb − 2
+√
+−1σa
+b = 0;
+Bn
+nb +
+√
+−1σnb = 0;
+Bnb
+n −
+√
+−1σb
+n = 0.
+Since σ[αβ] = 0 and Bγ
+[αβ] = Bγ
+αβ, then the equation (10) gives the following relations:
+σab = 0;
+σnb = 0;
+σa
+b =
+√
+−1(
+√
+2Ban
+b
++ δa
+b ).
+(11)
+Then from the above discussion, we can establishing the theorem below.
+Theorem 5.4. If the Hermitian manifold has the class of Kenmotsu type as a hypersurface,
+then the second quadratic form has components agree with the equation (11).
+On the other hand, we can establish a relation between the components of the Rieman-
+nian curvature tensors of the almost Hermitian manifold and its hypersurfaces.
+Suppose that Ri
+jkl are the components of the Riemannian curvature tensor of the almost
+Hermitian manifold, N 2n and �Ri
+jkl are the components of the Riemannian curvature tensor
+of its hypersurface M 2n−1. Then from the second group of cartan structure equations, we
+have
+dωi
+j = ωi
+k ∧ ωk
+j + 1
+2Ri
+jkl ωk ∧ ωl;
+dθi
+j = θi
+k ∧ θk
+j + 1
+2
+�Ri
+jkl θk ∧ θl;
+where ωi
+j and θi
+j are the Riemannian connection forms of N and M respectively. Whereas,
+ωk and θk are the dual A-frames on AG-structure spaces of N and M respectively. More-
+over, from [5], we have
+θi = Ci
+j ωj;
+ωi = �Ci
+j θj;
+θi
+j = Ci
+k ωk
+r �Cr
+j ;
+ωi
+j = �Ci
+k θk
+r Cr
+j ;
+where C = (Ci
+j) and C−1 = ( �Ci
+j) were defined in [5]. Then the substitution of the above
+relations in the second group of cartan structure equations, we conclude the following
+theorem:
+Theorem 5.5. If Ri
+jkl and �Rq
+rst are the components of the Riemannian curvature tensor of
+the almost Hermitian manifold (N 2n, J, g) and its hypersurface (M 2n−1, Φ, ξ, η, g) respec-
+tively, then they are related as follow:
+Ri
+jkl = �Ci
+q �Rq
+rst Cr
+j Cs
+k Ct
+l .
+
+12
+Mohammed Y. Abass and Habeeb M. Abood
+References
+[1] Abood H. M. and Abass M. Y.: A study of new class of almost contact metric manifolds of
+Kenmotsu type. Tamkang Journal of Mathematics 52 (2) (2021) 253–266-.
+[2] Abood H. M. and Al-Hussaini F. H. : Constant curvature of a locally conformal almost
+cosymplectic manifold. In: AIP Conference Proceedings2086. 2019, 030003.
+[3] Alegre P. and Carriazo A.: Submanifolds of generalized Sasakian space forms. Taiwanese Journal
+of Mathematics 13 (3) (2009) 923–941-.
+[4] Banaru M. B.: Geometry of 6-dimensional Hermitian manifolds of the octave algebra. Journal of
+Mathematical Sciences 207 (3) (2015) 354–388-.
+[5] Banaru M. B. and Kirichenko V. F.: Almost contact metric structures on the hypersurface of
+almost Hermitian manifolds. Journal of Mathematical Sciences 207 (4) (2015) 513–537-.
+[6] Chen B. -Y.: Differential geometry of warped product manifolds and submanifolds. World
+Scientific Publishing Co. Pte. Ltd., Singapore (2017).
+[7] Chinea D. and Gonzalez C.: A classification of almost contact metric manifolds. Annali di
+Matematica Pura ed Applicata 156 (1) (1990) 15–36-.
+[8] De U. C. and Suh Y. J.: On weakly semiconformally symmetric manifolds. Acta Mathematica
+Hungarica 157 (2) (2019) 503–521-.
+[9] Goldberg S. I.: Totally geodesic hypersufaces of Kaehler manifolds. Pacific Journal of
+Mathematics 27 (2) (1968) 275–281-.
+[10] Gray A. and Hervella L. M.: The sixteen classes of almost Hermitian manifolds and their linear
+invariants. Annali di Matematica Pura ed Applicata 123 (1) (1980) 35–58-.
+[11] Ignatochkina L. A. and Morozov P. B.: The transformations induced by conformal
+transformations on T 1-bundle. Journal of Basrah Researches ((Sciences)) 37 (4C) (2011) 8–15-.
+[12] Ignatochkina L. A. and Morozov P. B.: Induced transformations for almost Hermitian structure of
+linear extensions. Chebyshevskii Sbornik 18 (2) (2017) 124–133-.
+[13] Kenmotsu K.: A class of almost contact Riemannian manifolds. Tohoku Mathematical Journal 24
+(1) (1972) 93–103-.
+[14] Kirichenko V. F.: Differential-geometric structures on manifolds (in Russian). Moscow State
+Pedagogical University, Moscow (2003).
+[15] Kirichenko V. F. and Dondukova N. N.: Contactly geodesic transformations of almost-contact
+metric structures. Mathematical Notes 80 (2) (2006) 204–213-.
+[16] Kirichenko V. F. and Kharitonova S. V.: On the geometry of normal locally conformal almost
+cosymplectic manifolds. Mathematical Notes 91 (1) (2012) 34–45-.
+[17] Nikiforova A. V. and Ignatochkina L. A.: The transformations induced on hypersurfaces of almost
+Hermitian manifolds. Journal of Basrah Researches ((Sciences)) 37 (4C) (2011) 1–7-.
+[18] Pitis G.: Geometry of Kenmotsu Manifolds. Editura Universitatii Transilvania, Brasov (2007).
+[19] Shaikh A. A. and Kundu H.: On equivalency of various geometric structures. Journal of Geometry
+105 (1) (2014) 139–165-.
+[20] Sular S. and ¨Ozg¨ur C.: On some submanifolds of Kenmotsu manifolds. Chaos, Solitons and
+Fractals 42 (4) (2009) 1990–1995-.
+
+Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu
+type
+13
+[21] Umnova S. V.: Geometry of Kenmotsu manifolds and their generalizations (in Russian) (2002).
+[22] Yıldız A. and De U. C.: A classification of (κ, µ)-contact metric manifolds. Communications of
+the Korean Mathematical Society 27 (2) (2012) 327–339-.
+Received: Received date
+Accepted for publication: Accepted date
+Communicated by: Handling Editor
+
+Communications in Mathematics n (2023), no. m, 00–1
+DOI: https://doi.org/10.46298/cm.ABCD
+©2023 First Author, Second Author and Third Author
+This is an open access article licensed under the CC BY-SA 4.0
+1
+Title of the paper
+First Author, Second Author and Third Author
+Abstract. Abstract of the paper...
+Body of the paper ...
+References
+[1] Lastname1 F. N.: Title1. Journal1 Volume1 (Number1) (Year1) Pages1.
+[2] Lastname2 F. N. and Lastname3 F. N.: Title2. Publisher2 (Year2).
+[3] Lastname4 F. N., Lastname5 F. N. and Lastname6 F. N.: Title3. In: Journal3Volume3. Year3,
+Pages3.
+[4] Lastname7 F. N.: Title4. . In: Editor4. Series4 Volume4, Publisher4 (Year4).
+[5] Lastname8 F. N. and Lastname9 F. N.: Title5. ArXiv:id.
+[6] Lastname10 F. N., Lastname11 F. N. and Lastname12 F. N.: Title6 (Year6).
+Received: Received date
+Accepted for publication: Accepted date
+Communicated by: Handling Editor
+MSC 2020: AMS classification ... (see https://mathscinet.ams.org/mathscinet/msc/msc2020.html)
+Keywords: Keyword one, Keyword two, Keyword three, ...
+Affiliation:
+First author’s Name – Physical professional address of the First Author...
+E-mail: email@first.author.com
+Second author’s name – Physical professional address of the Second Author...
+E-mail: email@second.author.com
+Third author’s name – Physical professional address of the Third Author...
+E-mail: email@third.author.com
+arXiv:2301.11686v1  [math.DG]  27 Jan 2023
+
+=P sciences
\ No newline at end of file
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new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf,len=556
+page_content='Communications in Mathematics n (2023), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' m, 00–13  DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='46298/cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='ABCD  ©2023 Mohammed Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abass and Habeeb M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abood  This is an open access article licensed under the CC BY-SA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='0  1  Generalized curvature tensor and the hypersurfaces of the  Hermitian manifold for the class of Kenmotsu type  Mohammed Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abass and Habeeb M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abood  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' This paper determined the components of the generalized curvature ten-  sor for the class of Kenmotsu type and established the mentioned class is η-Einstein  manifold when the generalized curvature tensor is flat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' the converse holds true un-  der suitable conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' It also introduced the notion of generalized Φ-holomorphic  sectional (GΦSH-) curvature tensor and thus found the necessary and sufficient con-  ditions for the class of Kenmotsu type to be of constant GΦSH-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' In addi-  tion, the notion of Φ-generalized semi-symmetric was introduced and its relationship  with the class of Kenmotsu type and η-Einstein manifold established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Furthermore,  this paper generalized the notion of the manifold of constant curvature and deduced  its relationship with the aforementioned ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' It finally showed that the class of  Kenmotsu type exists as a hypersurface of the Hermitian manifold and derived a  relation between the components of the Riemannian curvature tensors of the almost  Hermitian manifold and its hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  1  Introduction  The notion of generalized curvature tensor was introduced by Shaikh and Kundu [19]  to generalize famous curvature tensors such as conformal curvature tensor, concircular  tensor, and conharmonic tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Yildiz and De [22] introduced and studied Φ-projectively  semisymmetric and Φ-Weyl semisymmetric while Kenmotsu [13] and Kirichenko and Khari-  tonova [16] discussed Φ-holomorphic sectional curvature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' On the other hand, in-  vestigation of the geometry of the submanifolds of the Riemannian manifold has won the  MSC 2020: AMS classification 53C25, 53D10, 53D15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Keywords: Almost contact metric manifolds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Einstein manifold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Kenmotsu manifold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Φ-holomorphic  sectional curvature tensor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' hypersurfaces on Hermitian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Affiliation:  Mohammed Yousif Abass – Department of Mathematics, College of Science, University of  Basrah, Basrah, Iraq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  E-mail: mohammedyousif42@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='com  Habeeb Mtashar Abood – Department of Mathematics, College of Education for Pure  Sciences, University of Basrah, Basrah, Iraq  E-mail: iraqsafwan2006@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='com  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='11686v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='DG]  27 Jan 2023  =P sciences2  Mohammed Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abass and Habeeb M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abood  interest of authors such as Alegre and Carriazo [3], Sular and ¨Ozg¨ur [20] and Chen [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  The special subject in the study of the geometry of submanifolds is the hypersurface of the  Riemannian manifolds, which has been discussed by Goldberg [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' We concentrated on the  geometry of the hypersurfaces of the almost Hermitian manifolds that have almost contact  structures on the associated G-structure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The last mentioned topic was studied by  Banaru and Kirichenko [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Moreover, Ignatochkina [12], Ignatochkina and Morozov [11],  and Nikiforova and Ignatochkina [17] studied the transformations and conformal transfor-  mations on hypersurfaces induced from almost Hermitian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  The aim of this article is organized according to the differential geometry of the gen-  eralized curvature tensor of the almost contact metric manifolds, especially the class of  Kenmotsu type and the class of Kenmotsu type as a hypersurface of the Hermitian mani-  fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  2  Preliminaries  We use the notations M 2n+1, X(M) and ∇ to denote the smooth manifold M of  dimension 2n + 1, the Lie algebra of smooth vector fields of M, and the Riemannian  connection respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' [14] A smooth manifold M 2n+1 with the quadruple (Φ, ξ, η, g) is called  an almost contact metric manifold or briefly ACR-manifold, where Φ : X(M) → X(M),  ξ ∈ X(M), g is the Riemannian metric and η(·) = g(·, ξ), such that  Φ(ξ) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  η(ξ) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  η ◦ Φ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Φ2 = −id + η ⊗ ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  g(ΦX, ΦY ) = g(X, Y ) − η(X)η(Y );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  ∀X, Y ∈ X(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  In the present article, we fix the components of the Riemannian metric g of ACR-  manifold M 2n+1 as follows:  g00 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  ga0 = gab = gˆaˆb = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  gˆab = δa  b ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  gij = gji,  (1)  where a, b = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=', n, ˆa = a + n and i, j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=', 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Moreover, the components of the  endomorphism Φ are given by  Φ0  0 = Φa  ˆb = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Φa  b =  √  −1δa  b ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Φj  i = −Φ  ˆi  ˆj,  (2)  where ˆˆi = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' So, for all X, Y ∈ X(M), we have  X = Xiεi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  g(X, Y ) = gijXiY j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Φ(X) = Φi  jXjεi,  where Xi ∈ C∞(M) and (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' ε0 = ξ, ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=', ε2n) is an A-frame over M 2n+1 such that p ∈ M,  εa =  1  √  2(id − √−1Φ)ea, εˆa =  1  √  2(id + √−1Φ)ea, and {ξ, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=', en, Φe1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=', Φen} is a basis  of X(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The set of all such A-frame that given above is called an associated G-structure  space (AG-structure space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' For more detail, we refer to the citation [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu  type  3  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' [1] A class of ACR-manifold such that the following identity:  ∇X(Φ)Y − ∇ΦX(Φ)ΦY = −η(Y )ΦX,  ∀ X, Y ∈ X(M)  hold is called a class of Kenmotsu type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' [1] On the AG-structure space, the class of Kenmotsu type satisfies the fol-  lowing relations:  Aad  [bc] − Bad  [cb] − Bah  [b B  d  |h|c] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Aacd  b  − Ba[cd]  b + Ba[c  hB|h|d]  b = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Aa  [bcd] = 0  A[bc]  ad + B  [cb]  ad  + B  [b  ah  B|h|c]  d = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Ab  acd + B  b  a[cd] − B  h  a[c B  b  |h|d] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  A[bcd]  a  = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  where [·| · |·] denotes the anti-symmetric operator of the involving indexes except | · | and  c, d, h = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  We denote by R, r, Q the Riemann curvature tensor, Ricci tensor and Ricci operator  of ACR-manifold respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' [1] The components of R for the class of Kenmotsu type over the AG-  structure space are given by  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Ra  0c0 = −δa  c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Ra  �bcd = 2(Bab  [cd] − δa  [c δb  d]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Ra  �bc �d = Babd  c − Bab  h Bhd  c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Ra  bcd = 2Aa  bcd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Ra  bc �d = Aad  bc − Bah  c B  d  bh  − δa  c δd  b,  where R(X, Y )Z = Ri  jklXkY lZjεi, k, l = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=', 2n and the remaining components of R  are given by the first Bianchi identity or the conjugate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Ri  jkl = Rˆi  ˆjˆkˆl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' ˆ0 = 0) to the  above components or identical to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' [1] The components of r of the class for Kenmotsu type over the AG-  structure space are as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' r00 = −2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  rab = −2Ac  abc + B  c  cab − B  h  ca  B  c  hb ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' ra0 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  r�ab = −2(nδa  b + Bca  [bc]) + Aac  cb − Bah  b B  c  ch ,  where r(X, Y ) = rijXiY j, rij = rji and the remaining components of r are conjugate to  the above components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' [1] An ACR-manifold (M 2n+1, Φ, ξ, η, g) with Ricci tensor r, is called  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Einstein manifold, if rij = λgij, where λ is an Einstein constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' η-Einstein manifold, if rij = λgij + µηiηj, where λ, µ are scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' has Φ-invariant property, if ra0 = rab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  4  Mohammed Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abass and Habeeb M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abood  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' [19] The projective, concircular and generalized curvature tensors of type  (4, 0) on ACR-manifold (M 2n+1, Φ, ξ, η, g) are defined by the following formulas respec-  tively:  P(X, Y, Z, W) = R(X, Y, Z, W) − 1  2n{g(X, Z)r(Y, W) − g(X, W)r(Y, Z)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  C(X, Y, Z, W) = R(X, Y, Z, W) −  s  2n(2n + 1){g(X, Z)g(Y, W) − g(X, W)g(Y, Z)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  B(X, Y, Z, W) = a0R(X, Y, Z, W) + a1{g(X, Z)r(Y, W) − g(X, W)r(Y, Z) + r(X, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  g(Y, W) − r(X, W)g(Y, Z)} + 2a2s{g(X, Z)g(Y, W) − g(X, W)g(Y, Z)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  for all X, Y, Z, W ∈ X(M), where s is the scalar curvature, a0, a1, a2 are scalars and for  any tensor T of type (3, 1), we get T(X, Y, Z, W) = g(T(Z, W)Y, X) a tensor of type (4,  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  We can rewrite the above tensors on AG-structure space as follows:  Pijkl = Rijkl − 1  2n{gik rjl − gil rjk};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  (3)  Cijkl = Rijkl −  s  2n(2n + 1){gik gjl − gil gjk};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  (4)  Bijkl = a0Rijkl + a1{gik rjl − gil rjk + rik gjl − ril gjk} + 2a2s{gik gjl − gil gjk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  (5)  We note that the generalized curvature tensor B satisfies the first Bianchi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  3  Properties of Generalized Curvature Tensor  In this section, we shall investigate some properties of the generalized curvature tensor  on the class of Kenmotsu type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' On AG-structure space, the components of generalized curvature tensor are  given by  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Ba0b0 = a1 rab;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Bˆa0b0 = −(a0 + 2na1 − 2a2s)δa  b + a1 rˆab;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Bˆabcd = 2a0 Aa  bcd + a1{δa  c rbd − δa  d rbc};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Bˆabc ˆd = a0(Aad  bc − Bah  c B  d  bh ) + a1{δa  c Qd  b + δd  b Qa  c} + (2a2s − a0)δa  c δd  b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Bˆaˆbcd = 2a0 Bab  [cd] + 4a1 δ[a  [c Qb]  d] + 2(2a2s − a0) δ[a  [c δb]  d];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  and the remaining components are identical to zero or given by the first Bianchi identity  or the conjugate to the above components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu  type  5  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Since r(X, Y ) = g(X, QY ), then rij = gikQk  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Consquently, regarding the equation  (1), we have  rˆab = gˆakQk  b = gˆa0Q0  b + gˆacQc  b + gˆaˆcQˆc  b = Qa  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Since B defined on the class of Kenmotsu type, then the substitutions of the values of  Rijkl = Rˆi  jkl and gij from the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='4 and the equation (1) in the equation (5), we  get the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The class of Kenmotsu type (M 2n+1, Φ, ξ, η, g) has flat generalized curvature  tensor if and only if, M is η-Einstein manifold with λ =  1  a1(a0 + 2na1 − 2a2s), Aa  bcd = 0,  µ = −(2n + λ), Aad  bc = Bah  c B  d  bh  + a1  a0µ δa  cδd  b and Bab  [cd] =  a1  a0µ δa  [cδb  d], provided that  a0, a1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Suppose that M 2n+1 has flat generalized curvature tensor with a0 ̸= 0 and a1 ̸= 0,  then Bijkl = 0 and from the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1, we have  rab = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  rˆab = 1  a1  (a0 + 2na1 − 2a2s)δa  b ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Aa  bcd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Then according to the Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='6, we get λ =  1  a1(a0 + 2na1 − 2a2s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Since M is the  class of Kenmotsu type, then from the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='5, we have r00 = −2n = λ + µ and this  gives µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Again, the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' item 4 gives Aad  bc = Bah  c B  d  bh  + a1  a0µ δa  c δd  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Moreover,  the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' item 5 gives Bab  [cd] = a1  a0µ δa  [c δb  d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The converse is also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Now, we introduce the notion of generalized Φ-holomorphic sectional (GΦHS-) curva-  ture tensor as follows:  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' A GΦHS-curvature tensor S of ACR-manifold (M 2n+1, Φ, ξ, η, g) is a map  defined by  S(X) = B(ΦX, X, X, ΦX)  (g(X, X))2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  ∀ X ∈ ker(η);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  X ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Moreover, M is called of pointwise constant GΦHS-curvature if S(X) = γ and γ does not  depend on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Clearly that, GΦHS-curvature tensor is Φ-holomorphic sectional (ΦHS-) curvature  tensor if and only if, a0 = 1, and a1 = a2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Therefore, we can drive the necessary and  sufficient condition of ACR-manifold to be has pointwise constant GΦHS-curvature on  AG-structure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' An ACR-manifold (M 2n+1, Φ, ξ, η, g) has pointwise constant GΦHS- cur-  vature if and only if, on AG-structure space, the generalized curvature tensor B of M  satisfies  B(a d)  (bc) = γ  2  �δad  bc ,  where �δad  bc = δa  b δd  c + δa  cδd  b and (··) denotes the symmetric operator of the including indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  6  Mohammed Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abass and Habeeb M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abood  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Since the tensor B has the same properties of Riemannian curvature tensor R, then  we can follow the same proof was found in [14] or equivalently in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The class of Kenmotsu type (M 2n+1, Φ, ξ, η, g) has pointwise constant GΦ  HS-curvature if and only if, on AG-structure space, M satisfies the following equality:  Aad  bc = B  [ad]  bc  − B  a  hb  Bdh  c − 2a1  a0  δ(a  (b Qd)  c) + γ − 2a2s + a0  2a0  �δad  bc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Suppose that M is the class of Kenmotsu type and has pointwise constant GΦ  HS-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Regarding the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='4 and the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' item 4, we get  A(ad)  (bc) = B(a|h|  (b B  d)  c)h  − 2a1  a0  δ(a  (b Qd)  c) + γ − 2a2s + a0  2a0  �δad  bc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  The above equation can be rewritten as follows:  A(ad)  (bc) = −B  (a  h(b  Bd)h  c) − 2a1  a0  δ(a  (b Qd)  c) + γ − 2a2s + a0  2a0  �δad  bc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Since Aad  bc = A[ad]  [bc] + A[ad]  (bc) + A(ad)  [bc] + A(ad)  (bc) , then taking into account the Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='3 and the  above result, we attain the requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Recently, Yıldız and De [22] introduced the notions of Φ-projectively semisymmetric  and Φ-Weyl semisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Regarding these ideas, we can introduce the following defini-  tion:  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' An ACR-manifold (M 2n+1, Φ, ξ, η, g) is called Φ-generalized semi-symmetric  if B(Z, W) · Φ = 0, for all Z, W ∈ X(M), or equivalently  B(X, ΦY, Z, W) + B(ΦX, Y, Z, W) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  ∀ X, Y, Z, W ∈ X(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' On AG-structure space, the ACR-manifold (M 2n+1, Φ, ξ, η, g) is Φ-generalized  semi (ΦGS-) symmetric if and only if,  Ba0b0 = Bˆa0b0 = Ba0bc = B�a0bc = Ba0�bc = Babcd = Bˆaˆbcd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' According to the Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='6, we have M is Φ-generalized semi-symmetric if and  only if,  B(X, ΦY, Z, W) + B(ΦX, Y, Z, W) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  ∀ X, Y, Z, W ∈ X(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  On the AG-structure space, the above identity equivalent to the following:  Biqkl Φq  j + Btjkl Φt  i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  q, t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=', 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  If we take  (i, j, k, l) = (a, 0, b, 0), (ˆa, 0, b, 0), (a, 0, b, c), (�a, 0, b, c), (a, 0,�b, c), (a, b, c, d), (ˆa,ˆb, c, d),  and using the equation (2), we obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu  type  7  It is not hard to conclude the following:  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The ACR-manifold (M 2n+1, Φ, ξ, η, g) of flat generalized curvature tensor is  usually ΦGS-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The class of Kenmotsu type (M 2n+1, Φ, ξ, η, g) has flat generalized curvature  tensor if and only if, M is ΦGS-symmetric with Aa  bcd = 0 and Aad  bc = Bah  c B  d  bh  + a1  a0µ δa  cδd  b,  where µ = − 1  a1(a0 + 4na1 − 2a2s), provided that a0, a1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Suppose that M is the class of Kenmotsu type and it has flat generalized curvature  tensor, then from the Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='8, M is ΦGS-symmetric and from the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1, we  get the other conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Conversely, If M is ΦGS-symmetric with the above conditions then according to the  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='7 and the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1, M has flat generalized curvature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The class of Kenmotsu type (M 2n+1, Φ, ξ, η, g) is ΦGS-symmetric if and  only if, M is η-Einstein manifold with λ =  1  a1(a0 + 2na1 − 2a2s), µ = −(2n + λ) and  Bab  [cd] = a1  a0µ δa  [cδb  d], provided that a0, a1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Suppose that M is Φ-generalized semi-symmetric class of Kenmotsu type, then  from the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='7 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1, we have  rab = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  rˆab = 1  a1  (a0 + 2na1 − 2a2s)δa  b ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Bab  [cd] = − 1  a0  (a0 + 4na1 − 2a2s)δa  [cδb  d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Regarding the Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='6 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='5, we attain the values of λ and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  The converse is verified directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The class of Kenmotsu type (M 2n+1, Φ, ξ, η, g) is ΦGS-symmetric and has  GΦHS-curvature if and only if, M is η-Einstein manifold with λ =  1  a1(a0 + 2na1 − 2a2s),  µ = −(2n + λ), Bab  [cd] = a1  a0µδa  [cδb  d], and Aad  bc =  γ  2a0 �δad  bc − B  a  hb  Bdh  c + a1  a0µδa  b δd  c, provided  that a0, a1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Suppose that M is the class of Kenmotsu type, then the necessary and sufficient  conditions of the present corollary are satisfy from the Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  4  Generalized Curvature Tensor Related with Another Tensors  In this section, we introduce a generalization of the notion of ACR-manifold of constant  curvature that used by Abood and Al-Hussaini [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' We shall show this idea in the following  definition:  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' An ACR-manifold (M 2n+1, Φ, ξ, η, g) is said to be has constant generalized  curvature κ if the following identity holds:  B(X, Y, Z, W) = κ{g(X, Z)g(Y, W) − g(X, W)g(Y, Z)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  ∀ X, Y, Z, W ∈ X(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  8  Mohammed Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abass and Habeeb M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abood  On the AG-structure space, the Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1 equivalent to the identity below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Bijkl = κ{gik gjl − gil gjk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  (6)  Directly, regarding the Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='7 and the definition of the conharmonic  curvature tensor (see [8]), we have the following result:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Suppose that M 2n+1 is an ACR-manifold of constant generalized curvature  κ = 2a2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Then M has flat conharmonic curvature tensor if and only if, a0 = 1 and  a1 = −  1  2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' An ACR-manifold (M 2n+1, Φ, ξ, η, g) has constant generalized curvature κ  if and only if, on the AG-structure space, B has the following components:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Bˆa0b0 = κ δa  b ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Bˆabc ˆd = κ δa  cδd  b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Bˆaˆbcd = 2κ δa  [cδb  d];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  and the remaining components are identical to zero or establishing from the above compo-  nents by the first Bianchi identity or by taking the conjugate operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The result follows from the equation (6) by taking  (i, j, k, l) = (ˆa, 0, b, 0), (ˆa, b, c, ˆd), (ˆa,ˆb, c, d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  and using the equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The ACR-manifold (M 2n+1, Φ, ξ, η, g) is ΦGS-symmetric if and only if, M  has constant generalized curvature κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The claim of this theorem is achieving from the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='7 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' If an ACR-manifold (M 2n+1, Φ, ξ, η, g) has constant generalized curvature  κ, then M has pointwise constant GΦHS-curvature equal to γ = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The allegation of the present theorem occurs from the Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The class of Kenmotsu type (M 2n+1, Φ, ξ, η, g) has constant generalized cur-  vature κ if and only if, M is η-Einstein manifold with λ =  1  a1(a0 + 2na1 − 2a2s + κ),  Aa  bcd = 0, µ = −(2n + λ), Aad  bc = Bah  c B  d  bh  + a1  a0µ δa  cδd  b and Bab  [cd] = a1  a0µ δa  [cδb  d], provided  that a0, a1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The assertion of this theorem can be happen, if we combining the results of the  Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu  type  9  Now, we find the geometric properties of ACR-manifold if the generalized curvature  tensor, the concircular tensor and the projective tensor are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Suppose that (M 2n+1, Φ, ξ, η, g) is an ACR-manifold satisfies the following condition:  B(X, Y, Z, W) = a0  3 {P(X, Y, Z, W) − P(Y, X, Z, W) + C(X, Y, Z, W)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  (7)  Regarding the equations (3), (4) and (5), we can write the equation (7) on AG-structure  space as follows:  (a1 + a0  6n){gik rjl −gil rjk +rik gjl −ril gjk}+(2a2 +  a0  6n(2n + 1))s{gik gjl −gil gjk} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' (8)  The contracting of the equation (8), that is multiply it by gik ( the components of g−1 on  AG-structure space), we can deduce that  rjl = −(α + 2nβ)s  (2n − 1)α gjl,  (9)  where α = a1 + a0  6n and β = 2a2 +  a0  6n(2n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Moreover, the contracting of the equation (9)  gives a0 + 4na1 + 4n(2n + 1)a2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Then we can state the following theorem:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Any ACR-manifold (M 2n+1, Φ, ξ, η, g) which satisfies the identity (7) is an  Einstein manifold with a0 + 4na1 + 4n(2n + 1)a2 = 0, provided that α ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Moreover, if  M is the class of Kenmotsu type then s = 2n(2n−1)α  α+2nβ , provided that α + 2nβ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The first part of this theorem is obvious from the above discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Now, if M is  the class of Kenmotsu type then from the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='5, we have r00 = −2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Then the  result is establishing from the equations (1) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  5  The Hypersurfaces of the Hermitian Manifold  Suppose that (M 2n−1, Φ, ξ, η, g) is an ACR-manifold, then there exists an almost com-  plex structure J on M × R defined by J(X, f d  dt) = (ΦX − fξ, η(X) d  dt), where X ∈ X(M),  t ∈ R and f is a smooth function on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' The Riemannian metric h on M × R is defined by  h((X, f1  d  dt), (Y, f2  d  dt)) = g(X, Y ) + f1 f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  ∀ X, Y ∈ X(M);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  f1, f2 ∈ C∞(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  The structure on M × R is Hermitian if and only if, the structure on M is normal (see  [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Since the class of Kenmotsu type is normal because its the class C3 ⊕C4 ⊕C5, where  C5 is taken here to be α-Kenmotsu manifold with α = 1 (see [7] for more detail about the  classes C3 and C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Then the product manifold of the class of Kenmotsu type and the real  line is Hermitian (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' W3 ⊕ W4, see [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Now, we discuss the opposite problem, that is, if (N 2n, J, h) is the Hermitian manifold,  then can be find a hypersuface of N which is the class of Kenmotsu type?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' We depend on  the citation [5] for the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Suppose that a, b, c = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=', n−1 and σij = σji;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' i, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=', 2n−1 are the components  of the second quadratic form as mentioned in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  10  Mohammed Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abass and Habeeb M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abood  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' [5] An ACR-manifold which a hypersuface of an almost Hermitian manifold  has the following first family of the Cartan structure equations:  dωa = ωa  b ∧ ωb + Bab  c  ωc ∧ ωb + Babc ωb ∧ ωc + (  √  2Ban  b  +  √  −1σa  b )ωb ∧ ω  + (  √  −1σab −  √  2 �Bnab − 1  √  2Bab  n − 1  √  2  �Babn)ωb ∧ ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  dωa = −ωb  a ∧ ωb + Bc  ab ωc ∧ ωb + Babc ωb ∧ ωc + (  √  2Bb  an −  √  −1σb  a)ωb ∧ ω  − (  √  −1σab +  √  2 �Bnab + 1  √  2Bn  ab + 1  √  2  �Babn)ωb ∧ ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  dω =  √  2Bnab ωa ∧ ωb +  √  2Bnab ωa ∧ ωb + (  √  2Bna  b  −  √  2Ba  nb − 2  √  −1σa  b )ωb ∧ ωa  + ( �Bnbn + Bn  nb +  √  −1σnb)ω ∧ ωb + ( �Bnbn + Bnb  n −  √  −1σb  n)ω ∧ ωb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  From Banaru [4], we see that the Hermitian manifold N satisfies Bαβγ = Bαβγ = 0,  where α, β, γ = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=', n, then the Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='1 reduce to the following form:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' An ACR-manifold which a hypersuface of the Hermitian manifold has the  following first family of the Cartan structure equations:  dωa = ωa  b ∧ ωb + Bab  c  ωc ∧ ωb + (  √  2Ban  b  +  √  −1σa  b )ωb ∧ ω + (  √  −1σab − 1  √  2Bab  n )ωb ∧ ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  dωa = −ωb  a ∧ ωb + Bc  ab ωc ∧ ωb + (  √  2Bb  an −  √  −1σb  a)ωb ∧ ω − (  √  −1σab + 1  √  2Bn  ab)ωb ∧ ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  dω = (  √  2Bna  b  −  √  2Ba  nb − 2  √  −1σa  b )ωb ∧ ωa + (Bn  nb +  √  −1σnb)ω ∧ ωb  + (Bnb  n −  √  −1σb  n)ω ∧ ωb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Regarding Abood and Abass [1], we note that the class of Kenmotsu type satisfies the  following theorem:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' [1] The class of Kenmotsu type M 2n−1 has the following first group of Cartan  structure equations:  dωa = ωa  b ∧ ωb + Bab  c ωc ∧ ωb − ωa ∧ ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  dωa = −ωb  a ∧ ωb + B  c  ab  ωc ∧ ωb − ωa ∧ ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  dω = 0,  where Bab  c and B  c  ab  are the components of the first Kirichenko’s tensor as explained in  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Now, if the class of Kenmotsu type M 2n−1 is a hypersurface of the Hermitian manifold  N 2n, then the cartan structure equations that mentioned in the Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='3 must  Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu  type  11  be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Then we get  Bab  c = Bab  c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  √  2Ban  b  +  √  −1σa  b = −δa  b ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  √  −1σab − 1  √  2Bab  n = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Bc  ab = B  c  ab ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  √  2Bb  an −  √  −1σb  a = −δb  a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  √  −1σab + 1  √  2Bn  ab = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  (10)  √  2Bna  b  −  √  2Ba  nb − 2  √  −1σa  b = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Bn  nb +  √  −1σnb = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Bnb  n −  √  −1σb  n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Since σ[αβ] = 0 and Bγ  [αβ] = Bγ  αβ, then the equation (10) gives the following relations:  σab = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  σnb = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  σa  b =  √  −1(  √  2Ban  b  + δa  b ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  (11)  Then from the above discussion, we can establishing the theorem below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' If the Hermitian manifold has the class of Kenmotsu type as a hypersurface,  then the second quadratic form has components agree with the equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  On the other hand, we can establish a relation between the components of the Rieman-  nian curvature tensors of the almost Hermitian manifold and its hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Suppose that Ri  jkl are the components of the Riemannian curvature tensor of the almost  Hermitian manifold, N 2n and �Ri  jkl are the components of the Riemannian curvature tensor  of its hypersurface M 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Then from the second group of cartan structure equations, we  have  dωi  j = ωi  k ∧ ωk  j + 1  2Ri  jkl ωk ∧ ωl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  dθi  j = θi  k ∧ θk  j + 1  2  �Ri  jkl θk ∧ θl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  where ωi  j and θi  j are the Riemannian connection forms of N and M respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Whereas,  ωk and θk are the dual A-frames on AG-structure spaces of N and M respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' More-  over, from [5], we have  θi = Ci  j ωj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  ωi = �Ci  j θj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  θi  j = Ci  k ωk  r �Cr  j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  ωi  j = �Ci  k θk  r Cr  j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  where C = (Ci  j) and C−1 = ( �Ci  j) were defined in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Then the substitution of the above  relations in the second group of cartan structure equations, we conclude the following  theorem:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' If Ri  jkl and �Rq  rst are the components of the Riemannian curvature tensor of  the almost Hermitian manifold (N 2n, J, g) and its hypersurface (M 2n−1, Φ, ξ, η, g) respec-  tively, then they are related as follow:  Ri  jkl = �Ci  q �Rq  rst Cr  j Cs  k Ct  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  12  Mohammed Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abass and Habeeb M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Abood  References  [1] Abood H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Abass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': A study of new class of almost contact metric manifolds of  Kenmotsu type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Tamkang Journal of Mathematics 52 (2) (2021) 253–266-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [2] Abood H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Al-Hussaini F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' : Constant curvature of a locally conformal almost  cosymplectic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' In: AIP Conference Proceedings2086.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' 2019, 030003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [3] Alegre P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Carriazo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Submanifolds of generalized Sasakian space forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Taiwanese Journal  of Mathematics 13 (3) (2009) 923–941-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [4] Banaru M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Geometry of 6-dimensional Hermitian manifolds of the octave algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Journal of  Mathematical Sciences 207 (3) (2015) 354–388-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [5] Banaru M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Kirichenko V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Almost contact metric structures on the hypersurface of  almost Hermitian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Journal of Mathematical Sciences 207 (4) (2015) 513–537-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [6] Chen B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' -Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Differential geometry of warped product manifolds and submanifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' World  Scientific Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Pte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=', Singapore (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [7] Chinea D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Gonzalez C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': A classification of almost contact metric manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Annali di  Matematica Pura ed Applicata 156 (1) (1990) 15–36-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [8] De U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Suh Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': On weakly semiconformally symmetric manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Acta Mathematica  Hungarica 157 (2) (2019) 503–521-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [9] Goldberg S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Totally geodesic hypersufaces of Kaehler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Pacific Journal of  Mathematics 27 (2) (1968) 275–281-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [10] Gray A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Hervella L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': The sixteen classes of almost Hermitian manifolds and their linear  invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Annali di Matematica Pura ed Applicata 123 (1) (1980) 35–58-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [11] Ignatochkina L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Morozov P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': The transformations induced by conformal  transformations on T 1-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Journal of Basrah Researches ((Sciences)) 37 (4C) (2011) 8–15-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [12] Ignatochkina L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Morozov P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Induced transformations for almost Hermitian structure of  linear extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Chebyshevskii Sbornik 18 (2) (2017) 124–133-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [13] Kenmotsu K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': A class of almost contact Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Tohoku Mathematical Journal 24  (1) (1972) 93–103-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [14] Kirichenko V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Differential-geometric structures on manifolds (in Russian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Moscow State  Pedagogical University, Moscow (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [15] Kirichenko V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Dondukova N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Contactly geodesic transformations of almost-contact  metric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Mathematical Notes 80 (2) (2006) 204–213-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [16] Kirichenko V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Kharitonova S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': On the geometry of normal locally conformal almost  cosymplectic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Mathematical Notes 91 (1) (2012) 34–45-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [17] Nikiforova A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Ignatochkina L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': The transformations induced on hypersurfaces of almost  Hermitian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Journal of Basrah Researches ((Sciences)) 37 (4C) (2011) 1–7-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [18] Pitis G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Geometry of Kenmotsu Manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Editura Universitatii Transilvania, Brasov (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [19] Shaikh A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Kundu H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': On equivalency of various geometric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Journal of Geometry  105 (1) (2014) 139–165-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [20] Sular S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and ¨Ozg¨ur C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': On some submanifolds of Kenmotsu manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Chaos, Solitons and  Fractals 42 (4) (2009) 1990–1995-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu  type  13  [21] Umnova S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Geometry of Kenmotsu manifolds and their generalizations (in Russian) (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [22] Yıldız A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and De U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': A classification of (κ, µ)-contact metric manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Communications of  the Korean Mathematical Society 27 (2) (2012) 327–339-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
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+page_content='  References  [1] Lastname1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Title1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
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+page_content='  [2] Lastname2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Lastname3 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Title2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' Publisher2 (Year2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  [3] Lastname4 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
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+page_content=' In: Journal3Volume3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
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+page_content=' In: Editor4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
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+page_content='  [5] Lastname8 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Lastname9 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' and Lastname12 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content=': Title6 (Year6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='  Received: Received date  Accepted for publication: Accepted date  Communicated by: Handling Editor  MSC 2020: AMS classification .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFJT4oBgHgl3EQf9C1W/content/2301.11686v1.pdf'}
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+A GOA-BASED TRAJECTORY TRACKING FTC
+1
+A GOA-based Fault-tolerant Trajectory Tracking
+Control for an Underwater Vehicle of Multi-thruster
+System without Actuator Saturation
+Danjie Zhu, Member, IEEE, Lei Wang, Hua Zhang and Simon X. Yang, Senior Member, IEEE
+Abstract—This paper proposes an intelligent fault-tolerant
+control (FTC) strategy to tackle the trajectory tracking prob-
+lem of an underwater vehicle (UV) under thruster damage
+(power loss) cases and meanwhile resolve the actuator saturation
+brought by the vehicle’s physical constraints. In the proposed
+control strategy, the trajectory tracking component is formed
+by a refined backstepping algorithm that controls the velocity
+variation and a sliding mode control deducts the torque/force
+outputs; the fault-tolerant component is established based on a
+Grasshopper Optimization Algorithm (GOA), which provides fast
+convergence speed as well as satisfactory accuracy of deducting
+optimized reallocation of the thruster forces to compensate for
+the power loss in different fault cases. Simulations with or
+without environmental perturbations under different fault cases
+and comparisons to other traditional FTCs are presented, thus
+verifying the effectiveness and robustness of the proposed GOA-
+based fault-tolerant trajectory tracking design.
+Note to Practitioners: This paper is motivated by the actuator
+saturation problem that exists in the trajectory tracking of
+an underwater vehicle (UV) when encountering power loss
+of the thruster system. The fault-tolerance trajectory tracking
+performance is affected by physical constraints of the vehi-
+cle when using the traditional methods as they may deduct
+excessive kinematic/dynamic requirements during the control
+process, thus inducing the deviation of the tracking trajectory.
+Therefore, the refined backstepping as well as the grasshopper
+optimization (GOA) are combined to eliminate the excess, where
+the refined backstepping is used to alleviate the speed jumps
+(kinematic outputs) and the GOA is to control the propulsion
+forces (dynamic outputs) when facing thruster fault cases. This
+innovates the industrial practitioners that the control design of
+the vehicle can be improved to avoid the tracking deviation
+brought by unsatisfied driving commands under fault cases
+through embedding optimization algorithms. Moreover, for the
+specific type of UV studied in this paper used for dam detection,
+simulations regarding practical dam detection such as the 3D
+polygonal line trajectory tracking and the frequently occurring
+UV single-fault cases are chosen, which can serve as references
+for practitioners working in the related field. In the future,
+underwater experiments of the UV will be investigated, with more
+effects of the practical environment involved.
+Index Terms—actuator saturation, backstepping control, fault-
+tolerant control, grasshopper optimization, trajectory tracking,
+underwater vehicle.
+I. INTRODUCTION
+This work is supported by the Natural Sciences and Engineering Research
+Council (NSERC) and the National Key Project of Research and Development
+Program of China. Danjie Zhu and Simon X. Yang are with Advanced
+Robotics and Intelligent System (ARIS) Laboratory, School of Engineer-
+ing, University of Guelph, Guelph, ON. N1G2W1, Canada (e-mail:{danjie;
+syang}@uoguelph.ca); Lei Wang and Hua Zhang are with the Underwater
+Engineering Institute, China Ship Scientific Research Center, Wuxi 214082,
+China (e-mail: 13771115826@163.com; zhanghua702@126.com).
+T
+HE study of vehicle controls has been extended to various
+conditions where fault tolerance is widely involved in
+the corresponding control design, which is denoted as fault-
+tolerant control (FTC) [1, 2]. Scientists have worked on the
+FTC study for decades in various fields such as crafts in the air
+or space, land vehicles and industrial manufacturing [3–10].
+In previous studies, the FTC is usually applied to alleviate
+abrupt errors and provides the most feasible solution when
+inevitable damages happen for equipment in different fields
+[11]. However, the research regarding the FTC on underwater
+vehicles (UV) has not been thoroughly investigated, due to
+the complexity brought by the underwater environment and
+the UV system [12–15].
+Corresponding studies on the FTC have been proposed in
+this century [3, 4, 6, 16]. Based on these studies, the design
+of the excessive number of thrusters compared to the number
+of degrees of freedom (DOF) is raised and accepted as a
+resolution to the UV FTC problem, which is called thruster
+reconfiguration [17, 18]. For example, when unexpected fault
+cases of the vehicle thrusters occur, the thrusters installed on
+the vehicle that exceeds the number of DOFs (six: surge, sway,
+heave, row, pitch and yaw) have enough flexible space to be
+regulated to provide the required propulsion at corresponding
+DOFs. To implement the thruster configuration theory in
+practical cases, the weighted pseudo-inverse matrix method
+has been proposed, where the fault cases are quantified as
+degrees of damage and serve as the inputs to form the thruster
+configuration model [19]. By this method, the process of the
+FTC is largely simplified, as the required thruster propulsion
+can be deducted directly through a weighted pseudo-inverse
+matrix model. Nevertheless, physical constraints of the thruster
+outputs are rarely considered, thus inducing the over-actuated
+vehicle issue [20, 21]. Additionally, among these studies, most
+of them work on eliminating the static errors induced by the
+fault cases. While in UV practical application, the realization
+of dynamic control on the vehicle’s outputs in a real-time
+manner, which commonly refers to the trajectory tracking
+control for underwater vehicles, is of crucial importance [22–
+24].
+Therefore, motivated by the over-actuated issue and mean-
+while realizing robustness for underwater vehicle trajectory
+tracking, optimization methods are combined with the tracking
+control to optimize the vehicle’s dynamic outputs within allow-
+able domains during the tracking procedure when encounter-
+ing fault cases. Among the current mainstream optimization
+algorithms, the genetic algorithm consumes a long time on
+arXiv:2301.01827v1  [cs.RO]  4 Jan 2023
+
+A GOA-BASED TRAJECTORY TRACKING FTC
+2
+iteration, which is not ideal for the UV FTC that requires
+both fast and feasible solutions [25–27]. The neural network
+is demanding on the choice of data inputs while the UV
+cannot provide when encountering fault cases, which shares
+the same concern with the greedy algorithm, as the greedy
+algorithm needs to decompose the data for processing [28–
+30]. Hence the swarm intelligence algorithm for optimization
+stands out to be a preferable method to tackle the FTC
+application of UVs due to its flexibility of data inputs and fast
+convergence speed [31–33]. Zhu’s group has applied Particle
+Swarm Optimization (PSO) based FTC on the unmanned
+underwater vehicle, though satisfactory torque outputs are
+achieved, the traditional PSO method shows poor real-time
+feedback, which does not conform to the online requirement
+of UV FTCs [34, 35]. In this study, an advanced swarm in-
+telligent method named Grasshopper Optimization Algorithm
+(GOA) is chosen based on its satisfactory balance between fast
+convergence and accuracy of obtaining optimization results as
+well as its simple implementation [36]. The fast convergence
+is realized by its simply updated iteration derived from the
+position of each search agent, which dramatically promotes
+optimizing efficiency and offers the possibility of real-time
+feedback for the UV FTC [37, 38]. At the same time, GOA
+can provide accurate fault-tolerant results within acceptable
+driving constraints through the limitations embedded in the
+optimization algorithm, thus performing adaptive in resolving
+the over-actuated issue of the UV FTC [39, 40].
+The contribution of the control strategy proposed in this
+paper is to combine an advanced swarm intelligence algorithm
+(GOA) with the fault-tolerant trajectory tracking control to
+resolve fault cases of a progressed UV without actuator satu-
+ration. By identifying and quantifying the degree of damage
+(power loss) of the multi-thruster system and then efficiently
+reallocating their forces through the GOA method, a feasible
+solution with satisfactory accuracy will be given in a fast
+convergence manner. The strategy establishes a systematic
+fault identification as well as efficient error elimination process
+for the UV with abrupt damages; and it first realizes the
+application of the GOA method in FTC of specific under-
+water vehicles with a multi-thruster system. Conventional
+methods such as the constrained control allocation method
+developed by Durham are based on the basic linear algebra
+concepts and a means to determine the bounding surface of
+the attainable moment space, yet the bounding surface is
+difficult to be addressed [41]. Commonly applied methods
+used for constructing UV FTC such as T-approximation or
+S-approximation cannot thoroughly resolve the problem of
+actuator saturation due to the inevitable errors brought by the
+vehicle constraints [34]. Therefore, the fast convergence speed
+and ideal accuracy of the GOA method help to amend the
+commands given by the dynamic controller in time, which
+accomplishes real-time FTC on the UV trajectory tracking
+[36].
+The rest of the paper is organized as follows. First, the
+kinematic and dynamic models of an advanced UV, ”YuLong”,
+are introduced and the torque-force transition/normalization
+is defined based on the UV thruster configuration. Next,
+the fault-tolerant trajectory tracking problem of the UV is
+described, with its restrictions explained. The grasshopper
+optimization-based fault-tolerant trajectory tracking control is
+then proposed, where a refined backstepping algorithm is
+applied to form the kinematic control component; a sliding
+mode control works as the dynamic control component; and
+the grasshopper optimization algorithm is used to form the
+fault-tolerant component by reconfiguring the thruster force
+outputs to eliminate the errors produced by different damage
+degrees of the thrusters in fault cases. In the last section, the
+effectiveness of the proposed grasshopper optimization-based
+FTC is evaluated by tracking desired polygonal line or helix
+trajectories, under the condition that one thruster (single-fault)
+or two thrusters (double-fault) are supposed to be damaged.
+II. UV MODELS AND PROBLEM STATEMENT
+In this section, a typical type of UV named ”YuLong”
+is studied. Its robot models and trajectory tracking problem
+descriptions are given in the form of specific equations.
+A. Models of the ”YuLong” UV
+In this section, a typical type of UV named ”YuLong” is
+studied. Its robot models and fault-tolerant trajectory tracking
+problem descriptions are given in the form of specific equa-
+tions.
+1) Kinematic and Dynamic Model: ”YuLong” UV is one of
+the latest UV for dam detection, designed by the Underwater
+Engineering Institute of China Ship Scientific Research Center,
+whose diving depth reaches 3000m. Its rough sketch is shown
+in Fig. 1 and its multi-thruster system structure is shown in Fig.
+2. As a UV specially designed for detecting and maintaining
+a satisfactory condition of the dam in the deep-water area, the
+robustness and efficiency of the vehicle’s operation are crucial
+for dam detectors.
+Among the six degrees of freedom (DOF) of the UV,
+surge, sway, heave, roll, pitch and yaw, roll and pitch can be
+neglected because these two DOFs barely have an influence
+on the underwater vehicle during practical navigation (Fig. 1).
+Therefore, when establishing the trajectory tracking model to
+keep a controllable operation of the UV, usually four DOFs
+surge, sway, heave and yaw are involved. Based on the four
+involved DOFs, for the kinematic equation of the UV, the
+velocity vector v can be transformed into the time derivative
+of the trajectory vector ˙p as
+˙p =
+�
+���
+˙x
+˙y
+˙z
+˙ψ
+�
+��� = J(p)v ==
+�
+���
+cos ψ
+− sin ψ
+0
+0
+sin ψ
+cos ψ
+0
+0
+0
+0
+1
+0
+0
+0
+0
+1
+�
+���
+�
+���
+u
+v
+w
+r
+�
+��� ,
+(1)
+where J is a transformation matrix derived from the physical
+structure of the UV body, [u v w r]T represents the velocities
+at the chosen four axes of the UV (see Fig. 1).
+In an actual UV system, several complex and nonlin-
+ear forces such as hydrodynamic drag, damping, lift forces,
+Coriolis and centripetal forces, gravity and buoyancy forces,
+thruster forces, and environmental disturbances are acting on
+
+A GOA-BASED TRAJECTORY TRACKING FTC
+3
+Fig. 1. The reference frame and six degrees of freedom (x, y,
+z, k, m and n) of the ”YuLong” UV (Top view).
+the vehicle. Considering the origins and effect of the forces,
+a general dynamic model can be written as
+M ˙v + C(v)v + D(v)v + g(p) = τ ,
+(2)
+where M is the inertia matrix of the summation of rigid body
+and added mass; C(v) is the Coriolis and centripetal matrix
+of the summation of rigid body and added mass; D(v) is the
+quadratic and linear drag matrix; g(p) is the matrix of gravity
+and buoyancy; and τ is the torque vector of the thruster inputs.
+As mentioned in the previous section, in this study only four
+states are considered for the specific model YuLong UV. The
+torque vector of the thruster input is represented by
+τ =
+�τx
+τy
+τz
+τn
+�
+,
+(3)
+where x, y and z represent the linear displacements of the
+UV at surge, sway and heave directions, while n represents
+the angular displacement of the UV at yaw direction (see Fig.
+1).
+For
+the
+”Yulong”
+UV,
+the
+following
+param-
+eter
+values
+are
+assigned:
+inertia
+matrix
+M
+=
+[42 0 0 0;
+0 153 0 0;
+0 0 141 0;
+0 0 0 100]T ;
+Coriolis and centripetal matrix C(v) = J−1M ˙JJ−1, where
+J−1 represents the inverse matrix of J and ˙J represents
+the derivative of J and quadratic and linear drag matrix
+D(v) = [42+69u 0 0 0; 0 319+245v 0 0; 0 0 272+
+86w
+0;
+0
+0
+0
+33 + 4r]T . In addition, the gravity force
+applied on the vehicle is balanced off by the buoyancy force
+when the whole system sustains at an equilibrium status.
+2) Torque-force Transition and Normalization: According
+to the physical structure of the ”Yulong” vehicle propulsion
+system (see Fig. 2), the relation between its torque vector and
+the forces of the thrusters is
+τ =
+�
+���
+τx
+τy
+τz
+τn
+�
+��� =
+�
+���
+T1 + T2
+T7 + T8
+T3 + T4 + T5 + T6
+T1 + 1.4 × T8 − T2 − 1.4 × T7
+�
+��� ,
+(4)
+where T1, T2, T3, T4, T5, T6, T7, T8 are the forces produced
+by the eight thrusters installed around the vehicle body.
+The eight thrusters are of the same type and are supposed to
+have the same maximum force Tm. Therefore the maximum
+of the torque vector τm can be deducted based on Eq. (4) as
+τm =
+�
+���
+τxm
+τym
+τzm
+τnm
+�
+��� =
+�
+���
+2Tm
+2Tm
+4Tm
+(2 + 2.8)Tm
+�
+��� .
+(5)
+On the basis of Eq. (4), divide both sides of Eq. (5) by
+the maximum torques to restrict the output in a certain range
+of -1 to 1, and set τ = τ/τm, T = T/Tm, the vector is
+transformed into
+�
+���
+τx
+τy
+τz
+τn
+�
+��� =
+�
+���
+τx/τxm
+τy/τym
+τz/τzm
+τn/τnm
+�
+���
+=
+�
+���
+1
+2
+1
+2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+2
+1
+2
+0
+0
+1
+4
+1
+4
+1
+4
+1
+4
+0
+0
+5
+24
+− 5
+24
+0
+0
+0
+0
+− 7
+24
+7
+24
+�
+���
+�
+�����������
+T1/Tm
+T2/Tm
+T3/Tm
+T4/Tm
+T5/Tm
+T6/Tm
+T7/Tm
+T8/Tm
+�
+�����������
+= B
+�
+�����������
+T1/Tm
+T2/Tm
+T3/Tm
+T4/Tm
+T5/Tm
+T6/Tm
+T7/Tm
+T8/Tm
+�
+�����������
+= B
+�
+�����������
+T 1
+T 2
+T 3
+T 4
+T 5
+T 6
+T 7
+T 8
+�
+�����������
+.
+(6)
+The above equation has the compact form as
+τ = B T
+T = B
+−1 τ ,
+(7)
+where B
+−1 is the generalized inverse matrix of B.
+Therefore, the transition between thruster forces and torques
+is achieved and normalized. For all the torques and forces in
+Eq. (6), they are ranged from -1 to 1 (−1 ≤ τ ≤ 1 and
+−1 ≤ T ≤ 1) to perform a direct and simplified showcase
+during the tracking control process.
+Fig. 2. The thruster distribution in the ”Yulong” UV propulsion
+system. (a) The top view, (b) The lateral view.
+
+Roll (K)
+Yaw (N)
+Heave (z) 
+Surge (X)
+Pitch (M)
+Sway (Y)2800mm
+2800mm
+T5
+T3
+1000mm
+X
+T8
+1000mm
+1000mm
+Y
+T6
+T4
+T4
+T6
+(a)
+(b)A GOA-BASED TRAJECTORY TRACKING FTC
+4
+B. Problem Statement
+In this subsection, the fault-tolerant control problem on
+the thruster system of the ”Yulong” vehicle is modeled and
+explained. Requirements and constraints during the control
+process are introduced.
+1) Fault-tolerant Problem in UV Trajectory Tracking:
+To achieve the accuracy and efficient control of the UV,
+the fault cases of the thruster system should be taken into
+consideration, where one or more of the thrusters might get
+broken and corresponding controls are proposed to sustain the
+movement and posture of the vehicle as desired, by deriving
+the normalized desired torques τd. The control of the torque
+outputs τ is realized by the combined action of the thruster
+forces under fault conditions, which is allocated by approxi-
+mation/optimization methods. The approximation/optimization
+deducts the optimal reallocation when the thrusters’ working
+condition changes, and accomplishes the elimination of the
+torque errors during the process.
+For UV with a multi-thruster system, the ideal fault-tolerant
+control is realized by,
+||e|| = ||τd − τ|| → 0 ,
+(8)
+||θe|| = arccos
+τd • τ
+||τd|| · ||τ|| → 0 ,
+(9)
+where ||e|| is the magnitude error obtained by the vector norm
+of the difference between the desired and the actual torques of
+the vehicle, ||θe|| is the direction error that computes the arc
+cosine of the ratio between the multiplication of desired and
+actual torques and their vector norm. Besides the magnitude
+error, the direction error is also important to the trajectory
+tracking control of the underwater vehicle, as the vehicle’s
+movement is also determined by the direction displacement
+of its dynamic outputs (torques) along the axes. It is possible
+for the vehicle to have an acceptable magnitude dynamic error
+but meanwhile obtain excessive direction error, thus inducing
+non-ideal tracking results.
+Hence, when designing the fault-tolerant control, the error
+values should be as small as possible to obtain good control-
+ling results.
+2) Constraints on Fault-tolerant Control of the UV: In the
+actual application, the controlling effect is always restricted by
+the physical constraints of the vehicle. The UV cannot provide
+infinite driving inputs such as torques/forces to complete the
+navigation, thus resulting in the problem of actuator saturation.
+Therefore driving restrictions are always applied on the UV
+to achieve a reliable and controllable navigation process. The
+maximum forces of the vehicle thrusters, derived from the
+maximum torques that can be offered by the vehicle body, are
+the essential constraints of the vehicle’s controlling problem.
+To assess the influence of the constraints, maximum torques
+τm are introduced in the simulation part of this paper (see Fig.
+3). By the definition given in Eq. (6), normalized torques τ
+and normalized forces T are supposed to have the limits of
+-1 to 1. The two variables are used to quantify the effect of
+the constraints during the trajectory tracking process in the
+simulation part.
+III. GOA-BASED FAULT-TOLERANT TRAJECTORY
+TRACKING CONTROL (GFTC) DESIGN
+The basic control architecture of the system is illustrated in
+Fig. 3. The design of the control strategy consists of two parts:
+(1) a trajectory tracking component formed by an outer loop of
+auxiliary kinematic control based on the position errors of the
+UV and an inner loop of dynamic torque controller based on
+the velocity state vector; (2) a fault-tolerant control component
+that identifies the fault cases of the UV propulsion system and
+reallocates the thruster force outputs based on Grasshopper
+optimization algorithm to eliminate the effect brought by the
+fault cases. Additionally, maximum thruster force outputs Tm
+are applied (see the bold frame in Fig. 3). Specially for the
+simulation of underwater environmental perturbations, current
+disturbance on torque outputs is considered. Moreover, the
+environmental noise at the step of forming vehicle positions
+is also involved. Details of the control design will be presented
+in this section.
+A. Component of Trajectory Tracking Control
+In this section, the methods that form the kinematic and dy-
+namic control of the UV are explained, with detailed equations
+and stability analysis given.
+1) Error
+Restricted
+Backstepping
+Control:
+Suppose
+the
+maximum
+velocity
+vector
+of
+UV
+is
+vm = [vxm
+vym
+vzm
+vψm]T , the input e(t) is the
+error between the desired and actual trajectories. a transfer
+function is defined to process the input trajectory errors
+within an acceptable range,
+ve = f(e)µvm =
+e(t)
+|e(t)| + 1µvm ,
+(10)
+where vm represents the maximum velocities of the UV DOFs
+at their corresponding axes; µ is a positive constant chosen
+according to the variation requirement and is assigned with
+0.5 in this study.
+Therefore, when e(t) → 0, ve = f(e)µvm → 0; and
+when e(t) → ∞, ve = f(e)µvm → the restricted maximum
+velocity µvm. Based on the restriction, the transfer function
+ve limits its control output in a smaller domain and meanwhile
+provides faster convergence to the input errors.
+The processed outputs ve after the convergence shall not
+exceed the maximum possible value of the UV velocity, and
+the definition of f(e) provides a smooth transition of the UV
+at the beginning. Hence the speed-jump problem of the UV
+can be alleviated. The vector ve at the four DOFs can be
+written as ve = [vex
+vey
+vez
+ven]T . Additionally, in the
+backstepping method, control functions for each subsystem
+are designed based on the Lyapunov techniques and generated
+to form the complete control law [42]. Therefore, based on Eq.
+(1) and the definition of the backstepping method, the error
+variables in the control law of the backstepping method are
+
+A GOA-BASED TRAJECTORY TRACKING FTC
+5
+Fig. 3. Schematic of the proposed fault-tolerant trajectory tracking control designed for the UV.
+replaced by the restricted outputs processed in Eq. (10), the
+control law of the backstepping control can be derived as
+vc =
+�
+���
+uc
+vc
+wc
+rc
+�
+���
+(11)
+=
+�
+���
+k(vex cos ψ + vey sin ψ) + ud cos veψ − vd sin veψ
+k(−vex sin ψ + vey cos ψ) + ud sin veψ − vd cos veψ
+wd + kzvez
+rd + kψveψ
+�
+��� .
+where k, kz and kψ are positive constants.
+Then the processed control velocities vc are passed to the
+UV, where they are calculated to keep pace with the desired
+trajectory through the dynamic model of the UV. Additionally,
+the stability of the refined backstepping control can be proved
+by constructing a Lyapunov function Γ0 = 1
+2(e2
+x+e2
+y+e2
+z+e2
+ψ),
+whose derivative is less than and equal to zero (see Appendix
+A).
+2) Sliding Mode Control: To design the sliding mode
+control, the desired dynamics (s) should be introduced. Based
+on Eq. (2) where the UV dynamic system is of the second
+order for the velocity v, the dynamics can be designed as
+s =
+� d
+dt + λ
+�2 �
+evdt = ˙ev + 2λev + λ2
+�
+evdt,
+(12)
+where
+d
+dt is the derivative operator; ev represents the errors
+given by the control velocities (see Fig. 3), ev = vc − v; and
+λ > 0 is a positive parameter [43].
+Then take the derivative of s, we can get
+˙s = ¨ev + 2λ˙ev + λ2ev ,
+(13)
+where ˙ev = ˙vc − ˙v
+To keep the system states consistent with the desired dynam-
+ics, Eq. (13) should be equal to zero. This means the system
+states are on the sliding surface of the perfect tracking. At the
+same time, plug in the equation of the UV dynamic model
+(Eq. (2)),
+˙s = ¨ev + 2λ˙ev + λ2ev = 0
+¨ev + 2λ( ˙vc − ˙v) + λ2ev = 0
+¨ev + 2λ( ˙vc − (τ − Cv − Dv − g)M−1) + λ2ev = 0
+τ = M( ˙vc + ¨ev
+2λ + λ
+2 ev) + Cv + Dv + g .
+(14)
+The standard sliding mode control law is defined as
+τ = ˆτ + τc ,
+(15)
+where ˆτ represents the major control law, which is continuous
+and model-based. It is designed to maintain the trajectory
+consistently on the sliding surface. τc represents the switching
+control law, dealing with the model uncertainty. When the
+trajectory is getting out of control, τc is used to push the
+trajectory back to the sliding surface and continue satisfactory
+tracking. For Eq. (14), supposing a simplification ¨ev ≈ −k˙ev
+based on the error acceleration feedback control to reduce
+computation complexity, the estimated item in the major
+control law ˆτ can be deducted as
+ˆτ = ˆ
+M( ˙vc + −k˙ev
+2λ
++ λ
+2 ev) + ˆCv + ˆDv + ˆg ,
+(16)
+where ˆM, ˆC, ˆD, ˆg are the estimated values of M, C, D and
+g; approximate values can be obtained from the practical case
+respectively [44].
+The switching item τc in sliding mode control can be
+defined as
+τc = −K1s − K2|s|rsign(s) ,
+(17)
+where sign(s) is the nonlinear sign function of s; K1 and
+K2 are positive coefficients, K1 ≥ η + F and K2 ≥ η +
+F, η is the design parameter which is always chosen as a
+positive constant; 0 < r < 1; and F represents the upper
+bound of the difference between the system actual output and
+the estimation,
+F = |f(v) − ˆf(v)|.
+(18)
+Additionally, an adaptive variation term �τest is added to
+the control law, where ˙�τest = Γs and Γ represents a positive
+
+ Perturbation
+(Current
+disturbance)
+Fault-tolerant
+Trajectory tracking
+Backstep-
+Thruster system
+Desired
+pd
+Error
+Vc
+Ve
+"Lc
+UV
+p = J(p)v
+SMC
+(Constraints Tm)
+Restriction
+ping
+Trajectory
+Fault
+GOA
+identification
+Environmental Noise
+(Random error inputs)A GOA-BASED TRAJECTORY TRACKING FTC
+6
+constant. Hence the final sliding mode control law is defined
+as
+τ = ˆτ + �τest + τc
+= ˆ
+M( ˙vc + −k˙ev
+2λ
++ λ
+2 ev) + ˆCv + ˆDv + ˆg
++�τest − K1s − K2|s|rsign(s) .
+(19)
+Detailed proof of the SMC stability can be found in Ap-
+pendix B.
+B. Component of Fault-tolerant Control
+The fault-tolerant control design is mainly built on the
+adjustment of the forces required by the thruster system, where
+the forces operate together and provide the torques as desired.
+The adjustment of the thruster forces is deducted by the
+grasshopper optimization algorithm (GOA), which efficiently
+eliminates the errors brought by the fault of the propulsion
+system after fault identification. Details of the control strategy
+are presented in this section.
+1) Weighting Matrix: To quantify the degree of damage in
+the fault cases for the multi-thruster system, a weighting matrix
+W is introduced. The matrix W decides the service condition
+of the thruster, which is usually defined as a diagonal matrix,
+W =
+�
+�����������
+w1
+0
+0
+0
+0
+0
+0
+0
+0
+w2
+0
+0
+0
+0
+0
+0
+0
+0
+w3
+0
+0
+0
+0
+0
+0
+0
+0
+w4
+0
+0
+0
+0
+0
+0
+0
+0
+w5
+0
+0
+0
+0
+0
+0
+0
+0
+w6
+0
+0
+0
+0
+0
+0
+0
+0
+w7
+0
+0
+0
+0
+0
+0
+0
+0
+w8
+�
+�����������
+(20)
+where wj > 0 is the weight of the jth thruster. If all the
+thrusters are working in the desired condition with no power
+loss, W will be a unit matrix, meaning all wj = 1. If there is
+power loss for any of the thrusters, its corresponding weight
+will be reduced by the degree of the loss. For example, when
+T1 thruster attains 20% of power loss, w2 in the weighting
+matrix is assigned as 0.8 respectively.
+As the relation between the thruster forces and the vehicle
+torques at different states are defined and given in Eq. (7),
+the following transition between the torques and forces in the
+fault cases is defined as,
+τ = BWT,
+(21)
+where T is the control parameters in the UV case, which
+is deducted by the optimization method, such as the GOA
+method used in this study.
+Additionally, as a comparison to the GOA method, the
+weighted pseudo-inverse matrix method is used, which is
+determined based on the defined weighting matrix,
+T = B
++
+w τd = (WB
+T (BWB
+T )−1)τd,
+(22)
+where B
++
+w is the matrix that transmits the damage information
+to the propulsion system and meanwhile make the adjustment
+accordingly. Thus, the thruster force results under fault cases
+can be deducted. For example, if the thruster T1 can only
+provide 70% of power after encountering a power loss of 30%,
+the weighted pseudo-inverse matrix method will request larger
+output ( 1
+0.7× original force) from T1 such that the same force
+can be achieved after weakened by 30% of power loss.
+Then T-approximation or S-approximation methods are ap-
+plied for achieving force results within the range of the thruster
+force maximum, which is generally denoted as pseudo-inverse
+(P-I) matrix approximation. T-approximation restricts all nor-
+malized forces T between [-1, 1] by subtracting/adding the
+excessive part of the states whose value is larger than 1 or
+smaller than -1, where
+Tt =
+�
+�
+�
+T i,
+T i ∈ [−1, 1]
+1,
+T i > 1
+−1,
+T i < −1
+S-approximation realizes the limits of [-1, 1] by timing the
+reciprocal ratio of the largest normalized force for all states,
+where
+Ts =
+1
+max(T i)T, i = 1, 2, ..., 8.
+For example, in S-approximation, if the largest normalized
+force for one of the states reaches 2, all normalized forces
+will be multiplied by the ratio of 1
+2 to guarantee they do not
+exceed the limits of -1 to 1.
+In the simulation section of this study, T-approximation
+method is used as the typical pseudo-inverse (P-I) matrix
+approximation to work as a comparison of the proposed GOA-
+based FTC. The T-approximation has wider application in
+practical cases of the underwater vehicle FTC due to it gener-
+ally produces smaller errors compared to the S-approximation
+[34, 45].
+2) Grasshopper Optimization Algorithm: The grasshopper
+optimization algorithm is newly raised in 2017 [36]. As a
+developed algorithm based on the theory of swarm intelligence
+that imitates the activity of grasshoppers, GOA shows better
+performance than the traditional swarm intelligence algorithms
+due to that it finds a satisfactory balance between fast speed of
+convergence and accuracy based on its form switch between
+“adults” and “larvae”. The fast convergence is realized when
+GOA searches globally based on the position of each agent
+under its ”adult” form, which explores on a large scale in
+an attractive manner among the agents; while the accuracy is
+achieved by shrinking the range and keeping a repulsive zone
+based on the best agent under ”larvae” form, which avoids the
+local minimum.
+According to the movement of the grasshopper groups, a
+mathematical model can be defined to describe their swarming
+behavior [46, 47]
+Xi =
+N
+�
+j=1,j̸=i
+s(|xj − xi|)xj − xi
+dij
+− Gi + Ai,
+(23)
+where Xi represents the next position of the ith grasshopper;
+s(r) is the social interaction function where it is optimized as
+s(r) = 0.5e−r/1.5 − e−r. The item |xj − xi| is the distance
+between the current position of the ith and jth grasshopper,
+
+A GOA-BASED TRAJECTORY TRACKING FTC
+7
+(xj − xi)/dij is the unit vector pointing from the position of
+the ith grasshopper to the jth grasshopper. Gi represents the
+Gravity force at the ith grasshopper; Ai is the wind advection
+that is assumed to be always towards the target, Ti.
+Based on the assumptions made in this control case, where
+the gravity force is neglected and the wind force is always
+towards the target, Eq. (23) can be converted into
+Xd
+i = c(
+N
+�
+j=1,j̸=i
+cubd − lbd
+2
+s(|xj − xi|)xj − xi
+dij
+) + Td (24)
+where c is a decreasing coefficient that shrinks the comfort
+zone, repulsion zone and attraction zone, which is determined
+as c = cmax − l(cmax − cmin)/L, cmax is the maximum value,
+cmin is the minimum value, l indicates the current iteration,
+and L is the maximum number of iterations. In this work,
+we assign cmax = 1 and cmin = 0.00001 by trial and error.
+The variable ubd represents the upper bound of the case while
+the lbd represents the lower bound, which are 1 and -1 in
+this design. Td is the desired solution of the current iteration.
+These parameters are used to attain the fast convergence of
+the optimization, by increasing the speed of updating the local
+solution in relation to the increment of the iteration times, thus
+leading to the efficient searching result of the GOA method
+[48].
+The pseudocode of the GOA applied on the UV thruster
+forces reallocation can be concluded as follows (see Algo-
+rithm 1), with the fitness evaluation substituted by the error
+evaluation, given in Eq.s (8) and (9).
+The flow of the proposed GOA method can be concluded
+as follows:
+1) First initialize the swarm, with 10 groups of eight random
+numbers between -1 and 1 representing the normalized eight-
+thruster group and each group is regarded as a search agent;
+2) Next calculate the fitness of each search agent and address
+the agent with the minimum errors as the best based on the
+objective function combined by Eq. (8) and (9), which is ||e||+
+||θe|| → 0 , the constraints for each agent are set between [-1,
+1];
+3) Update the parameter c according to the iteration time to
+accelerate the convergence. If iteration time reaches maximum
+then stop, otherwise continue the update of c;
+4) Update positions (values) of search agents based on Eq.
+(24) and compare the fitness of the updated agents with the
+agent of the best fitness. If the updated fitness turns out to
+be better, updates the position of the agents, otherwise do not
+update.
+5) Update the iteration time, and repeat the loop from step 3).
+C. Component of Perturbations
+As the ”Yulong” UV model is designed for dam detection,
+which usually operates at the shoreside underwater condition,
+the perturbation of currents can be considered in a regular
+combination form of wave functions as [49–51]
+τp = Ap1 cos(ωp1t) sin(ωp2t) + Ap2 cos (ωp3t) sin (ωp4t),
+(25)
+where Ap1, Ap2 and ωp1 to ωp4 are random coefficients, which
+are chosen to synthesize the randomness of the currents to
+appropriately address the underwater environmental perturba-
+tion. Ap1 and Ap2 are assigned within the range of 10% of the
+torque outputs at four axes, for example, if τx output is about
+100N, the assignment range will be [−10, 10]. Coefficients
+ωp1 to ωp4 are chosen with the range of -1 to 1.
+In addition, considering the effect of environmental noise
+that produces perturbation to the data transfer at the stage
+of forming positions, an random error input is given in the
+simulation. The random error is supposed to be within [-0.1,
+0.1] and filtered by sensors, which corresponds to the practical
+UV case.
+IV. SIMULATION RESULTS AND ANALYSIS
+In this section, the polygonal line and helix trajectory
+tracking simulation results of the proposed FTC and con-
+ventional approximation methods are presented and analyzed.
+Fault cases of single-fault (one thruster broken) and double-
+fault (two thrusters broken) are applied due to their frequent
+occurrence.
+A. Helix Tracking
+In this section, one of the thrusters T1 is supposed to be
+broken, where 100% of power is lost. The initial position of
+the desired helix trajectory is set at (0, 0, 0, 0), while the initial
+position of the control trajectories is set at (0, 10, 0, 0). The
+difference of the initial positions is given to test the correction
+ability of the two tracking strategies when they start with a
+certain amount of deviation at one of the axes, i.e. the y axis.
+Assuming the desired trajectory is given as xd = 10 sin 0.2t,
+yd = 10 − 10 cos 0.2t, zd = 0.5t and ψd = 0.2t with the
+simulation time continuing for 50 seconds.
+The GFTC (in red dash) tracking result quickly eliminates
+the initial error and follows the desired helix trajectory till
+the end of the simulation yet the T1 thruster is supposed to be
+completely broken (Fig. 4(a)). The P-I approximation (in blue)
+cannot coincide with the desired helix under the single-fault
+case, where the deviation of abrupt variations is created at the
+
+Algorithm 1: GOA algorithm embedded in the fault-tolerant component
+Input: a: desired normalized torque matrix
+W: weighting matrix (fault identification)
+Output: T: A vector containing the optimal allocation of the normalized forces for
+eight thrusters
+Initialize the swarm Xi (i-1,2,---,10)
+Initialize the cmax, cmin, and maximum number of iterations L
+Calculate the fitness of each search agent to address the best search agent T with the
+minimum errors (Eq. s (8) and (9))
+while (l< L)
+Update c = cmax - l (cmax - cmin) / L
+for each search agent
+Update the value of the current search agent by Eq. (24)
+Retrieve the current search agent if it excess the limitation of
+[-1, 1]
+end for
+Update T if there is a better solution
+l=[+1
+end while
+Return TA GOA-BASED TRAJECTORY TRACKING FTC
+8
+Fig. 4. Helix tracking results using the GFTC with or
+without perturbations and the P-I approximation-based FTC
+under the single-fault case. (a) Comparison of trajectories,
+(b) Comparison of tracking errors, (c) Comparison of control
+velocities.
+beginning and trajectory distortions are produced throughout
+the whole process. Therefore when the dynamic constraints
+are considered, the P-I method fails to compensate for the
+power loss of a single thruster, which induces increasing
+errors and excessive velocities with abrupt jumps given in
+Fig.s 4(b) and (c); while the GFTC achieves smooth error and
+control velocity curves that indicate the satisfactory tracking
+performance of the method. Moreover, the ”GFTC-P” (in pink)
+results consider the effect of perturbations brought by the
+currents and environmental noise when addressing the position
+information for the vehicle. The GFTC-P result under the
+effect of perturbations sustains a similar tracking trajectory
+with the unaffected GFTC, which verifies the robustness of
+the proposed control.
+TABLE I. Maximum velocities of the GFTC with or without
+perturbations and the P-I based FTC under the single-fault
+case when tracking the helix
+uc (m/s)
+vc (m/s)
+wc (m/s)
+rc (m/s)
+GFTC
+2.0008
+-1.8978
+0.5002
+0.2016
+P-I
+4.9719
+-25
+0.6125
+1.0034
+GFTC-P
+2.0150
+-1.8944
+0.6404
+0.2836
+Tracking
+errors
+of
+the
+GFTC
+method
+(in
+red),
+P-I
+approximation-based FTC (in blue), single-fault case without
+FTC (in grey) and GFTC-P with the effect of perturbations
+(in pink) are given in Fig. 4(b). The error curve of the GFTC
+method eliminates the initial deviation and quickly converges
+to zero. While the error of the P-I approximation-based FTC
+presents obvious fluctuations and cannot be eliminated in
+all axes, furthermore, the P-I error curve attains even larger
+fluctuations compared to the case without FTC at the ψ
+axis, which supports the trajectory tracking performance given
+in Fig. 4(a). This shows the failure of P-I approximation
+FTC on keeping the desired helix tracking in the single-fault
+condition when dynamic constraints are applied, yet the GFTC
+method accomplishes the fault-tolerant trajectory tracking task
+with satisfactory kinematic outputs. This conclusion is also
+supported by the velocity variations in Fig. 4(c). The P-
+I method deducts largely excessive speeds at the x and y
+axes, where the maximum of 4.9719 m/s and -25 m/s are
+required, but the GFTC method satisfactorily restricts the
+velocity at x and y axes within the constraints of [-2, 2]m/s,
+with the maximum outputs at 2.0008 m/s and -1.8978 m/s
+(TABLE I). The GFTC also achieves much smoother velocity
+curves compared to the P-I method for all axes. Even when
+considering the perturbation effect in GFTC-P simulation,
+though small chattering is performed, the errors as well as the
+control velocities are successfully restricted in an acceptable
+range, with the maximum of 2.015m/s and -1.8944m/s at the x
+and y axes, and far less requirement of control velocity at the
+ψ axis. Hence the effectiveness of the proposed GFTC method
+in tracking the desired trajectory under the single-fault case is
+verified even when external perturbations are given.
+B. 3D Polygonal Line Tracking
+A 3D polygonal line is applied in this section as the
+reference tracking trajectory, as the ”YuLong” UV usually
+navigates in a movement similar to the polygonal line to detect
+the dam damage.
+The initial position of the desired trajectory is set at
+(0, 0, 0, 0), while the initial position of the control trajectories
+is set at (0, 2.5, 0, 0). A specific polygonal line function is
+applied and the simulation continues for 20 seconds:
+xd = t, 0 ≤ t ≤ 20,
+yd =
+�
+�
+�
+�
+�
+�
+�
+t,
+0 ≤ t ≤ 5
+5,
+5 < t ≤ 10
+t − 5,
+10 < t ≤ 15
+10,
+15 < t ≤ 20
+zd =
+�
+�
+�
+�
+�
+�
+�
+t,
+0 ≤ t ≤ 5
+5,
+5 < t ≤ 10
+t − 5,
+10 < t ≤ 15
+10,
+15 < t ≤ 20
+ψd = 0.2, 0 ≤ t ≤ 20.
+1) Single-fault Case: One of the thrusters T8 is supposed
+to be broken, with 100% of power lost. The tracking trajectory
+results are shown in Fig. 5(a). The GFTC (in red dash) has
+retained the polygonal line trajectory as desired after elimi-
+nating the initial error at the y axis, neglecting the power loss
+of the thruster. Moreover, the GFTC-P (in pink) results which
+consider the effect of perturbations sustain a similar tracking
+
+GFTC
+25 T
+GFTC-P
+P-I1
+No FTC
+20
+Desired
+15
+N
+10-
+51
+6
+(a)
+GFTC
+p-I
+GFTC-P
+No FTC
+8
+4 -
+e
+0.
+10
+20
+0
+30
+40
+50
+10
+20
+30
+40
+0
+50
+-10
+e
+V
+X
+-20 -
+-8
+0
+10
+40
+20
+30
+40
+50
+0
+10
+20
+30
+50
+0.4 -
+0.9
+No.2
+e
+..
+0.0.
+0.3
+10
+20
+30
+0
+40
+50
+0
+10
+20
+30
+40
+50
+1.0
+1.0
+3 0.5
+0.5
+e
+0.0
+0.0.
+-0.5.
+10
+20
+30
+40
+0
+50
+10
+20
+30
+40
+0
+50
+Time(s)
+Time(s)
+(b)
+(c)A GOA-BASED TRAJECTORY TRACKING FTC
+9
+Fig. 5. Polygonal line tracking results using the GFTC with
+or without perturbations and the P-I approximation-based
+FTC under the single-fault case. (a) Comparison of
+trajectories, (b) Comparison of tracking errors, (c)
+Comparison of control velocities.
+trajectory with the unaffected condition, which verifies the
+robustness of the proposed tracking control. The P-I method
+(in blue) fails to catch up with the desired trajectory especially
+at the turning point where larger dynamic inputs are needed.
+The P-I method cannot make up for the loss of the propulsive
+force when physical constraints (torque/force maximum) are
+involved, thus producing errors with large fluctuations as well
+as excessive control velocities presented in Fig.s 5(b) and (c).
+TABLE II. Maximum velocities under the single-fault case
+when tracking the polygonal Line
+uc (m/s)
+vc (m/s)
+wc (m/s)
+rc (m/s)
+GFTC
+1.1738
+0.8212
+1.0012
+0.0782
+P-I
+1.7278
+-5.0713
+1.2289
+0.2
+GFTC-P
+1.2410
+1.2188
+1.1192
+0.1035
+The errors at four axes are presented in Fig. 5(b), where
+the GFTC (in red) successfully eliminates the tracking errors.
+While for the P-I approximation-based FTC (in blue), its result
+fails to eliminate the error once the sharp turning is required
+by the trajectory, as the excessive dynamic outputs deducted
+by the P-I method cannot be satisfied when the dynamic
+constraints are applied, thus inducing the large trajectory
+deviation at the turning section. Velocity variations at four
+axes are shown in Fig. 5(c), the P-I method performs a sharp
+fluctuation and fails to retain the control velocity within the
+desired range at the y axis (see y axis in TABLE II). The
+velocity at the y axis of P-I method reaches a dramatic value
+of -5.0713 m/s, largely exceeding the desired range of the
+vehicle that is preset at -2m/s to 2m/s. At the same time,
+the GFTC successfully limits the kinematic outputs within
+the constraints and presents a smooth curve of much smaller
+fluctuations. In addition, when considering the perturbations in
+the GFTC-P simulation (in pink), though chattering issues are
+presented, errors are constrained within an acceptable range
+and kinematic outputs perform a smaller range compared to
+the P-I method at most axes, with the maximum of 1.2410m/s
+and 1.2188m/s at the x and y axes. These results indicate the
+effectiveness and robustness of the proposed GOA-based FTC.
+2) Double-fault Case: In this section, the effect of the
+GFTC, P-I approximation based FTC and GFTC consider-
+ing environmental perturbations are compared, supposing two
+thrusters (T1 and T8) of the propulsion system encounter
+power loss of 100%.
+The GFTC method (in red dash) successfully eliminates the
+initial errors at the y axis and retains the tracking trajectory
+as desired till the end (Fig. 6(a)). The GFTC-P (in pink)
+results under the effect of perturbations sustain a similar
+tracking trajectory with the unaffected GFTC trajectory, and at
+the second turning section it performs more smooth tracking
+curves than the first one, which verifies its robustness for
+tracking the desired polygonal line even under the double-fault
+case. At the same time, the P-I approximation based FTC fails
+to track the desired trajectory and even presents a much larger
+deviation compared to its single-fault case (Fig. 5(a)). This
+demonstrates that the GFTC method is capable of balancing
+off the tracking errors whenever the damage degree of power
+loss in the thruster system differs, thus proving the robustness
+of the proposed FTC.
+Similarly, as under the single-fault case, the error curve of
+GFTC method under double-fault case eliminates the initial
+deviation and quickly converges to zero (Fig. 6(b)). However,
+the P-I method presents fierce error vibrations in x, y and
+ψ axes compared to the single-fault case, which is even
+worse than the performance of double-fault case without FTC.
+This shows that the P-I approximation-based FTC is heavily
+affected by the damage degree of the UV propulsion system
+and it cannot balance off the error produced by the excessive
+power loss of the thrusters, e.g. the double-fault case. This
+conclusion is also supported by the velocity variations in
+Fig. 6(c), where the P-I method cannot be limited within
+the allowable range due to the thruster power loss, with
+excessive maximum velocities arriving at 17.5815m/s for the
+x axis, 25.9814m/s for the y axis and 1.4485m/s for the ψ
+axis given in TABLE III, resulting in the complete tracking
+failure shown in Fig. 6(a). The velocities of the GFTC method
+maintain within the allowable domain throughout the whole
+process, neglecting the change of fault cases. The GFTC-P
+simulations that involve perturbations in a practical underwater
+environment also perform successful restriction of the errors
+and the control velocities within the supposed range, with the
+maximum of 1.2473m/s and 0.9646m/s at the x and y axes.
+Therefore, the effectiveness of the proposed GFTC method in
+tracking a desired polygonal line is verified whenever single-
+fault or double-fault cases are applied.
+
+GFTC
+GFTC-P
+p-I
+10
+No FTC
+Desired
+8
+6
+N
+4
+12
+10
+8
+6
+0
+2
+5
++
+(a)
+P-1
+No FTC
+GFTC-P
+GFTC
+1.0
+2
+0.5
+e
+u
+0.0
+0.
+10
+15
+0
+20
+0
+5
+10
+15
+20
+1
+0
+y
+0.
+e
+.5
+-2 -
+5
+10
+15
+5
+0
+20
+10
+15
+0
+20
+0.5
+1.5
+1.0
+N
+...
+0.0
+e
+0.5 -
+0.01
+-0.5-
+:5
+5
+10
+15
+20
+10
+15
+0
+0
+20
+0.2
+3.0.2
+e
+0.0-
+0.0
+5
+10
+15
+0
+20
+5
+10
+0
+15
+20
+Time(s)
+Time(s)
+(c)
+(b)A GOA-BASED TRAJECTORY TRACKING FTC
+10
+Fig. 6. Polygonal line tracking results using the GFTC with
+or without perturbations and the P-I approximation-based
+FTC under the double-fault case. (a) Comparison of
+trajectories, (b) Comparison of tracking errors, (c)
+Comparison of control velocities.
+TABLE III. Maximum velocities under the double-fault case
+when tracking the polygonal line
+uc (m/s)
+vc (m/s)
+wc (m/s)
+rc (m/s)
+GFTC
+1.1710
+0.8158
+0.9989
+0.0782
+P-I
+17.5815
+25.9814
+1.2289
+1.4485
+GFTC-P
+1.2473
+0.9646
+1.0367
+-0.0885
+V. CONCLUSION
+In this paper, the fault-tolerant trajectory tracking problem
+for the ”Yulong” UV is resolved by a Grasshopper Optimiza-
+tion and backstepping & SMC-based cascade control (GFTC).
+The GFTC strategy applies a refined backstepping algorithm
+to restrict the kinematic outputs; and the Grasshopper Op-
+timization Algorithm (GOA) is used to achieve optimized
+thruster force reallocation within the allowable domain. When
+encountering fault cases in tracking the polygonal line or
+helix, the trajectory tracking errors of the GFTC are largely
+alleviated and the actuator saturation problem is eliminated,
+compared to the traditional FTCs such as the weighted pseudo-
+inverse matrix approximation-based methods. In addition, the
+robustness of the proposed FTC is also verified when environ-
+mental perturbations are involved, which serves as the basis
+of the experimental study on practical applications that will
+be extended in the future.
+APPENDIX A
+PROOF OF THE ERROR RESTRICTED BACKSTEPPING
+CONTROL STABILITY
+According to the Lyapunov stability theory, a special Lya-
+punov function Γ0 is chosen,
+Γ0 = 1
+2(e2
+x + e2
+y + e2
+z + e2
+ψ) .
+(26)
+By Eq.s (1) and (11), the derivative of Eq. (25) can be
+obtained to prove the stability of the backstepping system,
+˙Γ0 =ex ˙ex + ey ˙ey + ez ˙ez + eψ ˙eψ
+= ex ( ˙xd − ˙x) + ey ( ˙yd − ˙y)
++ ez( ˙zd − ˙z) + eψ ( ˙ψd − ˙ψ)
+= ex [(cos ψdud − sin ψdvd) − (cos ψuc − sin ψvc)]
++ ey [(sin ψdud + cos ψdvd) − (sin ψuc + cos ψvc)]
++ ez(wd − wc) + eψ (rd − rc)
+= ex [(cos ψdud − sin ψdvd)
+− (kvex + ud(cos ψ cos veψ − sin ψ sin veψ)
++ vd(sin ψ cos veψ − cos ψ sin veψ))]
++ ey [(sin ψdud + cos ψdvd)
+− (kvey + ud(sin ψ cos veψ + cos ψ sin veψ)
+− vd(sin ψ sin veψ − cos ψ cos veψ))]
++ ez(−kzvez) + eψ (−kψveψ)
+≤ −kexvex − keyvey − kzezvez − kψeψveψ .
+(27)
+According to the definition of ve, e(t) are of the same sign
+(see definition of Eq. (10)); k, kz, kψ are positive constants.
+The result of Eq. (26) is believed to be less than and equal to
+zero, which demonstrates the stability of the designed refined
+backstepping controller.
+APPENDIX B
+PROOF OF THE SMC STABILITY
+To prove the stability of the SMC, construct a Lyapunov
+function,
+V = 1
+4λsT Ms + 1
+2QT Γ−1Q ,
+(28)
+where Q = �τr − �τest and �τr = �
+M ˙vr + �Cvr + �Dv + �g.
+Previously we have given ev = vc − v, and s˙ev + 2λev +
+λ2 �
+evdt, such that two equations can be deducted as,
+v = vc − s − ˙ev − λ2 �
+evdt
+2λ
+,
+(29)
+˙v = ˙vc − ˙s − ¨ev − λ2ev
+2λ
+,
+(30)
+therefore the following items can be defined,
+vr = vc + ˙ev + λ2 �
+evdt
+2λ
+,
+(31)
+˙vr = ˙vc + ¨ev + λ2ev
+2λ
+.
+(32)
+
+ GFTC
+GFTC-P
+P-I
+No FTC
+10
+Desired
+8
+76
+N
+4
+72
+15
+10
+(a)
+GFTC
+P-I
+No FTC
+GFTC-P
+20
+10 -
+0
+5
++
+e
+-20 -
+-5 
+-40-
+5
+10
+15
+0
+20
+5
+10
+15
+0
+20
+20-
+-5 1
+e
+-10
+0
+-15-
+5
+10
+0
+15
+20
+0
+5
+10
+15
+20
+1.5
+0.5
+ 1.0
+N
+0.5 -
+e
+0.0
+0.01
+-0.5-
+0
+5
+10
+15
+20
+0
+5
+10
+15
+20
+5
+5
+0
+e
+-5 -
+-5.
+15
+0
+5
+10
+15
+10
+15
+20
+20
+0
+Time(s)
+Time(s)
+(b)
+(c)A GOA-BASED TRAJECTORY TRACKING FTC
+11
+By substituting into Eq. (2),
+M ˙s
+2λ + C s
+2λ
+= M( ˙vc + ¨ev + λ2ev
+2λ
+) + C(vc + ˙ev + λ2 �
+evdt
+2λ
+)
++Dv + g − τ = M ˙vr + Cvr + Dv + g − τ.
+(33)
+Based on previous definitions, the derivative of Eq. (27) can
+be simplified as,
+˙V = 1
+4λ(sT ˙Ms + ˙sT Ms + sT M˙s)
++1
+2
+˙QT Γ−1Q + 1
+2QT Γ−1 ˙Q
+= 1
+2λsT (M˙s + Cs) + 1
+2
+˙QT Γ−1Q + 1
+2QT Γ−1 ˙Q
+= sT (M ˙vr + Cvr + Dv + g − τ) + ˙QT Γ−1Q. (34)
+By substituting Eq. (19),
+˙V = sT (M ˙vr + Cvr + Dv + g − τ)
++(˙�τr − ˙�τest)T Γ−1Q
+= −sT (K1s + K2|s|rsign(s)) + (˙�τr)T Γ−1Q.
+(35)
+The dynamic item �τr is bounded due to the slow velocity
+of the underwater vehicle and sT (K1s + K2|s|rsign(s)) ≥
+(˙�τr)TΓ−1Q. When K1, K2 and Γ are assigned with large
+enough values at the design step, ˙V ≤ 0 can be achieved and
+V is ensured to be bounded, thus leading to the conclusion
+that Q is bounded. Then design a new Lyapunov function as
+V2 = 1
+4λsT Ms,
+(36)
+whose derivative can be deducted as,
+˙V2 = sT (Q − K1s − K2|s|rsign(s)),
+(37)
+where 0 < r < 1. Suppose ||Q|| < a,
+˙V2 ≤ 1
+2||s||2 + 1
+2a − λmin(K1)||s||2 − λmin(K2)||s||1+r,
+(38)
+choose K1 when λmin(K1) > 1
+2 + β, where β > 0,
+˙V2 ≤ −β||s||2 − λmin(K2)||s||1+r + 1
+2a,
+(39)
+which induces that the Lyapunov function converges to a range
+close to zero in a finite time and s converges to a range close to
+zero in a finite time. Therefore, the supposed condition of the
+Lyapunov theorem can be regarded as satisfied, thus proving
+the stability of the designed SMC.
+REFERENCES
+[1] T. Baraniuk, R. Simoni, and L. Weihmann, “Fault-tolerant architecture for AUVs,”
+in Proceedings of 2018 IEEE/OES Autonomous Underwater Vehicle Workshop,
+Porto, Portugal, 2018, pp. 1–6.
+[2] C. Huang, F. Naghdy, H. Du, and H. Huang, “Fault tolerant steer-by-wire systems:
+An overview,” Annual Reviews in Control, vol. 47, pp. 98–111, 2019.
+[3] X. Qi, J. Qi, D. Theilliol, Y. Zhang, J. Han, D. Song, and C. Hua, “A review on
+fault diagnosis and fault tolerant control methods for single-rotor aerial vehicles,”
+J. Intell. Robot. Syst., vol. 73, no. 1-4, pp. 535–555, Jan. 2014.
+[4] K. Lu, Y. Xia, C. Yu, and H. Liu, “Finite-time tracking control of rigid spacecraft
+under actuator saturations and faults,” IEEE Trans. Autom. Sci. Eng., vol. 13, no. 1,
+pp. 368–381, Jan. 2016.
+[5] Z. Mao, X. Yan, B. Jiang, and M. Chen, “Adaptive fault-tolerant sliding-mode
+control for high-speed trains with actuator faults and uncertainties,” IEEE Trans.
+Intell. Transp. Syst., vol. 21, no. 6, pp. 2449–2460, Jul. 2020.
+[6] R. T. Meyer, S. C. Johnson, R. A. DeCarlo, S. Pekarek, and S. D. Sudhoff, “Hybrid
+electric vehicle fault tolerant control,” J. Dyn. Syst-T. ASME, vol. 140, no. 2, pp.
+1–12, 2018.
+[7] R. Xiong, Q. Yu, and W. Shen, “Review on sensors fault diagnosis and fault-
+tolerant techniques for lithium ion batteries in electric vehicles,” in Proceedings of
+2018 IEEE Conference on Industrial Electronics and Applications (ICIEA), Wuhan,
+China, 2018, pp. 406–410.
+[8] J. Gong and M. Yang, “Evolutionary fault tolerance method based on virtual
+reconfigurable circuit with neural network architecture,” IEEE Transactions on
+Evolutionary Computation, vol. 22, no. 6, pp. 949–960, Dec. 2018.
+[9] G. Zhang, H. Zhang, X. Huang, J. Wang, H. Yu, and R. Graaf, “Active fault-
+tolerant control for electric vehicles with independently driven rear in-wheel motors
+against certain actuator faults,” IEEE Trans. Control Syst. Technol., vol. 24, no. 5,
+pp. 1557–1572, 2016.
+[10] X. Lang, T. Yang, Z. Wang, C. Wang, S. Bozhko, and P. Wheeler, “Fault
+tolerant control of advanced power generation center for more-electric aircraft
+applications,” IEEE Trans. Transp. Electrification, p. Article in press. DOI:
+dx.doi.org/10.1109/TTE.2021.3093506, 2021.
+[11] X. Cao, Y. Tian, X. Ji, and B. Qiu, “Fault-tolerant controller design for path
+following of the autonomous vehicle under the faults in braking actuators,” IEEE
+Trans. Transp. Electrification, vol. 7, no. 4, pp. 2530–2540, 2021.
+[12] M. Seto and K. Svendsen, Autonomous Underwater Vehicles: Design and practice.
+The Institution of Engineering and Technology, 2020, ch. Advanced AUV fault
+management, pp. 419–445.
+[13] J. Kadiyam, A. Parashar, S. Mohan, and D. Deshmukh, “Actuator fault-tolerant
+control study of an underwater robot with four rotatable thrusters,” Ocean Eng.,
+vol. 197, pp. 961–979, 2020.
+[14] G. Zhu, Y. Ma, Z. Li, R. Malekian, and M. Sotelo, “Event-triggered adaptive
+neural fault-tolerant control of underactuated msvs with input saturation,” IEEE
+Trans. Intell. Transp. Syst., vol. 23, no. 7, pp. 7045–7057, 2022.
+[15] J. Li, Z. Xu, D. Zhu, K. Dong, Z. Z. Tao Yan, and S. X. Yang, “Bio-inspired
+intelligence with applications to robotics: a survey,” Intelligence & Robotics, vol. 1,
+no. 1, pp. 58–83, 2021.
+[16] T. Li, G. Li, and Q. Zhao, “Adaptive fault-tolerant stochastic shape control with
+application to particle distribution control,” IEEE Trans. Syst. Man Cybern., Syst.,
+vol. 45, no. 12, pp. 1592–1604, Dec. 2015.
+[17] L. Martynova and M. Rozengauz, “Approach to reconfiguration of a motion control
+system for an autonomous underwater vehicle,” Gyroscopy and Navigation, vol. 11,
+no. 3, pp. 244–253, 2020.
+[18] X. Liu, M. Zhang, and F. Yao, “Adaptive fault tolerant control and thruster fault
+reconstruction for autonomous underwater vehicle,” Ocean Eng., vol. 155, pp. 10
+– 23, May 2018.
+[19] E. Omerdic and G. Roberts, “Thruster fault diagnosis and accommodation for open-
+frame underwater vehicles,” Control Eng. Pract., vol. 12, no. 12, pp. 1575–1598,
+Dec. 2004.
+[20] T. Podder, G. Antonelli, and N. Sarkar, “Fault tolerant control of an autonomous
+underwater vehicle under thruster redundancy: simulations and experiments,” in
+Proceedings of 2000 IEEE International Conference on Robotics and Automation
+(ICRA), vol. 2, San Francisco, CA, USA, 2000, pp. 1251–1256.
+[21] S. Zhang, Y. Wu, X. He, and Z. Liu, “Cooperative fault-tolerant control for a mobile
+dual flexible manipulator with output constraints,” IEEE Trans. Autom. Sci. Eng.,
+2021.
+[22] C. Shen, Y. Shi, and B. Buckham, “Trajectory tracking control of an autonomous
+underwater vehicle using lyapunov-based model predictive control,” IEEE Trans.
+Ind. Electron., vol. 65, no. 7, pp. 5796–5805, 2018.
+[23] Q. Wen, R. Kumar, and J. Huang, “Framework for optimal fault-tolerant control
+synthesis: maximize prefault while minimize post-fault behaviors,” IEEE Trans.
+Syst. Man Cybern., Syst., vol. 44, no. 8, pp. 1056–1066, Aug. 2014.
+[24] D. Zhu, T. Yan, and S. X. Yang, “Motion planning and tracking control of un-
+manned underwater vehicles: technologies, challenges and prospects,” Intelligence
+& Robotics, vol. 2, no. 3, pp. 200–222, 2022.
+[25] U. Sinha, M. Cashman, and M. Cohen, “Using a genetic algorithm to optimize
+configurations in a data-driven application,” in Proceedings of 12th International
+Symposium on Search-Based Software Engineering, Cham, Switzerland, 2020, pp.
+137–152.
+[26] S. Mohanty and S. Mohanty, “Genetic algorithm-based motif search problem: a
+review,” in Proceedings of 2020 3rd International Conference on Smart Computing
+and Informatics. Smart Innovation, Systems and Technologies (SIST), vol. 1,
+Singapore, Singapore, 2020, pp. 719–731.
+[27] D. Gong, J. Sun, and Z. Miao, “A set-based genetic algorithm for interval many-
+objective optimization problems,” IEEE Trans. Evol. Comput., vol. 22, no. 1, pp.
+47–60, Feb. 2018.
+[28] C. Lai, H. Ibrahim, M. Abdullah, J. Abdullah, S. Suandi, and A. Azman, “A liter-
+ature review on data conversion methods on EEG for convolution neural network
+applications,” in Proceedings of 2019 International Conference on Robotics, Vision,
+Signal Processing and Power Applications. Enabling Research and Innovation
+Towards Sustainability. Lecture Notes in Electrical Engineering (LNEE), Singapore,
+Singapore, 2019, pp. 521–527.
+[29] A. Asmara, “A possibility of artificial neural networks to be applied in the predictive
+test: a systematic literature review and study case,” J. Phys., Conf. Ser., vol. 1456,
+pp. 12 052–12 061, 2020.
+[30] S. Hulyalkar and H. Khanuja, “Optimal solution generation from reviews and
+micro-reviews using greedy algorithm,” in Proceedings of 2017 International
+Conference on Computing, Communication, Control and Automation (ICCUBEA),
+Pune, India, 2017, pp. 1–6.
+[31] P. Guo, M. Liu, and Z. Xue, “A PSO-based energy-efficient fault-tolerant static
+
+A GOA-BASED TRAJECTORY TRACKING FTC
+12
+scheduling algorithm for real-time tasks in clouds,” in Proceedings of 4th Inter-
+national Conference on Computer and Communications (ICCC), Chengdu, China,
+2018, pp. 2537–2541.
+[32] S. Celtek, A. Durdu, and M. Ali, “Real-time traffic signal control with swarm
+optimization methods,” Measurement, vol. 166, pp. 108 206–108 213, 2020.
+[33] J. Wu, C. Song, J. Ma, J. Wu, and G. Han, “Reinforcement learning and
+particle swarm optimization supporting real-time rescue assignments for multiple
+autonomous underwater vehicles,” IEEE Trans. Intell. Transp. Syst., vol. 23, pp.
+6807–6820, Jul. 2021.
+[34] D. Zhu, Q. Liu, and Z. Hu, “Fault-tolerant control algorithm of the manned subma-
+rine with multi-thruster based on quantum-behaved particle swarm optimisation,”
+Int. J. Control, vol. 84, no. 11, pp. 1817–1829, Nov. 2011.
+[35] W. Guo, J. Li, G. Chen, Y. Niu, and C. Chen, “A PSO-optimized real-time fault-
+tolerant task allocation algorithm in wireless sensor networks,” IEEE Trans. Parall.
+Distr. Syst., vol. 26, no. 12, pp. 3236–3249, 2015.
+[36] S. Saremi, S. Mirjalili, and A. Lewis, “Grasshopper optimisation algorithm: theory
+and application,” Adv. Eng. Softw., vol. 105, pp. 30–47, 2017.
+[37] G. Wang, A. Heidari, M. Wang, F. Kuang, W. Zhu, and H. Chen, “Chaotic arc
+adaptive grasshopper optimization,” IEEE Access, vol. 9, pp. 17 672–17 706, 2021.
+[38] J. Xia, Z. Wang, D. Yang, R. Li, G. Liang, H. Chen, A. A. Heidari, H. Tura-
+bieh, M. Mafarja, and Z. Pan, “Performance optimization of support vector
+machine with oppositional grasshopper optimization for acute appendicitis diag-
+nosis,” Computers in Biology and Medicine, vol. 143, p. Article in press. DOI:
+dx.doi.org/10.1016/j.compbiomed.2021.105206, 2022.
+[39] J. Wu, H. Wang, N. Li, P. Yao, Y. Huang, Z. Su, and Y. Yu, “Distributed trajectory
+optimization for multiple solar-powered UAVs target tracking in urban environment
+by adaptive grasshopper optimization algorithm,” Aerosp. Sci. Technol., vol. 70, pp.
+497–510, Nov. 2017.
+[40] P. Mishra, V. Goyal, and A. Shukla, Advances in Intelligent Computing and
+Communication. Lecture Notes in Networks and Systems.
+Springer, Singapore,
+2020, ch. An Improved Grasshopper Optimization Algorithm for Solving Numerical
+Optimization Problems, pp. 179–188.
+[41] W. Durham, “Constrained control allocation,” J. Guid. Control Dyn., vol. 16, no. 4,
+pp. 717–725, Jul. 1993.
+[42] D. Zhu and B. Sun, “The bio-inspired model based hybrid sliding-mode tracking
+control for unmanned underwater vehicles,” Eng. Appl. Artif. Intell., vol. 26, no. 10,
+pp. 2260–2269, Nov. 2013.
+[43] Y. Shtessel, D. Foreman, and C. Tournes, “Stability margins in traditional and
+second order sliding mode control,” in Proceedings of 2011 IEEE Conference on
+Decision and Control, Orlando, FL, USA, 2011, pp. 4604–4609.
+[44] J. Zand, “Enhanced navigation and tether management of inspection class remotely
+operated vehicles,” Master Thesis, Mechanical Engineering, University of Victoria,
+Victoria, Canada, 2005.
+[45] D. Zhu, L. Wang, Z. Hu, and S. X. Yang, “A grasshopper optimization-based fault-
+tolerant control algorithm for a human occupied submarine with the multi-thruster
+system,” Ocean Eng., vol. 242, 2021.
+[46] S. J. Simpson, A. R. Mccaffery, and B. F. H ¨AGELE, “A behavioural analysis of
+phase change in the desert locust,” Biological Reviews, vol. 74, no. 4, pp. 461–480,
+2010.
+[47] S. M. Rogers, T. Matheson, E. Despland, T. Dodgson, and S. J. Simpson,
+“Mechanosensory-induced behavioral gregarization in the desert locust schistocerca
+gregaria,” Journal of Experimental Biology, vol. 206, no. Pt 22, pp. 3991–4002,
+2003.
+[48] Y. Meraihi, A. Gabis, S. Mirjalili, and A. Ramdane-Cherif, “Grasshopper opti-
+mization algorithm: theory, variants, and applications,” IEEE Access, vol. 9, pp.
+50 001–50 024, 2021.
+[49] Y. Zhao, X. Qi, Y. Ma, Z. Li, R. Malekian, and M. Sotelo, “Path following opti-
+mization for an underactuated USV using smoothly-convergent deep reinforcement
+learning,” IEEE Trans. Intell. Transp. Syst., vol. 22, no. 10, pp. 6208–6220, 2021.
+[50] Z. Peng, D. Wang, Z. Chen, X. Hu, and W. Lan, “Adaptive dynamic surface control
+for formations of autonomous surface vehicles with uncertain dynamics,” IEEE
+Trans. Control Syst. Technol., vol. 21, no. 2, pp. 513–520, 2013.
+[51] P. Gong, Z. Yan, W. Zhang, and J. Tang, “Trajectory tracking control for au-
+tonomous underwater vehicles based on dual closed-loop of mpc with uncertain
+dynamics,” Ocean Eng., vol. 265, 2022.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf,len=1090
+page_content='A GOA-BASED TRAJECTORY TRACKING FTC  1  A GOA-based Fault-tolerant Trajectory Tracking  Control for an Underwater Vehicle of Multi-thruster  System without Actuator Saturation  Danjie Zhu, Member, IEEE, Lei Wang, Hua Zhang and Simon X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yang, Senior Member, IEEE  Abstract—This paper proposes an intelligent fault-tolerant  control (FTC) strategy to tackle the trajectory tracking prob-  lem of an underwater vehicle (UV) under thruster damage  (power loss) cases and meanwhile resolve the actuator saturation  brought by the vehicle’s physical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' In the proposed  control strategy, the trajectory tracking component is formed  by a refined backstepping algorithm that controls the velocity  variation and a sliding mode control deducts the torque/force  outputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' the fault-tolerant component is established based on a  Grasshopper Optimization Algorithm (GOA), which provides fast  convergence speed as well as satisfactory accuracy of deducting  optimized reallocation of the thruster forces to compensate for  the power loss in different fault cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Simulations with or  without environmental perturbations under different fault cases  and comparisons to other traditional FTCs are presented, thus  verifying the effectiveness and robustness of the proposed GOA-  based fault-tolerant trajectory tracking design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Note to Practitioners: This paper is motivated by the actuator  saturation problem that exists in the trajectory tracking of  an underwater vehicle (UV) when encountering power loss  of the thruster system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The fault-tolerance trajectory tracking  performance is affected by physical constraints of the vehi-  cle when using the traditional methods as they may deduct  excessive kinematic/dynamic requirements during the control  process, thus inducing the deviation of the tracking trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Therefore, the refined backstepping as well as the grasshopper  optimization (GOA) are combined to eliminate the excess, where  the refined backstepping is used to alleviate the speed jumps  (kinematic outputs) and the GOA is to control the propulsion  forces (dynamic outputs) when facing thruster fault cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' This  innovates the industrial practitioners that the control design of  the vehicle can be improved to avoid the tracking deviation  brought by unsatisfied driving commands under fault cases  through embedding optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Moreover, for the  specific type of UV studied in this paper used for dam detection,  simulations regarding practical dam detection such as the 3D  polygonal line trajectory tracking and the frequently occurring  UV single-fault cases are chosen, which can serve as references  for practitioners working in the related field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' In the future,  underwater experiments of the UV will be investigated, with more  effects of the practical environment involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Index Terms—actuator saturation, backstepping control, fault-  tolerant control, grasshopper optimization, trajectory tracking,  underwater vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' INTRODUCTION  This work is supported by the Natural Sciences and Engineering Research  Council (NSERC) and the National Key Project of Research and Development  Program of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Danjie Zhu and Simon X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yang are with Advanced  Robotics and Intelligent System (ARIS) Laboratory, School of Engineer-  ing, University of Guelph, Guelph, ON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' N1G2W1, Canada (e-mail:{danjie;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  syang}@uoguelph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='ca);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Lei Wang and Hua Zhang are with the Underwater  Engineering Institute, China Ship Scientific Research Center, Wuxi 214082,  China (e-mail: 13771115826@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' zhanghua702@126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  T  HE study of vehicle controls has been extended to various  conditions where fault tolerance is widely involved in  the corresponding control design, which is denoted as fault-  tolerant control (FTC) [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Scientists have worked on the  FTC study for decades in various fields such as crafts in the air  or space, land vehicles and industrial manufacturing [3–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  In previous studies, the FTC is usually applied to alleviate  abrupt errors and provides the most feasible solution when  inevitable damages happen for equipment in different fields  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' However, the research regarding the FTC on underwater  vehicles (UV) has not been thoroughly investigated, due to  the complexity brought by the underwater environment and  the UV system [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Corresponding studies on the FTC have been proposed in  this century [3, 4, 6, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Based on these studies, the design  of the excessive number of thrusters compared to the number  of degrees of freedom (DOF) is raised and accepted as a  resolution to the UV FTC problem, which is called thruster  reconfiguration [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' For example, when unexpected fault  cases of the vehicle thrusters occur, the thrusters installed on  the vehicle that exceeds the number of DOFs (six: surge, sway,  heave, row, pitch and yaw) have enough flexible space to be  regulated to provide the required propulsion at corresponding  DOFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' To implement the thruster configuration theory in  practical cases, the weighted pseudo-inverse matrix method  has been proposed, where the fault cases are quantified as  degrees of damage and serve as the inputs to form the thruster  configuration model [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' By this method, the process of the  FTC is largely simplified, as the required thruster propulsion  can be deducted directly through a weighted pseudo-inverse  matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Nevertheless, physical constraints of the thruster  outputs are rarely considered, thus inducing the over-actuated  vehicle issue [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Additionally, among these studies, most  of them work on eliminating the static errors induced by the  fault cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' While in UV practical application, the realization  of dynamic control on the vehicle’s outputs in a real-time  manner, which commonly refers to the trajectory tracking  control for underwater vehicles, is of crucial importance [22–  24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Therefore, motivated by the over-actuated issue and mean-  while realizing robustness for underwater vehicle trajectory  tracking, optimization methods are combined with the tracking  control to optimize the vehicle’s dynamic outputs within allow-  able domains during the tracking procedure when encounter-  ing fault cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Among the current mainstream optimization  algorithms, the genetic algorithm consumes a long time on  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='01827v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='RO]  4 Jan 2023  A GOA-BASED TRAJECTORY TRACKING FTC  2  iteration, which is not ideal for the UV FTC that requires  both fast and feasible solutions [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The neural network  is demanding on the choice of data inputs while the UV  cannot provide when encountering fault cases, which shares  the same concern with the greedy algorithm, as the greedy  algorithm needs to decompose the data for processing [28–  30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Hence the swarm intelligence algorithm for optimization  stands out to be a preferable method to tackle the FTC  application of UVs due to its flexibility of data inputs and fast  convergence speed [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhu’s group has applied Particle  Swarm Optimization (PSO) based FTC on the unmanned  underwater vehicle, though satisfactory torque outputs are  achieved, the traditional PSO method shows poor real-time  feedback, which does not conform to the online requirement  of UV FTCs [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' In this study, an advanced swarm in-  telligent method named Grasshopper Optimization Algorithm  (GOA) is chosen based on its satisfactory balance between fast  convergence and accuracy of obtaining optimization results as  well as its simple implementation [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The fast convergence  is realized by its simply updated iteration derived from the  position of each search agent, which dramatically promotes  optimizing efficiency and offers the possibility of real-time  feedback for the UV FTC [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' At the same time, GOA  can provide accurate fault-tolerant results within acceptable  driving constraints through the limitations embedded in the  optimization algorithm, thus performing adaptive in resolving  the over-actuated issue of the UV FTC [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The contribution of the control strategy proposed in this  paper is to combine an advanced swarm intelligence algorithm  (GOA) with the fault-tolerant trajectory tracking control to  resolve fault cases of a progressed UV without actuator satu-  ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' By identifying and quantifying the degree of damage  (power loss) of the multi-thruster system and then efficiently  reallocating their forces through the GOA method, a feasible  solution with satisfactory accuracy will be given in a fast  convergence manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The strategy establishes a systematic  fault identification as well as efficient error elimination process  for the UV with abrupt damages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' and it first realizes the  application of the GOA method in FTC of specific under-  water vehicles with a multi-thruster system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Conventional  methods such as the constrained control allocation method  developed by Durham are based on the basic linear algebra  concepts and a means to determine the bounding surface of  the attainable moment space, yet the bounding surface is  difficult to be addressed [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Commonly applied methods  used for constructing UV FTC such as T-approximation or  S-approximation cannot thoroughly resolve the problem of  actuator saturation due to the inevitable errors brought by the  vehicle constraints [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Therefore, the fast convergence speed  and ideal accuracy of the GOA method help to amend the  commands given by the dynamic controller in time, which  accomplishes real-time FTC on the UV trajectory tracking  [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' First, the  kinematic and dynamic models of an advanced UV, ”YuLong”,  are introduced and the torque-force transition/normalization  is defined based on the UV thruster configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Next,  the fault-tolerant trajectory tracking problem of the UV is  described, with its restrictions explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The grasshopper  optimization-based fault-tolerant trajectory tracking control is  then proposed, where a refined backstepping algorithm is  applied to form the kinematic control component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' a sliding  mode control works as the dynamic control component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' and  the grasshopper optimization algorithm is used to form the  fault-tolerant component by reconfiguring the thruster force  outputs to eliminate the errors produced by different damage  degrees of the thrusters in fault cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' In the last section, the  effectiveness of the proposed grasshopper optimization-based  FTC is evaluated by tracking desired polygonal line or helix  trajectories, under the condition that one thruster (single-fault)  or two thrusters (double-fault) are supposed to be damaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' UV MODELS AND PROBLEM STATEMENT  In this section, a typical type of UV named ”YuLong”  is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Its robot models and trajectory tracking problem  descriptions are given in the form of specific equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Models of the ”YuLong” UV  In this section, a typical type of UV named ”YuLong” is  studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Its robot models and fault-tolerant trajectory tracking  problem descriptions are given in the form of specific equa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  1) Kinematic and Dynamic Model: ”YuLong” UV is one of  the latest UV for dam detection, designed by the Underwater  Engineering Institute of China Ship Scientific Research Center,  whose diving depth reaches 3000m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Its rough sketch is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1 and its multi-thruster system structure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' As a UV specially designed for detecting and maintaining  a satisfactory condition of the dam in the deep-water area, the  robustness and efficiency of the vehicle’s operation are crucial  for dam detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Among the six degrees of freedom (DOF) of the UV,  surge, sway, heave, roll, pitch and yaw, roll and pitch can be  neglected because these two DOFs barely have an influence  on the underwater vehicle during practical navigation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Therefore, when establishing the trajectory tracking model to  keep a controllable operation of the UV, usually four DOFs  surge, sway, heave and yaw are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Based on the four  involved DOFs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' for the kinematic equation of the UV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' the  velocity vector v can be transformed into the time derivative  of the trajectory vector ˙p as  ˙p =  �  ���  ˙x  ˙y  ˙z  ˙ψ  �  ��� = J(p)v ==  �  ���  cos ψ  − sin ψ  0  0  sin ψ  cos ψ  0  0  0  0  1  0  0  0  0  1  �  ���  �  ���  u  v  w  r  �  ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (1)  where J is a transformation matrix derived from the physical  structure of the UV body,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' [u v w r]T represents the velocities  at the chosen four axes of the UV (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  In an actual UV system, several complex and nonlin-  ear forces such as hydrodynamic drag, damping, lift forces,  Coriolis and centripetal forces, gravity and buoyancy forces,  thruster forces, and environmental disturbances are acting on  A GOA-BASED TRAJECTORY TRACKING FTC  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The reference frame and six degrees of freedom (x, y,  z, k, m and n) of the ”YuLong” UV (Top view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Considering the origins and effect of the forces,  a general dynamic model can be written as  M ˙v + C(v)v + D(v)v + g(p) = τ ,  (2)  where M is the inertia matrix of the summation of rigid body  and added mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' C(v) is the Coriolis and centripetal matrix  of the summation of rigid body and added mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' D(v) is the  quadratic and linear drag matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' g(p) is the matrix of gravity  and buoyancy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' and τ is the torque vector of the thruster inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  As mentioned in the previous section, in this study only four  states are considered for the specific model YuLong UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The  torque vector of the thruster input is represented by  τ =  �τx  τy  τz  τn  �  ,  (3)  where x, y and z represent the linear displacements of the  UV at surge, sway and heave directions, while n represents  the angular displacement of the UV at yaw direction (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  For  the  ”Yulong”  UV,  the  following  param-  eter  values  are  assigned:  inertia  matrix  M  =  [42 0 0 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  0 153 0 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  0 0 141 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  0 0 0 100]T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Coriolis and centripetal matrix C(v) = J−1M ˙JJ−1, where  J−1 represents the inverse matrix of J and ˙J represents  the derivative of J and quadratic and linear drag matrix  D(v) = [42+69u 0 0 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 0 319+245v 0 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 0 0 272+  86w  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  0  0  0  33 + 4r]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' In addition, the gravity force  applied on the vehicle is balanced off by the buoyancy force  when the whole system sustains at an equilibrium status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  2) Torque-force Transition and Normalization: According  to the physical structure of the ”Yulong” vehicle propulsion  system (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2), the relation between its torque vector and  the forces of the thrusters is  τ =  �  ���  τx  τy  τz  τn  �  ��� =  �  ���  T1 + T2  T7 + T8  T3 + T4 + T5 + T6  T1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='4 × T8 − T2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='4 × T7  �  ��� ,  (4)  where T1, T2, T3, T4, T5, T6, T7, T8 are the forces produced  by the eight thrusters installed around the vehicle body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The eight thrusters are of the same type and are supposed to  have the same maximum force Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Therefore the maximum  of the torque vector τm can be deducted based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (4) as  τm =  �  ���  τxm  τym  τzm  τnm  �  ��� =  �  ���  2Tm  2Tm  4Tm  (2 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='8)Tm  �  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (5)  On the basis of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (4), divide both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (5) by  the maximum torques to restrict the output in a certain range  of -1 to 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' and set τ = τ/τm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' T = T/Tm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' the vector is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='transformed into  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='τx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='τy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='τz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='τn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='��� =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='τx/τxm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='τy/τym  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='τz/τzm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='τn/τnm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
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+page_content='�����������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T1/Tm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T2/Tm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T3/Tm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T4/Tm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T5/Tm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T6/Tm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T7/Tm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T8/Tm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='�����������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='= B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='�����������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='T 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='�����������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (6)  The above equation has the compact form as  τ = B T  T = B  −1 τ ,  (7)  where B  −1 is the generalized inverse matrix of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Therefore, the transition between thruster forces and torques  is achieved and normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' For all the torques and forces in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (6), they are ranged from -1 to 1 (−1 ≤ τ ≤ 1 and  −1 ≤ T ≤ 1) to perform a direct and simplified showcase  during the tracking control process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The thruster distribution in the ”Yulong” UV propulsion  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (a) The top view, (b) The lateral view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Roll (K)  Yaw (N)  Heave (z)  Surge (X)  Pitch (M)  Sway (Y)2800mm  2800mm  T5  T3  1000mm  X  T8  1000mm  1000mm  Y  T6  T4  T4  T6  (a)  (b)A GOA-BASED TRAJECTORY TRACKING FTC  4  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Problem Statement  In this subsection, the fault-tolerant control problem on  the thruster system of the ”Yulong” vehicle is modeled and  explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Requirements and constraints during the control  process are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  1) Fault-tolerant Problem in UV Trajectory Tracking:  To achieve the accuracy and efficient control of the UV,  the fault cases of the thruster system should be taken into  consideration, where one or more of the thrusters might get  broken and corresponding controls are proposed to sustain the  movement and posture of the vehicle as desired, by deriving  the normalized desired torques τd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The control of the torque  outputs τ is realized by the combined action of the thruster  forces under fault conditions, which is allocated by approxi-  mation/optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The approximation/optimization  deducts the optimal reallocation when the thrusters’ working  condition changes, and accomplishes the elimination of the  torque errors during the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  For UV with a multi-thruster system, the ideal fault-tolerant  control is realized by,  ||e|| = ||τd − τ|| → 0 ,  (8)  ||θe|| = arccos  τd • τ  ||τd|| · ||τ|| → 0 ,  (9)  where ||e|| is the magnitude error obtained by the vector norm  of the difference between the desired and the actual torques of  the vehicle, ||θe|| is the direction error that computes the arc  cosine of the ratio between the multiplication of desired and  actual torques and their vector norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Besides the magnitude  error, the direction error is also important to the trajectory  tracking control of the underwater vehicle, as the vehicle’s  movement is also determined by the direction displacement  of its dynamic outputs (torques) along the axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' It is possible  for the vehicle to have an acceptable magnitude dynamic error  but meanwhile obtain excessive direction error, thus inducing  non-ideal tracking results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Hence, when designing the fault-tolerant control, the error  values should be as small as possible to obtain good control-  ling results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  2) Constraints on Fault-tolerant Control of the UV: In the  actual application, the controlling effect is always restricted by  the physical constraints of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The UV cannot provide  infinite driving inputs such as torques/forces to complete the  navigation, thus resulting in the problem of actuator saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Therefore driving restrictions are always applied on the UV  to achieve a reliable and controllable navigation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The  maximum forces of the vehicle thrusters, derived from the  maximum torques that can be offered by the vehicle body, are  the essential constraints of the vehicle’s controlling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  To assess the influence of the constraints, maximum torques  τm are introduced in the simulation part of this paper (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' By the definition given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (6), normalized torques τ  and normalized forces T are supposed to have the limits of  1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The two variables are used to quantify the effect of  the constraints during the trajectory tracking process in the  simulation part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' GOA-BASED FAULT-TOLERANT TRAJECTORY  TRACKING CONTROL (GFTC) DESIGN  The basic control architecture of the system is illustrated in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The design of the control strategy consists of two parts:  (1) a trajectory tracking component formed by an outer loop of  auxiliary kinematic control based on the position errors of the  UV and an inner loop of dynamic torque controller based on  the velocity state vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (2) a fault-tolerant control component  that identifies the fault cases of the UV propulsion system and  reallocates the thruster force outputs based on Grasshopper  optimization algorithm to eliminate the effect brought by the  fault cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Additionally, maximum thruster force outputs Tm  are applied (see the bold frame in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Specially for the  simulation of underwater environmental perturbations, current  disturbance on torque outputs is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Moreover, the  environmental noise at the step of forming vehicle positions  is also involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Details of the control design will be presented  in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Component of Trajectory Tracking Control  In this section, the methods that form the kinematic and dy-  namic control of the UV are explained, with detailed equations  and stability analysis given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  1) Error  Restricted  Backstepping  Control:  Suppose  the  maximum  velocity  vector  of  UV  is  vm = [vxm  vym  vzm  vψm]T , the input e(t) is the  error between the desired and actual trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' a transfer  function is defined to process the input trajectory errors  within an acceptable range,  ve = f(e)µvm =  e(t)  |e(t)| + 1µvm ,  (10)  where vm represents the maximum velocities of the UV DOFs  at their corresponding axes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' µ is a positive constant chosen  according to the variation requirement and is assigned with  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5 in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Therefore, when e(t) → 0, ve = f(e)µvm → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' and  when e(t) → ∞, ve = f(e)µvm → the restricted maximum  velocity µvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Based on the restriction, the transfer function  ve limits its control output in a smaller domain and meanwhile  provides faster convergence to the input errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The processed outputs ve after the convergence shall not  exceed the maximum possible value of the UV velocity, and  the definition of f(e) provides a smooth transition of the UV  at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Hence the speed-jump problem of the UV  can be alleviated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The vector ve at the four DOFs can be  written as ve = [vex  vey  vez  ven]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Additionally, in the  backstepping method, control functions for each subsystem  are designed based on the Lyapunov techniques and generated  to form the complete control law [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Therefore, based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (1) and the definition of the backstepping method, the error  variables in the control law of the backstepping method are  A GOA-BASED TRAJECTORY TRACKING FTC  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Schematic of the proposed fault-tolerant trajectory tracking control designed for the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  replaced by the restricted outputs processed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (10), the  control law of the backstepping control can be derived as  vc =  �  ���  uc  vc  wc  rc  �  ���  (11)  =  �  ���  k(vex cos ψ + vey sin ψ) + ud cos veψ − vd sin veψ  k(−vex sin ψ + vey cos ψ) + ud sin veψ − vd cos veψ  wd + kzvez  rd + kψveψ  �  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  where k, kz and kψ are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Then the processed control velocities vc are passed to the  UV, where they are calculated to keep pace with the desired  trajectory through the dynamic model of the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Additionally,  the stability of the refined backstepping control can be proved  by constructing a Lyapunov function Γ0 = 1  2(e2  x+e2  y+e2  z+e2  ψ),  whose derivative is less than and equal to zero (see Appendix  A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  2) Sliding Mode Control: To design the sliding mode  control, the desired dynamics (s) should be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Based  on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (2) where the UV dynamic system is of the second  order for the velocity v, the dynamics can be designed as  s =  � d  dt + λ  �2 �  evdt = ˙ev + 2λev + λ2  �  evdt,  (12)  where  d  dt is the derivative operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' ev represents the errors  given by the control velocities (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 3), ev = vc − v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' and  λ > 0 is a positive parameter [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Then take the derivative of s, we can get  ˙s = ¨ev + 2λ˙ev + λ2ev ,  (13)  where ˙ev = ˙vc − ˙v  To keep the system states consistent with the desired dynam-  ics, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (13) should be equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' This means the system  states are on the sliding surface of the perfect tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' At the  same time, plug in the equation of the UV dynamic model  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (2)),  ˙s = ¨ev + 2λ˙ev + λ2ev = 0  ¨ev + 2λ( ˙vc − ˙v) + λ2ev = 0  ¨ev + 2λ( ˙vc − (τ − Cv − Dv − g)M−1) + λ2ev = 0  τ = M( ˙vc + ¨ev  2λ + λ  2 ev) + Cv + Dv + g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (14)  The standard sliding mode control law is defined as  τ = ˆτ + τc ,  (15)  where ˆτ represents the major control law, which is continuous  and model-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' It is designed to maintain the trajectory  consistently on the sliding surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' τc represents the switching  control law, dealing with the model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' When the  trajectory is getting out of control, τc is used to push the  trajectory back to the sliding surface and continue satisfactory  tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' For Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (14), supposing a simplification ¨ev ≈ −k˙ev  based on the error acceleration feedback control to reduce  computation complexity, the estimated item in the major  control law ˆτ can be deducted as  ˆτ = ˆ  M( ˙vc + −k˙ev  2λ  + λ  2 ev) + ˆCv + ˆDv + ˆg ,  (16)  where ˆM, ˆC, ˆD, ˆg are the estimated values of M, C, D and  g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' approximate values can be obtained from the practical case  respectively [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The switching item τc in sliding mode control can be  defined as  τc = −K1s − K2|s|rsign(s) ,  (17)  where sign(s) is the nonlinear sign function of s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' K1 and  K2 are positive coefficients, K1 ≥ η + F and K2 ≥ η +  F, η is the design parameter which is always chosen as a  positive constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 0 < r < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' and F represents the upper  bound of the difference between the system actual output and  the estimation,  F = |f(v) − ˆf(v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (18)  Additionally, an adaptive variation term �τest is added to  the control law, where ˙�τest = Γs and Γ represents a positive  Perturbation  (Current  disturbance)  Fault-tolerant  Trajectory tracking  Backstep-  Thruster system  Desired  pd  Error  Vc  Ve  "Lc  UV  p = J(p)v  SMC  (Constraints Tm)  Restriction  ping  Trajectory  Fault  GOA  identification  Environmental Noise  (Random error inputs)A GOA-BASED TRAJECTORY TRACKING FTC  6  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Hence the final sliding mode control law is defined  as  τ = ˆτ + �τest + τc  = ˆ  M( ˙vc + −k˙ev  2λ  + λ  2 ev) + ˆCv + ˆDv + ˆg  +�τest − K1s − K2|s|rsign(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (19)  Detailed proof of the SMC stability can be found in Ap-  pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Component of Fault-tolerant Control  The fault-tolerant control design is mainly built on the  adjustment of the forces required by the thruster system, where  the forces operate together and provide the torques as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The adjustment of the thruster forces is deducted by the  grasshopper optimization algorithm (GOA), which efficiently  eliminates the errors brought by the fault of the propulsion  system after fault identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Details of the control strategy  are presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  1) Weighting Matrix: To quantify the degree of damage in  the fault cases for the multi-thruster system, a weighting matrix  W is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The matrix W decides the service condition  of the thruster, which is usually defined as a diagonal matrix,  W =  �  �����������  w1  0  0  0  0  0  0  0  0  w2  0  0  0  0  0  0  0  0  w3  0  0  0  0  0  0  0  0  w4  0  0  0  0  0  0  0  0  w5  0  0  0  0  0  0  0  0  w6  0  0  0  0  0  0  0  0  w7  0  0  0  0  0  0  0  0  w8  �  �����������  (20)  where wj > 0 is the weight of the jth thruster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' If all the  thrusters are working in the desired condition with no power  loss, W will be a unit matrix, meaning all wj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' If there is  power loss for any of the thrusters, its corresponding weight  will be reduced by the degree of the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' For example, when  T1 thruster attains 20% of power loss, w2 in the weighting  matrix is assigned as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  As the relation between the thruster forces and the vehicle  torques at different states are defined and given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (7),  the following transition between the torques and forces in the  fault cases is defined as,  τ = BWT,  (21)  where T is the control parameters in the UV case, which  is deducted by the optimization method, such as the GOA  method used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Additionally, as a comparison to the GOA method, the  weighted pseudo-inverse matrix method is used, which is  determined based on the defined weighting matrix,  T = B  +  w τd = (WB  T (BWB  T )−1)τd,  (22)  where B  +  w is the matrix that transmits the damage information  to the propulsion system and meanwhile make the adjustment  accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Thus, the thruster force results under fault cases  can be deducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' For example, if the thruster T1 can only  provide 70% of power after encountering a power loss of 30%,  the weighted pseudo-inverse matrix method will request larger  output ( 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='7× original force) from T1 such that the same force  can be achieved after weakened by 30% of power loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Then T-approximation or S-approximation methods are ap-  plied for achieving force results within the range of the thruster  force maximum, which is generally denoted as pseudo-inverse  (P-I) matrix approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' T-approximation restricts all nor-  malized forces T between [-1, 1] by subtracting/adding the  excessive part of the states whose value is larger than 1 or  smaller than -1, where  Tt =  �  �  �  T i,  T i ∈ [−1, 1]  1,  T i > 1  −1,  T i < −1  S-approximation realizes the limits of [-1, 1] by timing the  reciprocal ratio of the largest normalized force for all states,  where  Ts =  1  max(T i)T, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  For example, in S-approximation, if the largest normalized  force for one of the states reaches 2, all normalized forces  will be multiplied by the ratio of 1  2 to guarantee they do not  exceed the limits of -1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  In the simulation section of this study, T-approximation  method is used as the typical pseudo-inverse (P-I) matrix  approximation to work as a comparison of the proposed GOA-  based FTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The T-approximation has wider application in  practical cases of the underwater vehicle FTC due to it gener-  ally produces smaller errors compared to the S-approximation  [34, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  2) Grasshopper Optimization Algorithm: The grasshopper  optimization algorithm is newly raised in 2017 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' As a  developed algorithm based on the theory of swarm intelligence  that imitates the activity of grasshoppers, GOA shows better  performance than the traditional swarm intelligence algorithms  due to that it finds a satisfactory balance between fast speed of  convergence and accuracy based on its form switch between  “adults” and “larvae”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The fast convergence is realized when  GOA searches globally based on the position of each agent  under its ”adult” form, which explores on a large scale in  an attractive manner among the agents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' while the accuracy is  achieved by shrinking the range and keeping a repulsive zone  based on the best agent under ”larvae” form, which avoids the  local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  According to the movement of the grasshopper groups, a  mathematical model can be defined to describe their swarming  behavior [46, 47]  Xi =  N  �  j=1,j̸=i  s(|xj − xi|)xj − xi  dij  − Gi + Ai,  (23)  where Xi represents the next position of the ith grasshopper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  s(r) is the social interaction function where it is optimized as  s(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5e−r/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5 − e−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The item |xj − xi| is the distance  between the current position of the ith and jth grasshopper,  A GOA-BASED TRAJECTORY TRACKING FTC  7  (xj − xi)/dij is the unit vector pointing from the position of  the ith grasshopper to the jth grasshopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Gi represents the  Gravity force at the ith grasshopper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Ai is the wind advection  that is assumed to be always towards the target, Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Based on the assumptions made in this control case, where  the gravity force is neglected and the wind force is always  towards the target, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (23) can be converted into  Xd  i = c(  N  �  j=1,j̸=i  cubd − lbd  2  s(|xj − xi|)xj − xi  dij  ) + Td (24)  where c is a decreasing coefficient that shrinks the comfort  zone, repulsion zone and attraction zone, which is determined  as c = cmax − l(cmax − cmin)/L, cmax is the maximum value,  cmin is the minimum value, l indicates the current iteration,  and L is the maximum number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' In this work,  we assign cmax = 1 and cmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='00001 by trial and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The variable ubd represents the upper bound of the case while  the lbd represents the lower bound, which are 1 and -1 in  this design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Td is the desired solution of the current iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  These parameters are used to attain the fast convergence of  the optimization, by increasing the speed of updating the local  solution in relation to the increment of the iteration times, thus  leading to the efficient searching result of the GOA method  [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The pseudocode of the GOA applied on the UV thruster  forces reallocation can be concluded as follows (see Algo-  rithm 1), with the fitness evaluation substituted by the error  evaluation, given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='s (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The flow of the proposed GOA method can be concluded  as follows:  1) First initialize the swarm, with 10 groups of eight random  numbers between -1 and 1 representing the normalized eight-  thruster group and each group is regarded as a search agent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  2) Next calculate the fitness of each search agent and address  the agent with the minimum errors as the best based on the  objective function combined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (8) and (9), which is ||e||+  ||θe|| → 0 , the constraints for each agent are set between [-1,  1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  3) Update the parameter c according to the iteration time to  accelerate the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' If iteration time reaches maximum  then stop, otherwise continue the update of c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  4) Update positions (values) of search agents based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (24) and compare the fitness of the updated agents with the  agent of the best fitness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' If the updated fitness turns out to  be better, updates the position of the agents, otherwise do not  update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  5) Update the iteration time, and repeat the loop from step 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Component of Perturbations  As the ”Yulong” UV model is designed for dam detection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  which usually operates at the shoreside underwater condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  the perturbation of currents can be considered in a regular  combination form of wave functions as [49–51]  τp = Ap1 cos(ωp1t) sin(ωp2t) + Ap2 cos (ωp3t) sin (ωp4t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (25)  where Ap1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Ap2 and ωp1 to ωp4 are random coefficients,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' which  are chosen to synthesize the randomness of the currents to  appropriately address the underwater environmental perturba-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Ap1 and Ap2 are assigned within the range of 10% of the  torque outputs at four axes, for example, if τx output is about  100N, the assignment range will be [−10, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Coefficients  ωp1 to ωp4 are chosen with the range of -1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  In addition, considering the effect of environmental noise  that produces perturbation to the data transfer at the stage  of forming positions, an random error input is given in the  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The random error is supposed to be within [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='1,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='1] and filtered by sensors, which corresponds to the practical  UV case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' SIMULATION RESULTS AND ANALYSIS  In this section, the polygonal line and helix trajectory  tracking simulation results of the proposed FTC and con-  ventional approximation methods are presented and analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Fault cases of single-fault (one thruster broken) and double-  fault (two thrusters broken) are applied due to their frequent  occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Helix Tracking  In this section, one of the thrusters T1 is supposed to be  broken, where 100% of power is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The initial position of  the desired helix trajectory is set at (0, 0, 0, 0), while the initial  position of the control trajectories is set at (0, 10, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The  difference of the initial positions is given to test the correction  ability of the two tracking strategies when they start with a  certain amount of deviation at one of the axes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' the y axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Assuming the desired trajectory is given as xd = 10 sin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2t,  yd = 10 − 10 cos 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2t, zd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5t and ψd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2t with the  simulation time continuing for 50 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The GFTC (in red dash) tracking result quickly eliminates  the initial error and follows the desired helix trajectory till  the end of the simulation yet the T1 thruster is supposed to be  completely broken (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The P-I approximation (in blue)  cannot coincide with the desired helix under the single-fault  case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' where the deviation of abrupt variations is created at the  Algorithm 1: GOA algorithm embedded in the fault-tolerant component  Input: a: desired normalized torque matrix  W: weighting matrix (fault identification)  Output: T: A vector containing the optimal allocation of the normalized forces for  eight thrusters  Initialize the swarm Xi (i-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='---,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='10)  Initialize the cmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' cmin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' and maximum number of iterations L  Calculate the fitness of each search agent to address the best search agent T with the  minimum errors (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' s (8) and (9))  while (l< L)  Update c = cmax - l (cmax - cmin) / L  for each search agent  Update the value of the current search agent by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (24)  Retrieve the current search agent if it excess the limitation of  [-1, 1]  end for  Update T if there is a better solution  l=[+1  end while  Return TA GOA-BASED TRAJECTORY TRACKING FTC  8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Helix tracking results using the GFTC with or  without perturbations and the P-I approximation-based FTC  under the single-fault case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (a) Comparison of trajectories,  (b) Comparison of tracking errors, (c) Comparison of control  velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  beginning and trajectory distortions are produced throughout  the whole process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Therefore when the dynamic constraints  are considered, the P-I method fails to compensate for the  power loss of a single thruster, which induces increasing  errors and excessive velocities with abrupt jumps given in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='s 4(b) and (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' while the GFTC achieves smooth error and  control velocity curves that indicate the satisfactory tracking  performance of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Moreover, the ”GFTC-P” (in pink)  results consider the effect of perturbations brought by the  currents and environmental noise when addressing the position  information for the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The GFTC-P result under the  effect of perturbations sustains a similar tracking trajectory  with the unaffected GFTC, which verifies the robustness of  the proposed control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Maximum velocities of the GFTC with or without  perturbations and the P-I based FTC under the single-fault  case when tracking the helix  uc (m/s)  vc (m/s)  wc (m/s)  rc (m/s)  GFTC  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0008  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='8978  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2016  P-I  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='9719  25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='6125  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0034  GFTC-P  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0150  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='8944  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='6404  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2836  Tracking  errors  of  the  GFTC  method  (in  red),  P-I  approximation-based FTC (in blue), single-fault case without  FTC (in grey) and GFTC-P with the effect of perturbations  (in pink) are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The error curve of the GFTC  method eliminates the initial deviation and quickly converges  to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' While the error of the P-I approximation-based FTC  presents obvious fluctuations and cannot be eliminated in  all axes, furthermore, the P-I error curve attains even larger  fluctuations compared to the case without FTC at the ψ  axis, which supports the trajectory tracking performance given  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' This shows the failure of P-I approximation  FTC on keeping the desired helix tracking in the single-fault  condition when dynamic constraints are applied, yet the GFTC  method accomplishes the fault-tolerant trajectory tracking task  with satisfactory kinematic outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' This conclusion is also  supported by the velocity variations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The P-  I method deducts largely excessive speeds at the x and y  axes, where the maximum of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='9719 m/s and -25 m/s are  required, but the GFTC method satisfactorily restricts the  velocity at x and y axes within the constraints of [-2, 2]m/s,  with the maximum outputs at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0008 m/s and -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='8978 m/s  (TABLE I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The GFTC also achieves much smoother velocity  curves compared to the P-I method for all axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Even when  considering the perturbation effect in GFTC-P simulation,  though small chattering is performed, the errors as well as the  control velocities are successfully restricted in an acceptable  range, with the maximum of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='015m/s and -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='8944m/s at the x  and y axes, and far less requirement of control velocity at the  ψ axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Hence the effectiveness of the proposed GFTC method  in tracking the desired trajectory under the single-fault case is  verified even when external perturbations are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 3D Polygonal Line Tracking  A 3D polygonal line is applied in this section as the  reference tracking trajectory, as the ”YuLong” UV usually  navigates in a movement similar to the polygonal line to detect  the dam damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The initial position of the desired trajectory is set at  (0, 0, 0, 0), while the initial position of the control trajectories  is set at (0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' A specific polygonal line function is  applied and the simulation continues for 20 seconds:  xd = t, 0 ≤ t ≤ 20,  yd =  �  �  �  �  �  �  �  t,  0 ≤ t ≤ 5  5,  5 < t ≤ 10  t − 5,  10 < t ≤ 15  10,  15 < t ≤ 20  zd =  �  �  �  �  �  �  �  t,  0 ≤ t ≤ 5  5,  5 < t ≤ 10  t − 5,  10 < t ≤ 15  10,  15 < t ≤ 20  ψd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2, 0 ≤ t ≤ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  1) Single-fault Case: One of the thrusters T8 is supposed  to be broken, with 100% of power lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The tracking trajectory  results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The GFTC (in red dash) has  retained the polygonal line trajectory as desired after elimi-  nating the initial error at the y axis, neglecting the power loss  of the thruster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Moreover, the GFTC-P (in pink) results which  consider the effect of perturbations sustain a similar tracking  GFTC  25 T  GFTC-P  P-I1  No FTC  20  Desired  15  N  10-  51  6  (a)  GFTC  p-I  GFTC-P  No FTC  8  4 -  e  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  10  20  0  30  40  50  10  20  30  40  0  50  10  e  V  X  20 -  8  0  10  40  20  30  40  50  0  10  20  30  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='4 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='9  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2  e  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='.  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='3  10  20  30  0  40  50  0  10  20  30  40  50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0  3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5  e  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  10  20  30  40  0  50  10  20  30  40  0  50  Time(s)  Time(s)  (b)  (c)A GOA-BASED TRAJECTORY TRACKING FTC  9  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Polygonal line tracking results using the GFTC with  or without perturbations and the P-I approximation-based  FTC under the single-fault case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (a) Comparison of  trajectories, (b) Comparison of tracking errors, (c)  Comparison of control velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  trajectory with the unaffected condition, which verifies the  robustness of the proposed tracking control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The P-I method  (in blue) fails to catch up with the desired trajectory especially  at the turning point where larger dynamic inputs are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The P-I method cannot make up for the loss of the propulsive  force when physical constraints (torque/force maximum) are  involved, thus producing errors with large fluctuations as well  as excessive control velocities presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='s 5(b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Maximum velocities under the single-fault case  when tracking the polygonal Line  uc (m/s)  vc (m/s)  wc (m/s)  rc (m/s)  GFTC  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='1738  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='8212  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0782  P-I  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='7278  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0713  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2289  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2  GFTC-P  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2410  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2188  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='1192  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='1035  The errors at four axes are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 5(b), where  the GFTC (in red) successfully eliminates the tracking errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  While for the P-I approximation-based FTC (in blue), its result  fails to eliminate the error once the sharp turning is required  by the trajectory, as the excessive dynamic outputs deducted  by the P-I method cannot be satisfied when the dynamic  constraints are applied, thus inducing the large trajectory  deviation at the turning section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Velocity variations at four  axes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 5(c), the P-I method performs a sharp  fluctuation and fails to retain the control velocity within the  desired range at the y axis (see y axis in TABLE II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The  velocity at the y axis of P-I method reaches a dramatic value  of -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0713 m/s, largely exceeding the desired range of the  vehicle that is preset at -2m/s to 2m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' At the same time,  the GFTC successfully limits the kinematic outputs within  the constraints and presents a smooth curve of much smaller  fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' In addition, when considering the perturbations in  the GFTC-P simulation (in pink), though chattering issues are  presented, errors are constrained within an acceptable range  and kinematic outputs perform a smaller range compared to  the P-I method at most axes, with the maximum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2410m/s  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2188m/s at the x and y axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' These results indicate the  effectiveness and robustness of the proposed GOA-based FTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  2) Double-fault Case: In this section, the effect of the  GFTC, P-I approximation based FTC and GFTC consider-  ing environmental perturbations are compared, supposing two  thrusters (T1 and T8) of the propulsion system encounter  power loss of 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The GFTC method (in red dash) successfully eliminates the  initial errors at the y axis and retains the tracking trajectory  as desired till the end (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 6(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The GFTC-P (in pink)  results under the effect of perturbations sustain a similar  tracking trajectory with the unaffected GFTC trajectory, and at  the second turning section it performs more smooth tracking  curves than the first one, which verifies its robustness for  tracking the desired polygonal line even under the double-fault  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' At the same time, the P-I approximation based FTC fails  to track the desired trajectory and even presents a much larger  deviation compared to its single-fault case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 5(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' This  demonstrates that the GFTC method is capable of balancing  off the tracking errors whenever the damage degree of power  loss in the thruster system differs, thus proving the robustness  of the proposed FTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Similarly, as under the single-fault case, the error curve of  GFTC method under double-fault case eliminates the initial  deviation and quickly converges to zero (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 6(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' However,  the P-I method presents fierce error vibrations in x, y and  ψ axes compared to the single-fault case, which is even  worse than the performance of double-fault case without FTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  This shows that the P-I approximation-based FTC is heavily  affected by the damage degree of the UV propulsion system  and it cannot balance off the error produced by the excessive  power loss of the thrusters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' the double-fault case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' This  conclusion is also supported by the velocity variations in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 6(c), where the P-I method cannot be limited within  the allowable range due to the thruster power loss, with  excessive maximum velocities arriving at 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5815m/s for the  x axis, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='9814m/s for the y axis and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='4485m/s for the ψ  axis given in TABLE III, resulting in the complete tracking  failure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The velocities of the GFTC method  maintain within the allowable domain throughout the whole  process, neglecting the change of fault cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' The GFTC-P  simulations that involve perturbations in a practical underwater  environment also perform successful restriction of the errors  and the control velocities within the supposed range, with the  maximum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2473m/s and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='9646m/s at the x and y axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Therefore, the effectiveness of the proposed GFTC method in  tracking a desired polygonal line is verified whenever single-  fault or double-fault cases are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  GFTC  GFTC-P  p-I  10  No FTC  Desired  8  6  N  4  12  10  8  6  0  2  5  +  (a)  P-1  No FTC  GFTC-P  GFTC  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5  e  u  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  10  15  0  20  0  5  10  15  20  1  0  y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  e  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5  2 -  5  10  15  5  0  20  10  15  0  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0  N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0  e  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5-  :5  5  10  15  20  10  15  0  0  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2  e  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0  5  10  15  0  20  5  10  0  15  20  Time(s)  Time(s)  (c)  (b)A GOA-BASED TRAJECTORY TRACKING FTC  10  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Polygonal line tracking results using the GFTC with  or without perturbations and the P-I approximation-based  FTC under the double-fault case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (a) Comparison of  trajectories, (b) Comparison of tracking errors, (c)  Comparison of control velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Maximum velocities under the double-fault case  when tracking the polygonal line  uc (m/s)  vc (m/s)  wc (m/s)  rc (m/s)  GFTC  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='1710  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='8158  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='9989  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0782  P-I  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5815  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='9814  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2289  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='4485  GFTC-P  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2473  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='9646  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0367  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0885  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' CONCLUSION  In this paper, the fault-tolerant trajectory tracking problem  for the ”Yulong” UV is resolved by a Grasshopper Optimiza-  tion and backstepping & SMC-based cascade control (GFTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The GFTC strategy applies a refined backstepping algorithm  to restrict the kinematic outputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' and the Grasshopper Op-  timization Algorithm (GOA) is used to achieve optimized  thruster force reallocation within the allowable domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' When  encountering fault cases in tracking the polygonal line or  helix, the trajectory tracking errors of the GFTC are largely  alleviated and the actuator saturation problem is eliminated,  compared to the traditional FTCs such as the weighted pseudo-  inverse matrix approximation-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' In addition, the  robustness of the proposed FTC is also verified when environ-  mental perturbations are involved, which serves as the basis  of the experimental study on practical applications that will  be extended in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  APPENDIX A  PROOF OF THE ERROR RESTRICTED BACKSTEPPING  CONTROL STABILITY  According to the Lyapunov stability theory, a special Lya-  punov function Γ0 is chosen,  Γ0 = 1  2(e2  x + e2  y + e2  z + e2  ψ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (26)  By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='s (1) and (11), the derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (25) can be  obtained to prove the stability of the backstepping system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='˙Γ0 =ex ˙ex + ey ˙ey + ez ˙ez + eψ ˙eψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='= ex ( ˙xd − ˙x) + ey ( ˙yd − ˙y)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='+ ez( ˙zd − ˙z) + eψ ( ˙ψd − ˙ψ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='= ex [(cos ψdud − sin ψdvd) − (cos ψuc − sin ψvc)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='+ ey [(sin ψdud + cos ψdvd) − (sin ψuc + cos ψvc)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='+ ez(wd − wc) + eψ (rd − rc)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='= ex [(cos ψdud − sin ψdvd)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='− (kvex + ud(cos ψ cos veψ − sin ψ sin veψ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='+ vd(sin ψ cos veψ − cos ψ sin veψ))]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='+ ey [(sin ψdud + cos ψdvd)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='− (kvey + ud(sin ψ cos veψ + cos ψ sin veψ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='− vd(sin ψ sin veψ − cos ψ cos veψ))]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='+ ez(−kzvez) + eψ (−kψveψ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='≤ −kexvex − keyvey − kzezvez − kψeψveψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (27)  According to the definition of ve, e(t) are of the same sign  (see definition of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (10));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' k, kz, kψ are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (26) is believed to be less than and equal to  zero, which demonstrates the stability of the designed refined  backstepping controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  APPENDIX B  PROOF OF THE SMC STABILITY  To prove the stability of the SMC, construct a Lyapunov  function,  V = 1  4λsT Ms + 1  2QT Γ−1Q ,  (28)  where Q = �τr − �τest and �τr = �  M ˙vr + �Cvr + �Dv + �g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Previously we have given ev = vc − v, and s˙ev + 2λev +  λ2 �  evdt, such that two equations can be deducted as,  v = vc − s − ˙ev − λ2 �  evdt  2λ  ,  (29)  ˙v = ˙vc − ˙s − ¨ev − λ2ev  2λ  ,  (30)  therefore the following items can be defined,  vr = vc + ˙ev + λ2 �  evdt  2λ  ,  (31)  ˙vr = ˙vc + ¨ev + λ2ev  2λ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (32)  GFTC  GFTC-P  P-I  No FTC  10  Desired  8  76  N  4  72  15  10  (a)  GFTC  P-I  No FTC  GFTC-P  20  10 -  0  5  +  e  20 -  5  40-  5  10  15  0  20  5  10  15  0  20  20-  5 1  e  10  0  15-  5  10  0  15  20  0  5  10  15  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0  N  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5 -  e  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='5-  0  5  10  15  20  0  5  10  15  20  5  5  0  e  5 -  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  15  0  5  10  15  10  15  20  20  0  Time(s)  Time(s)  (b)  (c)A GOA-BASED TRAJECTORY TRACKING FTC  11  By substituting into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (2),  M ˙s  2λ + C s  2λ  = M( ˙vc + ¨ev + λ2ev  2λ  ) + C(vc + ˙ev + λ2 �  evdt  2λ  )  +Dv + g − τ = M ˙vr + Cvr + Dv + g − τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (33)  Based on previous definitions, the derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (27) can  be simplified as,  ˙V = 1  4λ(sT ˙Ms + ˙sT Ms + sT M˙s)  +1  2  ˙QT Γ−1Q + 1  2QT Γ−1 ˙Q  = 1  2λsT (M˙s + Cs) + 1  2  ˙QT Γ−1Q + 1  2QT Γ−1 ˙Q  = sT (M ˙vr + Cvr + Dv + g − τ) + ˙QT Γ−1Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (34)  By substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' (19),  ˙V = sT (M ˙vr + Cvr + Dv + g − τ)  +(˙�τr − ˙�τest)T Γ−1Q  = −sT (K1s + K2|s|rsign(s)) + (˙�τr)T Γ−1Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  (35)  The dynamic item �τr is bounded due to the slow velocity  of the underwater vehicle and sT (K1s + K2|s|rsign(s)) ≥  (˙�τr)TΓ−1Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' When K1, K2 and Γ are assigned with large  enough values at the design step, ˙V ≤ 0 can be achieved and  V is ensured to be bounded, thus leading to the conclusion  that Q is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Then design a new Lyapunov function as  V2 = 1  4λsT Ms,  (36)  whose derivative can be deducted as,  ˙V2 = sT (Q − K1s − K2|s|rsign(s)),  (37)  where 0 < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Suppose ||Q|| < a,  ˙V2 ≤ 1  2||s||2 + 1  2a − λmin(K1)||s||2 − λmin(K2)||s||1+r,  (38)  choose K1 when λmin(K1) > 1  2 + β, where β > 0,  ˙V2 ≤ −β||s||2 − λmin(K2)||s||1+r + 1  2a,  (39)  which induces that the Lyapunov function converges to a range  close to zero in a finite time and s converges to a range close to  zero in a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Therefore, the supposed condition of the  Lyapunov theorem can be regarded as satisfied, thus proving  the stability of the designed SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  REFERENCES  [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Baraniuk, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Simoni, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Weihmann, “Fault-tolerant architecture for AUVs,”  in Proceedings of 2018 IEEE/OES Autonomous Underwater Vehicle Workshop,  Porto, Portugal, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Huang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Naghdy, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Du, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Huang, “Fault tolerant steer-by-wire systems:  An overview,” Annual Reviews in Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 47, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 98–111, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [3] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Qi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Qi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Theilliol, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Han, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Song, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Hua, “A review on  fault diagnosis and fault tolerant control methods for single-rotor aerial vehicles,”  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 73, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1-4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 535–555, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Lu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Xia, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Liu, “Finite-time tracking control of rigid spacecraft  under actuator saturations and faults,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Autom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 368–381, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [5] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Mao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Jiang, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Chen, “Adaptive fault-tolerant sliding-mode  control for high-speed trains with actuator faults and uncertainties,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Transp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2449–2460, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Meyer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Johnson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' DeCarlo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Pekarek, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Sudhoff, “Hybrid  electric vehicle fault tolerant control,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Syst-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' ASME, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 140, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  1–12, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Xiong, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yu, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Shen, “Review on sensors fault diagnosis and fault-  tolerant techniques for lithium ion batteries in electric vehicles,” in Proceedings of  2018 IEEE Conference on Industrial Electronics and Applications (ICIEA), Wuhan,  China, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 406–410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Gong and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yang, “Evolutionary fault tolerance method based on virtual  reconfigurable circuit with neural network architecture,” IEEE Transactions on  Evolutionary Computation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 949–960, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yu, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Graaf, “Active fault-  tolerant control for electric vehicles with independently driven rear in-wheel motors  against certain actuator faults,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Control Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 5,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1557–1572, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [10] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Lang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Bozhko, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wheeler, “Fault  tolerant control of advanced power generation center for more-electric aircraft  applications,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Transp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Electrification, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Article in press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' DOI:  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='1109/TTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='3093506, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [11] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Tian, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Ji, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Qiu, “Fault-tolerant controller design for path  following of the autonomous vehicle under the faults in braking actuators,” IEEE  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Transp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Electrification, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2530–2540, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Seto and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Svendsen, Autonomous Underwater Vehicles: Design and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  The Institution of Engineering and Technology, 2020, ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Advanced AUV fault  management, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 419–445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Kadiyam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Parashar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Mohan, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Deshmukh, “Actuator fault-tolerant  control study of an underwater robot with four rotatable thrusters,” Ocean Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=',  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 197, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 961–979, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [14] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Malekian, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Sotelo, “Event-triggered adaptive  neural fault-tolerant control of underactuated msvs with input saturation,” IEEE  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Transp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 7045–7057, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Xu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Dong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Tao Yan, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yang, “Bio-inspired  intelligence with applications to robotics: a survey,” Intelligence & Robotics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 58–83, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Li, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhao, “Adaptive fault-tolerant stochastic shape control with  application to particle distribution control,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Man Cybern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=',  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 45, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1592–1604, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Martynova and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Rozengauz, “Approach to reconfiguration of a motion control  system for an autonomous underwater vehicle,” Gyroscopy and Navigation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 11,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 244–253, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [18] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhang, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yao, “Adaptive fault tolerant control and thruster fault  reconstruction for autonomous underwater vehicle,” Ocean Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 155, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 10  – 23, May 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [19] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Omerdic and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Roberts, “Thruster fault diagnosis and accommodation for open-  frame underwater vehicles,” Control Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Pract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1575–1598,  Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [20] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Podder, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Antonelli, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Sarkar, “Fault tolerant control of an autonomous  underwater vehicle under thruster redundancy: simulations and experiments,” in  Proceedings of 2000 IEEE International Conference on Robotics and Automation  (ICRA), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2, San Francisco, CA, USA, 2000, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1251–1256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' He, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Liu, “Cooperative fault-tolerant control for a mobile  dual flexible manipulator with output constraints,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Autom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=',  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Shen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Shi, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Buckham, “Trajectory tracking control of an autonomous  underwater vehicle using lyapunov-based model predictive control,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 65, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 5796–5805, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [23] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Kumar, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Huang, “Framework for optimal fault-tolerant control  synthesis: maximize prefault while minimize post-fault behaviors,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Man Cybern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 44, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1056–1066, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yan, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yang, “Motion planning and tracking control of un-  manned underwater vehicles: technologies, challenges and prospects,” Intelligence  & Robotics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 200–222, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [25] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Sinha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Cashman, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Cohen, “Using a genetic algorithm to optimize  configurations in a data-driven application,” in Proceedings of 12th International  Symposium on Search-Based Software Engineering, Cham, Switzerland, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  137–152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Mohanty and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Mohanty, “Genetic algorithm-based motif search problem: a  review,” in Proceedings of 2020 3rd International Conference on Smart Computing  and Informatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Smart Innovation, Systems and Technologies (SIST), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1,  Singapore, Singapore, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 719–731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [27] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Gong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Sun, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Miao, “A set-based genetic algorithm for interval many-  objective optimization problems,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Evol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  47–60, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Lai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Ibrahim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Abdullah, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Abdullah, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Suandi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Azman, “A liter-  ature review on data conversion methods on EEG for convolution neural network  applications,” in Proceedings of 2019 International Conference on Robotics, Vision,  Signal Processing and Power Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Enabling Research and Innovation  Towards Sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Lecture Notes in Electrical Engineering (LNEE), Singapore,  Singapore, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 521–527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Asmara, “A possibility of artificial neural networks to be applied in the predictive  test: a systematic literature review and study case,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1456,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 12 052–12 061, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Hulyalkar and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Khanuja, “Optimal solution generation from reviews and  micro-reviews using greedy algorithm,” in Proceedings of 2017 International  Conference on Computing, Communication, Control and Automation (ICCUBEA),  Pune, India, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [31] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Liu, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Xue, “A PSO-based energy-efficient fault-tolerant static  A GOA-BASED TRAJECTORY TRACKING FTC  12  scheduling algorithm for real-time tasks in clouds,” in Proceedings of 4th Inter-  national Conference on Computer and Communications (ICCC), Chengdu, China,  2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2537–2541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Celtek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Durdu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Ali, “Real-time traffic signal control with swarm  optimization methods,” Measurement, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 166, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 108 206–108 213, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Song, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wu, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Han, “Reinforcement learning and  particle swarm optimization supporting real-time rescue assignments for multiple  autonomous underwater vehicles,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Transp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 23, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  6807–6820, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [34] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Liu, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Hu, “Fault-tolerant control algorithm of the manned subma-  rine with multi-thruster based on quantum-behaved particle swarm optimisation,”  Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 84, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1817–1829, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [35] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Guo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Niu, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Chen, “A PSO-optimized real-time fault-  tolerant task allocation algorithm in wireless sensor networks,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Parall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Distr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 3236–3249, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [36] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Saremi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Mirjalili, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Lewis, “Grasshopper optimisation algorithm: theory  and application,” Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 105, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 30–47, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [37] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Heidari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Kuang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Chen, “Chaotic arc  adaptive grasshopper optimization,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 17 672–17 706, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Xia, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Liang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Heidari, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Tura-  bieh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Mafarja, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Pan, “Performance optimization of support vector  machine with oppositional grasshopper optimization for acute appendicitis diag-  nosis,” Computers in Biology and Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 143, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Article in press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' DOI:  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='compbiomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='105206, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Li, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Su, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yu, “Distributed trajectory  optimization for multiple solar-powered UAVs target tracking in urban environment  by adaptive grasshopper optimization algorithm,” Aerosp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 70, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  497–510, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [40] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Mishra, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Goyal, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Shukla, Advances in Intelligent Computing and  Communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Lecture Notes in Networks and Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  Springer, Singapore,  2020, ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' An Improved Grasshopper Optimization Algorithm for Solving Numerical  Optimization Problems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 179–188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [41] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Durham, “Constrained control allocation,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Guid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Control Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 4,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 717–725, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [42] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhu and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Sun, “The bio-inspired model based hybrid sliding-mode tracking  control for unmanned underwater vehicles,” Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Artif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 10,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2260–2269, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [43] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Shtessel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Foreman, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Tournes, “Stability margins in traditional and  second order sliding mode control,” in Proceedings of 2011 IEEE Conference on  Decision and Control, Orlando, FL, USA, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 4604–4609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zand, “Enhanced navigation and tether management of inspection class remotely  operated vehicles,” Master Thesis, Mechanical Engineering, University of Victoria,  Victoria, Canada, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [45] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Hu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yang, “A grasshopper optimization-based fault-  tolerant control algorithm for a human occupied submarine with the multi-thruster  system,” Ocean Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 242, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [46] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Simpson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Mccaffery, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' H ¨AGELE, “A behavioural analysis of  phase change in the desert locust,” Biological Reviews, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 74, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 461–480,  2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [47] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Rogers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Matheson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Despland, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Dodgson, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Simpson,  “Mechanosensory-induced behavioral gregarization in the desert locust schistocerca  gregaria,” Journal of Experimental Biology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 206, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Pt 22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 3991–4002,  2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [48] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Meraihi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Gabis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Mirjalili, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Ramdane-Cherif, “Grasshopper opti-  mization algorithm: theory, variants, and applications,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  50 001–50 024, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [49] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Qi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Malekian, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Sotelo, “Path following opti-  mization for an underactuated USV using smoothly-convergent deep reinforcement  learning,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Transp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 6208–6220, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [50] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Peng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Hu, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Lan, “Adaptive dynamic surface control  for formations of autonomous surface vehicles with uncertain dynamics,” IEEE  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Control Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 513–520, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content='  [51] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Gong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Yan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' Tang, “Trajectory tracking control for au-  tonomous underwater vehicles based on dual closed-loop of mpc with uncertain  dynamics,” Ocean Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
+page_content=' 265, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQf3P6k/content/2301.01827v1.pdf'}
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+Dual-View Selective Instance Segmentation Network for
+Unstained Live Adherent Cells in Differential Interference
+Contrast Images
+Fei Pan†1, Yutong Wu†2, Kangning Cui1,2, Shuxun Chen3, Yanfang Li3, Yaofang Liu1,2, Adnan
+Shakoor4, Han Zhao3, Beijia Lu2, Shaohua Zhi5, Raymond Chan1,2, and Dong Sun∗3
+1Hong Kong Centre for Cerebro-Cardiovascular Health Engineering, Hong Kong, China
+2Department of Mathematics, City University of Hong Kong, Hong Kong, China
+3Department of Biomedical Engineering, City University of Hong Kong, Hong Kong, China
+4Control and Instrumentation Department, King Fahd University of Petroleum and Minerals,
+Dhahran, Saudi Arabia
+5Department of Health Technology and Informatics, the Hong Kong Polytechnic University,
+Hong Kong, China
+January 26, 2023
+Abstract
+Despite recent advances in data-independent and deep-learning algorithms, unstained live adherent cell
+instance segmentation remains a long-standing challenge in cell image processing. Adherent cells’ inherent visual
+characteristics, such as low-contrast structures, fading edges, and irregular morphology, have made it difficult
+to distinguish from one another, even by human experts, let alone computational methods. In this study, we
+developed a novel deep-learning algorithm called dual-view selective instance segmentation network (DVSISN)
+for segmenting unstained adherent cells in differential interference contrast (DIC) images. First, we used a
+dual-view segmentation (DVS) method with pairs of original and rotated images to predict the bounding box and
+its corresponding mask for each cell instance. Second, we used a mask selection (MS) method to filter the cell
+instances predicted by the DVS to keep masks closest to the ground truth only. The developed algorithm was
+trained and validated on our dataset containing 520 images and 12198 cells. Experimental results demonstrate that
+our algorithm achieves an APsegm of 0.555, which remarkably overtakes a benchmark by a margin of 23.6%. This
+study’s success opens up a new possibility of using rotated images as input for better prediction in cell images.
+Index Terms: adherent cells, DIC images, instance segmentation
+1
+Introduction
+The cell, the fundamental unit of life, is a complex of material metabolism, energy conversion, and information
+regulation. For a typical cell, whether a bacterial or an animal cell, water accounts for about 70% of its weight,
+which causes it transparent [1]. Consequently, when such a cell is observed under a bright-field microscope, the
+contrast is very weak, leading to poor image quality. So, it is best to use a phase contrast microscope or a differential
+interference contrast (DIC) microscope to observe live cells. The former, a phase contrast microscope, reveals more
+detail of a cell’s internal structures and discerns its attachments to nearby cells. While the latter, a DIC microscope,
+provides pseudo-three-dimensional images with a shadow-cast appearance.
+In addition to these two imaging modes, fluorescence microscopy is a commonly used approach for observing
+specific macromolecules, such as proteins and nucleic acids in cells in modern biological laboratories [2]. In a
+fluorescence microscope, a short-wavelength excitation light passing through the excitation filter irradiates the
+fluorescent molecules (fluorophores) marked in the sample to generate visible light of a particular wavelength
+that can be seen by the viewer or digitally captured using a complementary metal oxide semiconductor (CMOS)
+or charge-coupled device (CCD). However, fluorescence microscopy also brings several disadvantages, such as
+∗Corresponding Author: medsun@cityu.edu.hk; † The two authors contributed equally to this work.
+1
+
+photo-bleaching and photo-toxicity, so unstained microscopy is still the most common non-invasive approach for
+observing live cells [3].
+Concurrent with progress in optics and advances in imaging, cell image processing [4] has been in increasing
+demand in biomedical research. Typical tasks in cell image processing include image classification, image
+segmentation, object tracking, and augmented microscopy [5]. Here cell detection is a primary task, aiming to
+locate each cell’s positions using a bounding box. In contrast, cell instance segmentation is a more demanding task
+that aims at detecting each instance of different cells and generates its segmentation mask, even if they are of the
+same class in an image. Inaccuracies of cell instance segmentation can bring extensive consequences for diverse
+downstream applications, such as cell culture characteristics estimation [6], cell micromanipulation [7,8], digital
+pathology [9,10], and computer-aided diagnosis (CAD) [11,12].
+Although several convolutional neural networks (CNNs) [13–17] and relevant cell image datasets [18,19] have
+been proposed recently to solve this problem under various imaging circumstances, accurate instance segmentation
+of unstained live adherent cells in DIC images—a common situation in many biomedical experiments—remains
+unsolved. For computational researchers, this is mainly due to the lack of established datasets; but for biomedical
+researchers, it is primarily due to the lack of accurate and out-of-the-box algorithms. More specifically, the difficulty
+of instance segmentation for unstained adherent cells lies in four aspects, illustrated in Figs. 1 and 2. First, adherent
+cells’ morphology and orientations are heterogeneous. Second, sporadic individual cells’ edges usually fade into
+the image background. Third, a few cells are sick or dying, exhibiting unusual features. Fourth, adherent cells often
+gather together and thus make their bordering edges indistinguishable. These characteristics pose a prohibitive
+barrier in manual annotations, let alone establishing a high-quality dataset. Both early data-independent and recent
+deep-learning algorithms are primarily centered around fixed and stained histopathological images. As such,
+this study aims to fill the gap by providing a new instance segmentation algorithm dual-view selective instance
+segmentation network (DVSISN) for unstained live adherent cells in DIC images.
+Without bells and whistles, DVSISN surpasses several major state-of-the-art (SOTA) CNNs on our dataset.
+Particularly, DVSISN achieves 0.634 in APbbox and 0.555 in APsegm, approximately 10% to 20% better than its
+counterparts. Such an improvement is made by the following two methodological innovations in this study. (1) A
+dual-view segmentation (DVS) method is proposed to take combinations of original and rotated input images to
+increase the coverage of bounding boxes. (2) An mask selection (MS) method is proposed to keep the finest masks
+in a supervised way.
+The rest of this article is organized as follows. Section 2 briefly introduces image segmentation and the
+region-based convolutional neural network (R-CNN) family. Section 3 describes our cell image dataset. Section 4
+gives details of DVSISN. Section 5 reports quantitative comparisons of DVSISN against other SOTA algorithms.
+Finally, Section 6 concludes the study and discusses future work.
+2
+Related Work
+2.1
+Image Segmentation
+Image segmentation is partitioning an image into multiple segments or components. Depending on the complexity
+of the task, image segmentation can be divided into three main categories: 1) semantic segmentation, that is,
+classifying pixels with semantic labels; 2) instance segmentation, that is, identifying and segmenting individual
+objects; and 3) panoptic segmentation, that is, unifies semantic & instance segmentation [20].
+Semantic segmentation, also called scene labeling, predicts semantic labels for each pixel in an image. It has
+been a critical task in computer vision for decades, for which researchers have developed methods ranging from
+thresholding to SOTA CNNs [21–23]. These techniques are widely used in many applications, such as autonomous
+driving [24], remote sensing [25,26], and medical image processing [27,28].
+In recent years, instance segmentation has become one of computer vision’s most critical and problematic
+directions. Other than semantic segmentation, instance segmentation identifies and segments all instances in an
+image belonging to different categories. Existing R-CNN-based instance segmentation algorithms need two stages
+in general, i.e., detecting bounding boxes that contain objects and then predicting foreground masks for each region
+of interest (RoI) [29,30]. In contrast, other one-stage approaches adopt fully convolutional models for instance
+segmentation without an explicit feature localization, such as YOLACT [31]. Instance segmentation benefits
+applications in many fields, like robotics [32] and autonomous driving [33].
+Cell instance segmentation is strongly needed in biomedical applications, for example, cell micromanipulation
+[7,8], digital pathology [9,10], and CAD [11,12]. Cell instance segmentation aims to separate each cell instance
+and predict its corresponding class in input images. Although it has been a complicated task for a long time, several
+deep learning algorithms were proposed recently to increase the accuracy and robustness [13–16].
+2
+
+2.2
+R-CNN Family
+R-CNN [34] is the first algorithm to successfully apply deep learning to object detection. It can be divided into
+three main steps: (1) extracting and wrapping region proposals from each image, (2) computing CNN features for
+each warped patch, and (3) classifying each region and deleting redundant predictions.
+Despite its breakthrough advances, R-CNN still has several drawbacks, for example, high computational cost
+and multi-stage tuning. To further improve the efficacy of R-CNN, Fast R-CNN [35] feeds the whole image into a
+CNN to extract features and uses selective search [36] to reduce repeat computations. The wrap step is replaced by
+spatial pyramid pooling [37] to avoid distortions. Besides, Fast R-CNN adopts a multi-task loss to train the softmax
+classifier and the bounding box regressor end-to-end. Later, Faster R-CNN [38] applies a region proposal network
+(RPN) to optimize the quality of region proposals with lower computational costs. Rotated Cascade R-CNN [39]
+incorporates rotated bounding boxes to detect quadrangular and curved objects efficiently.
+Later, Mask R-CNN [29] extended Faster R-CNN by adding a mask branch to achieve instance segmentation.
+Mask R-CNN uses the RPN for each input image to search RoIs as Faster R-CNN did. Then the class and box
+offset are predicted for each RoI; in parallel, a binary mask that encodes the spatial layout of the contained object is
+generated. Soon, Mask Scoring R-CNN (MS R-CNN) [40] improves the inconsistency between the classification and
+binary mask quality of Mask R-CNN by adding a network block to compute the mask score. Cascade R-CNN [41]
+uses a multi-stage architecture based on Faster R-CNN that is trained with increasing intersection over union (IoU)
+thresholds stage by stage to balance the trade-off between performance and IoU threshold setting. Recently, Rotated
+Mask R-CNN has adopted a rotated bounding box representation to enhance the performance of Mask R-CNN on
+dense objects [42].
+3
+Dataset
+(a)
+(b)
+(c)
+(d)
+Fig. 1: The morphology of Swiss 3T3 mouse fibroblasts in 4 steps in the cell-spreading process on glass coverslips.
+Cells were fixed and shown after (a) 30 minutes, (b) 60 minutes, (c) 2 hours, and (d) 24 hours of attachment.
+SOURCE: Jonathan J. Rosen and Lloyd A. Culp, Exp. Cell Res. 107:141, 1977 [43] with permission from Elsevier.
+Most cells derived from vertebrates, such as birds and mammals, except for hematopoietic cells, germ cells, and
+a few others, are adherent cells. Adherent cells, as opposed to suspension cells, are anchorage-dependent and must
+be cultured on a tissue-culture-treated substrate to allow cell adhesion and spreading, as shown in Fig. 1. From
+the perspective of morphology, adherent cells can be classified into fibroblast-like and epithelial-like cells. The
+former is bipolar or multi-polar and usually has elongated shapes, while the latter is polygonal and grows as discrete
+patches [44]. Both cells have a highly irregular morphology compared with the spherical shape of suspensions cells,
+bringing considerable difficulties for an algorithm to detect, segment, track, and analyze [3,5,45].
+3
+
+(a-1) A DIC image of sparsely dis-
+tributed adherent cells.
+(b-1) A DIC image of densely dis-
+tributed adherent cells.
+(c-1) A DIC image of unhealthy adher-
+ent cells.
+(a-2) A fluorescence image of stained
+cells in Fig. 2(a-1).
+(b-2) A fluorescence image of stained
+cells in Fig. 2(b-1).
+(c-2) A fluorescence image of stained
+cells in Fig. 2(c-1).
+(a-3) A merged image of Figs. 2(a-1)
+and 2(a-2).
+(b-3) A merged image of Figs. 2(b-1)
+and 2(b-2).
+(c-3) A merged image of Figs. 2(c-1)
+and 2(c-2).
+(a-4) Annotated cell classes of Fig. 2(a-
+1).
+(b-4) Annotated cell classes of Fig. 2(b-
+1).
+(c-4) Annotated cell classes of Fig. 2(c-
+1).
+(a-5) Annotated ground truth of
+Fig. 2(a-1).
+(b-5) Annotated ground truth of
+Fig. 2(b-1).
+(c-5) Annotated ground truth of
+Fig. 2(c-1).
+Fig. 2: Representative microscopic images of HepG2 human liver cancer cells and their annotated ground truth.
+Nevertheless, adherent cells are transparent, so can hardly be observed under a light microscope unless stained.
+Conventionally, researchers usually use DIC microscopes because they can observe delicate structures in live or
+unstained specimens and render three-dimensional images with a sense of relief. The working principle is that a DIC
+microscope converts the phase difference of the object into amplitude changes through the interference of coherent
+4
+
+fading edgesunclear edgesunhealthy cells
+living cellsbad cell
+celt
+bad cell
+bad cell
+cell
+cellceii
+cell
+ccell
+cell
+cell:ll
+cell
+celll
+cell
+cell
+cell
+cell
+cell
+cell
+celicell
+cell
+cell
+cell
+cell
+cell
+cell
+cell
+cell
+cell
+cel
+cell
+cell
+icell
+cell
+celi
+cell
+celcett
+cell
+cell
+cell
+celll
+cell
+badcell
+cell
+cell
+cell
+cell
+cellbad cell
+bad cell
+bad cell
+bad celi
+ceul
+celllight beams between which the distance is relatively small, only 1 µm or less, inside and outside the sample.
+This study used HepG2 human liver cancer cells to provide cell images. Cells were cultured in the Dulbecco’s
+modified eagle medium (DMEM) (Gibco) supplemented with 10% fetal bovine serum (FBS) (Gibco), 100 U/mL of
+penicillin, and 100 U/mL of streptomycin in a 35 mm glass-bottom Petri dish (culture dish 801002, Wuxi NEST
+Biotechnology) and placed in a humidified atmosphere of 37 °C and 5% CO2. Calcein acetoxymethyl (AM), a
+commonly used fluorescent dye, was used to test cell viability and for short-term staining. Before image collection,
+5 µL 4 mmol calcein AM (L6037S, US Everbright Inc.) was taken from the refrigerator and restored to room
+temperature. Then it was mixed with 10 mL phosphate-buffered saline (PBS) to stain the cultured cells. Because
+the calcein AM emits 530 nm fluorescence when excited by a 488 nm laser, live cells stained with the calcein AM
+look green.
+After cell staining, the dish was transferred from the incubator to the inverted fluorescence microscope (Eclipse
+Ts2R-FL, Nikon). The microscope was equipped with a motorized XY stage (ProScan H117P1N4, Prior Scientific)
+and a CMOS camera (DigiRetina 16, Tucsen Photonics). A homemade control software [7, 46] first drove the
+motorized stage to move the dish (and the cultured cells) to predefined locations to capture DIC images and then
+drove the stage again to move the dish to the same locations to capture fluorescence images. 520 pairs of DIC and
+fluorescence images were captured under a 40× objective lens (CFI S Plan Fluor ELWD 40XC 228 MRH08430,
+Nikon). All images were RGB color images and resized to 1152 pixel × 863 pixel, representing approximately
+216.500 µm × 162.375 µm in the dish. Each pair of a DIC image [Figs. 2(a-1) to 2(c-1)] and its fluorescence
+counterpart [Figs. 2(a-2) to 2(c-2)] is merged for manual annotation [Figs. 2(a-3) to 2(c-3)]. Annotated images
+[Figs. 2(a-5) to 2(c-5)] can be read by the labelme software [47].
+Adherent cells can be roughly classified into two types from the perspective of cell health: healthy (live) and
+unhealthy (dead or loosely attached) cells, as indicated in Fig. 2(c-3). Healthy adherent cells usually adhere to
+the culture surface, having irregular morphology and looking completely green in the fluorescence images once
+stained by the calcein AM. As a comparison, some unhealthy cells, for example, dead cells, can hardly be stained by
+the calcein AM and only look negligibly green. Other unhealthy cells, though, can be successfully stained by the
+calcein AM but loosely adhere to the culture surface and are not ideal candidates for typical biomedical experiments,
+such as cell microinjection.
+In addition to classifying cells by how healthy they are, they can also be classified by how densely they grow,
+as shown in Figs. 2(a-1) and 2(b-1). Sparsely distributed cells [Fig. 2(a-1)] are relatively easy to recognize, but
+densely distributed cells [Fig. 2(b-1)] are difficult to be distinguished from one another even by humans, so only by
+live cell staining [Fig. 2(b-2)], can people distinguish individual cells clearly [Figs. 2(b-3) and 2(b-5)].
+4
+Dual-View Selective Instance Segmentation Network (DVSISN)
+Since adherent cells are elongated, often tightly closed to one another, and frequently at oblique angles. A natural
+doubt is that merely a horizontal bounding box cannot capture a sloping cell without including its adjacent cells,
+thus making predicting masks harder. For example, Mask R-CNN predicts binary masks for each RoI using a
+fully convolutional network (FCN) [22] that is sufficient for segmenting scattered objects. However, a preliminary
+experiment reveals that Mask R-CNN produces duplicate predictions of RoIs and causes the predicted masks of cell
+edges to be vague. Consequently, an intuitive question is that can we apply a rotation operation of 45° on input
+images before data augmentation?
+Fig. 3 shows an overview of our instance segmentation algorithm for adherent cells, a two-part trainable CNN.
+Its first part is a DVS responsible for producing binary segmentation masks with class labels. Its second part is
+an MS in charge of removing redundant cell instances and keeping the finest ones. Details of our algorithm are
+elucidated as follows.
+4.1
+Dual-View Segmentation (DVS)
+The DVS part extends the structure of Mask R-CNN, as shown in Fig. 3 (left). First, we augment each input image
+by rotating it by 45° and then pass the two views of the image to the backbone for feature extraction. RPN is then
+applied to generate region proposals from the extracted feature maps. RoIAlign [29] is used to align the input image
+and the feature maps properly. Second, the bounding box classification & regression, and mask segmentation are
+performed in parallel to predict the class, location, and profile of each object contained in bounding boxes. Third,
+we delete masks containing more than one component since cells are simply connected.
+The DVS part generates probability distribution p = (p0, . . . , pK) over K + 1 classes (p0 for background),
+bounding box regression offsets tk ∈ R4 for k = 1, . . . , K, and a binary mask ˆy ∈ RM×N of the ground truth class
+kgt for each RoI. Each RoI is labeled with a class kgt, a bounding box offsets vector v ∈ R4, and a binary mask
+5
+
+Fig. 3: Overview of our instance segmentation algorithm for unstained live adherent cells in DIC images, DVSISN.
+It consists of two parts, a DVS part (left) and an MS part (right). The DVS part takes a pair of original and rotated
+images as input and generates unfiltered bounding boxes and masks of identified cells as output. The middle
+visualizations show the bounding boxes and masks predicted by the DVS. Masks consisting of multiple pieces (that
+are not simply connected) are removed. After that, the MS part filters cell instances with high quality and generates
+the final prediction.
+y ∈ RM×N in training using the multi-task loss as in the Mask R-CNN:
+L = Lb + Lc + Lm,
+(1)
+where Lb(k, tk, v) = 1[k ≥ 1] �4
+i=1 d(tk
+i − vi) accounts for the bounding box regression loss with
+d(x) =
+�
+0.5x2,
+if |x| < 1,
+|x| − 0.5,
+otherwise,
+(2)
+Lc(p, k) = − log pk accounts for the classification loss of predictions of cell types, and Lm is the average binary
+cross-entropy loss only defined for kgt of each RoI:
+Lm(y, ˆy) = −
+�M
+i=1
+�N
+j=1 yi,j log s(ˆyi,j) + (1 − yi,j) log(1 − s(ˆyi,j))
+MN
+.
+(3)
+4.2
+Mask Selection (MS)
+The MS part consists of a ResNet classifier [48] and a post-processing module, as shown in Fig. 3 (right) and with
+details in Fig. 4. This part is responsible for removing unwanted bounding boxes generated by the DVS part, since
+feeding a pair of an original and rotated images into the DVS almost doubles the number of predicted bounding
+boxes, as each cell is often searched twice that leads to repeat detection of cells. Additionally, since the unsupervised
+non-maximum suppression (NMS) technique can efficiently remove duplicates only when cells are scattered, we
+relax its selection criteria by increasing the IoU threshold of NMS in DVS and add a supervised selection step,
+namely MS, to keep the predicated masks that are closest to the ground truth.
+A cell mask m produced by the DVS part is assigned a label ym = 1 if it has the maximum IoU with the ground
+truth. Otherwise, the cell mask is assigned a label ym = 0. Then these constructed cell masks and their binary
+labels are used to train a ResNet equipped with a cross-entropy loss to select appropriate masks. Finally, masks
+having the largest IoU (and also over 0.7) with other masks at “each spot” are preserved to prevent redundancies.
+As such, the MS is designed as a supervised selection step to keep the best mask predictions only.
+5
+Experimental Results
+This section shows a comparison of our algorithm to the SOTA algorithms along with ablations on our dataset.
+Our dataset contains 520 images and is randomly partitioned into three parts: 312 images for training, 104 for
+6
+
+Simply
+Connected
+ResNet Model
+Mask
+Conv
+Layers
+Post-processing
+cell 1.00
+bad cell 0.9s
+bad cell mask
+Bounding box
+Regression
+cell mask
+bounding box
+class
+RPN
+RolAlign
+SoftMax
+FC
+Layers
+Backbone
+Final Prediction
+Dual-View Segmentation
+Mask SelectionFig. 4: Flow chart of the MS part. As shown on the left, masks generated by the DVS are used to crop the DIC
+images as the input of the MS. The middle part shows the structure of the ResNet-34 model that implements ResNet
+architecture in 3 parts. The first part uses 7 × 7 filters followed by a max-pooling layer to extract features. Then, 4
+convolutional blocks and identity blocks are applied to use residual information, thus avoiding gradient vanishing.
+Finally, average pooling, flattening, and fully connected layers are used to decide if a mask will be kept. Then we
+delete redundant masks at “each spot” and produce the final instance segmentation of each DIC image.
+validation, and 104 for testing. Six metrics [49] that calculate the average precision (AP) of bounding boxes and
+masks with different thresholds are used to report the performances of evaluated algorithms, as shown in Table 1.
+Most experiments were conducted on two NVIDIA 2080 Ti GPUs.
+Table 1: Average Precision Metrics for Object Detection and Instance Segmentation
+Metrics
+Meaning
+APbbox
+AP at IoU = 0.50 : 0.05 : 0.95 (primary challenge metric) for object detection, i.e., drawing bounding
+boxes of detected objects.
+APbbox
+0.50
+AP at IoU = 0.50 (PASCALa VOC metric) for object detection.
+APbbox
+0.75
+AP at IoU = 0.75 (strict metric) for object detection.
+APsegm
+AP at IoU = 0.50 : 0.05 : 0.95 (primary challenge metric) for instance segmentation, i.e., generating
+individual masks of detected objects.
+APsegm
+0.50
+AP at IoU = 0.50 (PASCAL VOCb metric) for instance segmentation.
+APsegm
+0.75
+AP at IoU = 0.75 (strict metric) for instance segmentation.
+a PASCAL stands for pattern analysis, statistical modelling and computational learning [50].
+b VOC stands for visual object classes [50].
+5.1
+Implementation Details
+We implemented our algorithm based on the MMDetection toolbox [51] with the PyTorch framework [52]. DVSISN
+is trained in two stages. First, the backbone networks of the DVS are pre-trained on the COCO dataset [49] and
+tuned on our dataset. Second, The MS part is pre-trained on the ImageNet dataset [53], fine-tuned on the cell masks
+produced by the DVS, and constructed binary labels ym described in Section 4.2.
+Data augmentation techniques, such as flipping, padding, and resizing, are used to increase training samples in
+DVS. We assign each GPU two input images and use RPN to generate RoIs. An RoI is regarded as positive if its
+IoU with a ground truth is over 0.7. Moreover, the RPN anchors are constructed by 5 aspect ratios 0.3, 0.5, 1, 2, 3,
+with a fixed scale 8, representing the length of an anchor’s shortest side. ResNet-34 is used as the backbone of MS
+with a batch size of 32. We used stochastic gradient descent (SGD) with an initial learning rate of 0.05, a weight
+decay of 0.0001, a momentum of 0.9, and 500 iterations of warm-up.
+5.2
+Quantitative Results
+The quantitative results of adherent cell instance segmentation are shown in Table 2. Mask R-CNN [29], Cascade R-
+CNN [41], Mask Scoring R-CNN [40], InstaBoost [30], and YOLACT [31] equipped with backbones ResNet-50 [48],
+ResNet-101 [48], and ResNeXt-101 [54] were used to compare against our algorithm.
+7
+
+K4
+Kept Masks
+Layer
+Batch Normalization
+Convolutional Layer
+Zero Padding
+ Pooling
+Convolutional Block
+Flattening
+Fully Connected I
+ReLU
+Block
+Redundant Masks
+..·
+Removal
+Max I
+Identity E
+Final PredictionTable 2: Quantitative Results of Adherent Cell Instance Segmentation
+Algorithm
+Backbone
+APbbox
+APbbox
+0.50
+APbbox
+0.75
+APsegm
+APsegm
+0.50
+APsegm
+0.75
+Mask R-CNN [29]
+ResNet-50
+0.375
+0.761
+0.326
+0.415
+0.753
+0.439
+ResNet-101
+0.410
+0.771
+0.389
+0.414
+0.772
+0.486
+ResNeXt-101
+0.447
+0.768
+0.468
+0.431
+0.776
+0.454
+Cascade
+R-CNN [41]
+ResNet-50
+0.455
+0.776
+0.475
+0.437
+0.778
+0.483
+ResNeXt-101
+0.459
+0.764
+0.497
+0.443
+0.778
+0.492
+Mask Scoring
+R-CNN [40]
+ResNet-50
+0.437
+0.774
+0.438
+0.449
+0.793
+0.496
+ResNeXt-101
+0.438
+0.772
+0.467
+0.440
+0.770
+0.485
+InstaBoost [30]
+ResNet-50
+0.429
+0.749
+0.436
+0.419
+0.756
+0.443
+ResNeXt-101
+0.434
+0.739
+0.462
+0.429
+0.768
+0.467
+YOLACT [31]
+ResNet-50
+0.343
+0.688
+0.291
+0.329
+0.651
+0.293
+ResNet-101
+0.351
+0.709
+0.300
+0.335
+0.674
+0.316
+DVSISN
+ResNet-50
+0.609
+0.955
+0.648
+0.549
+0.889
+0.632
+ResNet-101
+0.634
+0.968
+0.686
+0.555
+0.892
+0.647
+Bold and underlined values indicate the best and the second-best performances. Italic values are the best
+performances reported by competitive algorithms.
+It can be observed clearly that the DVSISN outperforms all counterparts in terms of all six metrics. Its APbbox is
+over 15% better than its closest counterpart (Cascade R-CNN [41]) on all tested backbones. Additionally, its APbbox
+0.50
+is 0.968, implying that almost all cells are successfully detected, leading to a substantial improvement on the more
+strict metric APbbox
+0.75. Thanks to the nearly perfect cell detection, DVSISN’s instance segmentation performs nicely;
+DVSISN wins 13% more than the second-best algorithm (Mask Scoring R-CNN [40]) even on the most critical
+metric APsegm
+0.75.
+In contrast, the APbbox and APsegm of all the other algorithms are lower than 0.5, regardless of ResNet backbone
+choices, failing our expectations at the beginning of this study. Instead, DVSISN outperforms all other algorithms in
+all six metrics by over 10%. We can conclude that adopting the DVS and MS improves the performance of DVSISN
+remarkably.
+5.3
+Qualitative Results
+The qualitative results of adherent cell instance segmentation are displayed in Fig. 5. At first glance, it seems
+that these algorithms (Mask R-CNN [29], Cascade R-CNN [41], Mask Scoring R-CNN [40], InstaBoost [30],
+YOLACT [31]) can identify individual cells relatively well, but in fact, their inference details are not satisfactory.
+For example, as shown in Figs. 5(c-1) to 5(g-1), a few titled cells were always neglected, especially when cells
+were densely distributed. However, as shown in Figs. 5(c-2) to 5(g-2), there existed fragmented mask predictions,
+implying that a cell’s mask was mispredicted even if the cell was detected correctly. Furthermore, as shown in
+Figs. 5(c-3) to 5(g-4), overlapping mask predictions can be observed, which means that the NMS technique did not
+successfully filter a few unwanted masks. Last but not least, as shown in Figs. 5(c-5) to 5(g-5), an obvious but tilted
+cell in the upper left corner was neglected even though cells in the input image Fig. 5(a-5) are not densely distributed,
+meaning that the detection accuracy of these existing algorithms still has much room for improvement. Compared
+to these SOTA algorithms, our DVSISN demonstrates remarkable improvement. It can accurately detect cells in
+both sparsely and densely distributed situations. Based on accurate detection, mask predictions can be achieved.
+5.4
+Ablation Study
+The ablation study for different backbones with and without DVS or MS is listed in Table 3. In Table 3, DVSISN† is
+equipped with DVS only, while DVSISN‡ is equipped with MS only.
+The performance of DVSISN† indicates that the DVS improves APbbox and APsegm from 3% to 5% compared
+to the Mask R-CNN. Although the APbbox
+0.50 and APsegm
+0.50 of the DVSISN† approximate those of the Mask R-CNN,
+APbbox
+0.75 and APsegm
+0.75 of the DVSISN† wins by a large margin, especially with a ResNet-50 as the backbone (around
+10%), implying that DVSISN† makes better high-quality predictions than the Mask R-CNN.
+The performance of DVSISN‡ shows that the MS can efficiently select high-quality masks in a supervised way,
+thus indirectly improving the quality of their corresponding bounding boxes. DVSISN‡ equipped with a ResNet-50
+8
+
+Input
+(a-1)
+(a-2)
+(a-3)
+(a-4)
+(a-5)
+Ground truth
+(b-1)
+(b-2)
+(b-3)
+(b-4)
+(b-5)
+Mask R-CNN
+(c-1)
+(c-2)
+(c-3)
+(c-4)
+(c-5)
+Cascade R-CNN
+(d-1)
+(d-2)
+(d-3)
+(d-4)
+(d-5)
+MS R-CNN
+(e-1)
+(e-2)
+(e-3)
+(e-4)
+(e-5)
+InstaBoost
+(f-1)
+(f-2)
+(f-3)
+(f-4)
+(f-5)
+YOLACT
+(g-1)
+(g-2)
+(g-3)
+(g-4)
+(g-5)
+DVSISN
+(h-1)
+(h-2)
+(h-3)
+(h-4)
+(h-5)
+Fig. 5: Qualitative inference results of adherent cell images. Our algorithm outperforms the other counterparts.
+backbone outperforms Mask R-CNN in both APbbox and APsegm, but the one with ResNet-101 only surpasses Mask
+R-CNN in APbbox slightly while failing in APsegm.
+9
+
+celu
+cell
+cell
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+cell
+cell
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+cell/1.00Table 3: Quantitative Results of the Ablation Study
+Algorithm
+Backbone
+DVS
+MS
+APbbox
+APbbox
+0.50
+APbbox
+0.75
+APsegm
+APsegm
+0.50
+APsegm
+0.75
+Mask R-
+CNN [29]
+ResNet-50
+✗
+✗
+0.375
+0.761
+0.326
+0.415
+0.753
+0.439
+ResNet-101
+✗
+✗
+0.410
+0.771
+0.389
+0.414
+0.772
+0.486
+DVSISN†
+ResNet-50
+✓
+✗
+0.426
+0.750
+0.448
+0.449
+0.732
+0.533
+ResNet-101
+✓
+✗
+0.450
+0.799
+0.469
+0.463
+0.794
+0.520
+DVSISN‡
+ResNet-50
+✗
+✓
+0.499
+0.761
+0.529
+0.455
+0.739
+0.513
+ResNet-101
+✗
+✓
+0.450
+0.830
+0.424
+0.413
+0.788
+0.466
+DVSISN
+ResNet-50
+✓
+✓
+0.609
+0.955
+0.648
+0.549
+0.889
+0.632
+ResNet-101
+✓
+✓
+0.634
+0.968
+0.686
+0.555
+0.892
+0.647
+ResNet-50 and ResNet-101 are used as backbones in the experiments. Mask R-CNN is used as a benchmark.
+DVSISN† only adopts the DVS. DVSISN‡ only adopts the MS. DVSISN adopts both DVS and MS.
+A full DVSISN equipped with both DVS and MS performs better than DVSISN† and DVSISN‡. It is because
+the DVS generates sufficient cell-aligning bounding boxes, and then the MS keeps only bounding boxes associated
+with well-predicted masks. In a nutshell, using DVS or MS alone brings a slight improvement, but using them
+together brings remarkable advancement.
+6
+Conclusion
+In this study, we developed a new algorithm called DVSISN for segmenting unstained live adherent cells in DIC
+images. Experimental results demonstrate that the DVSISN outperforms major SOTA algorithms by a large margin,
+approximately 10% to 20%, in terms of APbbox and APsegm. Such an advantage can be attributed to two novel
+methods—DVS and MS—that take combinations of original and rotated views as input to capture cell instances as
+much as possible and select the finest instances in a supervised way. Ablation studies further confirmed that the
+DVS could squeeze bounding boxes to better align with cell instances of various orientations, and the MS can keep
+high-quality masks that improve the AP of masks, thus indirectly improving the quality of bounding boxes. In short,
+our DVSISN is an accurate and robust algorithm for adherent cell segmentation.
+We plan to integrate the DVSISN into our cell micromanipulation system [7] to conduct intracellular deliveries
+to investigate biological and biophysical reactions [55, 56]. Meanwhile, we plan to implicitly merge the DVS
+part into the training of RPN to reduce the computational cost in training [57]. We also plan to test quadrilateral
+bounding boxes to check final performance [58].
+Acknowledgment
+This work was partially conducted by using the computational facilities, CityU Burgundy, managed and provided by
+the Computing Services Center at the City University of Hong Kong.
+References
+[1] B. Alberts, D. Bray, K. Hopkin, A. D. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Essential Cell
+Biology, 4th ed.
+New York, NY: Garland Science, Oct. 2013. 1
+[2] G. Karp, J. Iwasa, and W. Marshall, Karp’s Cell and Molecular Biology: Concepts and Experiments, 8th ed.
+Wiley, Dec. 2015. 1
+[3] T. Vicar, J. Balvan, J. Jaros, F. Jug, R. Kolar, M. Masarik, and J. Gumulec, “Cell segmentation methods for
+label-free contrast microscopy: Review and comprehensive comparison,” BMC Bioinformatics, vol. 20, p.
+360, 2019. 2, 3
+[4] J. C. Vizcarra, E. A. Burlingame, C. B. Hug, Y. Goltsev, B. S. White, D. R. Tyson, and A. Sokolov, “A
+community-based approach to image analysis of cells, tissues and tumors,” Computerized Medical Imaging
+and Graphics, vol. 95, p. 102013, Jan. 2022. 2
+10
+
+[5] E. Moen, D. Bannon, T. Kudo, W. Graf, M. Covert, and D. V. Valen, “Deep learning for cellular image
+analysis,” Nature Methods, vol. 16, no. 12, pp. 1233–1246, Dec. 2019. 2, 3
+[6] N. Jaccard, L. D. Griffin, A. Keser, R. J. Macown, A. Super, F. S. Veraitch, and N. Szita, “Automated method
+for the rapid and precise estimation of adherent cell culture characteristics from phase contrast microscopy
+images,” Biotechnology and Bioengineering, vol. 111, no. 3, pp. 504–517, Mar. 2014. 2
+[7] F. Pan, Y. Jiao, S. Chen, L. Xing, and D. Sun, “Deep learning-enhanced dual-module large-throughput
+microinjection system for adherent cells,” IEEE Transactions on Automation Science and Engineering, pp.
+1–14, 2022. 2, 5, 10
+[8] W. Hu, Y. Ma, Z. Zhan, D. Hussain, and C. Hu, “Robotic intracellular electrochemical sensing for adherent
+cells,” Cyborg and Bionic Systems, vol. 2022, Sep. 2022. 2
+[9] V. Kumar, A. K. Abbas, and J. C. Aster, Robbins & Cotran Pathologic Basis of Disease, 10th ed.
+Elsevier,
+May 2020. 2
+[10] F. Xing and L. Yang, “Robust nucleus/cell detection and segmentation in digital pathology and microscopy
+images: A comprehensive review,” IEEE Reviews in Biomedical Engineering, vol. 9, pp. 234–263, 2016. 2
+[11] J. Yanase and E. Triantaphyllou, “A systematic survey of computer-aided diagnosis in medicine: Past and
+present developments,” Expert Systems with Applications, vol. 138, p. 112821, 2019. 2
+[12] K. Doi, “Computer-aided diagnosis in medical imaging: Historical review, current status and future potential,”
+Computerized Medical Imaging and Graphics, vol. 31, no. 4-5, pp. 198–211, 2007. 2
+[13] M. Pachitariu and C. Stringer, “Cellpose 2.0: How to train your own model,” Nature Methods, pp. 1–8, Nov.
+2022. 2
+[14] N. F. Greenwald, G. Miller, E. Moen, A. Kong, A. Kagel, T. Dougherty, C. C. Fullaway, B. J. McIntosh, K. X.
+Leow, M. S. Schwartz, C. Pavelchek, S. Cui, I. Camplisson, O. Bar-Tal, J. Singh, M. Fong, G. Chaudhry,
+Z. Abraham, J. Moseley, S. Warshawsky, E. Soon, S. Greenbaum, T. Risom, T. Hollmann, S. C. Bendall,
+L. Keren, W. Graf, M. Angelo, and D. Van Valen, “Whole-cell segmentation of tissue images with human-level
+performance using large-scale data annotation and deep learning,” Nature Biotechnology, vol. 40, no. 4, pp.
+555–565, Apr. 2022. 2
+[15] J. Yi, P. Wu, M. Jiang, Q. Huang, D. J. Hoeppner, and D. N. Metaxas, “Attentive neural cell instance
+segmentation,” Medical Image Analysis, vol. 55, pp. 228–240, 2019. 2
+[16] W. Wang, D. A. Taft, Y.-J. Chen, J. Zhang, C. T. Wallace, M. Xu, S. C. Watkins, and J. Xing, “Learn to
+segment single cells with deep distance estimator and deep cell detector,” Computers in Biology and Medicine,
+vol. 108, pp. 133–141, May 2019. 2
+[17] H.-F. Tsai, J. Gajda, T. F. W. Sloan, A. Rares, and A. Q. Shen, “Usiigaci: Instance-aware cell tracking in
+stain-free phase contrast microscopy enabled by machine learning,” SoftwareX, vol. 9, pp. 230–237, Jan. 2019.
+2
+[18] C. Edlund, T. R. Jackson, N. Khalid, N. Bevan, T. Dale, A. Dengel, S. Ahmed, J. Trygg, and R. Sj¨ogren,
+“LIVECell—A large-scale dataset for label-free live cell segmentation,” Nature Methods, vol. 18, no. 9, pp.
+1038–1045, Sep. 2021. 2
+[19] J. C. Caicedo, A. Goodman, K. W. Karhohs, B. A. Cimini, J. Ackerman, M. Haghighi, C. Heng, T. Becker,
+M. Doan, C. McQuin, M. Rohban, S. Singh, and A. E. Carpenter, “Nucleus segmentation across imaging
+experiments: The 2018 Data Science Bowl,” Nature Methods, vol. 16, no. 12, pp. 1247–1253, Dec. 2019. 2
+[20] S. Minaee, Y. Y. Boykov, F. Porikli, A. J. Plaza, N. Kehtarnavaz, and D. Terzopoulos, “Image segmentation
+using deep learning: A survey,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021. 2
+[21] X. Cai, R. Chan, and T. Zeng, “A two-stage image segmentation method using a convex variant of the
+Mumford–Shah model and thresholding,” SIAM Journal on Imaging Sciences, vol. 6, no. 1, pp. 368–390,
+2013. 2
+[22] J. Long, E. Shelhamer, and T. Darrell, “Fully convolutional networks for semantic segmentation,” in
+Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2015, pp.
+3431–3440. 2, 5
+11
+
+[23] O. Ronneberger, P. Fischer, and T. Brox, “U-Net: Convolutional networks for biomedical image segmentation,”
+in Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015, ser. Lecture Notes in
+Computer Science, N. Navab, J. Hornegger, W. M. Wells, and A. F. Frangi, Eds., vol. 9351.
+Munich,
+Germany: Springer International Publishing, 2015, pp. 234–241. 2
+[24] D. Feng, C. Haase-Sch¨utz, L. Rosenbaum, H. Hertlein, C. Glaeser, F. Timm, W. Wiesbeck, and K. Dietmayer,
+“Deep multi-modal object detection and semantic segmentation for autonomous driving: Datasets, methods,
+and challenges,” IEEE Transactions on Intelligent Transportation Systems, vol. 22, no. 3, pp. 1341–1360, 2020.
+2
+[25] S. L. Polk, K. Cui, R. J. Plemmons, and J. M. Murphy, “Diffusion and volume maximization-based clustering
+of highly mixed hyperspectral images,” arXiv preprint arXiv:2203.09992, 2022. 2
+[26] S. Camalan, K. Cui, V. P. Pauca, S. Alqahtani, M. Silman, R. Chan, R. J. Plemmons, E. N. Dethier, L. E.
+Fernandez, and D. A. Lutz, “Change detection of Amazonian alluvial gold mining using deep learning and
+Sentinel-2 imagery,” Remote Sensing, vol. 14, no. 7, p. 1746, 2022. 2
+[27] M. Zhang, X. Li, M. Xu, and Q. Li, “Automated semantic segmentation of red blood cells for sickle cell
+disease,” IEEE Journal of Biomedical and Health Informatics, vol. 24, no. 11, pp. 3095–3102, 2020. 2
+[28] Y. Liu, W. Wan, X. Zhang, S. Liu, Y. Liu, H. Liu, X. Zeng, W. Wang, and Q. Zhang, “Segmentation and
+automatic identification of vasculature in coronary angiograms,” Computational and Mathematical Methods
+in Medicine, vol. 2021, 2021. 2
+[29] K. He, G. Gkioxari, P. Doll´ar, and R. Girshick, “Mask R-CNN,” in Proceedings of the IEEE/CVF Conference
+on Computer Vision and Pattern Recognition (CVPR), 2017, pp. 2961–2969. 2, 3, 5, 7, 8, 10
+[30] H.-S. Fang, J. Sun, R. Wang, M. Gou, Y.-L. Li, and C. Lu, “InstaBoost: Boosting instance segmentation
+via probability map guided copy-pasting,” in Proceedings of the IEEE/CVF International Conference on
+Computer Vision (ICCV), 2019, pp. 682–691. 2, 7, 8
+[31] D. Bolya, C. Zhou, F. Xiao, and Y. J. Lee, “YOLACT: Real-time instance segmentation,” in Proceedings of
+the IEEE/CVF International Conference on Computer Vision (ICCV), 2019, pp. 9157–9166. 2, 7, 8
+[32] C. Xie, Y. Xiang, A. Mousavian, and D. Fox, “Unseen object instance segmentation for robotic environments,”
+IEEE Transactions on Robotics, vol. 37, no. 5, pp. 1343–1359, 2021. 2
+[33] D. Zhou, J. Fang, X. Song, L. Liu, J. Yin, Y. Dai, H. Li, and R. Yang, “Joint 3D instance segmentation and
+object detection for autonomous driving,” in Proceedings of the IEEE/CVF Conference on Computer Vision
+and Pattern Recognition (CVPR), 2020, pp. 1839–1849. 2
+[34] R. Girshick, J. Donahue, T. Darrell, and J. Malik, “Rich feature hierarchies for accurate object detection
+and semantic segmentation,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern
+Recognition (CVPR), 2014. 3
+[35] R. Girshick, “Fast R-CNN,” in Proceedings of the IEEE/CVF International Conference on Computer Vision
+(ICCV), 2015. 3
+[36] J. R. Uijlings, K. E. Van De Sande, T. Gevers, and A. W. Smeulders, “Selective search for object recognition,”
+International Journal of Computer Vision, vol. 104, no. 2, pp. 154–171, 2013. 3
+[37] K. He, X. Zhang, S. Ren, and J. Sun, “Spatial pyramid pooling in deep convolutional networks for visual
+recognition,” in Proceedings of the European Conference on Computer Vision (ECCV), ser. Lecture Notes in
+Computer Science, D. Fleet, T. Pajdla, B. Schiele, and T. Tuytelaars, Eds.
+Zurich, Switzerland: Springer
+International Publishing, Sep. 2014, pp. 346–361. 3
+[38] S. Ren, K. He, R. Girshick, and J. Sun, “Faster R-CNN: Towards real-time object detection with region
+proposal networks,” Advances in Neural Information Processing Systems, vol. 28, 2015. 3
+[39] Y. Zhu, C. Ma, and J. Du, “Rotated cascade R-CNN: A shape robust detector with coordinate regression,”
+Pattern Recognition, vol. 96, p. 106964, 2019. 3
+[40] Z. Huang, L. Huang, Y. Gong, C. Huang, and X. Wang, “Mask scoring R-CNN,” in Proceedings of the
+IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019, pp. 6409–6418. 3, 7, 8
+12
+
+[41] Z. Cai and N. Vasconcelos, “Cascade R-CNN: Delving into high quality object detection,” in Proceedings of
+the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), Salt Lake City, Utah, USA,
+2018, pp. 6154–6162. 3, 7, 8
+[42] S. Looi, “Rotated mask R-CNN: From bounding boxes to rotated bounding boxes,” 2019. 3
+[43] J. J. Rosen and L. A. Culp, “Morphology and cellular origins of substrate-attached material from mouse
+fibroblasts,” Experimental Cell Research, vol. 107, no. 1, pp. 139–149, 1977. 3
+[44] R. I. Freshney, Culture of Animal Cells: A Manual of Basic Technique and Specialized Applications, 7th ed.
+Hoboken, NJ: Wiley-Blackwell, 2016. 3
+[45] E. Meijering, “Cell segmentation: 50 years down the road,” IEEE Signal Processing Magazine, vol. 29, no. 5,
+pp. 140–145, 2012. 3
+[46] F. Pan, S. Chen, Y. Jiao, Z. Guan, A. Shakoor, and D. Sun, “Automated high-productivity microinjection
+system for adherent cells,” IEEE Robotics and Automation Letters, vol. 5, no. 2, pp. 1167–1174, 2020. 5
+[47] K. Wada, “Labelme: Image polygonal annotation with Python,” 2016. 5
+[48] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in Proceedings of the
+IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2016, pp. 770–778. 6, 7
+[49] T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Doll´ar, and C. L. Zitnick, “Microsoft
+COCO: Common objects in context,” in Proceedings of the European Conference on Computer Vision (ECCV).
+Springer, 2014, pp. 740–755. 7
+[50] M. Everingham, S. M. A. Eslami, L. Van Gool, C. K. I. Williams, J. Winn, and A. Zisserman, “The PASCAL
+visual object classes challenge: A retrospective,” International Journal of Computer Vision, vol. 111, no. 1,
+pp. 98–136, Jan. 2015. 7
+[51] K. Chen, J. Wang, J. Pang, Y. Cao, Y. Xiong, X. Li, S. Sun, W. Feng, Z. Liu, J. Xu, Z. Zhang, D. Cheng,
+C. Zhu, T. Cheng, Q. Zhao, B. Li, X. Lu, R. Zhu, Y. Wu, J. Dai, J. Wang, J. Shi, W. Ouyang, C. C. Loy, and
+D. Lin, “MMDetection: Open MMLab detection toolbox and benchmark,” arXiv preprint arXiv:1906.07155,
+2019. 7
+[52] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga,
+A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai,
+and S. Chintala, “PyTorch: An imperative style, high-performance deep learning library,” in Advances in
+Neural Information Processing Systems, vol. 32.
+Curran Associates, Inc., 2019. 7
+[53] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and F.-F. Li, “ImageNet: A large-scale hierarchical image
+database,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR),
+Miami, Florida, USA, Jun. 2009, pp. 248–255. 7
+[54] S. Xie, R. Girshick, P. Doll´ar, Z. Tu, and K. He, “Aggregated residual transformations for deep neural networks,”
+in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2017, pp.
+1492–1500. 7
+[55] V. I. P. Keizer, S. Grosse-Holz, M. Woringer, L. Zambon, K. Aizel, M. Bongaerts, F. Delille, L. Kolar-Znika,
+V. F. Scolari, S. Hoffmann, E. J. Banigan, L. A. Mirny, M. Dahan, D. Fachinetti, and A. Coulon, “Live-cell
+micromanipulation of a genomic locus reveals interphase chromatin mechanics,” Science, vol. 377, no. 6605,
+pp. 489–495, Jul. 2022. 10
+[56] M. P. Stewart, A. Sharei, X. Ding, G. Sahay, R. Langer, and K. F. Jensen, “In Vitro and Ex Vivo strategies for
+intracellular delivery,” Nature, vol. 538, no. 7624, pp. 183–192, Oct. 2016. 10
+[57] X. Xie, G. Cheng, J. Wang, X. Yao, and J. Han, “Oriented R-CNN for object detection,” in Proceedings of the
+IEEE/CVF International Conference on Computer Vision (ICCV), Oct. 2021, pp. 3520–3529. 10
+[58] Y. Zhou, X. Yang, G. Zhang, J. Wang, Y. Liu, L. Hou, X. Jiang, X. Liu, J. Yan, C. Lyu, W. Zhang, and
+K. Chen, “MMRotate: A rotated object detection benchmark using PyTorch,” in Proceedings of the 30th ACM
+International Conference on Multimedia, ser. MM ’22.
+New York, NY, USA: Association for Computing
+Machinery, Oct. 2022, pp. 7331–7334. 10
+13
+
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+page_content=' Shuxun Chen3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Yanfang Li3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
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+page_content=' Adnan  Shakoor4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Han Zhao3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Beijia Lu2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Shaohua Zhi5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Raymond Chan1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' and Dong Sun∗3  1Hong Kong Centre for Cerebro-Cardiovascular Health Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' China  2Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' City University of Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' China  3Department of Biomedical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' City University of Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' China  4Control and Instrumentation Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' King Fahd University of Petroleum and Minerals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Dhahran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Saudi Arabia  5Department of Health Technology and Informatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' the Hong Kong Polytechnic University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' China  January 26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2023  Abstract  Despite recent advances in data-independent and deep-learning algorithms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' unstained live adherent cell  instance segmentation remains a long-standing challenge in cell image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Adherent cells’ inherent visual  characteristics, such as low-contrast structures, fading edges, and irregular morphology, have made it difficult  to distinguish from one another, even by human experts, let alone computational methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' In this study, we  developed a novel deep-learning algorithm called dual-view selective instance segmentation network (DVSISN)  for segmenting unstained adherent cells in differential interference contrast (DIC) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' First, we used a  dual-view segmentation (DVS) method with pairs of original and rotated images to predict the bounding box and  its corresponding mask for each cell instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Second, we used a mask selection (MS) method to filter the cell  instances predicted by the DVS to keep masks closest to the ground truth only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' The developed algorithm was  trained and validated on our dataset containing 520 images and 12198 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Experimental results demonstrate that  our algorithm achieves an APsegm of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='555, which remarkably overtakes a benchmark by a margin of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' This  study’s success opens up a new possibility of using rotated images as input for better prediction in cell images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Index Terms: adherent cells, DIC images, instance segmentation  1  Introduction  The cell, the fundamental unit of life, is a complex of material metabolism, energy conversion, and information  regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' For a typical cell, whether a bacterial or an animal cell, water accounts for about 70% of its weight,  which causes it transparent [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Consequently, when such a cell is observed under a bright-field microscope, the  contrast is very weak, leading to poor image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' So, it is best to use a phase contrast microscope or a differential  interference contrast (DIC) microscope to observe live cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' The former, a phase contrast microscope, reveals more  detail of a cell’s internal structures and discerns its attachments to nearby cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' While the latter, a DIC microscope,  provides pseudo-three-dimensional images with a shadow-cast appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  In addition to these two imaging modes, fluorescence microscopy is a commonly used approach for observing  specific macromolecules, such as proteins and nucleic acids in cells in modern biological laboratories [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' In a  fluorescence microscope, a short-wavelength excitation light passing through the excitation filter irradiates the  fluorescent molecules (fluorophores) marked in the sample to generate visible light of a particular wavelength  that can be seen by the viewer or digitally captured using a complementary metal oxide semiconductor (CMOS)  or charge-coupled device (CCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' However, fluorescence microscopy also brings several disadvantages, such as  ∗Corresponding Author: medsun@cityu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' † The two authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  1  photo-bleaching and photo-toxicity, so unstained microscopy is still the most common non-invasive approach for  observing live cells [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Concurrent with progress in optics and advances in imaging, cell image processing [4] has been in increasing  demand in biomedical research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Typical tasks in cell image processing include image classification, image  segmentation, object tracking, and augmented microscopy [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Here cell detection is a primary task, aiming to  locate each cell’s positions using a bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' In contrast, cell instance segmentation is a more demanding task  that aims at detecting each instance of different cells and generates its segmentation mask, even if they are of the  same class in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Inaccuracies of cell instance segmentation can bring extensive consequences for diverse  downstream applications, such as cell culture characteristics estimation [6], cell micromanipulation [7,8], digital  pathology [9,10], and computer-aided diagnosis (CAD) [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Although several convolutional neural networks (CNNs) [13–17] and relevant cell image datasets [18,19] have  been proposed recently to solve this problem under various imaging circumstances, accurate instance segmentation  of unstained live adherent cells in DIC images—a common situation in many biomedical experiments—remains  unsolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' For computational researchers, this is mainly due to the lack of established datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' but for biomedical  researchers, it is primarily due to the lack of accurate and out-of-the-box algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' More specifically, the difficulty  of instance segmentation for unstained adherent cells lies in four aspects, illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' First, adherent  cells’ morphology and orientations are heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Second, sporadic individual cells’ edges usually fade into  the image background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Third, a few cells are sick or dying, exhibiting unusual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Fourth, adherent cells often  gather together and thus make their bordering edges indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' These characteristics pose a prohibitive  barrier in manual annotations, let alone establishing a high-quality dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Both early data-independent and recent  deep-learning algorithms are primarily centered around fixed and stained histopathological images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' As such,  this study aims to fill the gap by providing a new instance segmentation algorithm dual-view selective instance  segmentation network (DVSISN) for unstained live adherent cells in DIC images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Without bells and whistles, DVSISN surpasses several major state-of-the-art (SOTA) CNNs on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Particularly, DVSISN achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='634 in APbbox and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='555 in APsegm, approximately 10% to 20% better than its  counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Such an improvement is made by the following two methodological innovations in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' (1) A  dual-view segmentation (DVS) method is proposed to take combinations of original and rotated input images to  increase the coverage of bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' (2) An mask selection (MS) method is proposed to keep the finest masks  in a supervised way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  The rest of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Section 2 briefly introduces image segmentation and the  region-based convolutional neural network (R-CNN) family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Section 3 describes our cell image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Section 4  gives details of DVSISN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Section 5 reports quantitative comparisons of DVSISN against other SOTA algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Finally, Section 6 concludes the study and discusses future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  2  Related Work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='1  Image Segmentation  Image segmentation is partitioning an image into multiple segments or components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Depending on the complexity  of the task, image segmentation can be divided into three main categories: 1) semantic segmentation, that is,  classifying pixels with semantic labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2) instance segmentation, that is, identifying and segmenting individual  objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' and 3) panoptic segmentation, that is, unifies semantic & instance segmentation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Semantic segmentation, also called scene labeling, predicts semantic labels for each pixel in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' It has  been a critical task in computer vision for decades, for which researchers have developed methods ranging from  thresholding to SOTA CNNs [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' These techniques are widely used in many applications, such as autonomous  driving [24], remote sensing [25,26], and medical image processing [27,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  In recent years, instance segmentation has become one of computer vision’s most critical and problematic  directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Other than semantic segmentation, instance segmentation identifies and segments all instances in an  image belonging to different categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Existing R-CNN-based instance segmentation algorithms need two stages  in general, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=', detecting bounding boxes that contain objects and then predicting foreground masks for each region  of interest (RoI) [29,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' In contrast, other one-stage approaches adopt fully convolutional models for instance  segmentation without an explicit feature localization, such as YOLACT [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Instance segmentation benefits  applications in many fields, like robotics [32] and autonomous driving [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Cell instance segmentation is strongly needed in biomedical applications, for example, cell micromanipulation  [7,8], digital pathology [9,10], and CAD [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cell instance segmentation aims to separate each cell instance  and predict its corresponding class in input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Although it has been a complicated task for a long time, several  deep learning algorithms were proposed recently to increase the accuracy and robustness [13–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='2  R-CNN Family  R-CNN [34] is the first algorithm to successfully apply deep learning to object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' It can be divided into  three main steps: (1) extracting and wrapping region proposals from each image, (2) computing CNN features for  each warped patch, and (3) classifying each region and deleting redundant predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Despite its breakthrough advances, R-CNN still has several drawbacks, for example, high computational cost  and multi-stage tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' To further improve the efficacy of R-CNN, Fast R-CNN [35] feeds the whole image into a  CNN to extract features and uses selective search [36] to reduce repeat computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' The wrap step is replaced by  spatial pyramid pooling [37] to avoid distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Besides, Fast R-CNN adopts a multi-task loss to train the softmax  classifier and the bounding box regressor end-to-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Later, Faster R-CNN [38] applies a region proposal network  (RPN) to optimize the quality of region proposals with lower computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Rotated Cascade R-CNN [39]  incorporates rotated bounding boxes to detect quadrangular and curved objects efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Later, Mask R-CNN [29] extended Faster R-CNN by adding a mask branch to achieve instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Mask R-CNN uses the RPN for each input image to search RoIs as Faster R-CNN did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Then the class and box  offset are predicted for each RoI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' in parallel, a binary mask that encodes the spatial layout of the contained object is  generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Soon, Mask Scoring R-CNN (MS R-CNN) [40] improves the inconsistency between the classification and  binary mask quality of Mask R-CNN by adding a network block to compute the mask score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cascade R-CNN [41]  uses a multi-stage architecture based on Faster R-CNN that is trained with increasing intersection over union (IoU)  thresholds stage by stage to balance the trade-off between performance and IoU threshold setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Recently, Rotated  Mask R-CNN has adopted a rotated bounding box representation to enhance the performance of Mask R-CNN on  dense objects [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  3  Dataset  (a)  (b)  (c)  (d)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1: The morphology of Swiss 3T3 mouse fibroblasts in 4 steps in the cell-spreading process on glass coverslips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Cells were fixed and shown after (a) 30 minutes, (b) 60 minutes, (c) 2 hours, and (d) 24 hours of attachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  SOURCE: Jonathan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Rosen and Lloyd A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Culp, Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cell Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 107:141, 1977 [43] with permission from Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Most cells derived from vertebrates, such as birds and mammals, except for hematopoietic cells, germ cells, and  a few others, are adherent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Adherent cells, as opposed to suspension cells, are anchorage-dependent and must  be cultured on a tissue-culture-treated substrate to allow cell adhesion and spreading, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' From  the perspective of morphology, adherent cells can be classified into fibroblast-like and epithelial-like cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' The  former is bipolar or multi-polar and usually has elongated shapes, while the latter is polygonal and grows as discrete  patches [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Both cells have a highly irregular morphology compared with the spherical shape of suspensions cells,  bringing considerable difficulties for an algorithm to detect, segment, track, and analyze [3,5,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  3  (a-1) A DIC image of sparsely dis-  tributed adherent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  (b-1) A DIC image of densely dis-  tributed adherent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  (c-1) A DIC image of unhealthy adher-  ent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  (a-2) A fluorescence image of stained  cells in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(a-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  (b-2) A fluorescence image of stained  cells in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(b-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  (c-2) A fluorescence image of stained  cells in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
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+page_content='  (a-3) A merged image of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(a-1)  and 2(a-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  (b-3) A merged image of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(b-1)  and 2(b-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  (c-3) A merged image of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(c-1)  and 2(c-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  (a-4) Annotated cell classes of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(a-  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
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+page_content=' 2(b-  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
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+page_content='  (a-5) Annotated ground truth of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(a-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  (b-5) Annotated ground truth of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(b-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  (c-5) Annotated ground truth of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(c-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2: Representative microscopic images of HepG2 human liver cancer cells and their annotated ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Nevertheless, adherent cells are transparent, so can hardly be observed under a light microscope unless stained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Conventionally, researchers usually use DIC microscopes because they can observe delicate structures in live or  unstained specimens and render three-dimensional images with a sense of relief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' The working principle is that a DIC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
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+page_content='celllight beams between which the distance is relatively small,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' only 1 µm or less,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' inside and outside the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  This study used HepG2 human liver cancer cells to provide cell images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cells were cultured in the Dulbecco’s  modified eagle medium (DMEM) (Gibco) supplemented with 10% fetal bovine serum (FBS) (Gibco), 100 U/mL of  penicillin, and 100 U/mL of streptomycin in a 35 mm glass-bottom Petri dish (culture dish 801002, Wuxi NEST  Biotechnology) and placed in a humidified atmosphere of 37 °C and 5% CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Calcein acetoxymethyl (AM), a  commonly used fluorescent dye, was used to test cell viability and for short-term staining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Before image collection,  5 µL 4 mmol calcein AM (L6037S, US Everbright Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=') was taken from the refrigerator and restored to room  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Then it was mixed with 10 mL phosphate-buffered saline (PBS) to stain the cultured cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Because  the calcein AM emits 530 nm fluorescence when excited by a 488 nm laser, live cells stained with the calcein AM  look green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  After cell staining, the dish was transferred from the incubator to the inverted fluorescence microscope (Eclipse  Ts2R-FL, Nikon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' The microscope was equipped with a motorized XY stage (ProScan H117P1N4, Prior Scientific)  and a CMOS camera (DigiRetina 16, Tucsen Photonics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' A homemade control software [7, 46] first drove the  motorized stage to move the dish (and the cultured cells) to predefined locations to capture DIC images and then  drove the stage again to move the dish to the same locations to capture fluorescence images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 520 pairs of DIC and  fluorescence images were captured under a 40× objective lens (CFI S Plan Fluor ELWD 40XC 228 MRH08430,  Nikon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' All images were RGB color images and resized to 1152 pixel × 863 pixel, representing approximately  216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='500 µm × 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='375 µm in the dish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Each pair of a DIC image [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(a-1) to 2(c-1)] and its fluorescence  counterpart [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(a-2) to 2(c-2)] is merged for manual annotation [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(a-3) to 2(c-3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Annotated images  [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(a-5) to 2(c-5)] can be read by the labelme software [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Adherent cells can be roughly classified into two types from the perspective of cell health: healthy (live) and  unhealthy (dead or loosely attached) cells, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(c-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Healthy adherent cells usually adhere to  the culture surface, having irregular morphology and looking completely green in the fluorescence images once  stained by the calcein AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' As a comparison, some unhealthy cells, for example, dead cells, can hardly be stained by  the calcein AM and only look negligibly green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Other unhealthy cells, though, can be successfully stained by the  calcein AM but loosely adhere to the culture surface and are not ideal candidates for typical biomedical experiments,  such as cell microinjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  In addition to classifying cells by how healthy they are, they can also be classified by how densely they grow,  as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(a-1) and 2(b-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sparsely distributed cells [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(a-1)] are relatively easy to recognize, but  densely distributed cells [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(b-1)] are difficult to be distinguished from one another even by humans, so only by  live cell staining [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(b-2)], can people distinguish individual cells clearly [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2(b-3) and 2(b-5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  4  Dual-View Selective Instance Segmentation Network (DVSISN)  Since adherent cells are elongated, often tightly closed to one another, and frequently at oblique angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' A natural  doubt is that merely a horizontal bounding box cannot capture a sloping cell without including its adjacent cells,  thus making predicting masks harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' For example, Mask R-CNN predicts binary masks for each RoI using a  fully convolutional network (FCN) [22] that is sufficient for segmenting scattered objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' However, a preliminary  experiment reveals that Mask R-CNN produces duplicate predictions of RoIs and causes the predicted masks of cell  edges to be vague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Consequently, an intuitive question is that can we apply a rotation operation of 45° on input  images before data augmentation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3 shows an overview of our instance segmentation algorithm for adherent cells, a two-part trainable CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Its first part is a DVS responsible for producing binary segmentation masks with class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Its second part is  an MS in charge of removing redundant cell instances and keeping the finest ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Details of our algorithm are  elucidated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='1  Dual-View Segmentation (DVS)  The DVS part extends the structure of Mask R-CNN, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' First, we augment each input image  by rotating it by 45° and then pass the two views of the image to the backbone for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' RPN is then  applied to generate region proposals from the extracted feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' RoIAlign [29] is used to align the input image  and the feature maps properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Second, the bounding box classification & regression, and mask segmentation are  performed in parallel to predict the class, location, and profile of each object contained in bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Third,  we delete masks containing more than one component since cells are simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  The DVS part generates probability distribution p = (p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' , pK) over K + 1 classes (p0 for background),  bounding box regression offsets tk ∈ R4 for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' , K, and a binary mask ˆy ∈ RM×N of the ground truth class  kgt for each RoI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Each RoI is labeled with a class kgt, a bounding box offsets vector v ∈ R4, and a binary mask  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3: Overview of our instance segmentation algorithm for unstained live adherent cells in DIC images, DVSISN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  It consists of two parts, a DVS part (left) and an MS part (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' The DVS part takes a pair of original and rotated  images as input and generates unfiltered bounding boxes and masks of identified cells as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' The middle  visualizations show the bounding boxes and masks predicted by the DVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Masks consisting of multiple pieces (that  are not simply connected) are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' After that, the MS part filters cell instances with high quality and generates  the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  y ∈ RM×N in training using the multi-task loss as in the Mask R-CNN:  L = Lb + Lc + Lm,  (1)  where Lb(k, tk, v) = 1[k ≥ 1] �4  i=1 d(tk  i − vi) accounts for the bounding box regression loss with  d(x) =  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='5x2,  if |x| < 1,  |x| − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='5,  otherwise,  (2)  Lc(p, k) = − log pk accounts for the classification loss of predictions of cell types, and Lm is the average binary  cross-entropy loss only defined for kgt of each RoI:  Lm(y, ˆy) = −  �M  i=1  �N  j=1 yi,j log s(ˆyi,j) + (1 − yi,j) log(1 − s(ˆyi,j))  MN  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  (3)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='2  Mask Selection (MS)  The MS part consists of a ResNet classifier [48] and a post-processing module, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3 (right) and with  details in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' This part is responsible for removing unwanted bounding boxes generated by the DVS part, since  feeding a pair of an original and rotated images into the DVS almost doubles the number of predicted bounding  boxes, as each cell is often searched twice that leads to repeat detection of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Additionally, since the unsupervised  non-maximum suppression (NMS) technique can efficiently remove duplicates only when cells are scattered, we  relax its selection criteria by increasing the IoU threshold of NMS in DVS and add a supervised selection step,  namely MS, to keep the predicated masks that are closest to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  A cell mask m produced by the DVS part is assigned a label ym = 1 if it has the maximum IoU with the ground  truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Otherwise, the cell mask is assigned a label ym = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Then these constructed cell masks and their binary  labels are used to train a ResNet equipped with a cross-entropy loss to select appropriate masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Finally, masks  having the largest IoU (and also over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='7) with other masks at “each spot” are preserved to prevent redundancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  As such, the MS is designed as a supervised selection step to keep the best mask predictions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  5  Experimental Results  This section shows a comparison of our algorithm to the SOTA algorithms along with ablations on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Our dataset contains 520 images and is randomly partitioned into three parts: 312 images for training, 104 for  6  Simply  Connected  ResNet Model  Mask  Conv  Layers  Post-processing  cell 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='00  bad cell 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='9s  bad cell mask  Bounding box  Regression  cell mask  bounding box  class  RPN  RolAlign  SoftMax  FC  Layers  Backbone  Final Prediction  Dual-View Segmentation  Mask SelectionFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 4: Flow chart of the MS part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' As shown on the left, masks generated by the DVS are used to crop the DIC  images as the input of the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' The middle part shows the structure of the ResNet-34 model that implements ResNet  architecture in 3 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' The first part uses 7 × 7 filters followed by a max-pooling layer to extract features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Then, 4  convolutional blocks and identity blocks are applied to use residual information, thus avoiding gradient vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Finally, average pooling, flattening, and fully connected layers are used to decide if a mask will be kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Then we  delete redundant masks at “each spot” and produce the final instance segmentation of each DIC image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  validation, and 104 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Six metrics [49] that calculate the average precision (AP) of bounding boxes and  masks with different thresholds are used to report the performances of evaluated algorithms, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Most experiments were conducted on two NVIDIA 2080 Ti GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Table 1: Average Precision Metrics for Object Detection and Instance Segmentation  Metrics  Meaning  APbbox  AP at IoU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='50 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='05 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='95 (primary challenge metric) for object detection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=', drawing bounding  boxes of detected objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  APbbox  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='50  AP at IoU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='50 (PASCALa VOC metric) for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  APbbox  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='75  AP at IoU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='75 (strict metric) for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  APsegm  AP at IoU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='50 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='05 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='95 (primary challenge metric) for instance segmentation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=', generating  individual masks of detected objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  APsegm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='50  AP at IoU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='50 (PASCAL VOCb metric) for instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  APsegm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='75  AP at IoU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='75 (strict metric) for instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  a PASCAL stands for pattern analysis, statistical modelling and computational learning [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  b VOC stands for visual object classes [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='1  Implementation Details  We implemented our algorithm based on the MMDetection toolbox [51] with the PyTorch framework [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' DVSISN  is trained in two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' First, the backbone networks of the DVS are pre-trained on the COCO dataset [49] and  tuned on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Second, The MS part is pre-trained on the ImageNet dataset [53], fine-tuned on the cell masks  produced by the DVS, and constructed binary labels ym described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Data augmentation techniques, such as flipping, padding, and resizing, are used to increase training samples in  DVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' We assign each GPU two input images and use RPN to generate RoIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' An RoI is regarded as positive if its  IoU with a ground truth is over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Moreover, the RPN anchors are constructed by 5 aspect ratios 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
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+page_content='5, 1, 2, 3,  with a fixed scale 8, representing the length of an anchor’s shortest side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' ResNet-34 is used as the backbone of MS  with a batch size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' We used stochastic gradient descent (SGD) with an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='05, a weight  decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
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+page_content='9, and 500 iterations of warm-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='2  Quantitative Results  The quantitative results of adherent cell instance segmentation are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Mask R-CNN [29], Cascade R-  CNN [41], Mask Scoring R-CNN [40], InstaBoost [30], and YOLACT [31] equipped with backbones ResNet-50 [48],  ResNet-101 [48], and ResNeXt-101 [54] were used to compare against our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  7  K4  Kept Masks  Layer  Batch Normalization  Convolutional Layer  Zero Padding  Pooling  Convolutional Block  Flattening  Fully Connected I  ReLU  Block  Redundant Masks  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='.·  Removal  Max I  Identity E  Final PredictionTable 2: Quantitative Results of Adherent Cell Instance Segmentation  Algorithm  Backbone  APbbox  APbbox  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
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+page_content='647  Bold and underlined values indicate the best and the second-best performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Italic values are the best  performances reported by competitive algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  It can be observed clearly that the DVSISN outperforms all counterparts in terms of all six metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Its APbbox is  over 15% better than its closest counterpart (Cascade R-CNN [41]) on all tested backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Additionally, its APbbox  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='50  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='968, implying that almost all cells are successfully detected, leading to a substantial improvement on the more  strict metric APbbox  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Thanks to the nearly perfect cell detection, DVSISN’s instance segmentation performs nicely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  DVSISN wins 13% more than the second-best algorithm (Mask Scoring R-CNN [40]) even on the most critical  metric APsegm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
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+page_content='  In contrast, the APbbox and APsegm of all the other algorithms are lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='5, regardless of ResNet backbone  choices, failing our expectations at the beginning of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Instead, DVSISN outperforms all other algorithms in  all six metrics by over 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' We can conclude that adopting the DVS and MS improves the performance of DVSISN  remarkably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='3  Qualitative Results  The qualitative results of adherent cell instance segmentation are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' At first glance, it seems  that these algorithms (Mask R-CNN [29], Cascade R-CNN [41], Mask Scoring R-CNN [40], InstaBoost [30],  YOLACT [31]) can identify individual cells relatively well, but in fact, their inference details are not satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  For example, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 5(c-1) to 5(g-1), a few titled cells were always neglected, especially when cells  were densely distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' However, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 5(c-2) to 5(g-2), there existed fragmented mask predictions,  implying that a cell’s mask was mispredicted even if the cell was detected correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Furthermore, as shown in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 5(c-3) to 5(g-4), overlapping mask predictions can be observed, which means that the NMS technique did not  successfully filter a few unwanted masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Last but not least, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 5(c-5) to 5(g-5), an obvious but tilted  cell in the upper left corner was neglected even though cells in the input image Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 5(a-5) are not densely distributed,  meaning that the detection accuracy of these existing algorithms still has much room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Compared  to these SOTA algorithms, our DVSISN demonstrates remarkable improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' It can accurately detect cells in  both sparsely and densely distributed situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Based on accurate detection, mask predictions can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='4  Ablation Study  The ablation study for different backbones with and without DVS or MS is listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' In Table 3, DVSISN† is  equipped with DVS only, while DVSISN‡ is equipped with MS only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  The performance of DVSISN† indicates that the DVS improves APbbox and APsegm from 3% to 5% compared  to the Mask R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Although the APbbox  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='50 and APsegm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='50 of the DVSISN† approximate those of the Mask R-CNN,  APbbox  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='75 and APsegm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='75 of the DVSISN† wins by a large margin, especially with a ResNet-50 as the backbone (around  10%), implying that DVSISN† makes better high-quality predictions than the Mask R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  The performance of DVSISN‡ shows that the MS can efficiently select high-quality masks in a supervised way,  thus indirectly improving the quality of their corresponding bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' DVSISN‡ equipped with a ResNet-50  8  Input  (a-1)  (a-2)  (a-3)  (a-4)  (a-5)  Ground truth  (b-1)  (b-2)  (b-3)  (b-4)  (b-5)  Mask R-CNN  (c-1)  (c-2)  (c-3)  (c-4)  (c-5)  Cascade R-CNN  (d-1)  (d-2)  (d-3)  (d-4)  (d-5)  MS R-CNN  (e-1)  (e-2)  (e-3)  (e-4)  (e-5)  InstaBoost  (f-1)  (f-2)  (f-3)  (f-4)  (f-5)  YOLACT  (g-1)  (g-2)  (g-3)  (g-4)  (g-5)  DVSISN  (h-1)  (h-2)  (h-3)  (h-4)  (h-5)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 5: Qualitative inference results of adherent cell images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Our algorithm outperforms the other counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  backbone outperforms Mask R-CNN in both APbbox and APsegm, but the one with ResNet-101 only surpasses Mask  R-CNN in APbbox slightly while failing in APsegm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
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+page_content='00Table 3: Quantitative Results of the Ablation Study  Algorithm  Backbone  DVS  MS  APbbox  APbbox  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
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+page_content='75  APsegm  APsegm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
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+page_content='647  ResNet-50 and ResNet-101 are used as backbones in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Mask R-CNN is used as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  DVSISN† only adopts the DVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' DVSISN‡ only adopts the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' DVSISN adopts both DVS and MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  A full DVSISN equipped with both DVS and MS performs better than DVSISN† and DVSISN‡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' It is because  the DVS generates sufficient cell-aligning bounding boxes, and then the MS keeps only bounding boxes associated  with well-predicted masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' In a nutshell, using DVS or MS alone brings a slight improvement, but using them  together brings remarkable advancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  6  Conclusion  In this study, we developed a new algorithm called DVSISN for segmenting unstained live adherent cells in DIC  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Experimental results demonstrate that the DVSISN outperforms major SOTA algorithms by a large margin,  approximately 10% to 20%, in terms of APbbox and APsegm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Such an advantage can be attributed to two novel  methods—DVS and MS—that take combinations of original and rotated views as input to capture cell instances as  much as possible and select the finest instances in a supervised way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Ablation studies further confirmed that the  DVS could squeeze bounding boxes to better align with cell instances of various orientations, and the MS can keep  high-quality masks that improve the AP of masks, thus indirectly improving the quality of bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' In short,  our DVSISN is an accurate and robust algorithm for adherent cell segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  We plan to integrate the DVSISN into our cell micromanipulation system [7] to conduct intracellular deliveries  to investigate biological and biophysical reactions [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Meanwhile, we plan to implicitly merge the DVS  part into the training of RPN to reduce the computational cost in training [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' We also plan to test quadrilateral  bounding boxes to check final performance [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Acknowledgment  This work was partially conducted by using the computational facilities, CityU Burgundy, managed and provided by  the Computing Services Center at the City University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  References  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Alberts, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Bray, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hopkin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Johnson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Lewis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Raff, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Roberts, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Walter, Essential Cell  Biology, 4th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  New York, NY: Garland Science, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1  [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Karp, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Iwasa, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Marshall, Karp’s Cell and Molecular Biology: Concepts and Experiments, 8th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Wiley, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1  [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Vicar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Balvan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Jaros, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Jug, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Kolar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Masarik, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Gumulec, “Cell segmentation methods for  label-free contrast microscopy: Review and comprehensive comparison,” BMC Bioinformatics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  360, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2, 3  [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Vizcarra, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Burlingame, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hug, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Goltsev, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' White, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Tyson, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sokolov, “A  community-based approach to image analysis of cells, tissues and tumors,” Computerized Medical Imaging  and Graphics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 95, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 102013, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  10  [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Moen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Bannon, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Kudo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Graf, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Covert, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Valen, “Deep learning for cellular image  analysis,” Nature Methods, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1233–1246, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2, 3  [6] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Jaccard, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Griffin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Keser, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Macown, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Super, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Veraitch, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Szita, “Automated method  for the rapid and precise estimation of adherent cell culture characteristics from phase contrast microscopy  images,” Biotechnology and Bioengineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 111, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 504–517, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Pan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Jiao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Xing, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sun, “Deep learning-enhanced dual-module large-throughput  microinjection system for adherent cells,” IEEE Transactions on Automation Science and Engineering, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  1–14, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2, 5, 10  [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hussain, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hu, “Robotic intracellular electrochemical sensing for adherent  cells,” Cyborg and Bionic Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2022, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [9] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Abbas, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Aster, Robbins & Cotran Pathologic Basis of Disease, 10th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Elsevier,  May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Xing and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Yang, “Robust nucleus/cell detection and segmentation in digital pathology and microscopy  images: A comprehensive review,” IEEE Reviews in Biomedical Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 234–263, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Yanase and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Triantaphyllou, “A systematic survey of computer-aided diagnosis in medicine: Past and  present developments,” Expert Systems with Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 138, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 112821, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [12] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Doi, “Computer-aided diagnosis in medical imaging: Historical review, current status and future potential,”  Computerized Medical Imaging and Graphics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 4-5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 198–211, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Pachitariu and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Stringer, “Cellpose 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='0: How to train your own model,” Nature Methods, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1–8, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [14] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Greenwald, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Miller, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Moen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Kong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Kagel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Dougherty, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Fullaway, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' McIntosh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Leow, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Schwartz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Pavelchek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cui, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Camplisson, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Bar-Tal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Singh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Fong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Chaudhry,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Abraham, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Moseley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Warshawsky, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Soon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Greenbaum, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Risom, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hollmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Bendall,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Keren, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Graf, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Angelo, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Van Valen, “Whole-cell segmentation of tissue images with human-level  performance using large-scale data annotation and deep learning,” Nature Biotechnology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  555–565, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Yi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Jiang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Huang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hoeppner, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Metaxas, “Attentive neural cell instance  segmentation,” Medical Image Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 55, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 228–240, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [16] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Taft, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wallace, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Watkins, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Xing, “Learn to  segment single cells with deep distance estimator and deep cell detector,” Computers in Biology and Medicine,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 108, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 133–141, May 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [17] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Tsai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Gajda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sloan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Rares, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Shen, “Usiigaci: Instance-aware cell tracking in  stain-free phase contrast microscopy enabled by machine learning,” SoftwareX, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 230–237, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  2  [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Edlund, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Jackson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Khalid, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Bevan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Dale, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Dengel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Ahmed, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Trygg, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sj¨ogren,  “LIVECell—A large-scale dataset for label-free live cell segmentation,” Nature Methods, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  1038–1045, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Caicedo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Goodman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Karhohs, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cimini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Ackerman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Haghighi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Heng, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Becker,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Doan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' McQuin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Rohban, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Singh, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Carpenter, “Nucleus segmentation across imaging  experiments: The 2018 Data Science Bowl,” Nature Methods, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1247–1253, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Minaee, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Boykov, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Porikli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Plaza, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Kehtarnavaz, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Terzopoulos, “Image segmentation  using deep learning: A survey,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [21] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cai, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Chan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zeng, “A two-stage image segmentation method using a convex variant of the  Mumford–Shah model and thresholding,” SIAM Journal on Imaging Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 368–390,  2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Long, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Shelhamer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Darrell, “Fully convolutional networks for semantic segmentation,” in  Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  3431–3440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2, 5  11  [23] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Ronneberger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Fischer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Brox, “U-Net: Convolutional networks for biomedical image segmentation,”  in Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Lecture Notes in  Computer Science, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Navab, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hornegger, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wells, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Frangi, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 9351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Munich,  Germany: Springer International Publishing, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 234–241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Feng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Haase-Sch¨utz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Rosenbaum, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hertlein, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Glaeser, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Timm, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wiesbeck, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Dietmayer,  “Deep multi-modal object detection and semantic segmentation for autonomous driving: Datasets, methods,  and challenges,” IEEE Transactions on Intelligent Transportation Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1341–1360, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  2  [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Polk, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cui, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Plemmons, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Murphy, “Diffusion and volume maximization-based clustering  of highly mixed hyperspectral images,” arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='09992, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Camalan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cui, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Pauca, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Alqahtani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Silman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Chan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Plemmons, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Dethier, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Fernandez, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Lutz, “Change detection of Amazonian alluvial gold mining using deep learning and  Sentinel-2 imagery,” Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1746, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Xu, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Li, “Automated semantic segmentation of red blood cells for sickle cell  disease,” IEEE Journal of Biomedical and Health Informatics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3095–3102, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [28] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zeng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wang, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhang, “Segmentation and  automatic identification of vasculature in coronary angiograms,” Computational and Mathematical Methods  in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2021, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [29] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' He, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Gkioxari, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Doll´ar, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Girshick, “Mask R-CNN,” in Proceedings of the IEEE/CVF Conference  on Computer Vision and Pattern Recognition (CVPR), 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2961–2969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2, 3, 5, 7, 8, 10  [30] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Fang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Gou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Li, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Lu, “InstaBoost: Boosting instance segmentation  via probability map guided copy-pasting,” in Proceedings of the IEEE/CVF International Conference on  Computer Vision (ICCV), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 682–691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2, 7, 8  [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Bolya, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Xiao, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Lee, “YOLACT: Real-time instance segmentation,” in Proceedings of  the IEEE/CVF International Conference on Computer Vision (ICCV), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 9157–9166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2, 7, 8  [32] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Xiang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Mousavian, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Fox, “Unseen object instance segmentation for robotic environments,”  IEEE Transactions on Robotics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1343–1359, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [33] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Fang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Song, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Yin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Dai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Li, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Yang, “Joint 3D instance segmentation and  object detection for autonomous driving,” in Proceedings of the IEEE/CVF Conference on Computer Vision  and Pattern Recognition (CVPR), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1839–1849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2  [34] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Girshick, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Donahue, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Darrell, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Malik, “Rich feature hierarchies for accurate object detection  and semantic segmentation,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern  Recognition (CVPR), 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3  [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Girshick, “Fast R-CNN,” in Proceedings of the IEEE/CVF International Conference on Computer Vision  (ICCV), 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3  [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Uijlings, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Van De Sande, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Gevers, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Smeulders, “Selective search for object recognition,”  International Journal of Computer Vision, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 104, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 154–171, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3  [37] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Ren, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sun, “Spatial pyramid pooling in deep convolutional networks for visual  recognition,” in Proceedings of the European Conference on Computer Vision (ECCV), ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Lecture Notes in  Computer Science, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Fleet, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Pajdla, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Schiele, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Tuytelaars, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Zurich, Switzerland: Springer  International Publishing, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 346–361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3  [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Ren, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' He, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Girshick, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sun, “Faster R-CNN: Towards real-time object detection with region  proposal networks,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 28, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3  [39] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Ma, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Du, “Rotated cascade R-CNN: A shape robust detector with coordinate regression,”  Pattern Recognition, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 96, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 106964, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3  [40] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Gong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Huang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wang, “Mask scoring R-CNN,” in Proceedings of the  IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 6409–6418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3, 7, 8  12  [41] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cai and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Vasconcelos, “Cascade R-CNN: Delving into high quality object detection,” in Proceedings of  the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), Salt Lake City, Utah, USA,  2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 6154–6162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3, 7, 8  [42] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Looi, “Rotated mask R-CNN: From bounding boxes to rotated bounding boxes,” 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3  [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Rosen and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Culp, “Morphology and cellular origins of substrate-attached material from mouse  fibroblasts,” Experimental Cell Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 107, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 139–149, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3  [44] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Freshney, Culture of Animal Cells: A Manual of Basic Technique and Specialized Applications, 7th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Hoboken, NJ: Wiley-Blackwell, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3  [45] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Meijering, “Cell segmentation: 50 years down the road,” IEEE Signal Processing Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 5,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 140–145, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3  [46] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Pan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Jiao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Guan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Shakoor, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sun, “Automated high-productivity microinjection  system for adherent cells,” IEEE Robotics and Automation Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1167–1174, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 5  [47] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wada, “Labelme: Image polygonal annotation with Python,” 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 5  [48] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Ren, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sun, “Deep residual learning for image recognition,” in Proceedings of the  IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 6, 7  [49] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Lin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Maire, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Belongie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hays, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Perona, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Ramanan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Doll´ar, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zitnick, “Microsoft  COCO: Common objects in context,” in Proceedings of the European Conference on Computer Vision (ECCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Springer, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 740–755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 7  [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Everingham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Eslami, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Van Gool, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Winn, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zisserman, “The PASCAL  visual object classes challenge: A retrospective,” International Journal of Computer Vision, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 111, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 1,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 98–136, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 7  [51] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Pang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Xiong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sun, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Feng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Xu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cheng,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cheng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Lu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Dai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Shi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Ouyang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Loy, and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Lin, “MMDetection: Open MMLab detection toolbox and benchmark,” arXiv preprint arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='07155,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 7  [52] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Paszke, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Gross, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Massa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Lerer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Bradbury, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Chanan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Killeen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Lin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Gimelshein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Antiga,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Desmaison, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Kopf, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' DeVito, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Raison, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Tejani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Chilamkurthy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Steiner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Fang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Bai,  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Chintala, “PyTorch: An imperative style, high-performance deep learning library,” in Advances in  Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 7  [53] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Deng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Dong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Socher, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Li, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Li, “ImageNet: A large-scale hierarchical image  database,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR),  Miami, Florida, USA, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 248–255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 7  [54] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Xie, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Girshick, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Doll´ar, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Tu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' He, “Aggregated residual transformations for deep neural networks,”  in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  1492–1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 7  [55] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Keizer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Grosse-Holz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Woringer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zambon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Aizel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Bongaerts, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Delille, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Kolar-Znika,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Scolari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hoffmann, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Banigan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Mirny, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Dahan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Fachinetti, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Coulon, “Live-cell  micromanipulation of a genomic locus reveals interphase chromatin mechanics,” Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 377, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 6605,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 489–495, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 10  [56] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Stewart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sharei, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Ding, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Sahay, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Langer, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Jensen, “In Vitro and Ex Vivo strategies for  intracellular delivery,” Nature, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 538, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 7624, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 183–192, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 10  [57] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Xie, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Cheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Yao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Han, “Oriented R-CNN for object detection,” in Proceedings of the  IEEE/CVF International Conference on Computer Vision (ICCV), Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 3520–3529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 10  [58] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Yang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Hou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Jiang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Yan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Lyu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Zhang, and  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' Chen, “MMRotate: A rotated object detection benchmark using PyTorch,” in Proceedings of the 30th ACM  International Conference on Multimedia, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' MM ’22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content='  New York, NY, USA: Association for Computing  Machinery, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 7331–7334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
+page_content=' 10  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFJT4oBgHgl3EQfSSx0/content/2301.11499v1.pdf'}
diff --git a/edFIT4oBgHgl3EQfpCt4/content/tmp_files/2301.11321v1.pdf.txt b/edFIT4oBgHgl3EQfpCt4/content/tmp_files/2301.11321v1.pdf.txt
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@@ -0,0 +1,1772 @@
+Trajectory-Aware Eligibility Traces for
+Off-Policy Reinforcement Learning
+Brett Daley∗1,2, Martha White1,2,3, Christopher Amato4, and Marlos C. Machado1,2,3
+1Department of Computing Science, University of Alberta 2Alberta Machine Intelligence Institute
+3CIFAR Canada AI Chair 4Khoury College of Computer Sciences, Northeastern University
+Abstract
+Off-policy learning from multistep returns is crucial for sample-efficient reinforcement learning, but coun-
+teracting off-policy bias without exacerbating variance is challenging. Classically, off-policy bias is cor-
+rected in a per-decision manner: past temporal-difference errors are re-weighted by the instantaneous
+Importance Sampling (IS) ratio after each action via eligibility traces. Many off-policy algorithms rely
+on this mechanism, along with differing protocols for cutting the IS ratios to combat the variance of the
+IS estimator. Unfortunately, once a trace has been fully cut, the effect cannot be reversed. This has led
+to the development of credit-assignment strategies that account for multiple past experiences at a time.
+These trajectory-aware methods have not been extensively analyzed, and their theoretical justification
+remains uncertain. In this paper, we propose a multistep operator that can express both per-decision
+and trajectory-aware methods. We prove convergence conditions for our operator in the tabular setting,
+establishing the first guarantees for several existing methods as well as many new ones. Finally, we intro-
+duce Recency-Bounded Importance Sampling (RBIS), which leverages trajectory awareness to perform
+robustly across λ-values in an off-policy control task.
+1
+Introduction
+Reinforcement learning concerns an agent interacting with its environment through trial and error to max-
+imize its expected cumulative reward. One of the great challenges of reinforcement learning is the temporal
+credit assignment problem (Sutton, 1984): upon receiving a reward, which past actions should be held respon-
+sible and, hence, be reinforced? Basic temporal-difference (TD) methods assign credit to the immediately
+taken action (e.g., Watkins, 1989; Rummery and Niranjan, 1994), bootstrapping from previous experience
+to learn long-term dependencies. This process requires a large number of repetitions to generate effective
+behaviors from rewards, motivating research into multistep return estimation in which credit is distributed
+among multiple past actions according to some eligibility rule (e.g., Sutton, 1988).
+One challenge of multistep estimators is that they generally have higher variance than 1-step estima-
+tors (Kearns and Singh, 2000). This is exacerbated in the off-policy setting, where environment interaction
+is conducted according to a behavior policy that differs from the target policy for which returns are being
+estimated. The discrepancy between the two policies manifests mathematically as bias in the return estima-
+tion, which can be detrimental to learning if left unaddressed (Precup et al., 2000). Despite these challenges,
+off-policy learning is important for exploration and sample efficiency. The canonical bias-correction tech-
+nique is Importance Sampling (IS; Kahn and Harris, 1951), wherein the bias due to the differing policies is
+∗Corresponding author. Contact: brett.daley@ualberta.ca.
+1
+arXiv:2301.11321v1  [cs.LG]  26 Jan 2023
+
+eliminated by the product of their probability ratios (Precup et al., 2000). Although IS theoretically resolves
+the off-policy bias, it can suffer from extreme variance that makes it largely impractical.
+Directly managing the variance of the IS estimator has been a fruitful avenue for developing efficient off-
+policy algorithms. Past work has focused on modifying the individual IS ratios to reduce the variance of
+the full update: e.g., Tree Backup (Precup et al., 2000), Qπ(λ) (Harutyunyan et al., 2016), Retrace (Munos
+et al., 2016), ABQ (Mahmood et al., 2017), and C-trace (Rowland et al., 2020). All of these methods can
+be implemented online with per-decision rules (Precup et al., 2000) that determine how much to reduce, or
+cut, the IS ratio according to the current state-action pair. The re-weighted TD error is then broadcast to
+previous experiences using eligibility traces (Barto et al., 1983; Sutton, 1984). The decisions made by these
+algorithms are Markov in the sense that each iterative off-policy correction depends on only the current
+state-action pair. One issue with this is that it can lead to suboptimal decisions, since fully cutting a trace
+cannot be reversed later. In contrast, a trajectory-aware method can examine an entire sequence of past
+state-action pairs to make globally better decisions regarding credit assignment; for example, when a specific
+transition yields a high IS ratio, a trajectory-aware method can choose to not cut the trace if the product of
+all previous IS ratios remains small.
+Indeed, some existing off-policy methods already conduct offline bias correction in a trajectory-aware man-
+ner. Perhaps the simplest example is Truncated IS, where the IS ratio products are pre-calculated offline
+and then clipped to some finite value (see Section 4). More recently, Munos et al. (2016) suggested a recur-
+sive variant of Retrace that automatically relaxes the clipping bound when its historical trace magnitude
+becomes small; the authors conjectured that this could lead to faster learning. No theoretical analysis has
+been conducted on trajectory-aware algorithms such as these; their convergence properties are unknown,
+and the space of possible algorithms has not yet been fully explored.
+To better understand these algorithms, and to support new discoveries of efficient algorithms, we introduce
+a unifying theoretical perspective on per-decision and trajectory-aware off-policy corrections. We propose a
+multistep operator that accounts for arbitrary dependencies on past experiences, significantly generalizing the
+per-decision R operator introduced by Munos et al. (2016). We prove that our operator converges for policy
+evaluation and control. In the latter case, we remove the assumptions of increasingly greedy policies and pes-
+simistic initialization used by Munos et al. (2016), which has implications for per-decision methods. Finally,
+we derive a new method from our theory, Recency-Bounded Importance Sampling (RBIS), which performs
+favorably to other trajectory-aware methods across a wide range of λ-values in an off-policy control task.
+2
+Preliminaries
+We consider Markov Decision Processes (MDPs) of the form (S, A, P, R, γ). S and A are finite sets of states
+and actions, respectively. Letting ∆X denote the set of distributions over a set X, then P : S × A → ∆S is
+the transition function, R: S × A → R is the reward function, and γ ∈ [0, 1) is the discount factor. A policy
+π: S → ∆A determines an agent’s probability of selecting a given action in each state. A value function
+Q: S × A → R represents the agent’s estimate of the expected return achievable from each state-action pair.
+For a policy π, we define the operator
+(PπQ)(s, a) :=
+�
+s′∈S
+�
+a′∈A
+P(s′|s, a)π(a′|s′)Q(s′, a′).
+As a shorthand, we represent value functions and the reward function as vectors in Rn, where n = |S × A|.
+Linear operators such as Pπ can hence be interpreted as n × n square matrices that multiply these vectors,
+with repeated application corresponding to exponentiation: P t
+πQ = Pπ(P t−1
+π
+Q).
+In the policy evaluation setting, we seek to estimate the expected discounted returns for policy π, given
+by Qπ := �∞
+t=0 γtP t
+πR. The value function Qπ is the unique fixed point of the Bellman operator TπQ :=
+R + γPπQ, i.e., it uniquely solves the Bellman equation TπQπ = Qπ (Bellman, 1966). In the control setting,
+we seek to estimate the expected returns Q∗ under the optimal policy π∗. Q∗ is the unique fixed point of the
+2
+
+Figure 1:
+The Tightrope Problem. Starting from state s1, the agent must take a specific sequence of n
+actions to receive +1 reward.
+Bellman optimality operator (TQ)(s, a) := maxπ (TπQ)(s, a), i.e., it uniquely solves the Bellman optimality
+equation TQ∗ = Q∗. We are particularly interested in the off-policy learning case, where trajectories of the
+form (S0, A0), (S1, A1), (S2, A2), . . . are generated by interacting with the MDP using a behavior policy µ,
+where µ ̸= π. We define the TD error for policy π at time t as
+δπ
+t := Rt + γ
+�
+a′∈A
+π(a′|St+1)Q(St+1, a′) − Q(St, At),
+where Rt := R(St, At). Let ρk := π(Ak|Sk)
+µ(Ak|Sk) for brevity. Munos et al. (2016) introduced the off-policy operator
+(RQ)(s, a) := Q(s, a) + Eµ
+� ∞
+�
+t=0
+γt
+�
+t�
+k=1
+ck
+�
+δπ
+t
+����� (S0, A0) = (s, a)
+�
+,
+(1)
+where ck := c(Sk, Ak) ∈ [0, ρk] is a nonnegative coefficient. We refer to the product �t
+k=1 ck as the trace for
+(s, a) at time t. If any ck < ρk, we say that the trace has been (partially) cut. If any ck = 0, then we have
+fully cut the trace. If the trace is fully cut at t = 1, i.e., c1 = 0, then (RQ)(s, a) = Q(s, a) + E[δπ
+0|(S0, A0) =
+(s, a)] = R(s, a) + γE[�
+a′∈A π(a′|S1)Q(S1, a′)|(S0, A0) = (s, a)], which is the standard 1-step bootstrap
+target like in TD(0) (Sutton, 1988). Notice that each ck is Markov, as it depends only on the state-action
+pair (Sk, Ak) and is otherwise independent of the preceding trajectory. In other words, the update for R
+can be calculated per decision (Precup et al., 2000), thereby permitting an efficient online implementation
+with eligibility traces.
+3
+Trajectory-Aware Eligibility Traces
+While per-decision traces are convenient from a computational perspective, they require making choices
+about how much to cut the trace without considering the effects of previous choices.
+This can lead to
+suboptimal decisions; for example, if the trace is cut by setting ck = 0 at some timestep, then the effect
+cannot be reversed later. Regardless of whatever new experiences are encountered by the agent, experiences
+before time k will be ineligible for credit assignment, resulting in an opportunity cost. In fact, this exact
+phenomenon is why Watkins’ Q(λ) (Watkins, 1989) often learns more slowly than Peng’s Q(λ) (Peng and
+Williams, 1996), even though the former avoids off-policy bias (Sutton and Barto, 1998; Daley and Amato,
+2019; Kozuno et al., 2021). The same effect (but to a lesser extent) impacts Tree Backup and Retrace, where
+ck ≤ 1 always in Eq. (1), implying that the traces for past experiences can never increase.
+We illustrate this phenomenon in a small, deterministic MDP that we call the Tightrope Problem (see
+Figure 1). The environment consists of n sequential, non-terminal states with two actions a1, a2 available.
+The agent starts in state s1 and advances from si to si+1 whenever it takes action a1. If i = n, then the
+episode terminates and the agent receives +1 reward. Taking action a2 in any state immediately terminates
+the episode with no reward. Clearly, the optimal policy is to execute a1 regardless of the state.
+Now consider the following off-policy learning scenario. Suppose the agent’s behavior policy µ is uniform
+random, but the target policy π is ϵ-greedy with respect to a value function Q. For each state s, it follows
+that π(a|s) = 1 − ϵ if a = argmaxa′ Q(s, a′) and π(a|s) = ϵ otherwise. We assume ϵ is small in the sense
+that ϵ <
+1
+2, and that γ = 1. Suppose now that the agent successfully receives the +1 reward during an
+3
+
+episode, implying that it took action a1 on every timestep. We can compute the eligibility of the initial
+state-action pair (s1, a1) as an expression in the number k of incorrect actions in the greedy policy (i.e.,
+where argmaxa′ Q(s, a′) ̸= a1). Letting λ ∈ [0, 1] be a decay parameter, the standard IS estimator (which
+does not cut traces when λ = 1) provides an eligibility of
+�
+λ1 − ϵ
+1 / 2
+�n−k �
+λ
+ϵ
+1 / 2
+�k
+= λn[2(1 − ϵ)]n−k(2ϵ)k.
+(2)
+This value can be greater than 1 when k ≪ n, which suggests that the agent’s behavior should be heavily
+reinforced when the greedy policy agrees closely with the optimal policy; however, a per-decision method
+like Retrace, which cuts traces without considering the full trajectory (see Section 4), ultimately assigns a
+much lower eligibility:
+�
+λ min
+�
+1, 1 − ϵ
+1 / 2
+��n−k�
+λ min
+�
+1,
+ϵ
+1 / 2
+��k
+= λn(2ϵ)k.
+The eligibility now decays monotonically for every suboptimal action in the greedy policy, illuminating how
+per-decision trace cutting can lead to excessively small eligibilities, especially when ϵ is close to 0.
+This issue stems from the fact that Retrace is not aware of its past eligibilities, and it continues to decay them
+even when they already represent an underestimate compared to IS. This issue is not unique to Retrace, as
+it also affects other per-decision methods like Tree Backup. Instead, it would be better to have a trajectory-
+aware method that can actively adapt its trace-cutting behavior based on the magnitude of past eligibilities.
+One way to obtain a trajectory-aware method is to compute the exact IS product in Eq. (2), and then make
+adjustments to it to achieve certain properties (e.g., convergence and variance reduction). For example,
+Truncated IS (see Section 4) simply imposes a fixed bound on the IS estimator:
+λn min
+�
+1, [2(1 − ϵ)]n−k(2ϵ)k�
+.
+(3)
+Ignoring λ, Truncated IS reduces the eligibility only when it exceeds a pre-specified threshold, effectively
+avoiding trace cuts when the true IS estimate is small. In Section 6, we propose an algorithm, RBIS, which
+achieves a similar effect using a recursive, time-decaying threshold.
+As this example demonstrates, it can be advantageous to consider the agent’s past experiences to produce
+better decisions regarding credit assignment. One of our principal contributions is the proposal and analysis
+of an off-policy operator M that encompasses this possibility. Let Ft := (S0, A0), (S1, A1), . . . , (St, At). We
+define M such that
+(MQ)(s, a) := Q(s, a) + Eµ
+� ∞
+�
+t=0
+γtβtδπ
+t
+����� (S0, A0) = (s, a)
+�
+,
+(4)
+where βt := β(Ft) is a trace that generally depends on the history Ft. We define β0 := 1 to ensure that the
+first TD error, δπ
+0 , is applied. In Section 5, we characterize the values of βt for t ≥ 1 that lead to convergence.
+The major analytical challenge of our operator—and its main novelty—is the complex dependence on the
+sequence Ft. This makes the M operator difficult to analyze mathematically, as the terms in the series
+1 + γβ1 + γ2β2 + · · · generally share no common factors that would allow a recursive formula for eligibility
+traces. Some off-policy methods, however, cannot be described by factored traces, and therefore removing
+this assumption is necessary to understand existing algorithms (see Section 4), while also paving the way
+for new credit-assignment methods. In the special case where βt does factor into Markov coefficients, i.e.,
+βt = �t
+k=1 ck, then Eq. (4) reduces to Eq. (1), taking us back to the per-decision setting studied by Munos
+et al. (2016). M, therefore, unifies per-decision and trajectory-aware methods.
+4
+Unifying Off-Policy Algorithms
+The operator M is a strict generalization of the previous operator considered for trace-based methods,
+allowing us to express existing algorithms in this form. We provide a non-exhaustive list of examples below
+4
+
+with the corresponding βt used in M. For brevity, let Πt := �t
+k=1 ρk.
+Importance Sampling: βt = λtΠt (Kahn and Harris, 1951). The standard approach for correcting off-
+policy bias. Although it is the only unbiased estimator in this list (if λ = 1), it suffers from high variance,
+making it difficult to utilize.
+Qπ(λ): βt = λt (Harutyunyan et al., 2016). A straightforward algorithm that decays the TD errors by a
+fixed constant. The algorithm does not require explicitly knowing µ, which is desirable, but can diverge if π
+and µ differ too much (Harutyunyan et al., 2016, Theorem 1).
+Tree Backup: βt =�t
+k=1 λπ(Ak|Sk) (Precup et al., 2000). A method that automatically cuts traces accord-
+ing to the product of probabilities under π, which forms a conservative lower bound on the IS estimate. Tree
+Backup converges for any behavior policy µ, but it is not efficient since traces are cut excessively—especially
+in the on-policy case.
+Retrace: βt = �t
+k=1 λ min(1, ρk) (Munos et al., 2016). A convergent algorithm for arbitrary policies π
+and µ that remains efficient in the on-policy case because it does not cut traces (if λ = 1); however, the fact
+that βt never increases can cause the trace products to decay too quickly in practice (Mahmood et al., 2017;
+Rowland et al., 2020).
+All of the above can be analyzed using a per-decision operator. The next two, on the other hand, have
+weightings based on the entire trajectory. We use the theory for our general M operator to prove properties
+about these methods.
+Recursive Retrace: βt =λ min(1, βt−1ρt) (Munos et al., 2016). A modification to Retrace conjectured to
+lead to faster learning. It clips large products of ratios, rather than individual ratios. Its convergence for
+control is an open question, which we solve in Section 5.
+Truncated Importance Sampling: βt = λt min(1, Πt) (Ionides, 2008). A simple but effective method
+to combat the variance of IS. Variations of this algorithm have been applied in the reinforcement learning
+literature (e.g., Uchibe and Doya, 2004; Wawrzy´nski and Pacut, 2007; Wawrzy´nski, 2009; Wang et al., 2017),
+but, to our knowledge, its convergence in an MDP setting has not been studied. In Section 5.3, we show
+that it can diverge in at least one off-policy problem.
+5
+Convergence Analysis
+In this section, we study the convergence properties of the M operator for policy evaluation and control. It
+will be convenient to re-express Eq. (4) in vector notation for our analysis. To do this, let us first bring the
+expectation inside the sum, by linearity of expectation:
+(MQ)(s, a) = Q(s, a) +
+∞
+�
+t=0
+γtEµ
+�
+βtδπ
+t
+�� (S0, A0) = (s, a)
+�
+.
+(5)
+To write Eq. (5) in vector form, we define an operator Bt such that, for an arbitrary vector X in Rn,
+(BtX)(s, a):=Eµ
+�
+βtX(St, At)
+�� (S0, A0) = (s, a)
+�
+,
+(6)
+allowing us to express the M operator as
+MQ = Q +
+∞
+�
+t=0
+γtBt(TπQ − Q).
+(7)
+Bt is a linear operator and hence can be represented as a matrix in Rn×n, the elements of which are
+nonnegative. Each element of Bt, row-indexed by (s, a) and column-indexed by (s′, a′), has the form
+Bt((s, a), (s′, a′)) = Pr
+µ ((St, At)=(s′, a′)|(S0, A0)=(s, a)) × Eµ
+�
+βt
+��(S0, A0)=(s, a), (St, At)=(s′, a′)
+�
+.
+(8)
+5
+
+We justify this form in Appendix A. Note that B0 = I, the identity matrix, because of our earlier definition
+of β0 := 1. In the following sections, all inequalities involving vectors or matrices should be interpreted
+element wise. We let ∥X∥ := ∥X∥∞ for a matrix (or vector) X, which corresponds to the maximum absolute
+row sum of X. We also define 1 ∈ Rn to be the vector of ones, such that X1 gives the row sums of X.
+5.1
+Convergence for Policy Evaluation
+We start in the off-policy policy evaluation setting.
+Specifically, our goal is to prove that the repeated
+application of the M operator to an arbitrarily initialized vector Q ∈ Rn converges to Qπ.
+Condition 1. βt ≤ βt−1ρt, ∀ Ft, ∀ t ≥ 1.
+Theorem 2. If Condition 1 holds, then M is a contraction mapping with Qπ as its unique fixed point.
+Consequently, limi→∞ MiQ = Qπ, ∀ Q ∈ Rn.
+Proof. In Lemma 1 (Appendix B.1), we show that Qπ is a fixed point of M and that
+MQ − Qπ = Z(Q − Qπ),
+(9)
+where Z := �∞
+t=1 γt(Bt−1Pπ − Bt). In Lemma 2 (Appendix B.2), we also show that Z ≥ 0 and Z1 ≤ γ
+using the assumption that βt ≤ βt−1ρt, ∀ Ft, ∀ t ≥ 1 (Condition 1). Consequently, Z(Q − Qπ) is a vector
+whose components each comprise a nonnegative-weighted combination of the components of Q − Qπ, where
+the weights add up to at most γ. This means that ∥MQ − Qπ∥ ≤ γ∥Q − Qπ∥, and M is a contraction
+mapping. Its fixed point, Qπ, must therefore be unique by the Banach fixed-point theorem, implying that
+limi→∞ MiQ = Qπ for every Q ∈ Rn when γ < 1.
+Given that the ratio
+βt
+βt−1 is bounded by ρt (Condition 1), the M operator converges to Qπ. Intuitively,
+we can think of this ratio as the effective per-decision factor at time t; convergence is guaranteed whenever
+this factor is no greater than ρt, analogous to the convergence result for the R operator (Munos et al.,
+2016, Theorem 1). Our theorem implies the existence of a space of convergent trajectory-aware algorithms,
+because each trace βt can be chosen arbitrarily so long as it always satisfies the bound on this ratio.
+5.2
+Convergence for Control
+We now consider the more challenging setting of control. Given sequences of target policies (πi)i≥0 and
+behavior policies (µi)i≥0, we aim to show that the sequence of value functions (Qi)i≥0 given by Qi+1 := MiQi
+converges to Q∗. Here, Mi is the M operator defined for πi and µi.
+Compared to the convergence proof of the R operator (Munos et al., 2016, Theorem 2), the main novelty
+of our proof is the fact that the traces under M are not Markov. Consequently, we require new techniques
+to establish bounds on Q − Q∗, since Eq. (4) is not representable as an infinite geometric series and the
+summation therefore does not have a closed-form expression. We additionally relax two assumptions in the
+previous work, on initialization of the value function and on increasing greediness of the policy. We require
+only that the target policies become greedy in the limit. We say that a sequence of policies is greedy in the limit
+if TπiQi → TQi as i → ∞. We discuss the significance of these relaxations to the assumptions in Section 5.4.
+First, let Ci := �∞
+t=0 γtBt for the policies πi and µi, and write the M operator at iteration i as
+MiQ = Q + Ci(TπiQ − Q).
+(10)
+We now present our convergence theorem for control.
+6
+
+Theorem 3. Consider a sequence of target policies (πi)i≥0 and a sequence of arbitrary behavior policies
+(µi)i≥0. Let Q0 be an arbitrary vector in Rn and define the sequence Qi+1 := MiQi, where Mi is the
+operator defined by Eq. (10). Assume that (πi)i≥0 is greedy in the limit, and let ϵi ≥ 0 be the smallest
+constant such that TπiQi ≥ TQi − ϵi∥Qi∥1. If Condition 1 holds for all i, then
+∥MiQi − Q∗∥ ≤ γ∥Qi − Q∗∥ +
+ϵi
+1 − γ ∥Qi∥,
+(11)
+and, consequently, lim
+i→∞ Qi = Q∗.
+Proof (sketch; full proof in Appendix B.3). We define matrices Zi and Z∗
+i , which correspond to Z in Eq. (9)
+for target policies πi and π∗, respectively, and behavior policy µi. We then derive the inequalities
+Z∗
+i (Qi − Q∗) − ϵi∥Qi∥Ci1 ≤ MiQi − Q∗ ≤ Zi(Qi − Q∗),
+which together imply Eq. (11). Thus, Mi is nearly a contraction mapping with Q∗ as its unique fixed point,
+excepting the influence of the O(∥Qi∥) term. However, the greedy-in-the-limit target policies guarantee that
+ϵi → 0. Showing that ∥Qi∥ remains finite completes the proof because ∥Qi − Q∗∥ → 0 must follow.
+The convergence criteria for βt (Condition 1) is the same for both policy evaluation and control. In fact,
+the only additional assumption we need for control is the greedy-in-the-limit target policies. Crucially, the
+proof allows arbitrary behavior policies and an arbitrary value function initialization Q0, which we further
+discuss in Section 5.4.
+5.3
+Examples of Convergence and Divergence
+The generality of the M operator means that it provides convergence guarantees for a number of credit-
+assignment methods that we did not discuss in Section 4. These include variable or past-dependent λ-values
+(e.g., Watkins, 1989; Singh and Sutton, 1996; Yu et al., 2018). All of these can be represented in a common
+form and shown to satisfy Condition 1; convergence for policy evaluation and control for the instantiated
+trajectory-aware operator follows as a corollary, since Condition 1 is sufficient to apply Theorems 2 and 3.
+Any traces expressible in the form βt = �t
+k=1 λ(Fk)ρk, λ(Fk) ∈ [0, 1], satisfy Condition 1.
+Proof. βt = βt−1λ(Ft)ρt ≤ βt−1ρt.
+In Section 4, we also discussed two existing trajectory-aware methods whose convergence is unknown. We
+show that Recursive Retrace satisfies our required condition.
+Recursive Retrace satisfies Condition 1.
+Proof. For Recursive Retrace, βt = ctβt−1, where ct = λmin
+�
+1
+βt−1 , ρt
+�
+(Munos et al., 2016, Eq. 9). This
+means
+βt = λ min
+�
+1
+βt−1
+, ρt
+�
+βt−1
+= λ min (1, βt−1ρt)
+≤ βt−1ρt,
+(12)
+which is the bound required by Condition 1.
+Unfortunately, the traces for Truncated IS do not always satisfy the required bound.
+Truncated IS may
+violate Condition 1.
+Proof. We show this by providing a counterexample. Recall that Truncated IS has βt = λt min(1, Πt).
+Assume a trajectory Ft such that Πt−1 = 2 and
+1
+2 < ρt < λ.
+(It is straightforward to define an MDP,
+behavior policy, and target policy to create such a trajectory.) Because ρt >
+1
+Πt−1 , then Πt = Πt−1ρt > 1.
+Thus, βt = λt and βt−1 = λt−1, and Condition 1 is violated because
+βt
+βt−1 = λ ̸≤ ρt.
+7
+
+Because Theorems 2 and 3 cannot be applied, the precise conditions under which Truncated IS converges
+remains an open problem. We do know Condition 1 is sufficient for convergence, but it is unlikely to be strictly
+necessary. This is because our proofs of Theorems 2 and 3 use this assumption to guarantee that the matrix
+Z in Eq. (9) has nonnegative elements, making it straightforward to show that its row sums are sufficiently
+bounded to guarantee that M is a contraction mapping. However, M could remain a contraction mapping
+even when Z has negative elements, so long as ∥Z∥ < 1. It could theoretically be the case for Truncated IS
+that Z occasionally contains negative elements but ∥Z∥ is still bounded enough to permit convergence.
+Nevertheless, we are able to find at least one off-policy problem for which this is not true, implying that
+certain initializations of the value function could ultimately cause Truncated IS to diverge.
+Counterexample 4 (Off-Policy Truncated IS). Consider Truncated IS with λ = 1, so βt = min(1, Πt).
+Assume the MDP has one state and two actions: S = {s} and A = {a1, a2}, the behavior policy µ is uniform
+random, and π selects a1 with probability p ∈ (0, 1) and selects a2 otherwise. When p = 0.6 and γ = 0.94,
+then ∥Z∥ > 1.
+We provide details in Appendix C.1; note that many choices of p and γ make ∥Z∥ > 1, but we provide
+specific ones for the counterexample.
+Compared to the per-decision case, where Munos et al. (2016) showed that arbitrary trace cuts always
+produce a convergent algorithm, this result is surprising. Why would the analogous result—in which we
+ensure that βt ≤ Πt for all timesteps—not hold here? After all, clipping βt such that it never exceeds the IS
+estimate Πt would be expected to simply incur bias in the return estimation. For some insight, assume the
+following expectations are conditioned on (S0, A0) = (s, a), and observe that Eq. (4) is equivalent to
+Eµ
+� ∞
+�
+t=0
+γtβtRt
+�
++ Eµ
+� ∞
+�
+t=1
+γt(βt−1ρt − βt)Q(St, At)
+�
+.
+We show the derivation in Appendix A. The first term is a (partially) bias-corrected estimate of the dis-
+counted return. The second term is a weighted combination of value-function bootstraps, whose weights are
+nonnegative when Condition 1 is met. If the condition is violated on any timestep, then we may actually be
+subtracting bootstraps from the return estimate, which does not seem sensible. We believe this is related to
+the root cause of divergence in Counterexample 4; however, it remains open whether Condition 1 is necessary
+or merely sufficient.
+As our next counterexample example will demonstrate, this effect can even cause divergence in on-policy
+settings.
+Counterexample 5 (On-Policy Binary Traces). Assume the MDP has one state and two actions: S = {s}
+and A = {a1, a2}. Define a trajectory-aware method such that βt = 1 if At = a1 and βt = 0 if At = a2
+(without loss of generality). Assume π and µ are uniform random. When γ ≥ 2
+3, then ∥Z∥ ≥ 1.
+We provide details in Appendix C.2. Even though βt ≤ Πt = 1 always, we are able to produce a non-
+contraction. The method either fully cuts a trace or does not cut it at all, producing backups that consist of
+a sparse sum of on-policy TD errors. It is therefore surprising that divergence occurs. For the same reason
+we described above, the non-Markov nature of the trace appears to sometimes cause adverse bootstrap-
+ping effects; in this instance, the ability to examine the contents of each trajectory allows the method to
+strategically de-emphasize certain state-action pairs, ultimately producing a detrimental effect on learning.
+Notice that Condition 1 is indeed violated in this case because there is always some chance that βt = 1 after
+βt−1 = 0. If we add the restriction that βt−1 = 0 =⇒ βt = 0, i.e., we permanently cut the traces, then
+convergence is guaranteed by Theorem 2.
+8
+
+5.4
+Discussion
+In this section, we summarize our main theoretical contributions and their significance.
+We focused on
+characterizing the contraction properties of the M operator, both for policy evaluation and control, in the
+tabular setting. These results parallel those for the R operator underlying Retrace, where M is a strict
+generalization of R. These results indicate that using fixed-point updates, like dynamic programming and
+temporal difference learning updates, may have divergence issues. It does not, however, imply other algo-
+rithms, such as gradient-based algorithms, cannot find these fixed points. We show the fixed points still
+exist and are unbiased, but that algorithms based on iterating with the M operator might diverge.
+Removal of the Markov assumption. Removing the Markov (per-decision) assumption of the R operator
+(Munos et al., 2016) to enable trajectory-aware eligibility traces was our primary goal. When the trace
+factors are Markov, the operator Bt is independent of t, allowing the sum �∞
+t=0 γtBt to be reduced to
+�∞
+t=0(γPcµ)t for a linear operator Pcµ. The resulting geometric series can then be evaluated analytically, as
+was done by Munos et al. (2016). In our proofs, we avoided the Markov assumption by directly analyzing
+the infinite summation, which generally does not have a closed-form expression. Our work is the first to do
+this, establishing the first convergence guarantees for general trajectory-aware methods.
+Arbitrary initialization of the value function. We permit any initialization of Q0 in the control set-
+ting. In contrast, Munos et al. (2016) made the assumption that Tπ0Q0 − Q0 ≥ 0 in order to produce a
+lower bound on RiQi − Q∗, accomplished in practice by a pessimistic initialization of the value function:
+Q0(s, a) = −∥R∥ / (1 − γ), ∀ (s, a) ∈ S × A. Since R is a special case of our operator M where each trace
+βt factors into Markov coefficients, we deduce as a corollary that Retrace and all other algorithms described
+by R do not require pessimistic initialization for convergence.
+Greedy-in-the-limit policies.
+Our requirement of greedy-in-the-limit target policies in Theorem 3 is
+less restrictive than the increasingly greedy policies proposed by Munos et al. (2016).
+We need only
+limi→∞ TπiQi = TQi, and we do not force the sequence of target policies to satisfy Pπi+1Qi+1 ≥ PπiQi+1.
+This implies that the agent may target non-greedy policies for any finite period of time, as long as the policies
+do eventually become arbitrarily close to the greedy policy. As a corollary, increasingly greedy policies are
+not necessary for the optimal convergence of Retrace and other per-decision methods.
+6
+Recency-Bounded Importance Sampling
+Theorem 2 guarantees convergence to Qπ whenever Condition 1 holds, but we do not expect that all choices of
+coefficients that satisfy this condition will perform well in practice. At one extreme, if βt ≤ �t
+k=1 λ min(1, ρk)
+for every Ft, then we have a method that cuts coefficients more aggressively than Retrace does; it seems
+unlikely that such a method would learn faster than Retrace, or other per-decision methods. At the other
+extreme, when βt = Πt for every Ft, we recover the standard IS estimator, which suffers from high variance
+and is often ineffectual. We therefore know that it is possible to have a method that preserves traces too
+much, to the point of being detrimental. Thus, it is important to maintain some minimum efficiency by
+avoiding unnecessary cuts, yet also important to control the overall variance of the traces.
+Intuitively, we want something that falls between Retrace and IS in terms of trace cutting, in order to
+quickly backpropagate credit while still managing the overall variance. We further hypothesize that effective
+trajectory-aware methods will first compute βt−1ρt—i.e., the maximum trace permitted by Condition 1—and
+then apply some transformation that bounds its magnitude and manages the variance. This ensures that
+traces are cut only as needed.
+We propose one method, Recency-Bounded Importance Sampling (RBIS), which achieves this by cutting the
+traces only when they exceed an exponentially decaying threshold. Specifically, we define
+βt = min(λt, βt−1ρt).
+(RBIS)
+9
+
+It is easy to see that RBIS always converges, by construction.
+RBIS satisfies Condition 1.
+Proof. βt = min(λt, βt−1ρt) ≤ βt−1ρt.
+For further insight, we unroll the recursion to obtain
+min(λt, βt−1ρt) = min(λt, min(λt−1, βt−2ρt−1)ρt)
+· · ·
+= min(λt, λt−1ρt, λt−2ρt−1ρt, . . . , Πt).
+(13)
+RBIS effectively takes the minimum of all past, discounted n-step IS estimates. This reveals another property
+of RBIS: its traces are never less than those of Retrace because
+t�
+k=1
+λ min(1, ρk) ≤
+t�
+k=1
+min(λ, ρk)
+≤ λt−j
+t�
+k=t−j+1
+ρk, ∀ j ∈ {0, 1, . . . , t}.
+Since the inequality is true for all j, it is not possible for Retrace’s traces to exceed any of the arguments to
+the minimum function in Eq. (13). We have achieved exactly what we wanted earlier: a method that falls
+somewhere between Retrace and IS in regard to trace cutting. This does not automatically mean that RBIS
+will outperform Retrace, though, since preserving the magnitude of the trace βt too much can lead to high
+variance. However, we do expect RBIS to perform well in decision-making problems in which a few critical
+actions largely determine the long-term outcome of an episode. In such scenarios, the agent’s bottleneck to
+learning is its ability to assign meaningful credit to these critical actions over a potentially long time horizon.
+In order to test this empirically, we construct an environment called the Bifurcated Gridworld (see Figure 2).
+This is a 5 × 5 deterministic gridworld with walls arranged such that two unequal-length paths from the
+starting cell (S) to the goal cell (G) are available. The agent may move up, down, left, or right; taking any
+of these actions in the goal cell yields a reward of +1 and terminates the episode. The problem is discounted
+(γ = 0.9) to encourage the agent to learn the shorter path to the goal. Importantly, the action taken at the
+bifurcation (B) solely determines which path the agent follows, and quickly assigning credit to this state is
+paramount to learning the task.
+We compare RBIS against Retrace, Truncated IS, and Recursive Retrace when learning this task from off-
+policy data. Both behavior and target policies were ϵ-greedy with respect to the value function Q. The
+target policy used ϵ = 0.1. The behavior policy used a piecewise schedule: ϵ = 1 for the first 5 episodes and
+then ϵ = 0.2 afterwards. The policies were updated only at the end of each episode. The agents used online
+TD updates with eligibility traces (see Appendix D for pseudocode). At the end of each episode, the policies
+were updated, and then we evaluated the discounted return obtained by a near-greedy policy (ϵ = 0.05).
+The area under the curve (AUC) of each resulting learning curve was calculated, and the highest AUC
+achieved over a grid search of stepsizes was plotted for each λ-value (see Figure 2). We averaged the results
+over 1,000 independent trials and indicate the 95% confidence interval by the shaded regions. Appendix E
+contains additional experiment details and the learning curves. We also include the experiment code in the
+supplementary material.
+We make several observations regarding the results in Figure 2. First, the peak performance obtained by
+RBIS is significantly higher than that of the other three methods. This is notable because both Truncated
+IS and Recursive Retrace are also trajectory aware, indicating that different implementations of trajectory
+awareness are beneficial to varying degrees. In particular, the preservation of long-term eligibilities is not
+sufficient on its own to guarantee strong performance in general, as it appears that when and how much the
+traces are cut are important considerations as well. The role of λ as a decay hyperparameter is evidently
+critical for all methods to achieve their maximum performance, since λ = 1 never leads to the fastest learning.
+In fact, λ → 1 is especially catastrophic for Truncated IS, which we believe is related to the divergence issue
+10
+
+S
+B
+G
+Retrace
+Truncated IS
+RBIS
+Recursive
+Retrace
+Figure 2:
+The Bifurcated Gridworld environment. The agent starts in S and receives +1 reward for taking
+any action in G. The choice made at B greatly impacts the discounted return ultimately earned. We plot the
+AUC of the learning curve obtained by four off-policy methods across the λ-spectrum. The dashed horizontal
+lines mark the highest AUC achieved by each method.
+identified in Section 5.3. Finally, Retrace degrades less for larger λ, likely because it cuts traces more. It
+would be interesting to develop a trajectory-aware method that obtains the robustness of RBIS but also
+accounts for larger λ-values.
+7
+Conclusion
+In this work, we extended theory for per-decision eligibility traces to trajectory-aware traces. This extension
+allows us to consider a broader family of algorithms, with more flexibility in obtaining off-policy corrections.
+Specifically, we introduced the M operator, as a generalization of the R operator, and a sufficient condition
+to ensure convergence under M. Using our general result, we established the first convergence guarantee
+for an existing trajectory-aware method, Recursive Retrace, in the control setting. We also showed that
+Truncated IS may violate our condition and provided a counterexample showing that it can diverge.
+We also proposed a new trajectory-aware method, RBIS, that demonstrates one instance of how trajectory
+awareness can be utilized for faster learning in an off-policy control task. RBIS is able to outperform the
+other trajectory-aware methods that we tested in the Bifurcated Gridworld, suggesting that it possesses at
+least one unique property that is beneficial for long-term, off-policy credit assignment. It would be interesting
+to search for additional beneficial properties in future work, in order to better characterize off-policy methods
+that reliably lead to efficient and stable learning in challenging reinforcement learning environments.
+This work focused on convergence in expectation; a natural next step is to extend this result to the stochastic
+algorithms used in practice. Previous results for TD learning rely primarily on the properties of the expected
+update, with additional conditions on the noise in the update and appropriately annealed stepsizes (see
+Bertsekas and Tsitsiklis, 1996, Section 4.3). Similar analysis should be applicable, given that we know the
+expected update using the M operator is contraction mapping when Condition 1 is met.
+An important next step is extending these methods and results to function approximation. Incorporating
+these traces into deep reinforcement learning methods that rely on experience replay (Lin, 1992) should
+be straightforward. Multistep returns can be computed offline in the replay memory, and then randomly
+sampled in minibatches to train the neural network. Using a TD learning update, though, can suffer from
+convergence issues under function approximation and off-policy learning; this has been previously resolved
+by developing gradient-based updates (Sutton et al., 2009; Touati et al., 2018). An important next step is to
+11
+
+develop a gradient-based trajectory-aware algorithm. The contraction properties of the operator still impact
+the quality of the solution, as has been shown to be the case for other off-policy approaches (Patterson
+et al., 2022). The insights in this work, therefore, may provide insights on how to get quality solutions with
+gradient-based approaches.
+References
+Andrew G Barto, Richard S Sutton, and Charles W Anderson. Neuronlike adaptive elements that can solve
+difficult learning control problems. IEEE Transactions on Systems, Man, and Cybernetics, pages 834–846,
+1983.
+Richard Bellman. Dynamic programming. Science, 153(3731):34–37, 1966.
+Dimitri P Bertsekas and John N Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, 1996.
+Brett Daley and Christopher Amato. Reconciling λ-returns with experience replay. In Advances in Neural
+Information Processing Systems, pages 1133–1142, 2019.
+Anna Harutyunyan, Marc G Bellemare, Tom Stepleton, and R´emi Munos. Q(λ) with off-policy corrections.
+In International Conference on Algorithmic Learning Theory, pages 305–320, 2016.
+Edward L Ionides. Truncated importance sampling. Journal of Computational and Graphical Statistics, 17
+(2):295–311, 2008.
+Herman Kahn and Theodore E Harris. Estimation of particle transmission by random sampling. National
+Bureau of Standards: Applied Mathematics Series, 12:27–30, 1951.
+Michael J Kearns and Satinder P Singh. Bias-variance error bounds for temporal difference updates. In
+Conference on Learning Theory, pages 142–147, 2000.
+Tadashi Kozuno, Yunhao Tang, Mark Rowland, R´emi Munos, Steven Kapturowski, Will Dabney, Michal
+Valko, and David Abel. Revisiting Peng’s Q(λ) for modern reinforcement learning. arXiv:2103.00107,
+2021.
+Long-Ji Lin. Self-improving reactive agents based on reinforcement learning, planning and reaching. Machine
+Learning, 8:293–321, 1992.
+Ashique Rupam Mahmood, Huizhen Yu, and Richard S Sutton.
+Multi-step off-policy learning without
+importance sampling ratios. arXiv:1702.03006, 2017.
+R´emi Munos, Tom Stepleton, Anna Harutyunyan, and Marc G Bellemare.
+Safe and efficient off-policy
+reinforcement learning. In Advances in Neural Information Processing Systems, 2016.
+Andrew Patterson, Adam White, and Martha White. A generalized projected Bellman error for off-policy
+value estimation in reinforcement learning. Journal of Machine Learning Research, 2022.
+Jing Peng and Ronald J Williams. Incremental multi-step Q-Learning. Machine Learning, 22:226–232, 1996.
+Doina Precup, Richard S Sutton, and Satinder Singh. Eligibility traces for off-policy policy evaluation. In
+International Conference on Machine Learning, page 759–766, 2000.
+Mark Rowland, Will Dabney, and R´emi Munos. Adaptive trade-offs in off-policy learning. In International
+Conference on Artificial Intelligence and Statistics, pages 34–44, 2020.
+Gavin A Rummery and Mahesan Niranjan.
+On-line Q-Learning using connectionist systems.
+Technical
+report, Cambridge University, 1994.
+Satinder P Singh and Richard S Sutton. Reinforcement learning with replacing eligibility traces. Machine
+Learning, 22:123–158, 1996.
+12
+
+Richard S Sutton.
+Temporal Credit Assignment in Reinforcement Learning.
+PhD thesis, University of
+Massachusetts Amherst, 1984.
+Richard S Sutton. Learning to predict by the methods of temporal differences. Machine Learning, 3(1):9–44,
+1988.
+Richard S Sutton and Andrew G Barto. Reinforcement Learning: An Introduction. MIT Press, 1st edition,
+1998.
+Richard S Sutton, Hamid Reza Maei, Doina Precup, Shalabh Bhatnagar, David Silver, Csaba Szepesv´ari,
+and Eric Wiewiora. Fast gradient-descent methods for temporal-difference learning with linear function
+approximation. In International Conference on Machine Learning, pages 993–1000, 2009.
+Ahmed Touati, Pierre-Luc Bacon, Doina Precup, and Pascal Vincent. Convergent Tree Backup and Retrace
+with function approximation. In International Conference on Machine Learning, pages 4955–4964, 2018.
+Eiji Uchibe and Kenji Doya. Competitive-cooperative-concurrent reinforcement learning with importance
+sampling. In International Conference on Simulation of Adaptive Behavior: From Animals and Animats,
+pages 287–296, 2004.
+Ziyu Wang, Victor Bapst, Nicolas Heess, Volodymyr Mnih, Remi Munos, Koray Kavukcuoglu, and Nando
+de Freitas. Sample efficient actor-critic with experience replay. In International Conference on Learning
+Representations, 2017.
+Christopher John Cornish Hellaby Watkins. Learning from Delayed Rewards. PhD thesis, King’s College,
+Cambridge, 1989.
+Pawe�l Wawrzy´nski. Real-time reinforcement learning by sequential actor-critics and experience replay. Neural
+Networks, 22(10):1484–1497, 2009.
+Pawe�l Wawrzy´nski and Andrzej Pacut.
+Truncated importance sampling for reinforcement learning with
+experience replay. International Multiconference on Computer Science and Information Technology, pages
+305–315, 2007.
+Huizhen Yu, A Rupam Mahmood, and Richard S Sutton. On generalized Bellman equations and temporal-
+difference learning. Journal of Machine Learning Research, 19(1):1864–1912, 2018.
+13
+
+A
+M Operator Details
+In Section 5, we defined a linear operator Bt, where
+(BtX)(s, a) = Eµ
+�
+βtX(St, At)
+�� (S0, A0) = (s, a)
+�
+,
+(6)
+such that the expected-value version of our M operator,
+(MQ)(s, a) = Q(s, a) + Eµ
+� ∞
+�
+t=0
+γtβtδπ
+t
+����� (S0, A0) = (s, a)
+�
+(4)
+= Q(s, a) +
+∞
+�
+t=0
+γtEµ
+�
+βtδπ
+t
+�� (S0, A0) = (s, a)
+�
+,
+(5)
+is element-wise equivalent to the vector version,
+MQ = Q +
+∞
+�
+t=0
+γtBt(TπQ − Q).
+(7)
+We claimed that each element of Bt must have the form
+Bt((s, a), (s′, a′)) = Pr
+µ ((St, At) = (s′, a′) | (S0, A0) = (s, a)) × Eµ
+�
+βt
+�� (S0, A0) = (s, a), (St, At) = (s′, a′)
+�
+,
+(8)
+with (s, a) as the row index and (s′, a′) as the column index. This is because multiplying this matrix Bt
+with a vector X results in the same operation as the weighted expected value in Eq. (5):
+�
+s′,a′
+Bt((s, a), (s′, a′))X(s′, a′) = Eµ
+�
+Eµ
+�
+βt
+�� (S0, A0) = (s, a), (St, At)
+�
+· X(St, At)
+����� (S0, A0) = (s, a)
+�
+= Eµ
+�
+Eµ
+�
+βtX(St, At)
+�� (S0, A0) = (s, a), (St, At)
+� ���� (S0, A0) = (s, a)
+�
+= Eµ
+�
+βtX(St, At)
+�� (S0, A0) = (s, a)
+�
+.
+(14)
+So, when X is the expected TD error TπQ − Q, Eq. (7) becomes Eq. (5) exactly.
+M is a contraction mapping whenever βt ≤ βt−1ρt for all t (Condition 1), which Theorem 2 establishes.
+As we discussed in Section 5.3, violating this condition can sometimes cause M to no longer contract, even
+with on-policy updates. We can see one plausible reason for this by refactoring the definition of M. Let
+qt := Q(St, At) and vt := �
+a′∈A π(a′|St)Q(St, a′), so δπ
+t = Rt + γvt+1 − qt. Further, assume the following
+expectations are conditioned on (S0, A0) = (s, a). Eq. (4) is equivalent to
+(MQ)(s, a) = q0 + Eµ
+� ∞
+�
+t=0
+γtβt(Rt + γvt+1 − qt)
+�
+= q0 + Eµ
+� ∞
+�
+t=0
+γtβtRt +
+∞
+�
+t=1
+γtβt−1vt −
+∞
+�
+t=0
+γtβtqt
+�
+= Eµ
+� ∞
+�
+t=0
+γtβtRt +
+∞
+�
+t=1
+γtβt−1vt −
+∞
+�
+t=1
+γtβtqt
+�
+= Eµ
+� ∞
+�
+t=0
+γtβtRt
+�
++ Eµ
+� ∞
+�
+t=1
+γt(βt−1vt − βtqt)
+�
+= Eµ
+� ∞
+�
+t=0
+γtβtRt
+�
++ Eµ
+� ∞
+�
+t=1
+γt(βt−1ρt − βt)qt
+�
+,
+(15)
+14
+
+and we discussed in Section 5.3 that these two terms represent a biased return estimate and an infinite sum
+of weighted value-function bootstraps, respectively. In particular, this can be problematic if βt > βt−1ρt
+because the corresponding bootstrap’s weight becomes negative, causing it to get subtracted from the return
+estimate.
+B
+Additional Proofs
+B.1
+Proof of Lemma 1
+Lemma 1. Qπ is a fixed point of M; the difference between MQ and Qπ is given by
+MQ − Qπ = Z(Q − Qπ),
+(16)
+where Z := �∞
+t=1 γt(Bt−1Pπ − Bt).
+Proof. It is evident from Eq. (7) that Qπ is a fixed point of M because TπQπ − Qπ = 0, and so MQπ = Qπ.
+Therefore,
+MQ − Qπ = MQ − MQπ
+= Q +
+∞
+�
+t=0
+γtBt(TπQ − Q) − Qπ −
+∞
+�
+t=0
+γtBt(TπQπ − Qπ)
+= Q − Qπ +
+∞
+�
+t=0
+γtBt(TπQ − TπQπ) −
+∞
+�
+t=0
+γtBt(Q − Qπ)
+=
+∞
+�
+t=0
+γtBt(TπQ − TπQπ) −
+∞
+�
+t=1
+γtBt(Q − Qπ)
+=
+∞
+�
+t=0
+γt+1BtPπ(Q − Qπ) −
+∞
+�
+t=1
+γtBt(Q − Qπ)
+=
+� ∞
+�
+t=0
+γt+1BtPπ −
+∞
+�
+t=1
+γtBt
+�
+(Q − Qπ)
+=
+� ∞
+�
+t=1
+γt(Bt−1Pπ − Bt)
+�
+(Q − Qπ)
+= Z(Q − Qπ),
+which is the desired result.
+B.2
+Proof of Lemma 2
+Lemma 2. If Condition 1 holds, then Z has nonnegative elements and its row sums obey Z1 ≤ γ.
+Proof. Define the linear operator Dt := Bt−1Pπ − Bt and notice that Z = �∞
+t=1 γtDt. We will show that Dt
+comprises only nonnegative elements, and therefore so does Z. For any X ∈ Rn, observe that
+(DtX)(s, a) = Eµ
+�
+βt−1
+�
+St∈S
+�
+At∈A
+P(St|Ft−1)π(At|St)X(St, At)
+����� (S0, A0) = (s, a)
+�
+− Eµ
+�
+βtX(St, At)
+�� (S0, A0) = (s, a)
+�
+15
+
+= Eµ
+�
+βt−1
+�
+St∈S
+�
+At∈A
+P(St|Ft−1)π(At|St)X(St, At)
+����� (S0, A0) = (s, a)
+�
+− Eµ
+� �
+St∈S
+�
+At∈A
+P(St|Ft−1)µ(At|St)βtX(St, At)
+����� (S0, A0) = (s, a)
+�
+= Eµ
+� �
+St∈S
+P(St|Ft−1)
+�
+At∈A
+�
+π(At|St)βt−1 − µ(At|St)βt
+�
+X(St, At)
+����� (S0, A0) = (s, a)
+�
+.
+(17)
+Since we assumed that βt ≤ βt−1ρt in Condition 1, we have π(At|St)βt−1 − µ(At|St)βt ≥ 0, which implies
+that Dt ≥ 0. Furthermore, this holds for all t ≥ 1, so Z ≥ 0 follows immediately.
+To complete the proof, we show that the row sums of Z are bounded by γ. Recall that Pπ1 = 1. Hence,
+Z1 =
+∞
+�
+t=1
+γt(Bt−1Pπ − Bt)1
+=
+∞
+�
+t=1
+γt(Bt−11 − Bt1)
+=
+∞
+�
+t=0
+γt+1Bt1 −
+∞
+�
+t=1
+γtBt1
+= γ1 +
+∞
+�
+t=1
+γt+1Bt1 −
+∞
+�
+t=1
+γtBt1
+= γ1 − (1 − γ)
+∞
+�
+t=1
+γtBt1
+≤ γ1,
+(18)
+because Bt ≥ 0, ∀ t ≥ 1.
+B.3
+Proof of Theorem 3
+Theorem 3. Consider a sequence of target policies (πi)i≥0 and a sequence of arbitrary behavior policies
+(µi)i≥0. Let Q0 be an arbitrary vector in Rn and define the sequence Qi+1 := MiQi, where Mi is the
+operator defined by Eq. (10). Assume that (πi)i≥0 is greedy in the limit, and let ϵi ≥ 0 be the smallest
+constant such that TπiQi ≥ TQi − ϵi∥Qi∥1. If Condition 1 holds for all i, then
+∥MiQi − Q∗∥ ≤ γ∥Qi − Q∗∥ +
+ϵi
+1 − γ ∥Qi∥,
+(11)
+and, consequently, lim
+i→∞ Qi = Q∗.
+Proof. We first derive the following upper bound:
+TπiQi − TQ∗ = γPπiQi − γ max
+π
+PπQ∗ ≤ γPπi(Qi − Q∗).
+(19)
+From Eq. (10) and because Ci has nonnegative entries, we can deduce that
+MiQi − Q∗ = (I − Ci)(Qi − Q∗) + Ci(TπiQi − Q∗)
+(20)
+= (I − Ci)(Qi − Q∗) + Ci(TπiQi − TQ∗)
+≤ (I − Ci)(Qi − Q∗) + γCiPπi(Qi − Q∗)
+= Zi(Qi − Q∗),
+(21)
+16
+
+where Zi := I − Ci(I − γPπi). Notice that Zi is analogous to the matrix Z in Eq. (9) because, for policies
+πi and µi,
+I − Ci(I − γPπi) = I +
+∞
+�
+t=0
+γtBt(γPπi − I)
+= I +
+∞
+�
+t=0
+γt+1BtPπi −
+∞
+�
+t=0
+γtBt
+=
+∞
+�
+t=1
+γtBt−1Pπi −
+∞
+�
+t=1
+γtBt
+=
+∞
+�
+t=1
+γt(Bt−1Pπi − Bt).
+(22)
+Next, we derive the following lower bound:
+TQi − TQ∗ ≥ Tπ∗Qi − TQ∗ = γPπ∗(Qi − Q∗).
+(23)
+Additionally, for each policy πi, there exists some ϵi ≥ 0 such that TπiQi ≥ TQi − ϵi∥Qi∥1 (recall that we
+defined ϵi to be as small as possible). Starting again from Eq. (20), and noting that the elements of Ci are
+nonnegative, we obtain
+MiQi − Q∗ ≥ (I − Ci)(Qi − Q∗) + Ci(TQi − Q∗) − ϵi∥Qi∥Ci1
+= (I − Ci)(Qi − Q∗) + Ci(TQi − TQ∗) − ϵi∥Qi∥Ci1
+≥ (I − Ci)(Qi − Q∗) + γCiPπ∗(Qi − Q∗) − ϵi∥Qi∥Ci1
+= Z∗
+i (Qi − Q∗) − ϵi∥Qi∥Ci1,
+(24)
+where we have defined Z∗
+i := I − Ci(I − γPπ∗). By Lemma 2, since we assumed Condition 1 holds, both Zi
+and Z∗
+i have nonnegative elements and their row sums are bounded by γ. Therefore, when MiQi − Q∗ ≥ 0,
+Eq. (21) implies
+∥MiQi − Q∗∥ ≤ γ∥Qi − Q∗∥,
+(25)
+because element-wise inequality for nonnegative matrices implies the inequality holds also for their norms.
+When MiQi − Q∗ ≤ 0, we must use Eq. (24) and multiply both sides by −1 to get nonnegative matrices,
+giving
+∥MiQi − Q∗∥ ≤ γ∥Qi − Q∗∥ + ϵi∥Qi∥∥Ci∥
+≤ γ∥Qi − Q∗∥ +
+ϵi
+1 − γ ∥Qi∥,
+(26)
+because ∥Ci∥ ≤ �∞
+t=0 γt∥Pπi∥t = (1 − γ)−1. Since Eq. (26) is looser than Eq. (25), its bound holds in the
+worst case. It remains to show that this bound implies convergence to Q∗. Observe that
+γ∥Qi − Q∗∥ +
+ϵi
+1 − γ ∥Qi∥ ≤ γ∥Qi − Q∗∥ +
+ϵi
+1 − γ (∥Qi − Q∗∥ + ∥Q∗∥)
+=
+�
+γ +
+ϵi
+1 − γ
+�
+∥Qi − Q∗∥ +
+ϵi
+1 − γ ∥Q∗∥.
+(27)
+Our assumption of greedy-in-the-limit policies tells us that ϵi → 0 as i → ∞; thus, there must exist some
+iteration i∗ such that ϵi ≤ 1
+2(1 − γ)2, ∀ i ≥ i∗. Therefore, for i ≥ i∗,
+∥MiQi − Q∗∥ ≤ 1 + γ
+2
+∥Qi − Q∗∥ +
+ϵi
+1 − γ ∥Q∗∥.
+(28)
+If γ < 1, then 1
+2(1 + γ) < 1, and since ∥Q∗∥ is finite, we conclude that ∥Qi − Q∗∥ → 0 as i → ∞.
+17
+
+C
+Examples of Divergence
+C.1
+Counterexample 4: Off-Policy Truncated IS
+Our definitions of π and µ give us
+Pπ =
+�p
+1 − p
+p
+1 − p
+�
+,
+Pµ = 1
+2
+�1
+1
+1
+1
+�
+.
+(29)
+Recall that we assumed λ = 1. We define the following constant, using the definition of βt for Truncated IS:
+β(1)
+t
+:= E
+�
+βt
+�� (St, At) = (s, a1)
+�
+=
+�
+Ft
+Prµ(Ft | (St, At) = (s, a1)) · min
+�
+1, Prπ(Ft)
+Prµ(Ft)
+�
+=
+�
+Ft−1
+Prµ(Ft−1) min
+�
+1, Prπ(Ft−1) · p
+Prµ(Ft−1) · 1
+2
+�
+(30)
+=
+�
+Ft−1
+min (Prµ(Ft−1), 2p · Prπ(Ft−1))
+=
+�
+Ft−1
+min
+�
+1
+2t−1 , 2p · Prπ(Ft−1)
+�
+.
+(31)
+Eq. (30) is justified because the conditional probability of a trajectory ending in action a1 is just the
+probability of Ft−1 under µ, due to the 1-state (memoryless) MDP. We can simplify β(1)
+t
+further by using
+the binomial theorem to calculate Prπ(Ft−1) = pk(1 − p)t−1−k, where k ∈ [0, t − 1] is the number of times
+a1 is taken in Ft−1. There are
+�t−1
+k
+�
+trajectories with this same probability. Therefore,
+β(1)
+t
+=
+�
+Ft−1
+min
+�
+1
+2t−1 , 2p · Prπ(Ft−1)
+�
+=
+t−1
+�
+k=0
+�t − 1
+k
+�
+min
+�
+1
+2t−1 , 2p · pk(1 − p)t−1−k
+�
+.
+(32)
+Likewise, we can compute β(2)
+t
+by swapping p and 1 − p above. Let ⊙ denote element-wise multiplication.
+Using the fact that P t
+µ = Pµ, ∀ t ≥ 1, it follows that
+Bt = P t
+µ ⊙
+�
+β(1)
+t
+β(2)
+t
+β(1)
+t
+β(2)
+t
+�
+= 1
+2
+�
+β(1)
+t
+β(2)
+t
+β(1)
+t
+β(2)
+t
+�
+.
+(33)
+Using a computer program to calculate Z, assuming that p = 0.6 and γ = 0.94, we obtain
+Z =
+∞
+�
+t=1
+γt(Bt−1Pπ − Bt) ≈
+�0.704
+−0.436
+0.704
+−0.436
+�
+.
+(34)
+Therefore, ∥Z∥ ≈ 1.14, which is not a contraction, and the norm continues to increase for p > 0.6 or γ > 0.94.
+18
+
+C.2
+Counterexample 5: On-Policy Binary Traces
+The policy π is uniform random, so we have
+Pπ = 1
+2
+�1
+1
+1
+1
+�
+.
+(35)
+Let ⊙ denote element-wise multiplication. Because βt = 1 only when the trajectory Ft terminates in (s, a1)
+and βt = 0 otherwise, and since P t
+π = Pπ, ∀ t ≥ 1, we also have
+Bt = P t
+π ⊙
+�1
+0
+1
+0
+�
+= 1
+2
+�1
+0
+1
+0
+�
+.
+(36)
+Using a computer program to calculate Z, assuming that γ = 2
+3, we obtain
+Z =
+∞
+�
+t=1
+γt(Bt−1Pπ − Bt) = 1
+3
+�−1
+2
+−1
+2
+�
+.
+(37)
+Therefore, ∥Z∥ = 1, which is not a contraction, and the norm continues to increase for γ > 2
+3.
+D
+Implementation of Trajectory-Aware Eligibility Traces
+The implementation of trajectory-aware methods is closely related to that of backward-view TD(λ) in the
+tabular setting (see, e.g., Sutton and Barto, 1998, Chapter 7.3). On each timestep, an environment interaction
+is conducted according to the behavior policy µ. Then, the eligibilities for previously visited state-action
+pairs are modified, the eligibility for the current state-action pair is incremented, and the current TD error is
+applied to all state-action pairs in proportion to their eligibilities. The only difference in the trajectory-aware
+case is that the eligibilities are not modified by simply multiplying a constant decay factor γλ.
+Arbitrary, trajectory-dependent traces β(Ft), as studied in our theoretical results, can be complicated to
+implement. This stems from the fact that the timestep t in the M operator is defined relative to when
+the updated state-action pair was taken. In other words, each state-action pair (Sk, Ak) “disagrees” on the
+start of the current trajectory, generating its update from the unique sub-trajectory (Sk, Ak), . . . , (St, At).
+Implementing coefficients of this form would be possible using the general update
+Q(Sk, Ak) ← Q(Sk, Ak) + αγt−kβ((Sk, Ak), . . . , (St, At))δπ
+t ,
+(38)
+where α ∈ (0, 1] is the stepsize, but this would require repeatedly slicing the list of visited state-action
+pairs (S0, A0), . . . , (St, At). While this is certainly feasible, it does not easily accommodate vectorization or
+parallelization.
+Fortunately, this level of generality is rarely needed in practice, and specific optimizations can be made
+depending on the functional form of β. For example, Truncated IS defines β to be a pure function of the IS
+estimate Πt, which is useful because per-decision eligibility traces can be used to efficiently generate the IS
+estimates for every state-action pair visited during the episode. We demonstrate how this can be done in
+pseudocode (see Algorithm 1).
+Recursive methods like Recursive Retrace and RBIS, where βt explicitly depends on βt−1, require only two
+minor changes compared to Algorithm 1 for their implementations. These changes, which we highlight in blue
+for RBIS in Algorithm 2, correspond to the fact that the dynamic array Y is now used to store the previous
+trace βt−1 rather than the previous IS estimate Πt−1 at each timestep. The computational requirements for
+the methods remain nearly identical. The implementation for Recursive Retrace easily follows by changing
+line 10 of Algorithm 2 to
+Y (k) ← λ min(1, Y (k) · ρt).
+(39)
+19
+
+E
+Experiment Details and Learning Curves
+We conducted a grid search to find the best stepsize α for every λ-value for the four off-policy methods we
+evaluated in the Bifurcated Gridworld (Section 6). Using a training set of 1,000 trials, we searched over
+λ ∈ {0, 0.1, . . . , 1} and α ∈ {0.1, 0.3, 0.5, 0.7, 0.9}, for a total of 55 hyperparameter combinations. At the
+start of each trial, the initial value function Q was sampled from a zero-mean Gaussian distribution with
+standard deviation σ = 0.01. We trained each agent for 3,000 timesteps, allowing extra time to complete the
+final episode. We then generated learning curves by plotting the 100-episode moving average of these returns
+as a function of the number of timesteps and calculated their AUCs. In Table 1, we report the stepsize α
+that led to the highest average AUC for each λ-value. Then, using a separate test set of 1,000 trials to avoid
+bias in the search results, these α-values were used to generate the learning curves in Figure 3. The AUCs
+for these learning curves were finally used in the creation of the λ-sweep plot (Figure 2).
+Table 1:
+The best stepsizes found by our grid search in the Bifurcated Gridworld.
+λ
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Retrace
+0.9
+0.9
+0.9
+0.9
+0.9
+0.9
+0.9
+0.9
+0.7
+0.7
+0.5
+Truncated IS
+0.9
+0.9
+0.9
+0.9
+0.9
+0.9
+0.7
+0.5
+0.5
+0.5
+0.3
+Recursive Retrace
+0.9
+0.9
+0.9
+0.9
+0.9
+0.9
+0.9
+0.9
+0.7
+0.5
+0.5
+RBIS
+0.9
+0.9
+0.9
+0.9
+0.9
+0.7
+0.7
+0.7
+0.7
+0.7
+0.5
+20
+
+0
+500
+1000
+1500
+2000
+2500
+3000
+Timesteps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Discounted Return
+Bifurcated Gridworld ( = 0)
+Retrace
+Truncated IS
+Recursive Retrace
+RBIS
+0
+500
+1000
+1500
+2000
+2500
+3000
+Timesteps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Discounted Return
+Bifurcated Gridworld ( = 0.1)
+Retrace
+Truncated IS
+Recursive Retrace
+RBIS
+0
+500
+1000
+1500
+2000
+2500
+3000
+Timesteps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Discounted Return
+Bifurcated Gridworld ( = 0.2)
+Retrace
+Truncated IS
+Recursive Retrace
+RBIS
+0
+500
+1000
+1500
+2000
+2500
+3000
+Timesteps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Discounted Return
+Bifurcated Gridworld ( = 0.3)
+Retrace
+Truncated IS
+Recursive Retrace
+RBIS
+0
+500
+1000
+1500
+2000
+2500
+3000
+Timesteps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Discounted Return
+Bifurcated Gridworld ( = 0.4)
+Retrace
+Truncated IS
+Recursive Retrace
+RBIS
+0
+500
+1000
+1500
+2000
+2500
+3000
+Timesteps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Discounted Return
+Bifurcated Gridworld ( = 0.5)
+Retrace
+Truncated IS
+Recursive Retrace
+RBIS
+0
+500
+1000
+1500
+2000
+2500
+3000
+Timesteps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Discounted Return
+Bifurcated Gridworld ( = 0.6)
+Retrace
+Truncated IS
+Recursive Retrace
+RBIS
+0
+500
+1000
+1500
+2000
+2500
+3000
+Timesteps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Discounted Return
+Bifurcated Gridworld ( = 0.7)
+Retrace
+Truncated IS
+Recursive Retrace
+RBIS
+0
+500
+1000
+1500
+2000
+2500
+3000
+Timesteps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Discounted Return
+Bifurcated Gridworld ( = 0.8)
+Retrace
+Truncated IS
+Recursive Retrace
+RBIS
+0
+500
+1000
+1500
+2000
+2500
+3000
+Timesteps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Discounted Return
+Bifurcated Gridworld ( = 0.9)
+Retrace
+Truncated IS
+Recursive Retrace
+RBIS
+0
+500
+1000
+1500
+2000
+2500
+3000
+Timesteps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Discounted Return
+Bifurcated Gridworld ( = 1)
+Retrace
+Truncated IS
+Recursive Retrace
+RBIS
+Figure 3:
+Learning curves for the λ-values we tested in the Bifurcated Gridworld environment. The dashed
+black line indicates the optimal discounted return for this problem.
+21
+
+Algorithm 1 Truncated Importance Sampling
+1: Input: value function Q, stepsize α ∈ (0, 1]
+2: for each episode do
+3:
+Reset environment and observe state S0
+4:
+Reset dynamic array Y
+5:
+repeat {for t = 0, 1, 2, . . . }
+6:
+Take action At ∼ µ(·|St), receive reward Rt, and observe next state St+1
+7:
+ρt = π(At|St)
+µ(At|St)
+8:
+δt =
+�
+�
+�
+Rt − Q(St, At)
+if St+1 is terminal
+Rt − Q(St, At) + γ �
+a′∈A π(a′|St+1)Q(St+1, a′)
+else
+9:
+for k = 0, . . . , t − 1 do
+10:
+Y (k) ← Y (k) · ρt
+11:
+end for
+12:
+Y (t) ← 1
+13:
+for k = 0, . . . , t do
+14:
+z ← (γλ)t−k min(1, Y (k))
+15:
+Q(Sk, Ak) ← Q(Sk, Ak) + αzδt
+16:
+end for
+17:
+until St+1 is terminal
+18: end for
+Algorithm 2 Recency-Bounded Importance Sampling (RBIS)
+1: Input: value function Q, stepsize α ∈ (0, 1]
+2: for each episode do
+3:
+Reset environment and observe state S0
+4:
+Reset dynamic array Y
+5:
+repeat {for t = 0, 1, 2, . . . }
+6:
+Take action At ∼ µ(·|St), receive reward Rt, and observe next state St+1
+7:
+ρt = π(At|St)
+µ(At|St)
+8:
+δt =
+�
+�
+�
+Rt − Q(St, At)
+if St+1 is terminal
+Rt − Q(St, At) + γ �
+a′∈A π(a′|St+1)Q(St+1, a′)
+else
+9:
+for k = 0, . . . , t − 1 do
+10:
+Y (k) ← min(λt−k, Y (k) · ρt)
+11:
+end for
+12:
+Y (t) ← 1
+13:
+for k = 0, . . . , t do
+14:
+z ← γt−kY (k)
+15:
+Q(Sk, Ak) ← Q(Sk, Ak) + αzδt
+16:
+end for
+17:
+until St+1 is terminal
+18: end for
+22
+
diff --git a/edFIT4oBgHgl3EQfpCt4/content/tmp_files/load_file.txt b/edFIT4oBgHgl3EQfpCt4/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ed1c91c0b9f6930cec613af35287575c642a65b5
--- /dev/null
+++ b/edFIT4oBgHgl3EQfpCt4/content/tmp_files/load_file.txt
@@ -0,0 +1,855 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf,len=854
+page_content='Trajectory-Aware Eligibility Traces for  Off-Policy Reinforcement Learning  Brett Daley∗1,2, Martha White1,2,3, Christopher Amato4, and Marlos C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Machado1,2,3  1Department of Computing Science, University of Alberta 2Alberta Machine Intelligence Institute  3CIFAR Canada AI Chair 4Khoury College of Computer Sciences, Northeastern University  Abstract  Off-policy learning from multistep returns is crucial for sample-efficient reinforcement learning, but coun-  teracting off-policy bias without exacerbating variance is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Classically, off-policy bias is cor-  rected in a per-decision manner: past temporal-difference errors are re-weighted by the instantaneous  Importance Sampling (IS) ratio after each action via eligibility traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Many off-policy algorithms rely  on this mechanism, along with differing protocols for cutting the IS ratios to combat the variance of the  IS estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Unfortunately, once a trace has been fully cut, the effect cannot be reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This has led  to the development of credit-assignment strategies that account for multiple past experiences at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  These trajectory-aware methods have not been extensively analyzed, and their theoretical justification  remains uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In this paper, we propose a multistep operator that can express both per-decision  and trajectory-aware methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We prove convergence conditions for our operator in the tabular setting,  establishing the first guarantees for several existing methods as well as many new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Finally, we intro-  duce Recency-Bounded Importance Sampling (RBIS), which leverages trajectory awareness to perform  robustly across λ-values in an off-policy control task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  1  Introduction  Reinforcement learning concerns an agent interacting with its environment through trial and error to max-  imize its expected cumulative reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' One of the great challenges of reinforcement learning is the temporal  credit assignment problem (Sutton, 1984): upon receiving a reward, which past actions should be held respon-  sible and, hence, be reinforced?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Basic temporal-difference (TD) methods assign credit to the immediately  taken action (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', Watkins, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Rummery and Niranjan, 1994), bootstrapping from previous experience  to learn long-term dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This process requires a large number of repetitions to generate effective  behaviors from rewards, motivating research into multistep return estimation in which credit is distributed  among multiple past actions according to some eligibility rule (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', Sutton, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  One challenge of multistep estimators is that they generally have higher variance than 1-step estima-  tors (Kearns and Singh, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This is exacerbated in the off-policy setting, where environment interaction  is conducted according to a behavior policy that differs from the target policy for which returns are being  estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The discrepancy between the two policies manifests mathematically as bias in the return estima-  tion, which can be detrimental to learning if left unaddressed (Precup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Despite these challenges,  off-policy learning is important for exploration and sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The canonical bias-correction tech-  nique is Importance Sampling (IS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Kahn and Harris, 1951), wherein the bias due to the differing policies is  ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Contact: brett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='daley@ualberta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='11321v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='LG]  26 Jan 2023  eliminated by the product of their probability ratios (Precup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Although IS theoretically resolves  the off-policy bias, it can suffer from extreme variance that makes it largely impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Directly managing the variance of the IS estimator has been a fruitful avenue for developing efficient off-  policy algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Past work has focused on modifying the individual IS ratios to reduce the variance of  the full update: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', Tree Backup (Precup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2000), Qπ(λ) (Harutyunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2016), Retrace (Munos  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2016), ABQ (Mahmood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2017), and C-trace (Rowland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' All of these methods can  be implemented online with per-decision rules (Precup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2000) that determine how much to reduce, or  cut, the IS ratio according to the current state-action pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The re-weighted TD error is then broadcast to  previous experiences using eligibility traces (Barto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Sutton, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The decisions made by these  algorithms are Markov in the sense that each iterative off-policy correction depends on only the current  state-action pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' One issue with this is that it can lead to suboptimal decisions, since fully cutting a trace  cannot be reversed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In contrast, a trajectory-aware method can examine an entire sequence of past  state-action pairs to make globally better decisions regarding credit assignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' for example, when a specific  transition yields a high IS ratio, a trajectory-aware method can choose to not cut the trace if the product of  all previous IS ratios remains small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Indeed, some existing off-policy methods already conduct offline bias correction in a trajectory-aware man-  ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Perhaps the simplest example is Truncated IS, where the IS ratio products are pre-calculated offline  and then clipped to some finite value (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' More recently, Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (2016) suggested a recur-  sive variant of Retrace that automatically relaxes the clipping bound when its historical trace magnitude  becomes small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' the authors conjectured that this could lead to faster learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' No theoretical analysis has  been conducted on trajectory-aware algorithms such as these;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' their convergence properties are unknown,  and the space of possible algorithms has not yet been fully explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  To better understand these algorithms, and to support new discoveries of efficient algorithms, we introduce  a unifying theoretical perspective on per-decision and trajectory-aware off-policy corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We propose a  multistep operator that accounts for arbitrary dependencies on past experiences, significantly generalizing the  per-decision R operator introduced by Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We prove that our operator converges for policy  evaluation and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In the latter case, we remove the assumptions of increasingly greedy policies and pes-  simistic initialization used by Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (2016), which has implications for per-decision methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Finally,  we derive a new method from our theory, Recency-Bounded Importance Sampling (RBIS), which performs  favorably to other trajectory-aware methods across a wide range of λ-values in an off-policy control task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  2  Preliminaries  We consider Markov Decision Processes (MDPs) of the form (S, A, P, R, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' S and A are finite sets of states  and actions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Letting ∆X denote the set of distributions over a set X, then P : S × A → ∆S is  the transition function, R: S × A → R is the reward function, and γ ∈ [0, 1) is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' A policy  π: S → ∆A determines an agent’s probability of selecting a given action in each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' A value function  Q: S × A → R represents the agent’s estimate of the expected return achievable from each state-action pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  For a policy π, we define the operator  (PπQ)(s, a) :=  �  s′∈S  �  a′∈A  P(s′|s, a)π(a′|s′)Q(s′, a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  As a shorthand, we represent value functions and the reward function as vectors in Rn, where n = |S × A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Linear operators such as Pπ can hence be interpreted as n × n square matrices that multiply these vectors,  with repeated application corresponding to exponentiation: P t  πQ = Pπ(P t−1  π  Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  In the policy evaluation setting, we seek to estimate the expected discounted returns for policy π, given  by Qπ := �∞  t=0 γtP t  πR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The value function Qπ is the unique fixed point of the Bellman operator TπQ :=  R + γPπQ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', it uniquely solves the Bellman equation TπQπ = Qπ (Bellman, 1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In the control setting,  we seek to estimate the expected returns Q∗ under the optimal policy π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Q∗ is the unique fixed point of the  2  Figure 1:  The Tightrope Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Starting from state s1, the agent must take a specific sequence of n  actions to receive +1 reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Bellman optimality operator (TQ)(s, a) := maxπ (TπQ)(s, a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', it uniquely solves the Bellman optimality  equation TQ∗ = Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We are particularly interested in the off-policy learning case, where trajectories of the  form (S0, A0), (S1, A1), (S2, A2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' are generated by interacting with the MDP using a behavior policy µ,  where µ ̸= π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We define the TD error for policy π at time t as  δπ  t := Rt + γ  �  a′∈A  π(a′|St+1)Q(St+1, a′) − Q(St, At),  where Rt := R(St, At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Let ρk := π(Ak|Sk)  µ(Ak|Sk) for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (2016) introduced the off-policy operator  (RQ)(s, a) := Q(s, a) + Eµ  � ∞  �  t=0  γt  �  t�  k=1  ck  �  δπ  t  ����� (S0, A0) = (s, a)  �  ,  (1)  where ck := c(Sk, Ak) ∈ [0, ρk] is a nonnegative coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We refer to the product �t  k=1 ck as the trace for  (s, a) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' If any ck < ρk, we say that the trace has been (partially) cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' If any ck = 0, then we have  fully cut the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' If the trace is fully cut at t = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', c1 = 0, then (RQ)(s, a) = Q(s, a) + E[δπ  0|(S0, A0) =  (s, a)] = R(s, a) + γE[�  a′∈A π(a′|S1)Q(S1, a′)|(S0, A0) = (s, a)], which is the standard 1-step bootstrap  target like in TD(0) (Sutton, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Notice that each ck is Markov, as it depends only on the state-action  pair (Sk, Ak) and is otherwise independent of the preceding trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In other words, the update for R  can be calculated per decision (Precup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2000), thereby permitting an efficient online implementation  with eligibility traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  3  Trajectory-Aware Eligibility Traces  While per-decision traces are convenient from a computational perspective, they require making choices  about how much to cut the trace without considering the effects of previous choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  This can lead to  suboptimal decisions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' for example, if the trace is cut by setting ck = 0 at some timestep, then the effect  cannot be reversed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Regardless of whatever new experiences are encountered by the agent, experiences  before time k will be ineligible for credit assignment, resulting in an opportunity cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In fact, this exact  phenomenon is why Watkins’ Q(λ) (Watkins, 1989) often learns more slowly than Peng’s Q(λ) (Peng and  Williams, 1996), even though the former avoids off-policy bias (Sutton and Barto, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Daley and Amato,  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Kozuno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The same effect (but to a lesser extent) impacts Tree Backup and Retrace, where  ck ≤ 1 always in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (1), implying that the traces for past experiences can never increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  We illustrate this phenomenon in a small, deterministic MDP that we call the Tightrope Problem (see  Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The environment consists of n sequential, non-terminal states with two actions a1, a2 available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  The agent starts in state s1 and advances from si to si+1 whenever it takes action a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' If i = n, then the  episode terminates and the agent receives +1 reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Taking action a2 in any state immediately terminates  the episode with no reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Clearly, the optimal policy is to execute a1 regardless of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Now consider the following off-policy learning scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Suppose the agent’s behavior policy µ is uniform  random, but the target policy π is ϵ-greedy with respect to a value function Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' For each state s, it follows  that π(a|s) = 1 − ϵ if a = argmaxa′ Q(s, a′) and π(a|s) = ϵ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We assume ϵ is small in the sense  that ϵ <  1  2, and that γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Suppose now that the agent successfully receives the +1 reward during an  3  episode, implying that it took action a1 on every timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We can compute the eligibility of the initial  state-action pair (s1, a1) as an expression in the number k of incorrect actions in the greedy policy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=',  where argmaxa′ Q(s, a′) ̸= a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Letting λ ∈ [0, 1] be a decay parameter, the standard IS estimator (which  does not cut traces when λ = 1) provides an eligibility of  �  λ1 − ϵ  1 / 2  �n−k �  λ  ϵ  1 / 2  �k  = λn[2(1 − ϵ)]n−k(2ϵ)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (2)  This value can be greater than 1 when k ≪ n, which suggests that the agent’s behavior should be heavily  reinforced when the greedy policy agrees closely with the optimal policy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' however, a per-decision method  like Retrace, which cuts traces without considering the full trajectory (see Section 4), ultimately assigns a  much lower eligibility:  �  λ min  �  1, 1 − ϵ  1 / 2  ��n−k�  λ min  �  1,  ϵ  1 / 2  ��k  = λn(2ϵ)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  The eligibility now decays monotonically for every suboptimal action in the greedy policy, illuminating how  per-decision trace cutting can lead to excessively small eligibilities, especially when ϵ is close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  This issue stems from the fact that Retrace is not aware of its past eligibilities, and it continues to decay them  even when they already represent an underestimate compared to IS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This issue is not unique to Retrace, as  it also affects other per-decision methods like Tree Backup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Instead, it would be better to have a trajectory-  aware method that can actively adapt its trace-cutting behavior based on the magnitude of past eligibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  One way to obtain a trajectory-aware method is to compute the exact IS product in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (2), and then make  adjustments to it to achieve certain properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', convergence and variance reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' For example,  Truncated IS (see Section 4) simply imposes a fixed bound on the IS estimator:  λn min  �  1, [2(1 − ϵ)]n−k(2ϵ)k�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (3)  Ignoring λ, Truncated IS reduces the eligibility only when it exceeds a pre-specified threshold, effectively  avoiding trace cuts when the true IS estimate is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In Section 6, we propose an algorithm, RBIS, which  achieves a similar effect using a recursive, time-decaying threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  As this example demonstrates, it can be advantageous to consider the agent’s past experiences to produce  better decisions regarding credit assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' One of our principal contributions is the proposal and analysis  of an off-policy operator M that encompasses this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Let Ft := (S0, A0), (S1, A1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' , (St, At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We  define M such that  (MQ)(s, a) := Q(s, a) + Eµ  � ∞  �  t=0  γtβtδπ  t  ����� (S0, A0) = (s, a)  �  ,  (4)  where βt := β(Ft) is a trace that generally depends on the history Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We define β0 := 1 to ensure that the  first TD error, δπ  0 , is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In Section 5, we characterize the values of βt for t ≥ 1 that lead to convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  The major analytical challenge of our operator—and its main novelty—is the complex dependence on the  sequence Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This makes the M operator difficult to analyze mathematically, as the terms in the series  1 + γβ1 + γ2β2 + · · · generally share no common factors that would allow a recursive formula for eligibility  traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Some off-policy methods, however, cannot be described by factored traces, and therefore removing  this assumption is necessary to understand existing algorithms (see Section 4), while also paving the way  for new credit-assignment methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In the special case where βt does factor into Markov coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=',  βt = �t  k=1 ck, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (4) reduces to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (1), taking us back to the per-decision setting studied by Munos  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' M, therefore, unifies per-decision and trajectory-aware methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  4  Unifying Off-Policy Algorithms  The operator M is a strict generalization of the previous operator considered for trace-based methods,  allowing us to express existing algorithms in this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We provide a non-exhaustive list of examples below  4  with the corresponding βt used in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' For brevity, let Πt := �t  k=1 ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Importance Sampling: βt = λtΠt (Kahn and Harris, 1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The standard approach for correcting off-  policy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Although it is the only unbiased estimator in this list (if λ = 1), it suffers from high variance,  making it difficult to utilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Qπ(λ): βt = λt (Harutyunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' A straightforward algorithm that decays the TD errors by a  fixed constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The algorithm does not require explicitly knowing µ, which is desirable, but can diverge if π  and µ differ too much (Harutyunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2016, Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Tree Backup: βt =�t  k=1 λπ(Ak|Sk) (Precup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' A method that automatically cuts traces accord-  ing to the product of probabilities under π, which forms a conservative lower bound on the IS estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Tree  Backup converges for any behavior policy µ, but it is not efficient since traces are cut excessively—especially  in the on-policy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Retrace: βt = �t  k=1 λ min(1, ρk) (Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' A convergent algorithm for arbitrary policies π  and µ that remains efficient in the on-policy case because it does not cut traces (if λ = 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' however, the fact  that βt never increases can cause the trace products to decay too quickly in practice (Mahmood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Rowland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  All of the above can be analyzed using a per-decision operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The next two, on the other hand, have  weightings based on the entire trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We use the theory for our general M operator to prove properties  about these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Recursive Retrace: βt =λ min(1, βt−1ρt) (Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' A modification to Retrace conjectured to  lead to faster learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' It clips large products of ratios, rather than individual ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Its convergence for  control is an open question, which we solve in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Truncated Importance Sampling: βt = λt min(1, Πt) (Ionides, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' A simple but effective method  to combat the variance of IS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Variations of this algorithm have been applied in the reinforcement learning  literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', Uchibe and Doya, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Wawrzy´nski and Pacut, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Wawrzy´nski, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2017),  but, to our knowledge, its convergence in an MDP setting has not been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='3, we show  that it can diverge in at least one off-policy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  5  Convergence Analysis  In this section, we study the convergence properties of the M operator for policy evaluation and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' It  will be convenient to re-express Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (4) in vector notation for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' To do this, let us first bring the  expectation inside the sum, by linearity of expectation:  (MQ)(s, a) = Q(s, a) +  ∞  �  t=0  γtEµ  �  βtδπ  t  �� (S0, A0) = (s, a)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (5)  To write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (5) in vector form, we define an operator Bt such that, for an arbitrary vector X in Rn,  (BtX)(s, a):=Eµ  �  βtX(St, At)  �� (S0, A0) = (s, a)  �  ,  (6)  allowing us to express the M operator as  MQ = Q +  ∞  �  t=0  γtBt(TπQ − Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (7)  Bt is a linear operator and hence can be represented as a matrix in Rn×n, the elements of which are  nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Each element of Bt, row-indexed by (s, a) and column-indexed by (s′, a′), has the form  Bt((s, a), (s′, a′)) = Pr  µ ((St, At)=(s′, a′)|(S0, A0)=(s, a)) × Eµ  �  βt  ��(S0, A0)=(s, a), (St, At)=(s′, a′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (8)  5  We justify this form in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Note that B0 = I, the identity matrix, because of our earlier definition  of β0 := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In the following sections, all inequalities involving vectors or matrices should be interpreted  element wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We let ∥X∥ := ∥X∥∞ for a matrix (or vector) X, which corresponds to the maximum absolute  row sum of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We also define 1 ∈ Rn to be the vector of ones, such that X1 gives the row sums of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='1  Convergence for Policy Evaluation  We start in the off-policy policy evaluation setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Specifically, our goal is to prove that the repeated  application of the M operator to an arbitrarily initialized vector Q ∈ Rn converges to Qπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' βt ≤ βt−1ρt, ∀ Ft, ∀ t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' If Condition 1 holds, then M is a contraction mapping with Qπ as its unique fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Consequently, limi→∞ MiQ = Qπ, ∀ Q ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In Lemma 1 (Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='1), we show that Qπ is a fixed point of M and that  MQ − Qπ = Z(Q − Qπ),  (9)  where Z := �∞  t=1 γt(Bt−1Pπ − Bt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In Lemma 2 (Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='2), we also show that Z ≥ 0 and Z1 ≤ γ  using the assumption that βt ≤ βt−1ρt, ∀ Ft, ∀ t ≥ 1 (Condition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Consequently, Z(Q − Qπ) is a vector  whose components each comprise a nonnegative-weighted combination of the components of Q − Qπ, where  the weights add up to at most γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This means that ∥MQ − Qπ∥ ≤ γ∥Q − Qπ∥, and M is a contraction  mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Its fixed point, Qπ, must therefore be unique by the Banach fixed-point theorem, implying that  limi→∞ MiQ = Qπ for every Q ∈ Rn when γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Given that the ratio  βt  βt−1 is bounded by ρt (Condition 1), the M operator converges to Qπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Intuitively,  we can think of this ratio as the effective per-decision factor at time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' convergence is guaranteed whenever  this factor is no greater than ρt, analogous to the convergence result for the R operator (Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=',  2016, Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Our theorem implies the existence of a space of convergent trajectory-aware algorithms,  because each trace βt can be chosen arbitrarily so long as it always satisfies the bound on this ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='2  Convergence for Control  We now consider the more challenging setting of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Given sequences of target policies (πi)i≥0 and  behavior policies (µi)i≥0, we aim to show that the sequence of value functions (Qi)i≥0 given by Qi+1 := MiQi  converges to Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Here, Mi is the M operator defined for πi and µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Compared to the convergence proof of the R operator (Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2016, Theorem 2), the main novelty  of our proof is the fact that the traces under M are not Markov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Consequently, we require new techniques  to establish bounds on Q − Q∗, since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (4) is not representable as an infinite geometric series and the  summation therefore does not have a closed-form expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We additionally relax two assumptions in the  previous work, on initialization of the value function and on increasing greediness of the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We require  only that the target policies become greedy in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We say that a sequence of policies is greedy in the limit  if TπiQi → TQi as i → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We discuss the significance of these relaxations to the assumptions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  First, let Ci := �∞  t=0 γtBt for the policies πi and µi, and write the M operator at iteration i as  MiQ = Q + Ci(TπiQ − Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (10)  We now present our convergence theorem for control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  6  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Consider a sequence of target policies (πi)i≥0 and a sequence of arbitrary behavior policies  (µi)i≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Let Q0 be an arbitrary vector in Rn and define the sequence Qi+1 := MiQi, where Mi is the  operator defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Assume that (πi)i≥0 is greedy in the limit, and let ϵi ≥ 0 be the smallest  constant such that TπiQi ≥ TQi − ϵi∥Qi∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' If Condition 1 holds for all i, then  ∥MiQi − Q∗∥ ≤ γ∥Qi − Q∗∥ +  ϵi  1 − γ ∥Qi∥,  (11)  and, consequently, lim  i→∞ Qi = Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Proof (sketch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' full proof in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We define matrices Zi and Z∗  i , which correspond to Z in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (9)  for target policies πi and π∗, respectively, and behavior policy µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We then derive the inequalities  Z∗  i (Qi − Q∗) − ϵi∥Qi∥Ci1 ≤ MiQi − Q∗ ≤ Zi(Qi − Q∗),  which together imply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Thus, Mi is nearly a contraction mapping with Q∗ as its unique fixed point,  excepting the influence of the O(∥Qi∥) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' However, the greedy-in-the-limit target policies guarantee that  ϵi → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Showing that ∥Qi∥ remains finite completes the proof because ∥Qi − Q∗∥ → 0 must follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  The convergence criteria for βt (Condition 1) is the same for both policy evaluation and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In fact,  the only additional assumption we need for control is the greedy-in-the-limit target policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Crucially, the  proof allows arbitrary behavior policies and an arbitrary value function initialization Q0, which we further  discuss in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='3  Examples of Convergence and Divergence  The generality of the M operator means that it provides convergence guarantees for a number of credit-  assignment methods that we did not discuss in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' These include variable or past-dependent λ-values  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', Watkins, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Singh and Sutton, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' All of these can be represented in a common  form and shown to satisfy Condition 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' convergence for policy evaluation and control for the instantiated  trajectory-aware operator follows as a corollary, since Condition 1 is sufficient to apply Theorems 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Any traces expressible in the form βt = �t  k=1 λ(Fk)ρk, λ(Fk) ∈ [0, 1], satisfy Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' βt = βt−1λ(Ft)ρt ≤ βt−1ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  In Section 4, we also discussed two existing trajectory-aware methods whose convergence is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We  show that Recursive Retrace satisfies our required condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Recursive Retrace satisfies Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' For Recursive Retrace, βt = ctβt−1, where ct = λmin  �  1  βt−1 , ρt  �  (Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2016, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This  means  βt = λ min  �  1  βt−1  , ρt  �  βt−1  = λ min (1, βt−1ρt)  ≤ βt−1ρt,  (12)  which is the bound required by Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Unfortunately, the traces for Truncated IS do not always satisfy the required bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Truncated IS may  violate Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We show this by providing a counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Recall that Truncated IS has βt = λt min(1, Πt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Assume a trajectory Ft such that Πt−1 = 2 and  1  2 < ρt < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (It is straightforward to define an MDP,  behavior policy, and target policy to create such a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=') Because ρt >  1  Πt−1 , then Πt = Πt−1ρt > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Thus, βt = λt and βt−1 = λt−1, and Condition 1 is violated because  βt  βt−1 = λ ̸≤ ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  7  Because Theorems 2 and 3 cannot be applied, the precise conditions under which Truncated IS converges  remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We do know Condition 1 is sufficient for convergence, but it is unlikely to be strictly  necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This is because our proofs of Theorems 2 and 3 use this assumption to guarantee that the matrix  Z in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (9) has nonnegative elements, making it straightforward to show that its row sums are sufficiently  bounded to guarantee that M is a contraction mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' However, M could remain a contraction mapping  even when Z has negative elements, so long as ∥Z∥ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' It could theoretically be the case for Truncated IS  that Z occasionally contains negative elements but ∥Z∥ is still bounded enough to permit convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Nevertheless, we are able to find at least one off-policy problem for which this is not true, implying that  certain initializations of the value function could ultimately cause Truncated IS to diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Counterexample 4 (Off-Policy Truncated IS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Consider Truncated IS with λ = 1, so βt = min(1, Πt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Assume the MDP has one state and two actions: S = {s} and A = {a1, a2}, the behavior policy µ is uniform  random, and π selects a1 with probability p ∈ (0, 1) and selects a2 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' When p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='6 and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='94,  then ∥Z∥ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  We provide details in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' note that many choices of p and γ make ∥Z∥ > 1, but we provide  specific ones for the counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Compared to the per-decision case, where Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (2016) showed that arbitrary trace cuts always  produce a convergent algorithm, this result is surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Why would the analogous result—in which we  ensure that βt ≤ Πt for all timesteps—not hold here?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' After all, clipping βt such that it never exceeds the IS  estimate Πt would be expected to simply incur bias in the return estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' For some insight, assume the  following expectations are conditioned on (S0, A0) = (s, a), and observe that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (4) is equivalent to  Eµ  � ∞  �  t=0  γtβtRt  �  + Eµ  � ∞  �  t=1  γt(βt−1ρt − βt)Q(St, At)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  We show the derivation in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The first term is a (partially) bias-corrected estimate of the dis-  counted return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The second term is a weighted combination of value-function bootstraps, whose weights are  nonnegative when Condition 1 is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' If the condition is violated on any timestep, then we may actually be  subtracting bootstraps from the return estimate, which does not seem sensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We believe this is related to  the root cause of divergence in Counterexample 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' however, it remains open whether Condition 1 is necessary  or merely sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  As our next counterexample example will demonstrate, this effect can even cause divergence in on-policy  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Counterexample 5 (On-Policy Binary Traces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Assume the MDP has one state and two actions: S = {s}  and A = {a1, a2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Define a trajectory-aware method such that βt = 1 if At = a1 and βt = 0 if At = a2  (without loss of generality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Assume π and µ are uniform random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' When γ ≥ 2  3, then ∥Z∥ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  We provide details in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Even though βt ≤ Πt = 1 always, we are able to produce a non-  contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The method either fully cuts a trace or does not cut it at all, producing backups that consist of  a sparse sum of on-policy TD errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' It is therefore surprising that divergence occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' For the same reason  we described above, the non-Markov nature of the trace appears to sometimes cause adverse bootstrap-  ping effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' in this instance, the ability to examine the contents of each trajectory allows the method to  strategically de-emphasize certain state-action pairs, ultimately producing a detrimental effect on learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Notice that Condition 1 is indeed violated in this case because there is always some chance that βt = 1 after  βt−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' If we add the restriction that βt−1 = 0 =⇒ βt = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', we permanently cut the traces, then  convergence is guaranteed by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='4  Discussion  In this section, we summarize our main theoretical contributions and their significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  We focused on  characterizing the contraction properties of the M operator, both for policy evaluation and control, in the  tabular setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' These results parallel those for the R operator underlying Retrace, where M is a strict  generalization of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' These results indicate that using fixed-point updates, like dynamic programming and  temporal difference learning updates, may have divergence issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' It does not, however, imply other algo-  rithms, such as gradient-based algorithms, cannot find these fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We show the fixed points still  exist and are unbiased, but that algorithms based on iterating with the M operator might diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Removal of the Markov assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Removing the Markov (per-decision) assumption of the R operator  (Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2016) to enable trajectory-aware eligibility traces was our primary goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' When the trace  factors are Markov, the operator Bt is independent of t, allowing the sum �∞  t=0 γtBt to be reduced to  �∞  t=0(γPcµ)t for a linear operator Pcµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The resulting geometric series can then be evaluated analytically, as  was done by Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In our proofs, we avoided the Markov assumption by directly analyzing  the infinite summation, which generally does not have a closed-form expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Our work is the first to do  this, establishing the first convergence guarantees for general trajectory-aware methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Arbitrary initialization of the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We permit any initialization of Q0 in the control set-  ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In contrast, Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (2016) made the assumption that Tπ0Q0 − Q0 ≥ 0 in order to produce a  lower bound on RiQi − Q∗, accomplished in practice by a pessimistic initialization of the value function:  Q0(s, a) = −∥R∥ / (1 − γ), ∀ (s, a) ∈ S × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Since R is a special case of our operator M where each trace  βt factors into Markov coefficients, we deduce as a corollary that Retrace and all other algorithms described  by R do not require pessimistic initialization for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Greedy-in-the-limit policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Our requirement of greedy-in-the-limit target policies in Theorem 3 is  less restrictive than the increasingly greedy policies proposed by Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  We need only  limi→∞ TπiQi = TQi, and we do not force the sequence of target policies to satisfy Pπi+1Qi+1 ≥ PπiQi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  This implies that the agent may target non-greedy policies for any finite period of time, as long as the policies  do eventually become arbitrarily close to the greedy policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' As a corollary, increasingly greedy policies are  not necessary for the optimal convergence of Retrace and other per-decision methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  6  Recency-Bounded Importance Sampling  Theorem 2 guarantees convergence to Qπ whenever Condition 1 holds, but we do not expect that all choices of  coefficients that satisfy this condition will perform well in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' At one extreme, if βt ≤ �t  k=1 λ min(1, ρk)  for every Ft, then we have a method that cuts coefficients more aggressively than Retrace does;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' it seems  unlikely that such a method would learn faster than Retrace, or other per-decision methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' At the other  extreme, when βt = Πt for every Ft, we recover the standard IS estimator, which suffers from high variance  and is often ineffectual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We therefore know that it is possible to have a method that preserves traces too  much, to the point of being detrimental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Thus, it is important to maintain some minimum efficiency by  avoiding unnecessary cuts, yet also important to control the overall variance of the traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Intuitively, we want something that falls between Retrace and IS in terms of trace cutting, in order to  quickly backpropagate credit while still managing the overall variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We further hypothesize that effective  trajectory-aware methods will first compute βt−1ρt—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', the maximum trace permitted by Condition 1—and  then apply some transformation that bounds its magnitude and manages the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This ensures that  traces are cut only as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  We propose one method, Recency-Bounded Importance Sampling (RBIS), which achieves this by cutting the  traces only when they exceed an exponentially decaying threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Specifically, we define  βt = min(λt, βt−1ρt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (RBIS)  9  It is easy to see that RBIS always converges, by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  RBIS satisfies Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' βt = min(λt, βt−1ρt) ≤ βt−1ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  For further insight, we unroll the recursion to obtain  min(λt, βt−1ρt) = min(λt, min(λt−1, βt−2ρt−1)ρt)  · ·  = min(λt, λt−1ρt, λt−2ρt−1ρt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' , Πt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (13)  RBIS effectively takes the minimum of all past, discounted n-step IS estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This reveals another property  of RBIS: its traces are never less than those of Retrace because  t�  k=1  λ min(1, ρk) ≤  t�  k=1  min(λ, ρk)  ≤ λt−j  t�  k=t−j+1  ρk, ∀ j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' , t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Since the inequality is true for all j, it is not possible for Retrace’s traces to exceed any of the arguments to  the minimum function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We have achieved exactly what we wanted earlier: a method that falls  somewhere between Retrace and IS in regard to trace cutting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This does not automatically mean that RBIS  will outperform Retrace, though, since preserving the magnitude of the trace βt too much can lead to high  variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' However, we do expect RBIS to perform well in decision-making problems in which a few critical  actions largely determine the long-term outcome of an episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In such scenarios, the agent’s bottleneck to  learning is its ability to assign meaningful credit to these critical actions over a potentially long time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  In order to test this empirically, we construct an environment called the Bifurcated Gridworld (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  This is a 5 × 5 deterministic gridworld with walls arranged such that two unequal-length paths from the  starting cell (S) to the goal cell (G) are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The agent may move up, down, left, or right;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' taking any  of these actions in the goal cell yields a reward of +1 and terminates the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The problem is discounted  (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9) to encourage the agent to learn the shorter path to the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Importantly, the action taken at the  bifurcation (B) solely determines which path the agent follows, and quickly assigning credit to this state is  paramount to learning the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  We compare RBIS against Retrace, Truncated IS, and Recursive Retrace when learning this task from off-  policy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Both behavior and target policies were ϵ-greedy with respect to the value function Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The  target policy used ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The behavior policy used a piecewise schedule: ϵ = 1 for the first 5 episodes and  then ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='2 afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The policies were updated only at the end of each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The agents used online  TD updates with eligibility traces (see Appendix D for pseudocode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' At the end of each episode, the policies  were updated, and then we evaluated the discounted return obtained by a near-greedy policy (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  The area under the curve (AUC) of each resulting learning curve was calculated, and the highest AUC  achieved over a grid search of stepsizes was plotted for each λ-value (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We averaged the results  over 1,000 independent trials and indicate the 95% confidence interval by the shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Appendix E  contains additional experiment details and the learning curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We also include the experiment code in the  supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  We make several observations regarding the results in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' First, the peak performance obtained by  RBIS is significantly higher than that of the other three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This is notable because both Truncated  IS and Recursive Retrace are also trajectory aware, indicating that different implementations of trajectory  awareness are beneficial to varying degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In particular, the preservation of long-term eligibilities is not  sufficient on its own to guarantee strong performance in general, as it appears that when and how much the  traces are cut are important considerations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The role of λ as a decay hyperparameter is evidently  critical for all methods to achieve their maximum performance, since λ = 1 never leads to the fastest learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  In fact, λ → 1 is especially catastrophic for Truncated IS, which we believe is related to the divergence issue  10  S  B  G  Retrace  Truncated IS  RBIS  Recursive  Retrace  Figure 2:  The Bifurcated Gridworld environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The agent starts in S and receives +1 reward for taking  any action in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The choice made at B greatly impacts the discounted return ultimately earned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We plot the  AUC of the learning curve obtained by four off-policy methods across the λ-spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The dashed horizontal  lines mark the highest AUC achieved by each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  identified in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Finally, Retrace degrades less for larger λ, likely because it cuts traces more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' It  would be interesting to develop a trajectory-aware method that obtains the robustness of RBIS but also  accounts for larger λ-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  7  Conclusion  In this work, we extended theory for per-decision eligibility traces to trajectory-aware traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This extension  allows us to consider a broader family of algorithms, with more flexibility in obtaining off-policy corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Specifically, we introduced the M operator, as a generalization of the R operator, and a sufficient condition  to ensure convergence under M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Using our general result, we established the first convergence guarantee  for an existing trajectory-aware method, Recursive Retrace, in the control setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We also showed that  Truncated IS may violate our condition and provided a counterexample showing that it can diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  We also proposed a new trajectory-aware method, RBIS, that demonstrates one instance of how trajectory  awareness can be utilized for faster learning in an off-policy control task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' RBIS is able to outperform the  other trajectory-aware methods that we tested in the Bifurcated Gridworld, suggesting that it possesses at  least one unique property that is beneficial for long-term, off-policy credit assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' It would be interesting  to search for additional beneficial properties in future work, in order to better characterize off-policy methods  that reliably lead to efficient and stable learning in challenging reinforcement learning environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  This work focused on convergence in expectation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' a natural next step is to extend this result to the stochastic  algorithms used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Previous results for TD learning rely primarily on the properties of the expected  update, with additional conditions on the noise in the update and appropriately annealed stepsizes (see  Bertsekas and Tsitsiklis, 1996, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Similar analysis should be applicable, given that we know the  expected update using the M operator is contraction mapping when Condition 1 is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  An important next step is extending these methods and results to function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Incorporating  these traces into deep reinforcement learning methods that rely on experience replay (Lin, 1992) should  be straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Multistep returns can be computed offline in the replay memory, and then randomly  sampled in minibatches to train the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Using a TD learning update, though, can suffer from  convergence issues under function approximation and off-policy learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' this has been previously resolved  by developing gradient-based updates (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Touati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' An important next step is to  11  develop a gradient-based trajectory-aware algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The contraction properties of the operator still impact  the quality of the solution, as has been shown to be the case for other off-policy approaches (Patterson  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The insights in this work, therefore, may provide insights on how to get quality solutions with  gradient-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  References  Andrew G Barto, Richard S Sutton, and Charles W Anderson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Neuronlike adaptive elements that can solve  difficult learning control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' IEEE Transactions on Systems, Man, and Cybernetics, pages 834–846,  1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Richard Bellman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Science, 153(3731):34–37, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Dimitri P Bertsekas and John N Tsitsiklis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Neuro-Dynamic Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Athena Scientific, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Brett Daley and Christopher Amato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Reconciling λ-returns with experience replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In Advances in Neural  Information Processing Systems, pages 1133–1142, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Anna Harutyunyan, Marc G Bellemare, Tom Stepleton, and R´emi Munos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Q(λ) with off-policy corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  In International Conference on Algorithmic Learning Theory, pages 305–320, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Edward L Ionides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Truncated importance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Journal of Computational and Graphical Statistics, 17  (2):295–311, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Herman Kahn and Theodore E Harris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Estimation of particle transmission by random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' National  Bureau of Standards: Applied Mathematics Series, 12:27–30, 1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Michael J Kearns and Satinder P Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Bias-variance error bounds for temporal difference updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In  Conference on Learning Theory, pages 142–147, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Tadashi Kozuno, Yunhao Tang, Mark Rowland, R´emi Munos, Steven Kapturowski, Will Dabney, Michal  Valko, and David Abel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Revisiting Peng’s Q(λ) for modern reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='00107,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Long-Ji Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Self-improving reactive agents based on reinforcement learning, planning and reaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Machine  Learning, 8:293–321, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Ashique Rupam Mahmood, Huizhen Yu, and Richard S Sutton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Multi-step off-policy learning without  importance sampling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' arXiv:1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='03006, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  R´emi Munos, Tom Stepleton, Anna Harutyunyan, and Marc G Bellemare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Safe and efficient off-policy  reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In Advances in Neural Information Processing Systems, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Andrew Patterson, Adam White, and Martha White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' A generalized projected Bellman error for off-policy  value estimation in reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Journal of Machine Learning Research, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Jing Peng and Ronald J Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Incremental multi-step Q-Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Machine Learning, 22:226–232, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Doina Precup, Richard S Sutton, and Satinder Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Eligibility traces for off-policy policy evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In  International Conference on Machine Learning, page 759–766, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Mark Rowland, Will Dabney, and R´emi Munos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Adaptive trade-offs in off-policy learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In International  Conference on Artificial Intelligence and Statistics, pages 34–44, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Gavin A Rummery and Mahesan Niranjan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  On-line Q-Learning using connectionist systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Technical  report, Cambridge University, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Satinder P Singh and Richard S Sutton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Reinforcement learning with replacing eligibility traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Machine  Learning, 22:123–158, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  12  Richard S Sutton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Temporal Credit Assignment in Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  PhD thesis, University of  Massachusetts Amherst, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Richard S Sutton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Learning to predict by the methods of temporal differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Machine Learning, 3(1):9–44,  1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Richard S Sutton and Andrew G Barto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Reinforcement Learning: An Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' MIT Press, 1st edition,  1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Richard S Sutton, Hamid Reza Maei, Doina Precup, Shalabh Bhatnagar, David Silver, Csaba Szepesv´ari,  and Eric Wiewiora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Fast gradient-descent methods for temporal-difference learning with linear function  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In International Conference on Machine Learning, pages 993–1000, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Ahmed Touati, Pierre-Luc Bacon, Doina Precup, and Pascal Vincent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Convergent Tree Backup and Retrace  with function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In International Conference on Machine Learning, pages 4955–4964, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Eiji Uchibe and Kenji Doya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Competitive-cooperative-concurrent reinforcement learning with importance  sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In International Conference on Simulation of Adaptive Behavior: From Animals and Animats,  pages 287–296, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Ziyu Wang, Victor Bapst, Nicolas Heess, Volodymyr Mnih, Remi Munos, Koray Kavukcuoglu, and Nando  de Freitas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Sample efficient actor-critic with experience replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In International Conference on Learning  Representations, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Christopher John Cornish Hellaby Watkins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Learning from Delayed Rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' PhD thesis, King’s College,  Cambridge, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Pawe�l Wawrzy´nski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Real-time reinforcement learning by sequential actor-critics and experience replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Neural  Networks, 22(10):1484–1497, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Pawe�l Wawrzy´nski and Andrzej Pacut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Truncated importance sampling for reinforcement learning with  experience replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' International Multiconference on Computer Science and Information Technology, pages  305–315, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Huizhen Yu, A Rupam Mahmood, and Richard S Sutton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' On generalized Bellman equations and temporal-  difference learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Journal of Machine Learning Research, 19(1):1864–1912, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  13  A  M Operator Details  In Section 5, we defined a linear operator Bt, where  (BtX)(s, a) = Eµ  �  βtX(St, At)  �� (S0, A0) = (s, a)  �  ,  (6)  such that the expected-value version of our M operator,  (MQ)(s, a) = Q(s, a) + Eµ  � ∞  �  t=0  γtβtδπ  t  ����� (S0, A0) = (s, a)  �  (4)  = Q(s, a) +  ∞  �  t=0  γtEµ  �  βtδπ  t  �� (S0, A0) = (s, a)  �  ,  (5)  is element-wise equivalent to the vector version,  MQ = Q +  ∞  �  t=0  γtBt(TπQ − Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (7)  We claimed that each element of Bt must have the form  Bt((s, a), (s′, a′)) = Pr  µ ((St, At) = (s′, a′) | (S0, A0) = (s, a)) × Eµ  �  βt  �� (S0, A0) = (s, a), (St, At) = (s′, a′)  �  ,  (8)  with (s, a) as the row index and (s′, a′) as the column index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This is because multiplying this matrix Bt  with a vector X results in the same operation as the weighted expected value in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (5):  �  s′,a′  Bt((s, a), (s′, a′))X(s′, a′) = Eµ  �  Eµ  �  βt  �� (S0, A0) = (s, a), (St, At)  �  X(St, At)  ����� (S0, A0) = (s, a)  �  = Eµ  �  Eµ  �  βtX(St, At)  �� (S0, A0) = (s, a), (St, At)  � ���� (S0, A0) = (s, a)  �  = Eµ  �  βtX(St, At)  �� (S0, A0) = (s, a)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (14)  So, when X is the expected TD error TπQ − Q, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (7) becomes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (5) exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  M is a contraction mapping whenever βt ≤ βt−1ρt for all t (Condition 1), which Theorem 2 establishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  As we discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='3, violating this condition can sometimes cause M to no longer contract, even  with on-policy updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We can see one plausible reason for this by refactoring the definition of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Let  qt := Q(St, At) and vt := �  a′∈A π(a′|St)Q(St, a′), so δπ  t = Rt + γvt+1 − qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Further, assume the following  expectations are conditioned on (S0, A0) = (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (4) is equivalent to  (MQ)(s, a) = q0 + Eµ  � ∞  �  t=0  γtβt(Rt + γvt+1 − qt)  �  = q0 + Eµ  � ∞  �  t=0  γtβtRt +  ∞  �  t=1  γtβt−1vt −  ∞  �  t=0  γtβtqt  �  = Eµ  � ∞  �  t=0  γtβtRt +  ∞  �  t=1  γtβt−1vt −  ∞  �  t=1  γtβtqt  �  = Eµ  � ∞  �  t=0  γtβtRt  �  + Eµ  � ∞  �  t=1  γt(βt−1vt − βtqt)  �  = Eµ  � ∞  �  t=0  γtβtRt  �  + Eµ  � ∞  �  t=1  γt(βt−1ρt − βt)qt  �  ,  (15)  14  and we discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='3 that these two terms represent a biased return estimate and an infinite sum  of weighted value-function bootstraps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In particular, this can be problematic if βt > βt−1ρt  because the corresponding bootstrap’s weight becomes negative, causing it to get subtracted from the return  estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  B  Additional Proofs  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='1  Proof of Lemma 1  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Qπ is a fixed point of M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' the difference between MQ and Qπ is given by  MQ − Qπ = Z(Q − Qπ),  (16)  where Z := �∞  t=1 γt(Bt−1Pπ − Bt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' It is evident from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (7) that Qπ is a fixed point of M because TπQπ − Qπ = 0, and so MQπ = Qπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Therefore,  MQ − Qπ = MQ − MQπ  = Q +  ∞  �  t=0  γtBt(TπQ − Q) − Qπ −  ∞  �  t=0  γtBt(TπQπ − Qπ)  = Q − Qπ +  ∞  �  t=0  γtBt(TπQ − TπQπ) −  ∞  �  t=0  γtBt(Q − Qπ)  =  ∞  �  t=0  γtBt(TπQ − TπQπ) −  ∞  �  t=1  γtBt(Q − Qπ)  =  ∞  �  t=0  γt+1BtPπ(Q − Qπ) −  ∞  �  t=1  γtBt(Q − Qπ)  =  � ∞  �  t=0  γt+1BtPπ −  ∞  �  t=1  γtBt  �  (Q − Qπ)  =  � ∞  �  t=1  γt(Bt−1Pπ − Bt)  �  (Q − Qπ)  = Z(Q − Qπ),  which is the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='2  Proof of Lemma 2  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' If Condition 1 holds, then Z has nonnegative elements and its row sums obey Z1 ≤ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Define the linear operator Dt := Bt−1Pπ − Bt and notice that Z = �∞  t=1 γtDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We will show that Dt  comprises only nonnegative elements, and therefore so does Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' For any X ∈ Rn, observe that  (DtX)(s, a) = Eµ  �  βt−1  �  St∈S  �  At∈A  P(St|Ft−1)π(At|St)X(St, At)  ����� (S0, A0) = (s, a)  �  − Eµ  �  βtX(St, At)  �� (S0, A0) = (s, a)  �  15  = Eµ  �  βt−1  �  St∈S  �  At∈A  P(St|Ft−1)π(At|St)X(St, At)  ����� (S0, A0) = (s, a)  �  − Eµ  � �  St∈S  �  At∈A  P(St|Ft−1)µ(At|St)βtX(St, At)  ����� (S0, A0) = (s, a)  �  = Eµ  � �  St∈S  P(St|Ft−1)  �  At∈A  �  π(At|St)βt−1 − µ(At|St)βt  �  X(St, At)  ����� (S0, A0) = (s, a)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (17)  Since we assumed that βt ≤ βt−1ρt in Condition 1, we have π(At|St)βt−1 − µ(At|St)βt ≥ 0, which implies  that Dt ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Furthermore, this holds for all t ≥ 1, so Z ≥ 0 follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  To complete the proof, we show that the row sums of Z are bounded by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Recall that Pπ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Hence,  Z1 =  ∞  �  t=1  γt(Bt−1Pπ − Bt)1  =  ∞  �  t=1  γt(Bt−11 − Bt1)  =  ∞  �  t=0  γt+1Bt1 −  ∞  �  t=1  γtBt1  = γ1 +  ∞  �  t=1  γt+1Bt1 −  ∞  �  t=1  γtBt1  = γ1 − (1 − γ)  ∞  �  t=1  γtBt1  ≤ γ1,  (18)  because Bt ≥ 0, ∀ t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='3  Proof of Theorem 3  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Consider a sequence of target policies (πi)i≥0 and a sequence of arbitrary behavior policies  (µi)i≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Let Q0 be an arbitrary vector in Rn and define the sequence Qi+1 := MiQi, where Mi is the  operator defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Assume that (πi)i≥0 is greedy in the limit, and let ϵi ≥ 0 be the smallest  constant such that TπiQi ≥ TQi − ϵi∥Qi∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' If Condition 1 holds for all i, then  ∥MiQi − Q∗∥ ≤ γ∥Qi − Q∗∥ +  ϵi  1 − γ ∥Qi∥,  (11)  and, consequently, lim  i→∞ Qi = Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We first derive the following upper bound:  TπiQi − TQ∗ = γPπiQi − γ max  π  PπQ∗ ≤ γPπi(Qi − Q∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (19)  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (10) and because Ci has nonnegative entries, we can deduce that  MiQi − Q∗ = (I − Ci)(Qi − Q∗) + Ci(TπiQi − Q∗)  (20)  = (I − Ci)(Qi − Q∗) + Ci(TπiQi − TQ∗)  ≤ (I − Ci)(Qi − Q∗) + γCiPπi(Qi − Q∗)  = Zi(Qi − Q∗),  (21)  16  where Zi := I − Ci(I − γPπi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Notice that Zi is analogous to the matrix Z in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (9) because, for policies  πi and µi,  I − Ci(I − γPπi) = I +  ∞  �  t=0  γtBt(γPπi − I)  = I +  ∞  �  t=0  γt+1BtPπi −  ∞  �  t=0  γtBt  =  ∞  �  t=1  γtBt−1Pπi −  ∞  �  t=1  γtBt  =  ∞  �  t=1  γt(Bt−1Pπi − Bt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (22)  Next, we derive the following lower bound:  TQi − TQ∗ ≥ Tπ∗Qi − TQ∗ = γPπ∗(Qi − Q∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (23)  Additionally, for each policy πi, there exists some ϵi ≥ 0 such that TπiQi ≥ TQi − ϵi∥Qi∥1 (recall that we  defined ϵi to be as small as possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Starting again from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (20), and noting that the elements of Ci are  nonnegative, we obtain  MiQi − Q∗ ≥ (I − Ci)(Qi − Q∗) + Ci(TQi − Q∗) − ϵi∥Qi∥Ci1  = (I − Ci)(Qi − Q∗) + Ci(TQi − TQ∗) − ϵi∥Qi∥Ci1  ≥ (I − Ci)(Qi − Q∗) + γCiPπ∗(Qi − Q∗) − ϵi∥Qi∥Ci1  = Z∗  i (Qi − Q∗) − ϵi∥Qi∥Ci1,  (24)  where we have defined Z∗  i := I − Ci(I − γPπ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' By Lemma 2, since we assumed Condition 1 holds, both Zi  and Z∗  i have nonnegative elements and their row sums are bounded by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Therefore, when MiQi − Q∗ ≥ 0,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (21) implies  ∥MiQi − Q∗∥ ≤ γ∥Qi − Q∗∥,  (25)  because element-wise inequality for nonnegative matrices implies the inequality holds also for their norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  When MiQi − Q∗ ≤ 0, we must use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (24) and multiply both sides by −1 to get nonnegative matrices,  giving  ∥MiQi − Q∗∥ ≤ γ∥Qi − Q∗∥ + ϵi∥Qi∥∥Ci∥  ≤ γ∥Qi − Q∗∥ +  ϵi  1 − γ ∥Qi∥,  (26)  because ∥Ci∥ ≤ �∞  t=0 γt∥Pπi∥t = (1 − γ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (26) is looser than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (25), its bound holds in the  worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' It remains to show that this bound implies convergence to Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Observe that  γ∥Qi − Q∗∥ +  ϵi  1 − γ ∥Qi∥ ≤ γ∥Qi − Q∗∥ +  ϵi  1 − γ (∥Qi − Q∗∥ + ∥Q∗∥)  =  �  γ +  ϵi  1 − γ  �  ∥Qi − Q∗∥ +  ϵi  1 − γ ∥Q∗∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (27)  Our assumption of greedy-in-the-limit policies tells us that ϵi → 0 as i → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' thus, there must exist some  iteration i∗ such that ϵi ≤ 1  2(1 − γ)2, ∀ i ≥ i∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Therefore, for i ≥ i∗,  ∥MiQi − Q∗∥ ≤ 1 + γ  2  ∥Qi − Q∗∥ +  ϵi  1 − γ ∥Q∗∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (28)  If γ < 1, then 1  2(1 + γ) < 1, and since ∥Q∗∥ is finite, we conclude that ∥Qi − Q∗∥ → 0 as i → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  17  C  Examples of Divergence  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='1  Counterexample 4: Off-Policy Truncated IS  Our definitions of π and µ give us  Pπ =  �p  1 − p  p  1 − p  �  ,  Pµ = 1  2  �1  1  1  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (29)  Recall that we assumed λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We define the following constant, using the definition of βt for Truncated IS:  β(1)  t  := E  �  βt  �� (St, At) = (s, a1)  �  =  �  Ft  Prµ(Ft | (St, At) = (s, a1)) · min  �  1, Prπ(Ft)  Prµ(Ft)  �  =  �  Ft−1  Prµ(Ft−1) min  �  1, Prπ(Ft−1) · p  Prµ(Ft−1) · 1  2  �  (30)  =  �  Ft−1  min (Prµ(Ft−1), 2p · Prπ(Ft−1))  =  �  Ft−1  min  �  1  2t−1 , 2p · Prπ(Ft−1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (31)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' (30) is justified because the conditional probability of a trajectory ending in action a1 is just the  probability of Ft−1 under µ, due to the 1-state (memoryless) MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We can simplify β(1)  t  further by using  the binomial theorem to calculate Prπ(Ft−1) = pk(1 − p)t−1−k, where k ∈ [0, t − 1] is the number of times  a1 is taken in Ft−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' There are  �t−1  k  �  trajectories with this same probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Therefore,  β(1)  t  =  �  Ft−1  min  �  1  2t−1 , 2p · Prπ(Ft−1)  �  =  t−1  �  k=0  �t − 1  k  �  min  �  1  2t−1 , 2p · pk(1 − p)t−1−k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (32)  Likewise, we can compute β(2)  t  by swapping p and 1 − p above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Let ⊙ denote element-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Using the fact that P t  µ = Pµ, ∀ t ≥ 1, it follows that  Bt = P t  µ ⊙  �  β(1)  t  β(2)  t  β(1)  t  β(2)  t  �  = 1  2  �  β(1)  t  β(2)  t  β(1)  t  β(2)  t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (33)  Using a computer program to calculate Z, assuming that p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='6 and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='94, we obtain  Z =  ∞  �  t=1  γt(Bt−1Pπ − Bt) ≈  �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='704  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='436  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='704  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='436  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (34)  Therefore, ∥Z∥ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='14, which is not a contraction, and the norm continues to increase for p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='6 or γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  18  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='2  Counterexample 5: On-Policy Binary Traces  The policy π is uniform random, so we have  Pπ = 1  2  �1  1  1  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (35)  Let ⊙ denote element-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Because βt = 1 only when the trajectory Ft terminates in (s, a1)  and βt = 0 otherwise, and since P t  π = Pπ, ∀ t ≥ 1, we also have  Bt = P t  π ⊙  �1  0  1  0  �  = 1  2  �1  0  1  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (36)  Using a computer program to calculate Z, assuming that γ = 2  3, we obtain  Z =  ∞  �  t=1  γt(Bt−1Pπ − Bt) = 1  3  �−1  2  −1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (37)  Therefore, ∥Z∥ = 1, which is not a contraction, and the norm continues to increase for γ > 2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  D  Implementation of Trajectory-Aware Eligibility Traces  The implementation of trajectory-aware methods is closely related to that of backward-view TD(λ) in the  tabular setting (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=', Sutton and Barto, 1998, Chapter 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' On each timestep, an environment interaction  is conducted according to the behavior policy µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Then, the eligibilities for previously visited state-action  pairs are modified, the eligibility for the current state-action pair is incremented, and the current TD error is  applied to all state-action pairs in proportion to their eligibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The only difference in the trajectory-aware  case is that the eligibilities are not modified by simply multiplying a constant decay factor γλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Arbitrary, trajectory-dependent traces β(Ft), as studied in our theoretical results, can be complicated to  implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' This stems from the fact that the timestep t in the M operator is defined relative to when  the updated state-action pair was taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In other words, each state-action pair (Sk, Ak) “disagrees” on the  start of the current trajectory, generating its update from the unique sub-trajectory (Sk, Ak), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' , (St, At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Implementing coefficients of this form would be possible using the general update  Q(Sk, Ak) ← Q(Sk, Ak) + αγt−kβ((Sk, Ak), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' , (St, At))δπ  t ,  (38)  where α ∈ (0, 1] is the stepsize, but this would require repeatedly slicing the list of visited state-action  pairs (S0, A0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' , (St, At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' While this is certainly feasible, it does not easily accommodate vectorization or  parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Fortunately, this level of generality is rarely needed in practice, and specific optimizations can be made  depending on the functional form of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' For example, Truncated IS defines β to be a pure function of the IS  estimate Πt, which is useful because per-decision eligibility traces can be used to efficiently generate the IS  estimates for every state-action pair visited during the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We demonstrate how this can be done in  pseudocode (see Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Recursive methods like Recursive Retrace and RBIS, where βt explicitly depends on βt−1, require only two  minor changes compared to Algorithm 1 for their implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' These changes, which we highlight in blue  for RBIS in Algorithm 2, correspond to the fact that the dynamic array Y is now used to store the previous  trace βt−1 rather than the previous IS estimate Πt−1 at each timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The computational requirements for  the methods remain nearly identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The implementation for Recursive Retrace easily follows by changing  line 10 of Algorithm 2 to  Y (k) ← λ min(1, Y (k) · ρt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  (39)  19  E  Experiment Details and Learning Curves  We conducted a grid search to find the best stepsize α for every λ-value for the four off-policy methods we  evaluated in the Bifurcated Gridworld (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Using a training set of 1,000 trials, we searched over  λ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' , 1} and α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9}, for a total of 55 hyperparameter combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' At the  start of each trial, the initial value function Q was sampled from a zero-mean Gaussian distribution with  standard deviation σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We trained each agent for 3,000 timesteps, allowing extra time to complete the  final episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' We then generated learning curves by plotting the 100-episode moving average of these returns  as a function of the number of timesteps and calculated their AUCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' In Table 1, we report the stepsize α  that led to the highest average AUC for each λ-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' Then, using a separate test set of 1,000 trials to avoid  bias in the search results, these α-values were used to generate the learning curves in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The AUCs  for these learning curves were finally used in the creation of the λ-sweep plot (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  Table 1:  The best stepsizes found by our grid search in the Bifurcated Gridworld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  λ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  1  Retrace  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='5  Truncated IS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='3  Recursive Retrace  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='5  RBIS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='5  20  0  500  1000  1500  2000  2500  3000  Timesteps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='6  Discounted Return  Bifurcated Gridworld ( = 0)  Retrace  Truncated IS  Recursive Retrace  RBIS  0  500  1000  1500  2000  2500  3000  Timesteps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='1)  Retrace  Truncated IS  Recursive Retrace  RBIS  0  500  1000  1500  2000  2500  3000  Timesteps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='2)  Retrace  Truncated IS  Recursive Retrace  RBIS  0  500  1000  1500  2000  2500  3000  Timesteps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='3)  Retrace  Truncated IS  Recursive Retrace  RBIS  0  500  1000  1500  2000  2500  3000  Timesteps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='4)  Retrace  Truncated IS  Recursive Retrace  RBIS  0  500  1000  1500  2000  2500  3000  Timesteps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='5)  Retrace  Truncated IS  Recursive Retrace  RBIS  0  500  1000  1500  2000  2500  3000  Timesteps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='6)  Retrace  Truncated IS  Recursive Retrace  RBIS  0  500  1000  1500  2000  2500  3000  Timesteps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='7)  Retrace  Truncated IS  Recursive Retrace  RBIS  0  500  1000  1500  2000  2500  3000  Timesteps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='8)  Retrace  Truncated IS  Recursive Retrace  RBIS  0  500  1000  1500  2000  2500  3000  Timesteps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='6  Discounted Return  Bifurcated Gridworld ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='9)  Retrace  Truncated IS  Recursive Retrace  RBIS  0  500  1000  1500  2000  2500  3000  Timesteps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+page_content='6  Discounted Return  Bifurcated Gridworld ( = 1)  Retrace  Truncated IS  Recursive Retrace  RBIS  Figure 3:  Learning curves for the λ-values we tested in the Bifurcated Gridworld environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' The dashed  black line indicates the optimal discounted return for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content='  21  Algorithm 1 Truncated Importance Sampling  1: Input: value function Q, stepsize α ∈ (0, 1]  2: for each episode do  3:  Reset environment and observe state S0  4:  Reset dynamic array Y  5:  repeat {for t = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' }  6:  Take action At ∼ µ(·|St), receive reward Rt, and observe next state St+1  7:  ρt = π(At|St)  µ(At|St)  8:  δt =  �  �  �  Rt − Q(St, At)  if St+1 is terminal  Rt − Q(St, At) + γ �  a′∈A π(a′|St+1)Q(St+1, a′)  else  9:  for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' , t − 1 do  10:  Y (k) ← Y (k) · ρt  11:  end for  12:  Y (t) ← 1  13:  for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' , t do  14:  z ← (γλ)t−k min(1, Y (k))  15:  Q(Sk, Ak) ← Q(Sk, Ak) + αzδt  16:  end for  17:  until St+1 is terminal  18: end for  Algorithm 2 Recency-Bounded Importance Sampling (RBIS)  1: Input: value function Q, stepsize α ∈ (0, 1]  2: for each episode do  3:  Reset environment and observe state S0  4:  Reset dynamic array Y  5:  repeat {for t = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' }  6:  Take action At ∼ µ(·|St), receive reward Rt, and observe next state St+1  7:  ρt = π(At|St)  µ(At|St)  8:  δt =  �  �  �  Rt − Q(St, At)  if St+1 is terminal  Rt − Q(St, At) + γ �  a′∈A π(a′|St+1)Q(St+1, a′)  else  9:  for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' , t − 1 do  10:  Y (k) ← min(λt−k, Y (k) · ρt)  11:  end for  12:  Y (t) ← 1  13:  for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
+page_content=' , t do  14:  z ← γt−kY (k)  15:  Q(Sk, Ak) ← Q(Sk, Ak) + αzδt  16:  end for  17:  until St+1 is terminal  18: end for  22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFIT4oBgHgl3EQfpCt4/content/2301.11321v1.pdf'}
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+1 
+ 
+ 
+ 
+ 
+The BlueWalker 3 Satellite Has Faded 
+ 
+Anthony Mallama, Richard E. Cole and Scott Tilley 
+ 
+2022 December 26 
+ 
+Contact: anthony.mallama@gmail.com 
+ 
+ 
+Abstract 
+Observations of BlueWalker 3 (BW3) beginning on December 8 of this year 
+indicate that its apparent brightness had decreased. We postulate that the orbital 
+beta angle and resultant solar power considerations required an adjustment to 
+the satellite attitude around that time. So, the nominally zenith facing side of the 
+flat-panel shaped spacecraft, which supports the solar array, was tilted toward 
+the Sun. Consequently, the nadir side, which is seen by observers on the ground, 
+was mostly dark. Thus, BW3 has generally appeared faint and on some occasions 
+was not seen at all. The amount of fading was up to 4 magnitudes. Numerical 
+modeling indicates that the amount of tilt was in the range 13° to 16°. This 
+situation indicates the improvement in the appearance of BW3 from the ground 
+that can be achieved with small tilts of the spacecraft. Satellite operators and 
+astronomers can jointly address the adverse impact of bright satellites on celestial 
+observations based on this finding.  
+ 
+ 
+ 
+
+2 
+ 
+ 
+1. Introduction 
+BlueWalker 3 (BW3) is the prototype for a new constellation of satellites. Astronomers are concerned 
+because this type of spacecraft unfolds into an extremely large size on-orbit and becomes very bright. 
+The International Astronomical Union issued a press release regarding the adverse affect of BW3 and its 
+constellation on astronomical research and on the appearance of the night sky. Mroz et al. (2022) and 
+Halferty et al. (2022) present photometric results for other artificial satellites that document their 
+impact on scientific observations. 
+Mallama et al. (2022) reported that the apparent visual magnitude of BW3 is most often between 2.0 
+and 3.0. The average brightness when seen near zenith at the beginning and ending of astronomical 
+twilight is magnitude 1.4. Those findings were based on observations made between the time that the 
+spacecraft unfolded and November 24 of this year.  
+The magnitudes recorded more recently are significantly fainter. This paper documents that fading and 
+suggests an explanation. Section2 describes a hypothesis for the dimming which relates to the 
+spacecraft attitude.  Section3 summarizes recent observations when BW3 was faint or was not seen at 
+all. Section 4 compares the observations to a model of the satellite’s brightness that accounts for the 
+attitude. Section 5 presents the conclusions and suggests how the adverse impact of BW3 on 
+astronomical observations can be ameliorated. 
+ 
+ 
+2. Spacecraft attitude and orbital beta angle 
+BW3 unfolded into a 64 square meter flat-panel antenna on orbit. The nominal spacecraft attitude (yaw, 
+pitch and roll angles) has the two sides of the panel facing the zenith and nadir directions. The nadir side 
+contains radio communication elements while the zenith side supports the solar power array. During the 
+course of an orbit around the Earth, the solar array is exposed to sunlight about half the time. 
+In December of this year the satellite orbital plane became nearly perpendicular to the direction of the 
+Sun. The angle between the orbital plane and the solar direction is called beta (Versteeg and Cotton, 
+and Sumanth 2019). When the cosine of beta is small a zenith-nadir facing panel is almost edge-on to 
+the Sun throughout the orbit. Thus, solar power is severely curtailed unless the spacecraft attitude is 
+adjusted. The value of beta for BW3 and its cosine are plotted versus time in Figure 1. 
+ 
+
+3 
+ 
+ 
+Figure 1. The beta angle and its cosine are illustrated, and the minimum in the cosine 
+value on December 21 is indicated. 
+ 
+We hypothesize that the spacecraft attitude has been adjusted by tilting the nominally zenith-facing 
+solar array toward the Sun. Figure 2 illustrates the tilt which is mainly a roll angle adjustment. Tilting the 
+panel in this manner increases insolation on the array and, as a consequence, darkens the nadir side 
+that faces observers on the ground. Thus, sunlight strikes the nadir side from a direction that is more 
+parallel to the plane of the panel than usual. This reduces illumination of the panel and less light is 
+reflected toward the ground. So, the satellite will appear fainter. Additionally, sunlight may strike the 
+zenith side only, in which case the satellite will be invisible or nearly invisible from the ground. In the 
+‘nearly invisible’ case, some light may reflect from the edge of the panel and a small amount may leak 
+through the interstices between the articulated panel elements.  
+ 
+
+90
+Beta Angle (deg.)
+45
+0
+-45
+-90
+1.0
+December
+21
+Cosine (Beta Angle)
+0.8
+0.6
+0.4
+0.2
+0.0
+0
+30
+60
+90
+120
+Davs Since Launch4 
+ 
+ 
+Figure 2. (Left) This schematic shows the nominal spacecraft attitude where the flat- 
+panel is zenith-and-nadir facing. There is no insolation on the solar array in that case. 
+(Middle) This is an adjusted attitude where the zenith facing side of the panel is tilted 
+slightly toward the Sun and the nadir side receives less insolation. (Right) The panel is 
+tilted further. So, no sunlight reaches the nadir side and the satellite is invisible. (All) This 
+Sun-satellite-observer geometry would apply to a satellite seen near zenith at about the 
+time when astronomical twilight begins or ends. The beta angle is large but not exactly 
+90 degrees in these schematics. 
+ 
+3. Observations 
+The magnitudes used to study BW3 were obtained by the observers listed in Table 1. Most of the 
+measurements were made with the unaided eye or through binoculars. Visual magnitudes are 
+determined by comparing the brightness of the satellite to that of nearby reference stars. This proximity 
+accounts for variations in sky transparency and brightness. The method is described in more detail by 
+Mallama (2022). Some of the observations were derived from video recordings by Langbroek. He 
+transformed the red-sensitive magnitudes to the V-band using an empirical formula based on the 
+analysis of reference star measurements. 
+ 
+Table 1. Observer coordinates 
+Observer        Latitude  Longitude  Ht(m) 
+J. Barentine     32.234   -110.768   833 
+R. Cole          50.552     -4.735   100 
+ 
+K. Fetter        44.606    -75.691  
+S. Harrington    36.062    -91.688   185 
+M. Langbroek     52.154      4.491     0 
+M. Langbroek     52.139      4.499    -2 
+R. Lee           38.93     104.81    2082 
+P. Maley         33.811   -111.952   654 
+P. Maley         32.857   -113.220 
+P. Maley         34.6       33.0       0 
+A. Mallama       38.982    -76.763    43 
+A. Mallama       38.72     -75.08      0 
+A. Mallama       39.122    -77.891 
+R. McNaught     -32.27     149.16    610 
+J. Respler       40.330    -74.445 
+R. Swaney        41.403    -81.512 
+
+Zenith-Nadir Facing
+Tilted SlightlyTowardSun
+Tilted More Toward Sun
+To Sun
+To Sun
+To Sun
+Observer
+Observer
+Observer5 
+ 
+S. Tilley        49.434   -123.668    40 
+S. Tilley        49.418   -123.642     1 
+E. Visser        53.109      6.108    46 
+A. Worley        41.474    -81.519   351 
+J. Worley        41.474    -81.519   351 
+B. Young         36.139    -95.983   201 
+B. Young         35.831    -96.141   330 
+ 
+Figure 3 shows the light curve for BW3 after the satellite unfolded its large antenna panel. The 
+magnitudes are adjusted to a standard distance of 1000 km. The average brightness before the fading 
+that began around December 8 was magnitude 3.05. These data are referred to as regular brightness in 
+the graph. The average observed brightness during the fainter period is magnitude 5.72 and that does 
+not include the times when the satellite was too faint to be seen. 
+  
+ 
+Figure 3. The light curve of BW3 shows that observations beginning on December 8 are 
+all fainter than the average magnitude before that date. The ‘not seen’ symbols indicate 
+the magnitude of the faintest visible comparison star; so the satellite was fainter than 
+that value. 
+ 
+ 
+ 
+ 
+
+0
+adjusted to 1000 km
+2
+10!
+6
+Magnitude a
+8
+RegularBrightness
+Fainter Period -Seen
+10
+FainterPeriod-NotSeen
+60
+70
+80
+90
+100
+Days After Launch6 
+ 
+4. Comparison between observations and model 
+Figure 4 shows the BW3 spacecraft before launch. The antenna panel which can be seen from the 
+ground is visible. The solar panels are on the other face, but that face cannot be seen from the ground. 
+The antenna panel appears in the image as a diffusely reflecting surface, there are no significant areas of 
+specularly reflecting materials as there are on the Starlink spacecraft, both in the original Starlink design 
+and later updates. 
+ 
+ 
+ 
+Figure 4. The nadir face of the BlueWalker spacecraft during a pre-flight test at Cape 
+Canaveral. The general appearance is of a diffusely reflecting flat panel (photo courtesy 
+AST). 
+ 
+A numerical model for the brightness of BW3 was developed, building on experience from observing and 
+modeling Starlink spacecraft since 2020 (Cole 2020, 2021).  
+In the case of BW3, the numerical model uses a single Earth-facing flat surface that reflects light 
+diffusely. This diffuse reflection is well described by Lambert’s cosine law. More complex models of 
+reflection from the panel are available but not required to analyze the ‘fading’ discussed here. 
+The model takes account of the aspect angle of the panel with respect to the observer, its range and the 
+angle of the Sun illumination on the panel (which is different from the angle of the Sun at the observer).  
+The model was developed using observations made after full deployment of BW3. In that case the panel 
+was maintained in the model as nadir-facing and the only variable was an absolute brightness of the 
+panel. With suitable selection of that absolute brightness, a reasonable match was found between the 
+predictions of the model and the visual observations (Figure 5).  
+
+CA7 
+ 
+ 
+Figure 5. Comparison of the brightness predictions of the numerical model and 
+observations made in the period after the AST announcement of full deployment and 
+before December 7. The closer the points are to the diagonal line, the closer the model is 
+to the observed brightness. 
+ 
+The maximum brightness of BW3 is always in the zenith and is a strong function of the elevation of the 
+Sun, brighter when the Sun is further below the horizon at the observer. The brightest observations 
+were made when BW3 was in the zenith and entering or emerging from eclipse. 
+However, from December 8 of this year observations of BW3 were not well described by the same 
+model. The same comparison of model and observations (as in Figure 5) is displayed in Figure 6-i. Cases 
+where BW3 was below the limiting magnitude of the observation are shown with an arrow. 
+Clearly, the model no longer predicts the BW3 brightness to any level of accuracy. Some of the 
+observations are 3 or 4 magnitudes fainter than their prediction, or BW3 was not seen at all. Since the 
+surface material of the BW3 panel has not changed, the most likely explanation was a change in the 
+spacecraft attitude for the reasons discussed above, with a movement of the zenith axis towards the 
+Sun. A tilt of this sort reduces the angle of the Sun on the Earth-facing panel and reduces the amount of 
+sunlight that is scattered, thus making BW3 fainter (Figure 2).  
+
+Observations in period 2022-11-15to 2022-12-07
+Observed Magnitude
+2
+3
+Maximummodeledbrightness
+Model too bright
+2
+3
+5
+Model too faint
+68 
+ 
+A tilt was added as a parameter of the model and a range of tilt angles investigated. A best fit of the 
+predictions to the data was achieved for a tilt of 13° but tilts up to 16° are a reasonable fit to the data 
+(Figure 6-ii). Tilts of less than 13° do not fit the data. With this deduction, a small number of the 
+observations had been made when the Earth-facing panel was not illuminated by the Sun at all, that is 
+the Sun was shining on the solar-panel side. In these cases, the BW3 brightness was at the limit of 
+observations using binoculars, magnitudes fainter than 7 or 8. It is hypothesized that this limited solar 
+flux is being reflected from the edges of the panel or the joints between the panel elements. A simple 
+model was added that fitted this small number of observations. 
+ 
+ 
+Figure 6 i) Comparison of the brightness 
+predictions of the numerical model and the 
+observations taken in the period December 8 to 
+22, using the same model parameters as in Figure 
+5. 
+Figure 6 ii) The same data using a model that tilts 
+the BW3 solar panel 13° towards the Sun. A 
+simple model has been added to deal with cases 
+when the Sun is only illuminating the zenith side 
+of the panel. 
+ 
+Using this model, brightness maps can be created for the whole sky. Individual maps are required for 
+each elevation of the Sun at the observer as this affects the appearance of the satellite across the whole 
+sky. Figure 7 displays polar projection maps for Sun elevations of -18° (end of astronomical twilight) and 
+-12° (end of nautical twilight), in each case with no tilt and with a tilt of 13°. The azimuth of the Sun has 
+been standardized to 90°. The following can be noted: 
+1. In all the maps BW3 is in eclipse over part of the sky opposite the Sun – the anti-Sun direction 
+(white areas) 
+2. In the untilted skymaps (left of Figure 7) BW3 is brighter when the Sun is further below the 
+horizon due to the greater angle of the Sun on the Earth-facing antenna panel. Otherwise, the 
+appearance of BW3 is similar for the two Sun elevation cases. 
+
+Observations in period 2022-12-08to 2022-12-22
+Observed Magnitude
+2
+4
+5
+6
+7
+8
+9
+10
+Model too bright
+2
+3
+4
+Modeled Magnitude
+5
+7
+8
+BW3 Tilt  degrees
+6
+Model too faint
+10Observations in period 2022-12-08to 2022-12-22
+Observed Magnitude
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Model too bright
+2
+3
+4
+Modeled Magnitude
+5
+7
+8
+BW3 Tilt 13 degrees
+6
+Model too faint
+109 
+ 
+3. In the tilted skymaps (right of Figure 7) the appearance of BW3 is more complex and also 
+different for the two Sun elevations. In the pro-Sun direction the observer sees the un-
+illuminated antenna panel and thus BW3 is faint, below 7th magnitude. In the anti-Sun direction 
+BW3 is much fainter than the untilted cases, with the effect more pronounced when the Sun 
+elevation is -12°. This is because a tilt of 13° very significantly reduces the Sun illumination on 
+the antenna panel across the whole sky. 
+  
+ 
+Figure 7: Skymaps of BW3 brightness for sun elevations of -18° and -12°, no tilt (left) and 
+with a sunward tilt of 13° (right). A polar projection is used, north at the top and west at 
+the right. The Sun azimuth of 90° is shown here but the relative appearance is the same 
+for any Sun azimuth. 
+ 
+5. Conclusions  
+We offer a hypothesis to explain the observed fading of the BlueWalker 3 satellite that began on 
+December 8 of this year. The orbital beta angle at that time resulted in low insolation on the zenith 
+facing side of the spacecraft which supports the solar array. Power considerations required an 
+adjustment to the satellite attitude such that the zenith side was tilted toward the Sun to increase 
+
+Sun elevation -18° BW3 Tilt 0°
+0
+Sun elevation -18° BW3 Tilt 13°
+0
+Maximum brightness 1.1mag
+Maximum brightness 2.7mag
+Magnitude
+D 9.00-10.00
+D 8.00-9.00
+-7.00-8.00
+10
+20 30 40 50 60 70 8090
+eclipse
+...
+10203040 50 6070
+eclipse
+6.00-7.00
+90
+270
+90
+ 270
+sun
+5.00-6.00
+Sun
+Azimuth
+Azimuth
+-4.00-5.00
+03.00-4.00
+2.00-3.00
+1.00-2.00
+0.00-1.00
+180
+180
+Sun elevation -12° BW3 Tilt 0°
+0
+Sun elevation -12° BW3 Tilt 13°
+Maximum brightness 1.6mag
+Maximum brightness 4.4mag
+Magnitude
+D 9.00-10.00
+8.00-9.00
+7.00-8.00
+1920 30 40 50 60 70 8090
+eclipse
+270
+1020 30 40 50 60 70 8090
+clipse
+06.00-7.00
+90
+90
+ 270
+sun
+5.00-6.00
+Azimuth
+sun
+Azimuth
+4.00-5.00
+3.00-4.00
+02.00-3.00
+1.00-2.00
+-0.00-1.00
+180
+18010 
+ 
+insolation. Consequently, the nadir side of the panel seen by observers on the ground was dark or only 
+weakly illuminated by the Sun.  
+While this situation appears to have arisen for operational reasons, it demonstrates that even a small 
+change of spacecraft attitude has a major effect on the brightness of BlueWalker 3 as viewed from the 
+ground, though this effect is less pronounced when the Sun is further below the horizon. Satellite 
+operators and astronomers can begin a constructive dialog based on this finding. 
+ 
+References  
+Cole, R.E. 2020. A sky brightness model for the Starlink ‘Visorsat’ spacecraft – I, Research Notes of the 
+American Astronomical Society, 4, 10, https://iopscience.iop.org/article/10.3847/2515-5172/abc0e9  
+Cole, R.E. 2021. A sky brightness model for the Starlink ‘Visorsat’ spacecraft. 
+https://arxiv.org/abs/2107.06026  
+Halferty, G., Reddy, V., Campbell, T., Battle, A. and Furaro, R. 2022. Photometric characterization and 
+trajectory accuracy of Starlink satellites: implications for ground-based astronomical surveys. 
+https://arxiv.org/abs/2208.03226. 
+Mallama, A., Cole, R.E., Harrington, S. and Maley, P.D. 2022. Visual magnitude of the BlueWalker 3 
+satellite. https://arxiv.org/abs/2211.09811. 
+Mallama, A., 2022. The method of visual satellite photometry. https://arxiv.org/abs/2208.07834. 
+Mroz, P., Otarola, A., Prince, T.A., Dekany, R., Duev, D.A., Graham, M.J., Groom, S.L., Masci, F.J. and 
+Medford, M.S. 2022. Impact of the SpaceX Starlink satellites on the Zwicky Transient Facility survey 
+observations. https://arxiv.org/abs/2201.05343. 
+Sumanth, R.M. 2019. Computation of eclipse time for low-Earth orbiting small 
+satellites. https://commons.erau.edu/cgi/viewcontent.cgi?article=1412&context=ijaaa 
+Versteeg, C. and Cotten, D.L. Preliminary thermal analysis of small satellites. 
+https://s3vi.ndc.nasa.gov/ssri-
+kb/static/resources/Preliminary_Thermal_Analysis_of_Small_Satellites.pdf 
+ 
+
diff --git a/gtAzT4oBgHgl3EQfov1j/content/tmp_files/load_file.txt b/gtAzT4oBgHgl3EQfov1j/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,341 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf,len=340
+page_content='1  The BlueWalker 3 Satellite Has Faded  Anthony Mallama, Richard E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Cole and Scott Tilley  2022 December 26  Contact: anthony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='mallama@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='com  Abstract Observations of BlueWalker 3 (BW3) beginning on December 8 of this year indicate that its apparent brightness had decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' We postulate that the orbital beta angle and resultant solar power considerations required an adjustment to the satellite attitude around that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' So, the nominally zenith facing side of the flat-panel shaped spacecraft, which supports the solar array, was tilted toward the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Consequently, the nadir side, which is seen by observers on the ground, was mostly dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Thus, BW3 has generally appeared faint and on some occasions was not seen at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The amount of fading was up to 4 magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Numerical modeling indicates that the amount of tilt was in the range 13° to 16°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' This situation indicates the improvement in the appearance of BW3 from the ground that can be achieved with small tilts of the spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Satellite operators and astronomers can jointly address the adverse impact of bright satellites on celestial observations based on this finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Introduction BlueWalker 3 (BW3) is the prototype for a new constellation of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Astronomers are concerned because this type of spacecraft unfolds into an extremely large size on-orbit and becomes very bright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The International Astronomical Union issued a press release regarding the adverse affect of BW3 and its constellation on astronomical research and on the appearance of the night sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Mroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' (2022) and Halferty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' (2022) present photometric results for other artificial satellites that document their impact on scientific observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Mallama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' (2022) reported that the apparent visual magnitude of BW3 is most often between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='0 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The average brightness when seen near zenith at the beginning and ending of astronomical twilight is magnitude 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Those findings were based on observations made between the time that the spacecraft unfolded and November 24 of this year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The magnitudes recorded more recently are significantly fainter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' This paper documents that fading and suggests an explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Section2 describes a hypothesis for the dimming which relates to the spacecraft attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Section3 summarizes recent observations when BW3 was faint or was not seen at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Section 4 compares the observations to a model of the satellite’s brightness that accounts for the attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Section 5 presents the conclusions and suggests how the adverse impact of BW3 on astronomical observations can be ameliorated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Spacecraft attitude and orbital beta angle BW3 unfolded into a 64 square meter flat-panel antenna on orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The nominal spacecraft attitude (yaw, pitch and roll angles) has the two sides of the panel facing the zenith and nadir directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The nadir side contains radio communication elements while the zenith side supports the solar power array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' During the course of an orbit around the Earth, the solar array is exposed to sunlight about half the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' In December of this year the satellite orbital plane became nearly perpendicular to the direction of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The angle between the orbital plane and the solar direction is called beta (Versteeg and Cotton, and Sumanth 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' When the cosine of beta is small a zenith-nadir facing panel is almost edge-on to the Sun throughout the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Thus, solar power is severely curtailed unless the spacecraft attitude is adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The value of beta for BW3 and its cosine are plotted versus time in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  3  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The beta angle and its cosine are illustrated, and the minimum in the cosine value on December 21 is indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  We hypothesize that the spacecraft attitude has been adjusted by tilting the nominally zenith-facing solar array toward the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Figure 2 illustrates the tilt which is mainly a roll angle adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Tilting the panel in this manner increases insolation on the array and, as a consequence, darkens the nadir side that faces observers on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Thus, sunlight strikes the nadir side from a direction that is more parallel to the plane of the panel than usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' This reduces illumination of the panel and less light is reflected toward the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' So, the satellite will appear fainter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Additionally, sunlight may strike the zenith side only, in which case the satellite will be invisible or nearly invisible from the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' In the ‘nearly invisible’ case, some light may reflect from the edge of the panel and a small amount may leak through the interstices between the articulated panel elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  90  Beta Angle (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=')  45  0 45 90  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='0  December  21  Cosine (Beta Angle)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='0  0  30  60  90  120  Davs Since Launch4  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' (Left) This schematic shows the nominal spacecraft attitude where the flat- panel is zenith-and-nadir facing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' There is no insolation on the solar array in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' (Middle) This is an adjusted attitude where the zenith facing side of the panel is tilted slightly toward the Sun and the nadir side receives less insolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' (Right) The panel is tilted further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' So, no sunlight reaches the nadir side and the satellite is invisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' (All) This Sun-satellite-observer geometry would apply to a satellite seen near zenith at about the time when astronomical twilight begins or ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The beta angle is large but not exactly 90 degrees in these schematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Observations The magnitudes used to study BW3 were obtained by the observers listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Most of the measurements were made with the unaided eye or through binoculars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Visual magnitudes are determined by comparing the brightness of the satellite to that of nearby reference stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' This proximity accounts for variations in sky transparency and brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The method is described in more detail by Mallama (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Some of the observations were derived from video recordings by Langbroek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' He transformed the red-sensitive magnitudes to the V-band using an empirical formula based on the analysis of reference star measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Observer coordinates Observer        Latitude  Longitude  Ht(m) J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Barentine     32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='234   -110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='768   833 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Cole          50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='552     -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='735   100  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Fetter        44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='606    -75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='691 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Harrington    36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='062    -91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='688   185 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Langbroek     52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='154      4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='491     0 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Langbroek     52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='139      4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='499    -2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Lee           38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='93     104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='81    2082 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Maley         33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='811   -111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='952   654 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Maley         32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='857   -113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='220 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Maley         34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='6       33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='0       0 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Mallama       38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='982    -76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='763    43 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Mallama       38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='72     -75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='08      0 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Mallama       39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='122    -77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='891 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' McNaught     -32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='27     149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='16    610 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Respler       40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='330    -74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='445 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Swaney        41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='403    -81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='512  Zenith Nadir Facing  Tilted SlightlyTowardSun  Tilted More Toward Sun  To Sun  To Sun  To Sun  Observer  Observer  Observer5  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Tilley        49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='434   -123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='668    40 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Tilley        49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='418   -123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='642     1 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Visser        53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='109      6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='108    46 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Worley        41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='474    -81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='519   351 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Worley        41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='474    -81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='519   351 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Young         36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='139    -95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='983   201 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Young         35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='831    -96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='141   330  Figure 3 shows the light curve for BW3 after the satellite unfolded its large antenna panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The magnitudes are adjusted to a standard distance of 1000 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The average brightness before the fading that began around December 8 was magnitude 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' These data are referred to as regular brightness in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The average observed brightness during the fainter period is magnitude 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='72 and that does not include the times when the satellite was too faint to be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The light curve of BW3 shows that observations beginning on December 8 are all fainter than the average magnitude before that date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The ‘not seen’ symbols indicate the magnitude of the faintest visible comparison star;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' so the satellite was fainter than that value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  0  adjusted to 1000 km  2  10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  6  Magnitude a  8  RegularBrightness  Fainter Period Seen  10  FainterPeriod NotSeen  60  70  80  90  100  Days After Launch6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Comparison between observations and model Figure 4 shows the BW3 spacecraft before launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The antenna panel which can be seen from the ground is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The solar panels are on the other face, but that face cannot be seen from the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The antenna panel appears in the image as a diffusely reflecting surface, there are no significant areas of specularly reflecting materials as there are on the Starlink spacecraft, both in the original Starlink design and later updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The nadir face of the BlueWalker spacecraft during a pre-flight test at Cape Canaveral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The general appearance is of a diffusely reflecting flat panel (photo courtesy AST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  A numerical model for the brightness of BW3 was developed, building on experience from observing and modeling Starlink spacecraft since 2020 (Cole 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' In the case of BW3, the numerical model uses a single Earth-facing flat surface that reflects light diffusely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' This diffuse reflection is well described by Lambert’s cosine law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' More complex models of reflection from the panel are available but not required to analyze the ‘fading’ discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The model takes account of the aspect angle of the panel with respect to the observer, its range and the angle of the Sun illumination on the panel (which is different from the angle of the Sun at the observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The model was developed using observations made after full deployment of BW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' In that case the panel was maintained in the model as nadir-facing and the only variable was an absolute brightness of the panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' With suitable selection of that absolute brightness, a reasonable match was found between the predictions of the model and the visual observations (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  CA7  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Comparison of the brightness predictions of the numerical model and observations made in the period after the AST announcement of full deployment and before December 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The closer the points are to the diagonal line, the closer the model is to the observed brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  The maximum brightness of BW3 is always in the zenith and is a strong function of the elevation of the Sun, brighter when the Sun is further below the horizon at the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The brightest observations were made when BW3 was in the zenith and entering or emerging from eclipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' However, from December 8 of this year observations of BW3 were not well described by the same model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The same comparison of model and observations (as in Figure 5) is displayed in Figure 6-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Cases where BW3 was below the limiting magnitude of the observation are shown with an arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Clearly, the model no longer predicts the BW3 brightness to any level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Some of the observations are 3 or 4 magnitudes fainter than their prediction, or BW3 was not seen at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Since the surface material of the BW3 panel has not changed, the most likely explanation was a change in the spacecraft attitude for the reasons discussed above, with a movement of the zenith axis towards the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' A tilt of this sort reduces the angle of the Sun on the Earth-facing panel and reduces the amount of sunlight that is scattered, thus making BW3 fainter (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  Observations in period 2022 11 15to 2022 12 07  Observed Magnitude  2  3  Maximummodeledbrightness  Model too bright  2  3  5  Model too faint  68  A tilt was added as a parameter of the model and a range of tilt angles investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' A best fit of the predictions to the data was achieved for a tilt of 13° but tilts up to 16° are a reasonable fit to the data (Figure 6-ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Tilts of less than 13° do not fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' With this deduction, a small number of the observations had been made when the Earth-facing panel was not illuminated by the Sun at all, that is the Sun was shining on the solar-panel side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' In these cases, the BW3 brightness was at the limit of observations using binoculars, magnitudes fainter than 7 or 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' It is hypothesized that this limited solar flux is being reflected from the edges of the panel or the joints between the panel elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' A simple model was added that fitted this small number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  Figure 6 i) Comparison of the brightness predictions of the numerical model and the observations taken in the period December 8 to 22, using the same model parameters as in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Figure 6 ii) The same data using a model that tilts the BW3 solar panel 13° towards the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' A simple model has been added to deal with cases when the Sun is only illuminating the zenith side of the panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  Using this model, brightness maps can be created for the whole sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Individual maps are required for each elevation of the Sun at the observer as this affects the appearance of the satellite across the whole sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Figure 7 displays polar projection maps for Sun elevations of -18° (end of astronomical twilight) and -12° (end of nautical twilight), in each case with no tilt and with a tilt of 13°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The azimuth of the Sun has been standardized to 90°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The following can be noted: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' In all the maps BW3 is in eclipse over part of the sky opposite the Sun – the anti-Sun direction (white areas) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' In the untilted skymaps (left of Figure 7) BW3 is brighter when the Sun is further below the horizon due to the greater angle of the Sun on the Earth-facing antenna panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Otherwise, the appearance of BW3 is similar for the two Sun elevation cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  Observations in period 2022 12 08to 2022 12 22  Observed Magnitude  2  4  5  6  7  8  9  10  Model too bright  2  3  4  Modeled Magnitude  5  7  8  BW3 Tilt  degrees  6  Model too faint  10Observations in period 2022 12 08to 2022 12 22  Observed Magnitude  2  3  4  5  6  7  8  9  10  Model too bright  2  3  4  Modeled Magnitude  5  7  8  BW3 Tilt 13 degrees  6  Model too faint  109  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' In the tilted skymaps (right of Figure 7) the appearance of BW3 is more complex and also different for the two Sun elevations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' In the pro-Sun direction the observer sees the un- illuminated antenna panel and thus BW3 is faint, below 7th magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' In the anti-Sun direction BW3 is much fainter than the untilted cases, with the effect more pronounced when the Sun elevation is -12°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' This is because a tilt of 13° very significantly reduces the Sun illumination on the antenna panel across the whole sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  Figure 7: Skymaps of BW3 brightness for sun elevations of -18° and -12°, no tilt (left) and with a sunward tilt of 13° (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' A polar projection is used, north at the top and west at the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The Sun azimuth of 90° is shown here but the relative appearance is the same for any Sun azimuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Conclusions We offer a hypothesis to explain the observed fading of the BlueWalker 3 satellite that began on December 8 of this year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The orbital beta angle at that time resulted in low insolation on the zenith facing side of the spacecraft which supports the solar array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Power considerations required an adjustment to the satellite attitude such that the zenith side was tilted toward the Sun to increase  Sun elevation -18° BW3 Tilt 0° 0 Sun elevation -18° BW3 Tilt 13° 0 Maximum brightness 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='1mag Maximum brightness 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='7mag Magnitude D 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 D 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 10 20 30 40 50 60 70 8090 eclipse .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' 10203040 50 6070 eclipse 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 90 270 90 270 sun 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 Sun Azimuth Azimuth -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 180 180 Sun elevation -12° BW3 Tilt 0° 0 Sun elevation -12° BW3 Tilt 13° Maximum brightness 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='6mag Maximum brightness 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='4mag Magnitude D 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 1920 30 40 50 60 70 8090 eclipse 270 1020 30 40 50 60 70 8090 clipse 06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 90 90 270 sun 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 Azimuth sun Azimuth 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='00 180 18010  insolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Consequently, the nadir side of the panel seen by observers on the ground was dark or only weakly illuminated by the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' While this situation appears to have arisen for operational reasons, it demonstrates that even a small change of spacecraft attitude has a major effect on the brightness of BlueWalker 3 as viewed from the ground, though this effect is less pronounced when the Sun is further below the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Satellite operators and astronomers can begin a constructive dialog based on this finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='  References Cole, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' A sky brightness model for the Starlink ‘Visorsat’ spacecraft – I, Research Notes of the American Astronomical Society, 4, 10, https://iopscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='org/article/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='3847/2515-5172/abc0e9 Cole, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' A sky brightness model for the Starlink ‘Visorsat’ spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='org/abs/2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='06026 Halferty, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', Reddy, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', Campbell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', Battle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' and Furaro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Photometric characterization and trajectory accuracy of Starlink satellites: implications for ground-based astronomical surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='03226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Mallama, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', Cole, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', Harrington, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' and Maley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Visual magnitude of the BlueWalker 3 satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='09811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Mallama, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' The method of visual satellite photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='07834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Mroz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', Otarola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', Prince, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', Dekany, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', Duev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', Graham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', Groom, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=', Masci, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' and Medford, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Impact of the SpaceX Starlink satellites on the Zwicky Transient Facility survey observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='org/abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='05343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Sumanth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Computation of eclipse time for low-Earth orbiting small satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' https://commons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='erau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='edu/cgi/viewcontent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='article=1412&context=ijaaa Versteeg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' and Cotten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' Preliminary thermal analysis of small satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content=' https://s3vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='ndc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
+page_content='gov/ssri- kb/static/resources/Preliminary_Thermal_Analysis_of_Small_Satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfov1j/content/2301.01601v1.pdf'}
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+1
+Predicting Learning Interactions in Social
+Learning Networks: A Deep Learning
+Enabled Approach
+Rajeev Sahay∗, Graduate Student Member, IEEE, Serena Nicoll∗, Student Member, IEEE,
+Minjun Zhang, Tsung-Yen Yang, Student Member, IEEE, Carlee Joe-Wong, Member, IEEE,
+Kerrie A. Douglas, Member, IEEE, and Christopher G. Brinton, Senior Member, IEEE
+Abstract—We consider the problem of predicting link for-
+mation in Social Learning Networks (SLN), a type of social
+network that forms when people learn from one another through
+structured interactions. While link prediction has been studied
+for general types of social networks, the evolution of SLNs over
+their lifetimes coupled with their dependence on which topics
+are being discussed presents new challenges for this type of
+network. To address these challenges, we develop a series of
+autonomous link prediction methodologies that utilize spatial
+and time-evolving network architectures to pass network state
+between space and time periods, and that models over three
+types of SLN features updated in each period: neighborhood-
+based (e.g., resource allocation), path-based (e.g., shortest path),
+and post-based (e.g., topic similarity). Through evaluation on six
+real-world datasets from Massive Open Online Course (MOOC)
+discussion forums and from Purdue University, we find that our
+method obtains substantial improvements over Bayesian models,
+linear classifiers, and graph neural networks, with AUCs typically
+above 0.91 and reaching 0.99 depending on the dataset. Our
+feature importance analysis shows that while neighborhood and
+path-based features contribute the most to the results, post-based
+features add additional information that may not always be
+relevant for link prediction.
+Index Terms—Deep learning, graph neural networks, link
+prediction, online social networks, social learning networks.
+I. INTRODUCTION
+O
+NLINE education has exploded in popularity over the
+past few years, with estimates of up to 80% of stu-
+dents having taken an online course [2]. The advent of the
+COVID-19 outbreak has significantly increased the number of
+online learners since 2020, which in turn has demonstrated
+online platforms’ viability as an additional tool in physical
+R. Sahay, S. Nicoll, M. Zhang, and C. G. Brinton are with the Elmore Fam-
+ily School of Electrical and Computer Engineering, Purdue University, West
+Lafayette, IN, 47907. E-mail: {sahayr,snicoll,zhan3624,cgb}@purdue.edu.
+K. A. Douglas is with the School of Engineering Education, Purdue
+University, West Lafayette, IN, 47907. E-mail: douglask@purdue.edu.
+T. Yang is with the Department of Electrical and Computer Engineering,
+Princeton University, Princeton, NJ 08544. E-mail: ty3@princeton.edu.
+C. Joe-Wong is with the Department of Electrical and Computer
+Engineering,
+Carnegie
+Mellon
+University,
+Pittsburgh,
+PA
+15213.
+E-
+mail:cjoewong@andrew.cmu.edu.
+∗R. Sahay and S. Nicoll contributed equally to this work.
+This work was supported in part by the Charles Koch Foundation.
+The code and four of the datasets used in this work are available at
+https://github.com/Jess-jpg-txt/sln-learning.
+A preliminary version of the material in this work appeared in the Pro-
+ceedings of the IEEE Conference on Computer Communications (INFOCOM)
+2018 [1].
+classrooms. This growth has not been without challenges,
+however; online learning has raised concerns about its apparent
+lack of quality control, extraordinarily low teacher-to-student
+ratios, and scarcity of high-quality teachers [2]. The COVID-
+19 pandemic has highlighted the lack of quality tools for
+both students and teachers across online learning providers,
+making navigation of these massive communities a daunting
+or impossible task.
+One way course providers have attempted to mitigate these
+problems is by establishing online forums where students can
+learn from each other, thus compensating for a lack of per-
+sonalized instruction by posting questions, replying with an-
+swers, and otherwise exchanging ideas. Massive Open Online
+Courses (MOOCs), as well as Q&A sites like Piazza, Quora,
+and StackOverflow, rely on forums extensively, generating a
+plethora of data about how users interact with one another
+online for learning purposes. These forums generate Social
+Learning Networks (SLNs) within communities of student
+users that evolve over time, facilitating peer-to-peer knowledge
+transfer in the absence of instructor intervention. Data-driven
+studies on the SLNs emerging from online learning forums
+have analyzed the benefits of social learning [3], [4] geared
+towards the ultimate goal of improving learning outcomes by,
+for example, proposing methods for instructor analytics [5]
+and news feed personalization [6].
+In this work, we are motivated by the following research
+question: Can link formation between learners in an SLN be
+predicted in advance? Such predictions would enable several
+new ways of improving online learning and forum experiences
+(e.g., encouraging early formation of learner groups or recom-
+mending that learners respond to newly-posted questions that
+they are expected to answer/contribute to later), thus helping
+to reduce the gap between in-person and online instruction.
+SLNs, however, pose two key challenges that differentiate
+them from standard time-evolving social networks [44]. First,
+the SLN for an online course forms around the specific
+educational processes of that course [8], [48]. With an SLN,
+users connect as a result of specific learning needs, and
+in response to events that are exogeneous to the discussion
+forum, e.g., the instructor releasing new content/assessments.
+On the other hand, homophily and pre-existing relationships
+are known to play a strong role in the evolution of standard
+social networks over time, which can provide initialization
+information for predicting learner interactions. An online SLN
+arXiv:2301.01606v1  [cs.SI]  3 Jan 2023
+
+2
+tied to a specific course, on the other hand, exhibits a “cold
+start” from a state of little-to-no observable network. Second,
+links in SLNs are defined much more arbitrarily compared
+to other graphs [6]. On social media sites, links between
+users are typically quantified with concrete metrics such as
+‘friendships’ or ‘follows,’ where the connection between two
+users is explicit and typically optional. In an SLN, by contrast,
+a link between two users should indicate a transfer/sharing of
+knowledge. Explicit connection metrics do not typically exist,
+and even if they did, they do not imply the users have shared
+information. As a result of these challenges, the prediction of
+link formation in SLNs cannot be easily solved using previous
+methods designed for general time-evolving graphs [43].
+In this work, we develop a link prediction methodology,
+specifically tailored for addressing the challenges associated
+with SLNs, which analyzes a set of features describing (i)
+learner pairs in an SLN and (ii) the evolution of learner
+interactions over time. Our methodology is deep learning-
+based, allowing consideration for both time-variable features
+and latent learner characteristics. We evaluate our methodol-
+ogy on data collected from four MOOC discussion forums
+from Coursera and two courses at Purdue University. We
+then investigate how our methodology can be used to make
+recommendations that may enhance the timing and quality of
+replies to discussion posts, thus encouraging interactions and
+improving learner experience in discussion-based forums.
+A. Related Work
+The link prediction problem has been studied extensively
+in the context of online and digitally-enabled social networks,
+due to its usefulness in generating recommendations such
+as friendships, follows, or other forms of interactions [8]–
+[11]. Several methods have been proposed for this problem,
+beginning with unsupervised approaches and eventually tran-
+sitioning to supervised methods in the past few years. In terms
+of unsupervised methods, [13] proposed using features based
+on node proximity and properties, while [14] and [15] applied
+a model to incorporate additional contextual and temporal fea-
+tures. On the other hand, supervised approaches have proposed
+random walk algorithms using labels to increase the likelihood
+of traversing formed links [16], while [17] and [18] proposed
+deriving features from exogenous sources and training models
+on them to predict future link formation. Previous work has
+additionally considered using supervised and unsupervised
+methods simultaneously for exploratory learning environments
+[19]. However, these works do not consider characteristics
+unique to social learning networks. Specifically, the potential
+dependence on discussion topics, and the need for time-series
+modeling is not explicitly modeled. Research into SLNs until
+this point has been largely theoretical, although [20] provides
+a first look into the application of deep learning-based link
+prediction algorithms in a classroom setting. Additionally,
+unsupervised approaches have demonstrated recent popularity
+for problems related classification of student behavior [12].
+Although the central focus of our research is concerned with
+SLNs, unlike these works, our strictly supervised models
+specifically consider student social characteristics for large
+classrooms.
+Other works on online social networks have considered
+problems related to link formation, e.g., predicting the
+strength/repetition (rather than existence) of future links [21]–
+[23], predicting link types [12], or examining the effects
+of student confusion on SLNs [24]. The methods used and
+developed include linear regression/classification on network
+features and user demographics [21], [25], latent variable
+modeling of learner interaction frequencies [12], and dynamic
+models to account for the disappearance and strengthening of
+links over time [18]. Our models utilize some similar network
+features, but we consider the different prediction objective
+of pinpointing when links will form. In fact, given its high
+observed quality, we consider a time-series version of [26] as
+a potential model.
+An SLN is fully described by several datasets that each
+capture the a subset of student behavior inside the associated
+course. Recent papers choose to focus on one or a couple
+of these datasets: e.g. Student video-watching behavior [5],
+student performance [27], [28], student physical behavior [29],
+or discussion forum data [30]–[33]. Our work is evaluated
+on a similar dataset to [32] in that it provides information
+gathered on student message passing behavior in a discussion
+forum. The models created in these other works fundamentally
+differ from our focus on individual student relationships. [30]
+focuses on making group predictions from clusters of similar
+students, while [33] models changes in student behavior at
+critical points (e.g., exams and holidays).
+Some recent works have focused on other aspects of differ-
+ent types of SLNs, e.g., MOOCs [12], [21], [35], Q&A sites
+[22], [36], and enterprise social networks [37], [38]. Our work
+is perhaps most similar to [2], [21] in that we study prediction
+for SLNs using topological features. The prediction objectives
+in these other works, however, are fundamentally different
+than our focus of predicting interactions between learners in
+that they seek to predict course grades via video-watching
+behaviors [35] and student knowledge-state via learner post
+and reply frequencies [36].
+B. Our Methodology and Contributions
+In this work, we propose a novel framework specifically
+tailored to perform link prediction in SLNs. Fig. 1 summarizes
+the main components of our methodology, which are further
+outlined in the following discussion.
+1) Input Feature Computation: We begin by extracting the
+discussion data from the considered forum to construct the
+SLN (Sec. II-A). Next, we engineer a set of features for
+each learner pair (Sec. II-B). Here, we define three groups
+of features that we consider: (i) neighborhood-based features
+that are determined from common neighborhoods, (ii) path-
+based features based on paths between learners, and (iii) post-
+based features that are determined from latent topic analysis of
+learner posts. Because a specific definition of what constitutes
+link formation between two users in an SLN does not exist, a
+key question when quantifying an SLN is how best to model
+learner interactions without loss of accuracy [6]. We address
+this through inference from forum data, with consideration for
+both quality of interaction [26] and timing.
+
+3
+Fig. 1: Summary of the application of our SLN link prediction framework in post-based courses.
+2) Prediction Model: The second component of our frame-
+work shown in Fig. 1 is the prediction model (Sec. II-C).
+We consider three different classes of predictors: (i) lin-
+ear classifiers, (ii) graph neural networks (GNN), and (iii)
+gradient-based deep neural network classifiers (specifically,
+Bayesian neural networks, fully connected neural networks,
+convolutional neural networks, recurrent neural networks,
+and convolutional recurrent neural networks). The success
+of Bayesian models in static link prediction problems [40]
+motivates us to consider their performance in the time-evolving
+SLN setting, while GNNs offer efficient learning over graphs
+without explicit feature engineering [46]. However, we develop
+our core methodology around deep learning-based classifiers,
+because, as we will show, explicit feature modeling paired
+with various layer types, which can extract spatial or temporal
+patterns from the SLN features, result in more robust and
+accurate SLN link prediction.
+3) Evaluation and Analytics:
+To assess the quality of
+our models, we train and evaluate our considered prediction
+models on four MOOC discussion forums and two Piazza
+discussion forums, using an unsupervised method as a baseline
+(Sec. II-C1). Through our evaluation, we also generate four
+types of analytics. The first analytic is feature importance,
+which quantifies the importance of each considered feature
+group. The second and third analytics quantify time-dependent
+model parameters, including closeness between time of link
+prediction and actual link formation as well as the relationship
+between features and the timing and quality of formed links.
+The fourth analytic explores the effects of varying classifica-
+tion architectures, where we anaylize the importance of dif-
+ferent architectures in different course types (e.g., quantitative
+vs. humanities). In addition to these analytics, we provide
+visualizations for instructors to interact with the results of our
+proposed framework and respond to changes in the course
+SLN. These visualizations encapsulate our analytics, allowing
+for interpretation by those not familiar with our model.
+Summary of Contributions: In summary, our contributions
+are (i) developing a link prediction framework for SLNs, which
+learns based on topological and post-based features of user
+discussions (Sec. II), (ii) demonstrating that the combination
+of our features with spatial pattern-capturing neural networks
+obtains the most robust SLN link prediction quality over six
+datasets, with AUCs above 0.90 in each case (Sec. III), and
+(iii) developing a set of analytics for SLN link formation based
+on our link prediction framework (Sec. IV).
+II. SOCIAL LEARNING NETWORK METHODOLOGY
+In this section, we formalize our SLN link prediction
+methodology. We first quantify an SLN from forum data (Sec.
+II-A) and define the particular features that are used as model
+inputs (Sec. II-B). We then develop unsupervised predictor,
+linear classifiers, GNNs, and deep learning classifiers (Sec.
+II-C) for link prediction.
+A. SLN Graph Model
+In order to define our features, we must first describe how
+link creation in an SLN model is inferred and quantified from
+online forum data.
+1) Online forums: The format of online forums differs by
+host site and by classroom needs. We identify two main types
+of forum structures to account for in our methodology.
+MOOC forum structure: A large online forum such as
+those hosted on Coursera is typically comprised of a series of
+threads, with each thread in turn being comprised of one or
+more posts. Each post is written by a single user. A post, in
+turn, can have one or more comments attached to it. Given
+the observation that SLN forum users do not abide by the
+designation of post vs. comment consistently [6], we will not
+distinguish between them, instead referring to them both as
+posts. This structure of thread posts is depicted in Fig. 2a.
+Q&A forum structure: Another format, implemented by
+Piazza, forces a “Question/Answer” thread structure. The
+forum is constructed from a series of questions and their
+responses, with allowance for follow-up questions and re-
+sponses. In contrast to traditional forums, a response on Piazza
+may have contributions from multiple users in the same block,
+rather than requiring a new comment from each user. Any
+question may have comments attached to it in the form
+of “follow-ups”, which can in turn generate new responses.
+Using the observation listed above from [6] again, we do not
+distinguish between types of follow-up responses and label
+all responses after the initial question as posts. This alternate
+structure of thread posts is depicted in Fig. 2b.
+2) Quantifying SLN link creation: A link (u, v) is observed
+between learner u and another learner v if, in a specific time
+interval, both u and v contribute to a post in the same thread
+(e.g., by either creating the initial post or contributing via a
+
+Input Feature Computation
+Prediction Model and Evaluation
+Analytics
+Application
+(Sec. II)
+(Sec. II and III)
+(Sec. III and IV)
+(Sec. V)
+Data through time I
+time
+Data through.time i
+ime
+Evaluation Result
+Visualizations
+Input feature
+i-1
+Neighborhood
+Evaluation Result i
+computation
+Features
+Network
+Data
+Topology
+Preprocessing
+Path Features
+Time Series
+Cross Validation
+Model Update
+Model State i
+Feature
+Features
+Importance
+Make Predictions4
+Forum Posts
+Topic Extraction
+Post Features
+Feature Analytics
+Recommendations
+Model State i-1
+Model Evaluation
+to Learmer4
+Fig. 2: Example of how posts in two different forum structures are
+divided into time periods and how SLN link creation between the
+learners authoring these posts is modeled. Fig. 2a (left): model for a
+Coursera forum. Fig. 2b (right): model for a Piazza forum.
+follow-up post). We use this as the criterion for establishing
+the link (u, v) in the SLN because it signifies the fact that
+learner u and learner v have exchanged ideas and interacted
+in the same thread within a specific time interval.
+To model the evolution of an SLN, we group its posts into
+different time intervals. Specifically, we divide all posts in a
+given thread into L equally spaced intervals. Fig. 2 illustrates
+this procedure for two example threads. We use yuv(i) as an
+indicator variable for the formation of link (u, v): yuv(i) = 1
+if a link between u and v has been created in any interval
+up to and including i, and yuv(i) = 0 otherwise. Thus, as in
+most social networks [38] [16], links persist over time in our
+SLN model. The SLN graph structure in any given interval
+i is then comprised of nodes corresponding to the learners u
+and edges (u, v) corresponding to links between them. For
+the purpose of predicting future responses, we consider this
+interaction to be bidirectional, i.e., the resulting SLN is an
+undirected graph. Formally, we define G(i) = [yuv(i)] as the
+binary adjacency matrix of the SLN during interval i; since
+links are bidirectional, G(i) is symmetric.
+We can also define subgraphs of G(i) focusing on particular
+students. Fig. 3 visualizes the neighborhood for an individual,
+randomly selected student at a particular time instance, where
+first and second degree connections are considered. In addi-
+tion to capturing detailed link-formation behavior evaluated
+later in this study, evaluating a visual representation from
+the perspective of a single student provides an intuition for
+individual student contributions and demonstrates the presence
+of “hub” students. The lack of multiple paths between students
+highlights the underlying sparse nature of G(i), requiring users
+to traverse one long path rather than choose from several short
+connections. Additionally, the relative small false positive rate
+(denoted by blue links in Fig. 3) demonstrates our framework’s
+efficacy for link prediction, as we will describe further in Sec.
+III-C.
+Two particular subsets of G(i) are of interest in the link
+prediction problem. We define
+Ω = (u, v) : u, v ∈ N(G), u ̸= v,
+(1)
+i.e., all possible learner pairs in the SLN. We then define
+two subsets of Ω : G(L), which is the set of formed links
+at the final time i = L (i.e., with yuv(L) = 1), and
+Gc(L) = Ω \ G(L), the complement graph of un-formed links
+(i.e., yuv(L) = 0). Note that |Gc(L)| ≫ |G(L)| for each
+dataset (i.e., most learners are never linked). This large class
+imbalance between formed and unformed links informs our
+link prediction framework in Sec. II-C.
+B. SLN Feature Engineering
+We now define our features, computed for each learner pair
+(u, v), u ̸= v. These quantities serve as the inputs to our
+prediction algorithms in Sec. II-C.
+Neighborhood-based Features: These features, as well as
+path-based features discussed next, are extracted from the
+topology of the graph. Letting N(G) be the set of nodes in
+the SLN G and Γu(i) ⊆ N(G) denote the set of neighbors
+of u at time i, the neighborhood-based features qualitatively
+measure the “similarity” of u and v’s neighborhoods [7]. They
+are quantified as follows:
+1) Jaccard coefficient:
+Jauv = |Γu(i) ∩ Γv(i)|/|Γu(i) ∪ Γv(i)|
+2) Adamic-Adar index:
+Aduv =
+�
+n∈Γu(i)∩Γv(i)
+1/log|Γn(i)|
+3) Resource allocation index:
+Reuv =
+�
+n∈Γu(i)∩Γv(i)
+1/|Γn(i)|
+4) Preferential attachment score:
+Pruv = |Γu(i)| · |Γv(i)|
+We let buv denote the vector of these features for pair (u, v).
+Note that a larger value of each of these features, roughly
+speaking, indicates that u and v share more common, low
+degree neighbors than they do with others.
+Path-based Features: These features measure the proximity
+of u and v in the SLN. They are as follows:
+5) Shortest path length (Lpuv): The length of the shortest
+path between u and v.
+6) Number of paths (Npuv): The number of shortest paths
+(i.e., of length Lp) between u and v.
+We let auv denote the vector of these features. Note that as
+Lp decreases, u and v become more closely connected, while
+a larger Np indicates more redundancy in these paths.
+Post-based Features: Besides topology-based attributes,
+learners’ interests in different course topics will also influence
+their probability of forming links in an SLN. In particular,
+we would expect those with similar topic interests to be more
+likely to post in the same thread, i.e., form links. We thus
+compare the topics of different learners’ posts to compute
+another feature that shows the learners’ similarity in interests.
+To do this, we apply the Latent Dirichlet Allocation (LDA)
+algorithm [39] on the dictionary of all course words (i.e., all
+unique words used in all the considered posts of a course) to
+extract a set, K, of latent topics across posts, and a model of
+
+a. Coursera
+b. Piazza
+u1
+Post 1
+u1
+Question
+(1,2)
+(1,2), (1,5)
+4
+(1,3)
+u2
+Reply 1
+(1,3)
+u2, u5
+Answer
+(2,3)
+(2,3)
+4
+u3
+Reply 2
+u3
+Follow Up 1
+i1
+u4
+Follow Up 2
+u4
+Reply 3
+(2,4)
+u2
+Follow Up 3
+i2
+u2
+Reply 4
+(1,2)
+u1
+Reply 5
+u1
+Follow Up 4
+i35
+Forum
+Course Title
+Beginning
+Duration
+Users
+Threads
+Learner Pairs
+Posts
+ml
+Machine Learning
+4/29/13
+12
+4263
+4217
+73315
+25481
+algo
+Algorithms: Design and Analysis I
+9/22/14
+13
+3013
+4656
+50006
+16276
+shake
+Shakespeare in Community
+4/22/15
+5
+958
+1389
+66217
+7484
+comp
+English Composition I
+7/01/13
+8
+1862
+1286
+20083
+8255
+f19
+Python for Data Science
+8/20/19
+18
+115
+669
+17000
+2013
+s20
+Python for Data Science
+1/17/20
+17
+290
+1129
+44964
+4955
+TABLE I: Descriptive metrics on our six considered forum datasets. The title, beginning date (m/dd/yy), duration (weeks), number of users,
+threads, learner pairs, and posts by the end. All courses were broken into 20 time instances.
+Fig. 3: A snapshot of the SLN graph model for a single user
+(represented by a unique ID string) and their close neighborhood.
+The visual demonstrates the lack of multiple paths between users,
+underlying the sparse nature of the graph.
+posts as a probability vector of these topics. In our application,
+we view each post as a separate “document,” since learners
+are likely to discuss many distinct topics over time. For each
+learner, u, we obtain the latent topic vector of their posts
+through time i as the average of their post vectors through
+i. We denote the set of topics for learner u that exceed a
+minimum threshold of coverage across their posts through time
+i as Ku(i). With this, we define the last feature which captures
+the number of common topics between learners u and v:
+7) Number of common topics (To): |Ku(i) ∩ Kv(i)|
+We use cuv as the time-series version of To, i.e., the number
+of common topics discussed by u and v.
+C. Link Prediction Methodology
+As discussed in Sec. II-B, the features extracted from the
+graph topology contain spatially and temporally correlated
+patterns between learner pairs. Therefore, we employ pre-
+diction models that are capable of exploiting these patterns
+for accurate link prediction. In this capacity, we consider the
+efficacy of four distinct deep learning architectures for our
+proposed framework: (i) the fully connected neural network
+(FCNN), which offers effective latent space prediction; (ii) the
+convolutional neural network (CNN), which is highly effective
+for processing spatially correlated patterns; (iii) the long-
+short-term memory (LSTM) based recurrent neural network
+(RNN), which is desirable for time-series modeling; (iv)
+the convolutional recurrent neural network (CRNN), which
+extracts both spatial and temporal correlations. As baselines to
+these methods, and to demonstrate the necessity of the afore-
+mentioned classifiers and their corresponding architectures, we
+compare our proposed deep learning prediction framework to
+five traditional prediction models: an unsupervised predictor,
+two linear prediction models (support vector machines and
+linear discriminant analysis), a graph neural network [45], and
+a Bayesian neural network [40].
+For a given pair of users (u, v), the input feature vector into
+each of the following models is given by euv = [buv, auv, cuv]
+while the target output is the link state yuv(i) ∈ {0, 1}. In the
+following, we describe the latent state of each model as well
+as their corresponding training procedures.
+1) Unsupervised Predictor: We begin by using a simple
+prediction algorithm as a benchmark for the parameter-based
+models described below. Choosing the feature most associated
+with link formation, we follow [16] and turn the resource
+allocation index (Re) feature into an unsupervised predictor.
+To do this, we compute Re for each (u, v) ∈ Ω, normalize the
+vector of values to [0, 1], and use this as ˆyuv(i).
+2) Linear Classifiers: Next, we consider two relatively sim-
+ple linear models for SLN link prediction: linear discriminant
+analysis (LinDA) and support vector machines (SVMs). Both
+models attempt to find a separating linear hyper-plane between
+learners who did and did not form links. However, both models
+are learned using different methodologies. Specifically, LinDA
+uses every sample during training and assumes samples in
+each class follow the same distribution and have the same
+covariance matrix whereas SVM makes no prior assumptions
+on the data’s distribution and aims to find a decision boundary
+using the points that result in the highest error.
+3) Graph Neural Networks (GNN): GNNs are a class of
+neural networks for learning over datasets expressed as graphs.
+They have been employed to perform link prediction on a
+variety of graph topologies [45], [46]. A potential advantage
+of GNNs in our setting would be obviating much of the
+feature engineering in Sec. II-B, as they can learn directly
+from the graph structure. Thus, we compare the efficacy of
+GNNs to our proposed method for predicting link formation
+in SLNs. Specifically, we adopt a two-layer convolutional
+GraphSAGE model [47], where node attributes of the SLN
+are self-generated during training. Here, the adjacency matrix
+of the SLN is used as input into the GraphSAGE model at a
+given time in order to predict future links.
+4) Deep Learning Classifiers: One potential limitation of
+linear classifiers is their small parameter space, which pre-
+vents learning intricate non-linear relationships between input
+features extracted from an SLN. GraphSAGE GNNs aim to
+address this challenge, but they lose the ability to model
+
+yourID
+1stdegreeconnections
+2nddegreeconnections
+Peersyouareencouragedtotalkto
+FormedConnections
+EncouragedFutureConnections6
+Features
+SNR
+Mean
+s.d
+Ja
+0.5741
+0.1467
+0.1818
+0.0224
+0.0345
+Ad
+0.8069
+2.6963
+2.6556
+0.2121
+0.4783
+Re
+0.8221
+0.2838
+0.3108
+0.0085
+0.0241
+Pr
+0.3478
+5413.9
+12436
+512.37
+1653.8
+Lp
+-0.7037
+0.8712
+0.3454
+1.6186
+0.7165
+Np
+-0.1603
+2.0779
+9.1893
+9.3004
+35.855
+To
+0.2019
+1.0201
+1.6955
+0.4904
+0.9276
+(a) ml
+Features
+SNR
+Mean
+s.d
+Ja
+0.6614
+0.2312
+0.2727
+0.0246
+0.0396
+Ad
+0.8254
+3.1919
+3.3436
+0.1748
+0.3116
+Re
+0.9411
+0.3503
+0.3355
+0.0092
+0.0268
+Pr
+0.3812
+1797.6
+3253.4
+270.87
+752.06
+Lp
+-0.6638
+0.7974
+0.3091
+1.4348
+0.6511
+Np
+-0.2191
+1.3389
+3.8776
+4.9092
+12.421
+To
+0.1668
+0.5875
+0.9624
+0.3364
+0.5426
+(b) algo
+Features
+SNR
+Mean
+s.d
+Ja
+0.2535
+0.1608
+0.2207
+0.0721
+0.1291
+Ad
+0.7276
+1.8286
+2.1686
+0.0956
+0.2131
+Re
+0.7648
+0.2959
+0.3434
+0.0045
+0.0376
+Pr
+0.3836
+1041.8
+2325.5
+38.123
+291.32
+Lp
+-0.7048
+0.9248
+0.3497
+1.8233
+0.9251
+Np
+-0.2498
+1.3182
+3.2174
+5.9579
+15.352
+To
+0.1258
+0.5703
+0.8587
+0.4039
+0.4637
+(c) comp
+Features
+SNR
+Mean
+s.d
+Ja
+0.3527
+0.1354
+0.1318
+0.0565
+0.0914
+Ad
+0.7148
+2.6913
+2.8538
+0.2612
+0.5453
+Re
+0.6648
+0.2934
+0.3647
+0.0143
+0.0551
+Pr
+0.4871
+1904.1
+3074.1
+142.67
+541.58
+Lp
+-0.7802
+0.9519
+0.2995
+1.7221
+0.6874
+Np
+-0.2414
+1.8512
+4.3331
+7.3385
+18.397
+To
+0.3151
+1.3249
+1.6287
+0.5906
+0.7009
+(d) shake
+Features
+SNR
+Mean
+s.d
+Ja
+0.5807
+0.1413
+0.1294
+0.0323
+0.0582
+Ad
+0.6414
+1.8429
+2.0376
+0.2099
+0.5084
+Re
+0.5999
+0.2633
+0.3315
+0.0241
+0.0673
+Pr
+0.6066
+360.11
+449.03
+32.847
+90.413
+Lp
+-1.1082
+1.3231
+0.3538
+2.1759
+0.4158
+Np
+-0.4079
+1.7306
+1.4857
+3.9584
+3.9746
+To
+0.6042
+2.6515
+2.8861
+0.3702
+0.8893
+(e) f19
+Features
+SNR
+Mean
+s.d
+Ja
+0.6901
+0.1341
+0.1088
+0.0266
+0.0468
+Ad
+0.6628
+2.5344
+2.8694
+0.2289
+0.6088
+Re
+0.6149
+0.2347
+0.3019
+0.0164
+0.0531
+Pr
+0.5902
+1109.7
+1469.8
+81.076
+273.16
+Lp
+-0.9782
+1.4292
+0.3748
+2.1761
+0.3887
+Np
+-0.2908
+2.9899
+2.9203
+6.0636
+7.6483
+To
+0.6691
+2.8634
+2.9075
+0.3679
+0.8221
+(f) s20
+TABLE II: Summary statistics – SNR, mean and standard deviation (s.d.) – for the network features of the two link groups. The top row
+for each feature corresponds to formed links (yuv(L) = 1), and the bottom to non-formed links (yuv(L) = 0). Taken individually, the
+neighborhood-based features Re and Ad have the strongest correlations with link formation, while the topic-based To tends to have the least.
+k
+Support
+Top 3 Words
+1
+0.1257
+class question svm
+2
+0.1078
+computer work image
+3
+0.0895
+gradient set lambda
+4
+0.0835
+code problem exercise
+5
+0.0741
+octave line column
+(a) ml
+k
+Support
+Top 3 Words
+1
+0.2287
+thought fast graphs
+2
+0.0872
+heap length max
+3
+0.0713
+algorithm time run
+4
+0.0684
+file sort merge
+5
+0.0676
+set problem line
+(b) algo
+k
+Support
+Top 3 Words
+1
+0.1141
+project composition https
+2
+0.0736
+annotated idea good
+3
+0.0541
+great word read
+4
+0.0486
+writ time read
+5
+0.0425
+feedback hope find
+(c) comp
+k
+Support
+Top 3 Words
+1
+0.2607
+shakespeare play time
+2
+0.1671
+family bad sentence
+3
+0.1185
+romeo juliet scene
+4
+0.1009
+time play text
+5
+0.0528
+love night dream
+(d) shake
+k
+Support
+Top 3 Words
+1
+0.1108
+readme want fix
+2
+0.0822
+standard test sample
+3
+0.0765
+dataset issue
+4
+0.0746
+https pip install
+5
+0.0688
+file git ngrams
+(e) f19
+k
+Support
+Top 3 Words
+1
+0.1369
+data correct question
+2
+0.0968
+true points array
+3
+0.0787
+test case import
+4
+0.0762
+error redirect prefix
+5
+0.0615
+point report fine
+(f) s20
+TABLE III: Summary of the top five topics extracted by LDA for
+each online discussion forum. For each course, the topics tend to be
+reasonably disjoint, with the exception of common words
+explicit features between node pairs. To mitigate each of these
+shortcomings, we propose a deep learning approach on specif-
+ically engineered features in which various characteristics of
+(u, v) (e.g., spatial and time-varying properties) are expected
+to be learned for stronger prediction performance.
+Specifically, we propose five deep architectures for link
+prediction: the Bayesian neural network (BNN), the fully
+connected neural network (FCNN), the convolutional neural
+network (CNN), the recurrent neural network (RNN), and the
+convolutional recurrent neural network (CRNN). Each model
+(excluding the Bayesian Neural Network) applies the Rectified
+Linear Unit (ReLU) activation function, given by σ(a) =
+max{0, a}, in its hidden layers followed by a two-unit output
+layer, which applies the softmax activation function, which
+allows for a probabilistic interpretation of link formation for
+a learner pair (u, v). The model architecture for each of our
+considered models are discussed below. The hyper-parameter
+selection of each model was empirically determined to best fit
+the diverse datasets utilized in Sec. III.
+Bayesian Neural Network (BNN): The Bayesian Network
+(BNet) model [40] defines the probability density of latent
+variable zuv as a Gaussian:
+P(zuv|euv) = N(wT euv, σ2),
+(2)
+where w is the weight vector and σ2 is the variance, both
+to be estimated when the model is trained. From this, yuv is
+estimated according to
+P(yuv = 1|zuv) = σ(φφφT zuv + b),
+(3)
+where φφφ and b are a vector and scalar, respectively, to be
+estimated during training, and σ(·) is the logistic sigmoid
+function given by σ(·) = 1/(1 + e−(·)).
+Our BNN architecture is composed of a hidden layer
+encoding the latent variable zuv. This hidden layer has 10
+
+7
+(a) Ja
+(b) Ad
+(c) Re
+(d) Pr
+(e) Np
+(f) Lp
+(g) To
+Fig. 4: Cumulative distribution functions (CDFs) for each of the seven feature vectors from s20. CDFs of non-formed links are marked in
+blue, and CDFs of formed links are shown in orange. These demonstrate that there is (a) an observable difference in distribution between
+the two populations for each feature and (b) an inverse relationship between number of shortest paths and shortest path length.
+units, each represents a normal distribution with weight wi
+and variance σ2. Following this hidden layer is a dense output
+layer with softmax activation function given in [40].
+Fully Connected Neural Network (FCNN): FCNNs are
+considered a higher dimensional non-linear extension of link
+classifiers. Such models can potentially represent more so-
+phisticated non-linear relationships for better link prediction.
+Our fully connected multi-layer artificial neural network is
+composed of two hidden layers each containing 128 units.
+Convolutional Neural Network (CNN): In addition to
+FCNN models, we also consider deep convolutional neural
+networks (CNNs), which in addition to providing a large
+parameter space for learning, capture spatial characteristics
+between features for each learning pair (u, v). In the domain of
+link prediction, capturing spatial correlations between signal
+features is especially important since the majority of features
+(e.g., buv and auv) are extracted from the topology of the SLN
+graph. Our proposed CNN for link prediction is composed of
+two convolutional layers with 64 3 × 1 feature maps and 32
+2 × 1 feature maps, respectively, followed by a 32-unit fully
+connected layer.
+Recurrent Neural Network (RNN): BNNs, FCNNs and
+CNNs, as well as linear classifiers, do not explicitly model
+the evolution of latent space variables over time based on euv.
+This could potentially provide useful information for modeling
+an SLN, particularly so that the predictor could respond to
+sudden changes in the input relative to the prior state. This
+may occur, for example, when the topic of the course shifts,
+which could be reflected in a sudden change in cuv.
+To address this challenge, we consider a long-short-term
+memory (LSTM) based RNN with input duv = [euv, huv(i −
+1)]T , where huv(0) = 0 and huv(i − 1) is the output vector
+from the previous time. We then define the interaction gate,
+relationship gain gate, and relationship fading gate vectors at
+each time interval, i, as
+guv(i) = ψ(Wgduv(i) + bg),
+(4)
+iuv(i) = σ(Widuv(i) + bi),
+(5)
+fuv(i) = σ(Wfduv(i) + bf),
+(6)
+respectively. Here, ψ(·) and σ(·) are the tanh and sigmoid
+functions, respectively, and the matrices Wg, Wi, and Wf
+as well as the vectors bg, bi, and bf contain parameters that
+are estimated during the model training procedure. Formally,
+the latent cell state, zuv(i), is updated as
+zuv = guv(i) ⊙ iuv(i) + zuv(i − 1) ⊙ fuv(i),
+(7)
+where ⊙ denotes element-wise matrix multiplication. An out-
+put gate, ouv(i), is then used to determine the factor to which
+each element of zuv(i) should be used in the definition of
+huv(i):
+ouv(i) = σ(woduv(i) + bo), huv(i) = σ(zuv(i) ⊙ ouv(i)).
+(8)
+With this, yuv(i) is estimated as
+P(yuv(i) = 1|zuv(i)) = σ(h1(i)),
+(9)
+where h1(i) is the first element of h(i). Our implemented
+RNN is composed of 64-cell LSTM layer followed by 128-
+unit fully connected layer.
+Convolutional Recurrent Neural Network (CRNN): Con-
+volutional recurrent neural networks contain both convolu-
+tional layers and recurrent LSTM layers. Although such mod-
+els are typically computationally costly to train, they capture
+both spatial and time-varying correlations between learner
+pair feature vectors, thus providing the advantages of high
+parameter deep learning models with CNNs and RNNs. Our
+proposed CRNN architecture consists of two convolutional
+layers, containing 64 3 × 1 and 32 2 × 1 feature maps
+respectively, followed by a 32-cell LSTM layer, and a 32 unit
+fully connected layer.
+
+1.0
+Cumulative Probability
+0.8
+0.6
+0.4
+0.2
+0.0
+0.00
+3.79
+7.57
+11.36
+15.14
+Feature
+eValue1.0
+Cumulative Probability
+0.8
+0.6
+0.4
+0.2
+0.1592
+0.3183
+0.4775
+0.6367
+Feature Value1.0
+Cumulative Probability
+0.8
+0.6
+0.4
+0.2
+0.0
+0.00
+8.48
+16.95
+25.43
+33.90
+Feature Value1.0
+Cumulative Probability
+0.8
+0.6
+0.4
+0.2
+0.847
+1.694
+2.542
+3.389
+Feature Value1.0
+Cumulative Probability
+0.8
+0.6
+0.4
+0.2
+0.0
+0
+2154
+4308
+6462
+8616
+Feature
+eValue1.0
+Cumulative Probability
+0.8
+0.6
+0.4
+0.2
+0.0
+1.0
+27.5
+54.0
+80.5
+107.0
+Feature
+eValue1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+1
+2
+3
+4
+5
+Feature
+eValue8
+5) Deep Learning Parameter Training: We train each deep
+learning algorithm using the Adam optimizer as well as the
+categorical cross entropy loss function, which for our link
+prediction setup is given by
+L = − 1
+N
+N
+�
+n=1
+2
+�
+j=1
+yjlog( ˆyj),
+(10)
+where N is the total number of samples being used to calculate
+the loss and ˆy is the probability of link formation. Each model
+uses a batch size of 64 as well as a learning rate of 0.001.
+Finally, each model is trained using 300 epochs, which is
+sufficient for convergence on each dataset but simultaneously
+allows for convergence at slightly different optima, resulting
+in robust and reliable evaluation when used with k-fold cross
+validation as further discussed in Sec. III-B.
+III. LINK PREDICTION EVALUATION
+In this section, we begin by describing our considered
+courses along with their corresponding datasets (Sec. III-A)
+as well as our model evaluation procedure (Sec III-B). We
+then evaluate our framework’s performance for predicting link
+formation (Sec. III-C) and examine the time-accuracy of our
+prediction model (Sec. III-D).
+A. Datasets
+We consider the SLNs formed in six courses: four Coursera-
+based MOOC courses and two traditional courses offered
+at Purdue University. The four MOOC courses – “Machine
+Learning” (ml), “Algorithms: Design and Analysis, Part 1”
+(algo), “English Composition I” (comp), and “Shakespeare in
+Community” (shake) – were selected to represent a diverse
+set of subjects: two quantitative in nature and two in the
+humanities. In addition, we also consider the course “Python
+for Data Science” hosted through Purdue University over two
+semesters: “Fall 2019” (f19) and “Spring 2020” (s20). The
+availability of data from two offerings of a single course
+provides a unique opportunity to evaluate behavior in a single
+course over multiple semesters. The s20 dataset is of particular
+interest because of its relation with the COVID-19 pandemic.
+Specifically, this course was held in-person from January -
+March, allowing students to begin forming in-person links,
+which carried into their relationship in the course’s SLN.
+However, with the pandemic forcing a transition to fully online
+learning, link formation between students became completely
+dependent on discussion forum communication. The inclusion
+of the f19 and s20 datasets, which differ both in size and
+in format, demonstrate our framework’s broad applicability
+to different online course formats in dynamic environments.
+Table I shows detailed metrics of the six considered datasets.
+Fig. 5 summarizes the graph topology at the termination of
+each course under evaluation in terms of five social network
+metrics: number of nodes, number of edges, shortest path
+lengths (i.e., the Lpuv feature), degree per node, and user
+clustering coefficients. The diverse nature of each course is
+evident from each of the shown metrics and particularly from
+the varying number of edges and nodes. We observe the largest
+differences between the Purdue f19 and s20 courses versus
+the MOOC courses: the f19 and s20 courses are significantly
+smaller in nodes/edges and also have significantly larger
+degree per node and clustering coefficients. We also observe
+the difference in both the number of edges and the average
+degree per node between the f19 and s20 courses, which
+demonstrates the increase in student utilization of discussion
+forums in the absence of in-person instruction.
+Next, we describe the SLNs in terms of the features in
+Sec. II-B. We make several observations on associations with
+link formation within and across datasets before evaluating the
+link-prediction portion of our proposed framework.
+1) Data Preparation: To obtain a representative set of
+student behavior from a course, and to ensure that data
+gathered from each source is uniformly formatted, we filter
+each considered dataset. Specifically, we remove the instruc-
+tors from the list of learners and remove all links formed
+between learners and instructors, since we are interested in
+developing models targeted towards peer-to-peer interaction,
+with the goal of requiring less direct instructor intervention.
+Furthermore, interactions before the beginning of a course are
+removed; only links formed during a course are considered.
+Both course-hosting sites offer an option for full anonymity
+to learners – posts made with anonymity are ignored, as we
+cannot make meaningful connections with unknown users.
+Enrolled learners who did not access the forum (i.e., an empty
+adjacency matrix), are not considered to remove confusion –
+a lack of behavior excludes a helpful metric for predicting
+future behavior. Such students would likely benefit from more
+traditional intervention. After filtering, less than 2% of the
+learner pairs in each dataset demonstrated a formed link.
+This underscores an extreme sparsity of learner pairs for link
+prediction; the methodology applied to avoid overfitting will
+be discussed further in Section III-B.
+2) Topic extraction: To obtain the post similarities cuv(i),
+we must first extract the topics, K, and distributions for
+each post according to the LDA algorithm discussed in Sec.
+II-B. Prior to building the dictionary of topics, all URLs,
+punctuations, and stopwords are removed from each post’s
+text and all words are stemmed. Table III summarizes the
+topic extraction results for each dataset using |K| = 20 topics;
+the top three words shown are from the five topics that have
+the highest supports across posts. We find that |K| = 20
+produces a set of topics that have reasonably large supports
+across posts while retaining granular information, i.e., able
+to convey differences between student posts. In our manual
+inspection, larger values of |K| lacked the support to generate
+informative features, while smaller values of |K| resulted in
+too much intersection between topics for a good understanding
+of content.
+B. Model Evaluation Procedure
+To evaluate the models proposed in Sec. II, we use the
+following metrics, training procedures, and evaluation criteria.
+1) Metrics: We use three metrics to evaluate prediction
+performance. First, we compute the overall Accuracy (ACC),
+
+9
+Fig. 5: Social network graph metrics on our datasets. We see the largest distinction in characteristics between the four MOOC courses and
+the two Purdue courses.
+Model
+ml
+algo
+shake
+comp
+f19
+s20
+Re
+AUC
+0.5005 ± 0.0004
+0.5188 ± 0.0322
+0.5061 ± 0.0034
+0.5167 ± 0.0266
+0.5689 ± 0.0401
+0.5238 ± 0.0121
+ACC
+0.5995 ± 0.0054
+0.8338 ± 0.0104
+0.8296 ± 0.0073
+0.8349 ± 0.0082
+0.9524 ± 0.0057
+0.9599 ± 0.0020
+BNet
+AUC
+0.9053 ± 0.0106
+0.9488 ± 0.0058
+0.8603 ± 0.0095
+0.8684 ± 0.0116
+0.7413 ± 0.0546
+0.7495 ± 0.0269
+ACC
+0.9175 ± 0.0066
+0.9805 ± 0.0019
+0.9472 ± 0.0035
+0.9492 ± 0.0026
+0.9600 ± 0.0053
+0.9672 ± 0.0013
+FCNN
+AUC
+0.9766 ± 0.0033
+0.9706 ± 0.0039
+0.9670 ± 0.0059
+0.9714 ± 0.0084
+0.8991 ± 0.0367
+0.8844 ± 0.0330
+ACC
+0.9782 ± 0.0027
+0.9871 ± 0.0029
+0.9853 ± 0.0019
+0.9850 ± 0.0022
+0.9688 ± 0.0037
+0.9729 ± 0.0022
+SVM
+AUC
+0.9122 ± 0.0027
+0.9523 ± 0.0050
+0.8982 ± 0.0071
+0.8618 ± 0.0071
+0.8437 ± 0.0343
+0.8203 ± 0.0113
+ACC
+0.9137 ± 0.0026
+0.9755 ± 0.0035
+0.9608 ± 0.0031
+0.9462 ± 0.0022
+0.9670 ± 0.0040
+0.9700 ± 0.0015
+LinDA
+AUC
+0.8486 ± 0.0056
+0.8361 ± 0.0064
+0.7521 ± 0.0116
+0.7331 ± 0.0123
+0.6940 ± 0.0146
+0.6692 ± 0.0205
+ACC
+0.8674 ± 0.0051
+0.9425 ± 0.0018
+0.9117 ± 0.0050
+0.9084 ± 0.0056
+0.9582 ± 0.0046
+0.9620 ± 0.0026
+RNN
+AUC
+0.9880 ± 0.0011
+0.9808 ± 0.0026
+0.9807 ± 0.0054
+0.9770 ± 0.0071
+0.8304 ± 0.0373
+0.8329 ± 0.0349
+ACC
+0.9890 ± 0.0010
+0.9902 ± 0.0013
+0.9906 ± 0.0019
+0.9877 ± 0.0030
+0.9653 ± 0.0040
+0.9710 ± 0.0024
+CNN
+AUC
+0.9881 ± 0.0019
+0.9817 ± 0.0029
+0.9754 ± 0.0057
+0.9763 ± 0.0055
+0.9187 ± 0.0318
+0.9221 ± 0.0169
+ACC
+0.9894 ± 0.0015
+0.9916 ± 0.0009
+0.9888 ± 0.0025
+0.9882 ± 0.0022
+0.9711 ± 0.0033
+0.9740 ± 0.0015
+CRNN
+AUC
+0.9680 ± 0.0094
+0.9704 ± 0.0087
+0.9608 ± 0.0066
+0.9725 ± 0.0070
+0.8903 ± 0.0468
+0.8845 ± 0.0347
+ACC
+0.9713 ± 0.0090
+0.9846 ± 0.0036
+0.9803 ± 0.0028
+0.9859 ± 0.0020
+0.9705 ± 0.0016
+0.9724 ± 0.0020
+GNN
+AUC
+0.9969 ± 0.0014
+0.9989 ± 0.0007
+0.9988 ± 0.0011
+0.9955 ± 0.0029
+0.7395 ± 0.0508
+0.5628 ± 0.1157
+ACC
+0.9967 ± 0.0008
+0.9988 ± 0.0007
+0.9965 ± 0.0068
+0.9958 ± 0.0019
+0.6557 ± 0.0751
+0.5500 ± 0.0997
+TABLE IV: Performance of each considered link prediction model. The CNN model is among the best performing model across all six
+datasets with respect to the AUC and ACC metrics. All results in bold highlight the best performing results. We see that the GNN results in
+strong performance on the four MOOC courses while performing poorly on the two Purdue courses, indicating that GNNs are effective for
+link prediction in large courses whereas our method delivers strong performance in small courses as well as large.
+or the fraction of predictions over all time that are correct. For
+iteration k, it is obtained as:
+1
+|Ωke| · L
+�
+(u,v)∈Ωk
+e
+L
+�
+i=1
+1{yuv(i) = ¯yuv(i)},
+(11)
+where yuv(i) ∈ {0, 1} is the binary prediction made based on
+˜yuv(i) and 1 is the indicator function. Second, we compute
+the Area Under the ROC Curve (AUC), which assesses the
+tradeoff between true and false positive rates for a classifier
+[5]. Third, we define a metric called Time Accuracy (TAC)
+to be the fraction of links that are predicted to form within
+a fixed window w of when they actually form (among those
+that eventually form). Letting nuv = mini{yuv(i) = 1} be
+the actual time at which link (u, v) ∈ Ωf
+k forms and ˜nuv =
+mini{˜yuv(i) = 1} the predicted time, the TAC is defined as
+1
+|Ωf
+k|
+�
+(u,v)∈Ωf
+k
+1{|˜nuv − nuv| ≤ w}
+(12)
+for iteration k, where Ωk
+f ⊂ Ωk
+e is the set of correctly predicted
+links in the test set that will eventually form. We compute
+the mean and standard deviation of each metric across three
+evaluation iterations.
+2) Training and Testing: k-fold cross validation is used to
+evaluate each predictor with k = 10. Following Sec. III-A, we
+again consider the link sets G(L) and Gc(L). Our objective
+is to train models capable of accurate link prediction despite
+the large class imbalance between G(L) and Gc(L) that will
+be observed during training and inference. To achieve this,
+we take an equal proportion of samples from both G(L) and
+Gc(L) to form each training fold, which, in turn, retains the
+overall class imbalance in the training set during each training
+iteration. The corresponding testing set of each training fold
+contains the same class imbalance. After each training fold, we
+calculate the metrics of interest on the respective testing set of
+the validation run. This sampling, along with the utilization of
+the AUC measurement, allows us to quantify the false alarm
+versus true positive rate, since the prediction accuracies on a
+poorly trained model could be very high due to the large class
+imbalance.
+In each of the k iterations, we consider a set of time intervals
+from which the model parameters are estimated considering
+each pair (u, v) ∈ Ωr
+k, using the procedures in Sec. III-B2.
+Then, for each (u, v) ∈ Ωe
+k, the inputs are used to make a
+prediction ˜yuv(i) ∈ [0, 1] of the link state yuv(i).
+C. Link Prediction Evaluation
+Table IV gives the overall performance of the baseline,
+linear, GNN, and deep learning models in terms of the AUC
+and ACC metrics. Overall, we see that the CNN consistently
+outperforms the other predictors for each considered dataset.
+In addition, the GNN achieves strong (comparable to the CNN)
+
+User Clustering Coefficients
+Shotest Path Length
+Number of Edges
+Number of Nodes
+Degree per Node
+30000
+9
+200
+10000
+2
+2910
+prediction performance on the four MOOC datasets, but it
+performs poorly on both f19 and s20, achieving AUCs of 0.74
+and 0.56 in f19 and s20, respectively. This behavior is con-
+sistent with observations in prior work [46] that GNNs require
+large datasets for effective generalization – a characteristic that
+MOOCs are able to provide (with at least 1,000 users in each
+case, see Table I) whereas the Purdue courses, f19 and s20,
+are not. Our explicit feature engineered methodology paired
+with a CNN classifier, on the other hand, is more robust against
+variations in SLN course size and type in comparison to the
+GNN.
+Of particular interest is the s20 dataset and its performance
+relative to the other five datasets. Because s20 was held
+partially in-person prior to the COVID-19 outbreak in March
+2020, the behavior represented includes both in-person and
+online interactions. Furthermore, it contains a rapid change
+in behavior midway through the semester that models must
+account for. It follows from the high accuracies and AUCs
+demonstrated by each deep-learning model on this dataset that
+our prediction model can be applied to hybrid-online courses
+with a similar level of accuracy to fully online courses. It also
+suggests that our proposed model is responsive to large-scale
+shifts in student behavior. From Table IV, we see neither the
+GNN nor the other baseline models are capable of capturing
+either of these desirable characteristics. As a result, we find
+that our proposed framework is capable of increasing both
+course quality and learner interactions during the pandemic;
+an attribute that can be leveraged to improve instruction in a
+post-pandemic course offering.
+Considering all courses, the CNN model has slightly higher
+performance across the metrics and datasets, reaching average
+AUCs between 0.92 and 0.99 and average ACCs between 0.97
+and 0.99. The AUC of Re is nearly random, but demonstrates
+a high accuracy in all cases because of the large class im-
+balance present. Similarly, the linear classifiers demonstrate
+high ACC values because of the large class imbalance as
+well. Although the Bayesian model consistently outperforms
+the baseline models, the lower accuracy and AUC relative
+to the CNN and CRNN models confirms our hypothesis from
+Sec. II that capturing spatial and temporal variance leads to
+improvement in the model. More specifically, the evolution
+of the state of an SLN between different time periods, both
+temporally and spatially, is important to predicting learner
+interactions; this aspect is effectively included in the LSTM-
+based CRNN. We further observe that the CNN model, capturing
+spatial variance, and the RNN capturing temporal variance,
+each perform similarly to the CRNN model for several datasets.
+This suggests that while spatial and temporal variance both
+individually assist in prediction, their combined usage may
+not result in significant performance improvements.
+Although an accurate prediction is most informative on the
+efficacy of a connection between learners, recommendations
+may also be supported by false predictions. If a high-accuracy
+model falsely predicts that two users will connect, we may
+infer that the formation of a link between these two users
+would be beneficial based on model parameters. Conversely,
+there is a strong correlation between false negative predictions
+and weak links between learners, implying that the benefits of
+forming a connection between two such users would be trivial
+compared to other, more highly-weighted connections.
+D. Early Detection of Link Formation
+The models proposed in Sec. III-C consider the ability to
+predict link formation in subsequent time intervals up until
+the end of the course. However, it does not consider links
+that will form at an earlier or later interval. These occurrences
+of a delay between link formation and prediction can lend
+additional information of importance to learners: if we can
+predict in advance which learners may form connections, we
+may encourage them to connect sooner, potentially resulting in
+a stronger connection or faster replies from learners expected
+to have delayed responses. On the other hand, if we find that
+a link forms much sooner than predicted by our model, this
+may indicate that learners would benefit from re-connecting
+on the current topic later in the course.
+To study these cases, we evaluate the TAC metric from
+Sec. III-B for our RNN, CNN, FCNN, and CRNN models; i.e.,
+we measure whether links form within a given window w of
+when they are predicted to. Note that the TAC metric was
+only calculated for the deep learning models, since they were
+consistently the best performing link formation predictors.
+The granular value of 20 time intervals used to generate the
+SLN graph model gives the predictive model access to more
+frequently updated features, and allows the model to respond
+quickly to changes in SLN behavior. Fig. 6 shows the TAC
+values as w is increased from 0 to 20 for several of our
+proposed deep learning prediction models. The sharp increase
+of each TAC curve for small w of each model – with the
+exception of the RNN – indicates that many links form close to
+when they are predicted to form, reinforcing our observations
+of model quality from other performance metrics in Sec. III-C.
+A window of w = 2, for example, is already sufficient for all
+six forums to reach a TAC of 0.5 or above.
+Observing Fig. 6e, which represents the TAC curve of the
+f19 dataset, it is clear that our TAC metric demonstrates
+a lower accuracy for small datasets but the performance of
+individual models has more variation. This is largely attributed
+to the smaller number of learner pairs contained in the f19
+dataset with which to train the model compared to a MOOC
+forum. However, with the exception of the RNN, we can
+observe the same curve shape and sharp initial increase present
+for larger datasets, indicating that TAC is both a consistent and
+useful evaluation metric of model performance. We can further
+observe in the ml, f19, and s20 datasets that the RNN model
+fails to correctly predict links consistently across datasets
+within a small interval of when they actually occur, further
+suggesting that spatial features play a more important role in
+the problem of link prediction, which is further discussed in
+Sec. IV-D.
+Furthermore, there are very few links with large w, once
+again reinforcing the results of other performance metrics. The
+small quantity of links with large w in each forum present a
+significant opportunity to recommend early formation of links
+(when predictions are early) and potential times for learners
+to reconnect (when predictions are late). Though there is less
+
+11
+(a) ml
+(b) algo
+(c) shake
+(d) comp
+(e) f19
+(f) s20
+Fig. 6: TAC with different windows w. The TAC curves all exhibit sharp increases initially, indicating many links form around the time they
+are predicted to. The links at higher w, on the other hand, indicate potential for recommending early link formation and future reconnection.
+room for change on links with smaller w, learners may be
+more willing to act on recommendations in these cases since
+they induce less modification to actual behavior [6]; after all, a
+learner may be reluctant to reach out to others on the basis of
+outdated threads or on the assumption that they will eventually
+collaborate.
+IV. LINK FORMATION ANALYTICS
+In this section, we consider several descriptive analytic
+tools and visualizations for instructors. We first describe the
+evolution of model parameters during prediction (Sec. IV-A).
+We then examine the correlations between features (Sec.
+IV-B) and analyze their individual and collective impact on
+prediction (Sec. IV-C). Finally, we analyze the importance of
+the predictor’s architecture in Sec. IV-D.
+A. Time-Series Variable Evolution
+Because the hidden layers of deep-learning models cannot
+be understood intuitively, we provide an alternate form of
+visualizing their behavior. It is possible to observe the de-
+cisions made by the deep learning model during prediction by
+investigating changes in state for each model gate over time,
+and making inferences about the final prediction from these
+observations. The stability exhibited by the gates over time
+supports the viability of early link formation prediction from
+Sec. III-D. To demonstrate this, we consider an example of
+how the CRNN LSTM layer parameters specified in Sec. II-C
+for deep learning prediction models evolve over time.
+By examining the relationship fading gate, f, in particular,
+we are able to demonstrate how the inputs from time interval
+i−1 affect the model output at time interval i, i.e., how much
+information is carried over from interval to interval. To do so,
+we choose a link (u, v) ∈ G(L) at random from algo, and
+feed euv(i) into the trained model for L = 20 to generate the
+predictions ˜yuv(i). The prediction has high accuracy on the
+chosen link, which forms within one time interval of when it
+is predicted to form.
+The neuron activation values for the gates g, i, f, o and the
+state z and output h are additionally considered and shown in
+Fig. 7. The vertical axis is the vector dimension (i.e., neuron
+number), and the horizontal is the time instance i. A few of the
+input gate dimensions, g, change at about the time the link is
+formed (around i = 17). These changes propagate through the
+network, causing the output, h, as well as some dimensions
+of the intermediate gates (e.g., f, i, and o) to change around
+i = 17 as well, thus forming an accurate prediction. The fact
+that i and f in particular tend to take extreme values indicates
+that the input, g, and prior state, z, are either fully passed or
+blocked.
+We also observe that several dimensions in z evolve gradu-
+ally over time, with several non-zero dimensions in f passing
+information across multiple time periods. This result helps
+explain why models using an LSTM layer in conjunction with
+other methods perform better than the Bayesian model: passing
+information from one time interval to another increases the
+prediction quality compared to only updating the input features
+at each time interval.
+B. Feature Correlations
+Investigating the relationship between individual features
+provides insights into the shape of an SLN in a different
+capacity than the predictions made by our deep-learning mod-
+els, and it provides an analytical tool with which instructors
+can monitor an online classroom. Table II summarizes the
+distributions of G(L) (top row) and Gc(L) (bottom row), with
+the top 5% of outliers removed. We show the means and
+standard deviations (s.d.) of each feature for both groups, as
+well as the signal-to-noise ratio (SNR) for each feature. The
+large difference in magnitude for both mean and s.d. between
+formed and unformed links indicates a clear difference in
+behavior between these two groups. The large gap in values
+reinforces the results of our predictive algorithms discussed in
+Sec. III-B. The SNR measures how effectively a feature can
+distinguish between the two groups, with a higher magnitude
+indicating more efficacy [41]. We make a few impactful
+observations for link prediction from these statistics:
+(i) Infrequent short paths: The length and number of shortest
+paths between learners are both negatively associated with link
+formation. The former is consistent with the intuition that
+learners who are closer together (i.e., smaller shortest path
+lengths) are more likely to form links. The latter, however,
+indicates that links are more likely to form when fewer such
+shortest paths exist, i.e., the paths should be unique. An
+interesting analogy can be drawn here to the small world
+phenomenon, where users can discover short paths in a social
+network even when only one or a few exist [7]; in other
+words, the presence of fewer short paths makes each of those
+neighboring connections more important and more likely to
+foster link creation.
+(ii) Low-degreed shared neighbors: In order of increasing
+SNR, Ja, Re and Ad are each positively associated with link
+formation. Each of these measures the common neighborhood
+of two learners, with increasing penalty placed on the degrees
+
+1.0
+0.8
+0.6
+TA(
+0.4
+crnn
+cnn
+0.2
+fcnn
+rnn
+0.0
+0
+2
+4
+6
+8
+10 12 14 16 18 20
+W1.0
+0.8:
+0.6
+TA(
+0.4
+crnn
+cnn
+0.2
+fcnn
+rnn
+0.0
+0
+2
+4
+6
+8
+10 12 14 16 18 20
+W1.0
+0.8
+0.6
+TAC
+0.4
+crnn
+cnn
+0.2
+fcnn
+rnn
+0.0
+0
+2
+4
+6
+8
+10 12 14 16 18 20
+W1.0
+0.8:
+0.6
+A
+0.4
+crnn
+cnn
+0.2
+fcnn
+rnn
+0.0
+0
+2
+4
+6
+8
+10 12 14 16 18 20
+W1.0
+0.8:
+0.6
+A
+0.4
+crnn
+cnn
+0.2
+fcnn
+rnn
+0.0
+0
+2
+4
+6
+8
+10 12 14 16 18 20
+W1.0
+0.8
+0.6
+A
+0.4
+crnn
+cnn
+0.2
+fcnn
+rnn
+0.0
+0
+2
+4
+6
+8
+10 12 14 16 18 20
+W12
+(a) fuv(i)
+(b) guv(i)
+(c) huv(i)
+(d) iuv(i)
+(e) ouv(i)
+(f) zuv(i)
+Fig. 7: Neuron activations of each gate fuv(i), guv(i), huv(i), iuv(i), ouv(i), and zuv(i) over time of the LSTM layer inside the CRNN
+model for two particular links (u, v) in algo. The fact that several gate dimensions are non-zero indicates that information is propagating
+across multiple time periods for prediction. The top row demonstrates activations for a link formed late in the course, and the bottom row
+demonstrates activations for an early-formed link.
+of these neighbors (i.e., Ja does not include degree at all,
+while Re is inversely proportional to it). The fact that Ad
+has the highest SNR, then, implies that shared neighbors with
+fewer links are more prone to facilitate link formation, which
+is consistent with the the point above on unique paths being
+more predictive.
+(iii) Low ceiling feature values: Taking the statistics present
+in Table II in conjunction with each feature’s cumulative
+distribution function (CDF), shown in Fig. 4, it is evident for
+several features including To and Pr that no learner pairs reach
+the maximum possible value for the feature. Most notably
+with respect to To, the maximum number of shared topics
+between two connected users is always less than 15 of the
+20 extracted topics. Given the highly connected nature of
+“hub” students that possess a large number of shortest path
+connections, it would be expected that the maximum number
+of shared topics would be 20. This discrepancy in number
+of shared topics suggests that hub students connect frequently
+with less-engaged students, but rarely interact with each other,
+creating smaller student ecosystems within the course centered
+around their knowledge dissemination. Another possibility
+is a difference in student knowledge state/engagement on
+particular topics, indicating that learners are more motivated to
+post about topics they are confident in or interested in learning
+and avoid topics they are not.
+(iv) Topology vs. post properties: Pr and To are both
+positively associated with link formation, as one would expect:
+those with higher degrees (Pr) and focusing on similar topics
+(To) should be more likely to interact in the discussions.
+Surprisingly, though, these features have lower SNRs than
+the other neighborhood-based features, indicating that the
+network topology drives link formation in an SLN more than
+individual learner properties like a learner’s tendency to post,
+for example, or topic interest. Furthermore, the SNR of To is
+higher in the less densely populated courses (f19 and s20),
+indicating that clearer signals may emerge around topics when
+there is less overall volume of discussion in the forums. This
+is consistent with the performance differential of the GNN
+model in link prediction on the large vs. small datasets, since
+it does not learn from topic features.
+(v) Quantitative vs. humanities courses: Among the four
+MOOC courses, Pr is higher in comp and shake (particularly
+shake) than in ml and algo. This is consistent with human-
+ities courses tending to invite more open-ended discussions,
+whereas quantitative courses have questions requiring explicit
+answers [6]. More learners would then be motivated to post in
+the forums of humanities courses – in fact, such participation
+may be a course requirement – leading to more links forming.
+Table I confirms the intuition that even with a smaller class
+size, comp and shake have a higher ratio of learner pairs to
+learners. The distinction between quantitative and humanities
+courses also helps explain which settings temporal behavior is
+helpful for link prediction, as we will discuss in Sec. IV-D.
+C. Feature Importance Analysis
+Recall in Sec. II-B that we define three groups of fea-
+tures: (i) Nei, which quantify the overlap between learner
+neighborhoods, (ii) Path, which are the length and number
+of shortest paths, and (iii) Post, or the similarity in what
+learners discuss. To complement the correlation analysis in
+Table II that was done for each feature individually, we now
+analyze the contribution of each feature type to the prediction
+quality of our CRNN model, by evaluating it using different
+input feature combinations.
+To evaluate smaller groups of features using our CNN
+and CRNN models, a modification in model architecture
+is required. Our implementation of the CRNN model for
+computing links with all features contained both a 3 × 1
+kernel layer and a 2 × 1 kernel layer. To classify samples
+using a subset of less than five of the seven features, the
+second convolutional layer using a 2 × 1 kernel was removed,
+leaving a single convolutional layer with a 3×1 kernel before
+the fully connected and output layers. This eliminates the
+issue of convolving a 1 × 1 output shape with an additional
+2 × 1 kernel without requiring zero-padding. Determining
+the individual and combined effects of each feature group
+allows identification of potentially redundant features, which
+can improve computational speed when updating predictions
+in real time.
+
+1.0000
+30
+0.8333
+neurons
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+0.5000
+0.3333
+10
+0.1667
+0.0000
+5
+10
+15
+time instance1.000
+30
+0.667
+neurons
+0.333
+20
+0.000
+-0.333
+10
+-0.667
+-1.000
+5
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+30
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+0.6667
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+5
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+30
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+neurons
+6
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+0
+-6
+10
+-12
+-18
+5
+10
+15
+time instance1.0000
+30
+0.8333
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+30
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+0.333
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+-0.667
+-1.000
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+30
+0.8333
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+0.6667
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+30
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+0.6667
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+0.1667
+0.0000
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+30
+0.8333
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+0.6667
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+0.5000
+0.3333
+10
+0.1667
+0.0000
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+30
+12
+neurons
+6
+20
+0
+-6
+10
+-12
+-18
+5
+10
+15
+time instance13
+Set
+ml
+algo
+shake
+comp
+f19
+s20
+Nei + Path
+AUC
+0.9487 ± 0.0241
+0.9647 ± 0.0091
+0.8978 ± 0.0303
+0.9609 ± 0.0093
+0.8945 ± 0.0330
+0.9035 ± 0.0261
+ACC
+0.9528 ± 0.0196
+0.9844 ± 0.0035
+0.9693 ± 0.0071
+0.9801 ± 0.0044
+0.9695 ± 0.0064
+0.9732 ± 0.0027
+Nei + Post
+AUC
+0.9398 ± 0.0011
+0.9399 ± 0.0015
+0.8541 ± 0.0024
+0.8922 ± 0.0078
+0.6735 ± 0.0519
+0.6346 ± 0.0118
+ACC
+0.9446 ± 0.0008
+0.9753 ± 0.0006
+0.9314 ± 0.0050
+0.9482 ± 0.0029
+0.9538 ± 0.0015
+0.9627 ± 0.0011
+Path + Post
+AUC
+0.9332 ± 0.0034
+0.9455 ± 0.0058
+0.9255 ± 0.0096
+0.9444 ± 0.0078
+0.8832 ± 0.0358
+0.8848 ± 0.0175
+ACC
+0.9418 ± 0.0031
+0.9659 ± 0.0028
+0.9650 ± 0.0038
+0.9736 ± 0.0039
+0.9679 ± 0.0051
+0.9736 ± 0.0022
+TABLE V: Performance of the CRNN Model with selected input feature groups. The top two highest performing groups for each course metric
+are bolded. The combinations of Nei + Path and Path + Post outperform Nei + Post consistently, indicating that while neighborhood-
+based features are most important for prediction, the other feature types contribute significantly to link prediction as well.
+Table V shows the results when each course is broken
+into 20 time periods. None of the combinations reach the
+performance of the original model with all input variables
+in Table IV, indicating that each feature group contributes
+to the prediction quality. The Nei + Path and Path + Post
+combinations show the highest overall performance across all
+six forums, indicating that the combination of Nei + Path has
+a confounding effect on the model – we would expect both
+Nei-based groups to share a higher AUC. Combining these
+values with the SNRs in Table II indicates that the Nei features
+contribute the most to model accuracy, followed by Post and
+then Path.
+If we compare the individual feature groups, we generally
+find that the Nei features perform the best, followed by Path,
+and then Post. This is consistent with the behavior of these
+features within groups as well. This ordering of Post and Path
+is opposite of the SNR magnitudes from Table II: here, the
+single feature To outperforms the combined impact of Path.
+Given that Table II is concerned with the eventual formation of
+links but not the time at which they form, we conjecture that
+in the absence of Nei, Post is more important to pinpointing
+the time of link formation while Path is more important to
+whether they form at all. After all, the timing of particular
+topic coverage should influence when learners interested in
+those topics connect.
+D. Model Architecture Analysis
+Here, we first analyze the importance of spatial pattern
+preserving convolutional layers and temporal pattern preserv-
+ing recurrent layers for link prediction in SLNs. We find
+that, in general, classification models that incorporate only
+spatial pattern dependencies (CNN) outperform models that
+only incorporate time dependencies (RNN), as shown in Table
+IV. This is consistent with Table V, where we find that SLN
+topology features (i.e., neighborhood and path-based features),
+which explain spatial relationships between links, are the
+most important for accurate link prediction. However, we also
+find that incorporating time dependencies into link prediction
+models (e,g., RNN and CRNN) obtains strong performance in
+large courses such as MOOCs, whereas these models become
+less accurate on small courses such as f19 and s20. Interest-
+ingly, although the RNN accurately predicts whether links will
+form (as shown in Table IV), they do not accurately predict
+when the links will form as shown from the TAC curves
+in Fig. 6, particularly on ml, f19, and s20. This behavior
+is consistent with such quantitative courses requiring short
+answers in fast time intervals whereas the humanities courses
+typically involve threads of discussion that persist over longer
+periods of time [8]. In Fig. 7, we further explored the efficacy
+of recurrent layers by visualizing the various gates of the CRNN
+in the algo course, where we saw that information propagates
+from multiple time periods to aid link prediction after spatial
+patterns have been identified. This reinforces that recurrent
+layers may carry long-term information for link prediction,
+but convolutional layers are more robust for in SLNs on both
+large and small courses.
+In addition, as shown in Table IV, convolutional GNNs
+achieve strong link prediction performance on each of the
+MOOC datasets. Rather than employing our explicitly de-
+fined model features, GraphSAGE embeds features across the
+SLN topology that exploit spatial patterns, hence resulting in
+strong performance for these datasets captured by the GNN’s
+convolutional layers. However, for the GNN to learn such
+discriminative features, it may require a large graph to train on
+[46], thus making the model less effective for smaller courses
+such as f19 and s20. Our proposed framework, in which we
+explicitly model features between node pairs, on the other
+hand, is better able to learn and generalize on the smaller
+datasets. More generally, these results indicate that in the
+SLN domain, informed feature engineering (i.e., using spatial
+features) paired with corresponding layers (i.e., convolutional)
+results in better trained models with less data than that required
+by GNNs. This is useful for generating analytics in the early
+stages of courses before a significant amount of links have
+formed (i.e., before interaction data has been observed) on the
+forums [5], [6].
+V. CONCLUSION
+In this work, we developed a link prediction framework
+specifically tailored to operate in social learning networks
+(SLNs) based on neighborhood-based, path-based, and post-
+based modeling features. Through evaluation in six different
+courses, we demonstrated our framework’s ability to perform
+accurate link prediction in a variety of learning environments.
+In particular, we examined the efficacy of our framework on a
+course forced online after approximately eight weeks of tradi-
+tional instruction due to the COVID-19 pandemic. In addition,
+we considered the SLNs formed in four Massive Open Online
+Courses (MOOCs) as well as one traditional undergraduate
+course, with a heavy reliance on student participation in an
+online discussion forum, offered through Purdue University.
+While our work establishes an initial framework and re-
+sults for link prediction in SLNs, many avenues remain for
+exploring the challenges of link prediction in this new type of
+online social network. One is additional feature engineering:
+other features that we did not consider – such as learners’
+
+14
+background knowledge, level of education, and personal goals
+– may also be associated with link formation, and may
+allow further improvements in link prediction quality. As
+demonstrated here, our proposed framework is applicable
+across multiple datasets; thus, additional evaluation variants
+on forums or classes with different structures, such as those
+present in K-12 education, may be beneficial.
+REFERENCES
+[1] T. Yang, C. G. Brinton and C. Joe-Wong, ”Predicting Learner Interactions
+in Social Learning Networks,” IEEE INFOCOM, 2018, pp. 1322-1330.
+[2] C. G. Brinton and M. Chiang, “Social Learning Networks: A Brief
+Survey,” IEEE CISS, 2014, pp. 1–6.
+[3] D. L. Miller, L. K. Soh, A. Samal, K. Kupzyk, G. Nugent, ”A Compar-
+ison of Educational Statistics and Data Mining Approaches to Identify
+Characteristics that Impact Online Learning,” in Journal of Educational
+Data Mining, vol. 7, no. 3, pp. 117-150, 2015.
+[4] L. F. Pendry and J. Salvatore, “Individual and Social Benefits of On-
+line Discussion Forums,” IEEE Trans. Learning Technol., vol. 50, pp.
+211–220, 2015.
+[5] C. G. Brinton and M. Chiang, “MOOC Performance Prediction via
+Clickstream Data and Social Learning Networks,” IEEE INFOCOM,
+2015, pp. 2299–2307.
+[6] C. G. Brinton, S. Buccapatnam, F. M. F. Wong, M. Chiang, and H.
+V. Poor, “Social learning networks: Efficiency optimization for mooc
+forums,” IEEE INFOCOM, 2016, pp. 1–9.
+[7] D. Liben-Nowell and J. Kleinberg, “The Link-Prediction Problem for
+Social Networks,” Journal of the Association for Information Science
+and Technology, vol. 58, no. 7, pp. 1019–1031, 2007.
+[8] C. G. Brinton, S. Buccapatnam, L. Zheng, D. Cao, A. S. Lan, F. M.
+F. Wong, S. Ha, M. Chiang, H. V. Poor, ”On the Efficiency of Online
+Social Learning Networks,” in IEEE/ACM Trans. Netw., vol. 26, no. 5,
+pp. 2076-2089, 2018.
+[9] C. Wu, X. Chen, W. Zhu, Y. Zhang, ”Socially-Driven Learning-Based
+Prefetching in Mobile Online Social Networks,” in IEEE/ACM Trans.
+Netw., vol. 25, no. 4, pp. 2320-2333, 2017.
+[10] F. M. F. Wong, Z. Liu, M. Chiang, ”On the Efficiency of Social
+Recommender Networks,” in IEEE/ACM Trans. Netw., vol. 24, no. 4,
+pp. 2512–2524, 2016.
+[11] A. Divakaran and A. Mohan. “Temporal Link Prediction: A Survey.”
+New Generation Computing, vol. 38, no. 1, pp. 213–258, 2020.
+[12] S. Lorenzen, N. Hjuler, and S. Alstrup, ”Tracking Behavioral Patterns
+Among Students in an Online Education System,” EDM, 2018, pp. 280-
+285.
+[13] S. Aghababaei and M. Makrehchi. “Interpolative Self-Training Approach
+for Link Prediction,” Intelligent Data Analysis, vol. 23, no. 6, pp.
+1379–1395, 2019.
+[14] C. P. Muniz, R. Goldschmidt, R. Choren, ”Combining contextual,
+temporal and topological information for unsupervised link prediction
+in social networks,” Knowledge-Based Systems, 2018, pp. 129–137.
+[15] Z. Jie, Y. Li and R. Liu, ”Social Network Group Identification based
+on Local Attribute Community Detection,” 3rd IEEE ITNEC, 2019, pp.
+443-447.
+[16] L. Backstrom and J. Leskovec, ”Supervised Random Walks: Predicting
+and Recommending Links in Social Networks,” Web Search and Data
+Mining, 2011.
+[17] F. Aghabozorgi and M. R. Khayyambashi, “A New Study of Using
+Temporality and Weights to Improve Similarity Measures for Link
+Prediction of Social Networks,” Journal of Intelligent & Fuzzy Systems,
+vol. 34, no. 4, pp. 2667–2678, 2018.
+[18] K. Chen, Y. Chen, Y. Li, J. Han. ”A supervised link prediction method
+for dynamic networks,” Journal of Intelligent & Fuzzy Systems, vol. 31,
+no. 1, pp. 291-299, 2016.
+[19] S. Amershi, C. Conati, ”Combining Unsupervised and Supervised Clas-
+sification to Build User Models for Exploratory Learning Environments,”
+in Journal of Educational Data Mining, vol. 1, no. 1, pp. 18-71, 2009.
+[20] N. Gurjar, “Leveraging Social Networks for Authentic Learning in
+Distance Learning Teacher Education,” TechTrends: Linking Research &
+Practice to Improve Learning, vol. 64, no. 4, pp. 666–677, 2020.
+[21] Y. Xu, C. F. Lynch, T. Barnes, ”How Many Friends Can You Make in
+a Week? Evolving Social Relationships in MOOCs over Time,” EDM,
+2018, pp. 97-103.
+[22] B. Cui, S.J. Yang , C.M. Homan ”Modeling Information Sharing Behav-
+ior on Q&A Forums,” Pacific-Asia Conference on Knowledge Discovery
+and Data Mining, Lecture Notes in Computer Science, 2017, vol 10235.
+[23] I. Koprulu, Y. Kim and N. B. Shroff, ”Battle of Opinions Over Evolving
+Social Networks,” in IEEE/ACM Trans. Netw., vol. 27, no. 2, pp. 532-545,
+2019.
+[24] D. Yang, R. Kraut, C. Rose, “Exploring the Effect of Student Confusion
+in Massive Open Online Courses,” in Journal of Educational Data
+Mining, vol. 8, no. 1, pp. 52-83, 2018.
+[25] S. Zhang, X. Liang, Y. Wei and X. Zhang, ”On Structural Features, User
+Social Behavior, and Kinship Discrimination in Communication Social
+Networks,” in IEEE Trans. Computat. Social Syst., vol. 7, no. 2, pp. 425-
+436, 2020.
+[26] R. Xiang, J. Neville, and M. Rogati, “Modeling Relationship Strength
+in Online Social Networks,” in WWW. ACM, 2010, pp. 981–990.
+[27] F. Dalipi, A. S. Imran and Z. Kastrati, ”MOOC dropout prediction
+using machine learning techniques: Review and research challenges,”
+2018 IEEE EDUCON, 2018, pp. 1007-1014.
+[28] M. Tsiakmaki, G. Kostopoulos, S. Kotsiantis, and O.Ragos. ”Transfer
+Learning from Deep Neural Networks for Predicting Student Perfor-
+mance.” Applied Sciences, 2020.
+[29] F. Yang, Z. Jiang, C. Wang, Y. Dai, Z. Jia and K. Hirota, ”Student Eye
+Gaze Tracking During MOOC Teaching,” SCIS, 2018, pp. 875-880.
+[30] Z. Papamitsiou and A. A. Economides, ”Motivating Students in Col-
+laborative Activities With Game-Theoretic Group Recommendations,” in
+IEEE Trans. Learn. Technol., vol. 13, no. 2, pp. 374-386, 2020.
+[31] O. Almatrafi and A. Johri, ”Systematic Review of Discussion Forums
+in Massive Open Online Courses (MOOCs),” in IEEE Trans. Learn.
+Technol., vol. 12, no. 3, pp. 413-428, 2019.
+[32] S. Joksimovi´c and J. Jovanovi´c and V. Kovanovi´c and D. Gaˇsevi´c and N.
+Miliki´c and A. Zouaq and J. P. van Staalduinen, ”Comprehensive Analysis
+of Discussion Forum Participation: From Speech Acts to Discussion
+Dynamics and Course Outcomes,” in IEEE Trans. Learn. Technol., vol.
+13, no. 1, pp. 38-51, 2020.
+[33] P. M. Moreno-Marcos, C. Alario-Hoyos, P. J. Mu˜noz-Merino, I. Est´evez-
+Ayres and C. D. Kloos, ”A Learning Analytics Methodology for Under-
+standing Social Interactions in MOOCs,” in IEEE Trans. Learn. Technol.,
+vol. 12, no. 4, pp. 442-455, 2019.
+[34] P. M. Moreno-Marcos, C. Alario-Hoyos, P. J. Mu˜noz-Merino and C.
+D. Kloos, ”Prediction in MOOCs: A Review and Future Research
+Directions,” in IEEE Trans. Learn. Technol., vol. 12, no. 3, pp. 384-401,
+2019.
+[35] A. Pigeau, O. Aubert, and Y. Prie, ”Success Prediction in MOOCs,”
+EDM, 2019, pp. 390-395.
+[36] F. Calefato, F. Lanubile and N. Novielli, ”Moving to Stack Overflow:
+Best-Answer Prediction in Legacy Developer Forums,” ACM/IEEE Inter-
+national Symposium, Sep. 2016, pp. 1-10.
+[37] A. Rezvanian and M. R. Meybodi. “A New Learning Automata-Based
+Sampling Algorithm for Social Networks.” International Journal of
+Communication Systems, vol. 30, no. 5, 2017.
+[38] K. Cheng, X. Guo, X. Cui, F. Shan, ”Dynamical Modeling, Analysis,
+and Control of Information Diffusion over Social Networks: A Deep
+Learning-Based Recommendation Algorithm in Social Network,” Dis-
+crete Dynamics in Nature & Society, Jul. 2020, pp. 1–8.
+[39] D. M. Blei, A. Y. Ng, and M. I. Jordan, “Latent Dirichlet Allocation,”
+JMLR, vol. 3, no. 3, pp. 993–1022, 2003.
+[40] D. Varshney, S. Kumar, V. Gupta, ”Predicting information diffusion
+probabilities in social networks: A Bayesian networks based approach,”
+Knowledge-Based Systems, vol. 133, pp. 66-76, 2017.
+[41] S. Y. Kung, Kernel Methods and Machine Learning, Cambridge Uni-
+versity Press, 2014.
+[42] M. Fire, L. Tenenboim, O. Lesser, R. Puzis, L. Rokach, and Y. Elovici,
+“Link Prediction in Social Networks using Computationally Efficient
+Topological Features,” in SocialCom., 2011, pp. 73–80.
+[43] A. Aswathy Divakaran and A. Mohan, “Temporal Link Prediction: A
+Survey,” New Gener. Comput., vol. 38, pp. 213–258, 2020.
+[44] W. Yu, W. Cheng, C. Aggarwal, H. Chen, and W. Wang, ”Link Prediction
+with Spatial and Temporal Consistency in Dynamic Networks,” IJCAI,
+2017.
+[45] M. Zhang and Y. Chen, ”Link prediction based on graph neural net-
+works,” NeurIPS, 2018.
+[46] J. Zhou, G. Cui, S. Hu, Z. Zhang, C. Yang, Z. Liu, L. Wang, C. Li, and
+M. Sun, “Graph neural networks: A review of methods and applications,”
+AI Open, vol. 1, pp. 57-81, 2020.
+[47] W. Hamilton, Z. Ying, and J. Leskovec, “Inductive Representation
+Learning on Large Graphs,” NeurIPS, vol. 30, 2017.
+[48] Z. He, Y. Wu, and X. You, “Research on optimisation of MOOC
+education model based on participatory visual teaching technology,”
+IEEE/ACM Trans. Netw., vol. 29, no. 4, 2019.
+
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+page_content='1  Predicting Learning Interactions in Social  Learning Networks: A Deep Learning  Enabled Approach  Rajeev Sahay∗, Graduate Student Member, IEEE, Serena Nicoll∗, Student Member, IEEE,  Minjun Zhang, Tsung-Yen Yang, Student Member, IEEE, Carlee Joe-Wong, Member, IEEE,  Kerrie A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Douglas, Member, IEEE, and Christopher G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Brinton, Senior Member, IEEE  Abstract—We consider the problem of predicting link for-  mation in Social Learning Networks (SLN), a type of social  network that forms when people learn from one another through  structured interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' While link prediction has been studied  for general types of social networks, the evolution of SLNs over  their lifetimes coupled with their dependence on which topics  are being discussed presents new challenges for this type of  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' To address these challenges, we develop a series of  autonomous link prediction methodologies that utilize spatial  and time-evolving network architectures to pass network state  between space and time periods, and that models over three  types of SLN features updated in each period: neighborhood-  based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', resource allocation), path-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', shortest path),  and post-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', topic similarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Through evaluation on six  real-world datasets from Massive Open Online Course (MOOC)  discussion forums and from Purdue University, we find that our  method obtains substantial improvements over Bayesian models,  linear classifiers, and graph neural networks, with AUCs typically  above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='91 and reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='99 depending on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our  feature importance analysis shows that while neighborhood and  path-based features contribute the most to the results, post-based  features add additional information that may not always be  relevant for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Index Terms—Deep learning, graph neural networks, link  prediction, online social networks, social learning networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' INTRODUCTION  O  NLINE education has exploded in popularity over the  past few years, with estimates of up to 80% of stu-  dents having taken an online course [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The advent of the  COVID-19 outbreak has significantly increased the number of  online learners since 2020, which in turn has demonstrated  online platforms’ viability as an additional tool in physical  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Sahay, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Nicoll, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Zhang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Brinton are with the Elmore Fam-  ily School of Electrical and Computer Engineering, Purdue University, West  Lafayette, IN, 47907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' E-mail: {sahayr,snicoll,zhan3624,cgb}@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Douglas is with the School of Engineering Education, Purdue  University, West Lafayette, IN, 47907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' E-mail: douglask@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Yang is with the Department of Electrical and Computer Engineering,  Princeton University, Princeton, NJ 08544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' E-mail: ty3@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Joe-Wong is with the Department of Electrical and Computer  Engineering,  Carnegie  Mellon  University,  Pittsburgh,  PA  15213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  E-  mail:cjoewong@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  ∗R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Sahay and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Nicoll contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  This work was supported in part by the Charles Koch Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  The code and four of the datasets used in this work are available at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='com/Jess-jpg-txt/sln-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  A preliminary version of the material in this work appeared in the Pro-  ceedings of the IEEE Conference on Computer Communications (INFOCOM)  2018 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  classrooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This growth has not been without challenges,  however;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' online learning has raised concerns about its apparent  lack of quality control, extraordinarily low teacher-to-student  ratios, and scarcity of high-quality teachers [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The COVID-  19 pandemic has highlighted the lack of quality tools for  both students and teachers across online learning providers,  making navigation of these massive communities a daunting  or impossible task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  One way course providers have attempted to mitigate these  problems is by establishing online forums where students can  learn from each other, thus compensating for a lack of per-  sonalized instruction by posting questions, replying with an-  swers, and otherwise exchanging ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Massive Open Online  Courses (MOOCs), as well as Q&A sites like Piazza, Quora,  and StackOverflow, rely on forums extensively, generating a  plethora of data about how users interact with one another  online for learning purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' These forums generate Social  Learning Networks (SLNs) within communities of student  users that evolve over time, facilitating peer-to-peer knowledge  transfer in the absence of instructor intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Data-driven  studies on the SLNs emerging from online learning forums  have analyzed the benefits of social learning [3], [4] geared  towards the ultimate goal of improving learning outcomes by,  for example, proposing methods for instructor analytics [5]  and news feed personalization [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  In this work, we are motivated by the following research  question: Can link formation between learners in an SLN be  predicted in advance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Such predictions would enable several  new ways of improving online learning and forum experiences  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', encouraging early formation of learner groups or recom-  mending that learners respond to newly-posted questions that  they are expected to answer/contribute to later), thus helping  to reduce the gap between in-person and online instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  SLNs, however, pose two key challenges that differentiate  them from standard time-evolving social networks [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' First,  the SLN for an online course forms around the specific  educational processes of that course [8], [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' With an SLN,  users connect as a result of specific learning needs, and  in response to events that are exogeneous to the discussion  forum, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', the instructor releasing new content/assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  On the other hand, homophily and pre-existing relationships  are known to play a strong role in the evolution of standard  social networks over time, which can provide initialization  information for predicting learner interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' An online SLN  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='01606v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='SI]  3 Jan 2023  2  tied to a specific course, on the other hand, exhibits a “cold  start” from a state of little-to-no observable network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Second,  links in SLNs are defined much more arbitrarily compared  to other graphs [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' On social media sites, links between  users are typically quantified with concrete metrics such as  ‘friendships’ or ‘follows,’ where the connection between two  users is explicit and typically optional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In an SLN, by contrast,  a link between two users should indicate a transfer/sharing of  knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Explicit connection metrics do not typically exist,  and even if they did, they do not imply the users have shared  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' As a result of these challenges, the prediction of  link formation in SLNs cannot be easily solved using previous  methods designed for general time-evolving graphs [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  In this work, we develop a link prediction methodology,  specifically tailored for addressing the challenges associated  with SLNs, which analyzes a set of features describing (i)  learner pairs in an SLN and (ii) the evolution of learner  interactions over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our methodology is deep learning-  based, allowing consideration for both time-variable features  and latent learner characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We evaluate our methodol-  ogy on data collected from four MOOC discussion forums  from Coursera and two courses at Purdue University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We  then investigate how our methodology can be used to make  recommendations that may enhance the timing and quality of  replies to discussion posts, thus encouraging interactions and  improving learner experience in discussion-based forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Related Work  The link prediction problem has been studied extensively  in the context of online and digitally-enabled social networks,  due to its usefulness in generating recommendations such  as friendships, follows, or other forms of interactions [8]–  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Several methods have been proposed for this problem,  beginning with unsupervised approaches and eventually tran-  sitioning to supervised methods in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In terms  of unsupervised methods, [13] proposed using features based  on node proximity and properties, while [14] and [15] applied  a model to incorporate additional contextual and temporal fea-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' On the other hand, supervised approaches have proposed  random walk algorithms using labels to increase the likelihood  of traversing formed links [16], while [17] and [18] proposed  deriving features from exogenous sources and training models  on them to predict future link formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Previous work has  additionally considered using supervised and unsupervised  methods simultaneously for exploratory learning environments  [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' However, these works do not consider characteristics  unique to social learning networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Specifically, the potential  dependence on discussion topics, and the need for time-series  modeling is not explicitly modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Research into SLNs until  this point has been largely theoretical, although [20] provides  a first look into the application of deep learning-based link  prediction algorithms in a classroom setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Additionally,  unsupervised approaches have demonstrated recent popularity  for problems related classification of student behavior [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Although the central focus of our research is concerned with  SLNs, unlike these works, our strictly supervised models  specifically consider student social characteristics for large  classrooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Other works on online social networks have considered  problems related to link formation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', predicting the  strength/repetition (rather than existence) of future links [21]–  [23], predicting link types [12], or examining the effects  of student confusion on SLNs [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The methods used and  developed include linear regression/classification on network  features and user demographics [21], [25], latent variable  modeling of learner interaction frequencies [12], and dynamic  models to account for the disappearance and strengthening of  links over time [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our models utilize some similar network  features, but we consider the different prediction objective  of pinpointing when links will form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In fact, given its high  observed quality, we consider a time-series version of [26] as  a potential model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  An SLN is fully described by several datasets that each  capture the a subset of student behavior inside the associated  course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Recent papers choose to focus on one or a couple  of these datasets: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Student video-watching behavior [5],  student performance [27], [28], student physical behavior [29],  or discussion forum data [30]–[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our work is evaluated  on a similar dataset to [32] in that it provides information  gathered on student message passing behavior in a discussion  forum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The models created in these other works fundamentally  differ from our focus on individual student relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' [30]  focuses on making group predictions from clusters of similar  students, while [33] models changes in student behavior at  critical points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', exams and holidays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Some recent works have focused on other aspects of differ-  ent types of SLNs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', MOOCs [12], [21], [35], Q&A sites  [22], [36], and enterprise social networks [37], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our work  is perhaps most similar to [2], [21] in that we study prediction  for SLNs using topological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The prediction objectives  in these other works, however, are fundamentally different  than our focus of predicting interactions between learners in  that they seek to predict course grades via video-watching  behaviors [35] and student knowledge-state via learner post  and reply frequencies [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our Methodology and Contributions  In this work, we propose a novel framework specifically  tailored to perform link prediction in SLNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1 summarizes  the main components of our methodology, which are further  outlined in the following discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  1) Input Feature Computation: We begin by extracting the  discussion data from the considered forum to construct the  SLN (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Next, we engineer a set of features for  each learner pair (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Here, we define three groups  of features that we consider: (i) neighborhood-based features  that are determined from common neighborhoods, (ii) path-  based features based on paths between learners, and (iii) post-  based features that are determined from latent topic analysis of  learner posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Because a specific definition of what constitutes  link formation between two users in an SLN does not exist, a  key question when quantifying an SLN is how best to model  learner interactions without loss of accuracy [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We address  this through inference from forum data, with consideration for  both quality of interaction [26] and timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1: Summary of the application of our SLN link prediction framework in post-based courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  2) Prediction Model: The second component of our frame-  work shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1 is the prediction model (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  We consider three different classes of predictors: (i) lin-  ear classifiers, (ii) graph neural networks (GNN), and (iii)  gradient-based deep neural network classifiers (specifically,  Bayesian neural networks, fully connected neural networks,  convolutional neural networks, recurrent neural networks,  and convolutional recurrent neural networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The success  of Bayesian models in static link prediction problems [40]  motivates us to consider their performance in the time-evolving  SLN setting, while GNNs offer efficient learning over graphs  without explicit feature engineering [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' However, we develop  our core methodology around deep learning-based classifiers,  because, as we will show, explicit feature modeling paired  with various layer types, which can extract spatial or temporal  patterns from the SLN features, result in more robust and  accurate SLN link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  3) Evaluation and Analytics:  To assess the quality of  our models, we train and evaluate our considered prediction  models on four MOOC discussion forums and two Piazza  discussion forums, using an unsupervised method as a baseline  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II-C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Through our evaluation, we also generate four  types of analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The first analytic is feature importance,  which quantifies the importance of each considered feature  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The second and third analytics quantify time-dependent  model parameters, including closeness between time of link  prediction and actual link formation as well as the relationship  between features and the timing and quality of formed links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  The fourth analytic explores the effects of varying classifica-  tion architectures, where we anaylize the importance of dif-  ferent architectures in different course types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', quantitative  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' humanities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In addition to these analytics, we provide  visualizations for instructors to interact with the results of our  proposed framework and respond to changes in the course  SLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' These visualizations encapsulate our analytics, allowing  for interpretation by those not familiar with our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Summary of Contributions: In summary, our contributions  are (i) developing a link prediction framework for SLNs, which  learns based on topological and post-based features of user  discussions (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II), (ii) demonstrating that the combination  of our features with spatial pattern-capturing neural networks  obtains the most robust SLN link prediction quality over six  datasets, with AUCs above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='90 in each case (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III), and  (iii) developing a set of analytics for SLN link formation based  on our link prediction framework (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' SOCIAL LEARNING NETWORK METHODOLOGY  In this section, we formalize our SLN link prediction  methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We first quantify an SLN from forum data (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  II-A) and define the particular features that are used as model  inputs (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We then develop unsupervised predictor,  linear classifiers, GNNs, and deep learning classifiers (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  II-C) for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' SLN Graph Model  In order to define our features, we must first describe how  link creation in an SLN model is inferred and quantified from  online forum data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  1) Online forums: The format of online forums differs by  host site and by classroom needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We identify two main types  of forum structures to account for in our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  MOOC forum structure: A large online forum such as  those hosted on Coursera is typically comprised of a series of  threads, with each thread in turn being comprised of one or  more posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Each post is written by a single user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' A post, in  turn, can have one or more comments attached to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Given  the observation that SLN forum users do not abide by the  designation of post vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' comment consistently [6], we will not  distinguish between them, instead referring to them both as  posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This structure of thread posts is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Q&A forum structure: Another format, implemented by  Piazza, forces a “Question/Answer” thread structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The  forum is constructed from a series of questions and their  responses, with allowance for follow-up questions and re-  sponses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In contrast to traditional forums, a response on Piazza  may have contributions from multiple users in the same block,  rather than requiring a new comment from each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Any  question may have comments attached to it in the form  of “follow-ups”, which can in turn generate new responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Using the observation listed above from [6] again, we do not  distinguish between types of follow-up responses and label  all responses after the initial question as posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This alternate  structure of thread posts is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  2) Quantifying SLN link creation: A link (u, v) is observed  between learner u and another learner v if, in a specific time  interval, both u and v contribute to a post in the same thread  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', by either creating the initial post or contributing via a  Input Feature Computation  Prediction Model and Evaluation  Analytics  Application  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II)  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II and III)  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III and IV)  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' V)  Data through time I  time  Data through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='time i  ime  Evaluation Result  Visualizations  Input feature  i-1  Neighborhood  Evaluation Result i  computation  Features  Network  Data  Topology  Preprocessing  Path Features  Time Series  Cross Validation  Model Update  Model State i  Feature  Features  Importance  Make Predictions4  Forum Posts  Topic Extraction  Post Features  Feature Analytics  Recommendations  Model State i-1  Model Evaluation  to Learmer4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2: Example of how posts in two different forum structures are  divided into time periods and how SLN link creation between the  learners authoring these posts is modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2a (left): model for a  Coursera forum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2b (right): model for a Piazza forum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  follow-up post).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We use this as the criterion for establishing  the link (u, v) in the SLN because it signifies the fact that  learner u and learner v have exchanged ideas and interacted  in the same thread within a specific time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  To model the evolution of an SLN, we group its posts into  different time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Specifically, we divide all posts in a  given thread into L equally spaced intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2 illustrates  this procedure for two example threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We use yuv(i) as an  indicator variable for the formation of link (u, v): yuv(i) = 1  if a link between u and v has been created in any interval  up to and including i, and yuv(i) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Thus, as in  most social networks [38] [16], links persist over time in our  SLN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The SLN graph structure in any given interval  i is then comprised of nodes corresponding to the learners u  and edges (u, v) corresponding to links between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' For  the purpose of predicting future responses, we consider this  interaction to be bidirectional, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', the resulting SLN is an  undirected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Formally, we define G(i) = [yuv(i)] as the  binary adjacency matrix of the SLN during interval i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' since  links are bidirectional, G(i) is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  We can also define subgraphs of G(i) focusing on particular  students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 3 visualizes the neighborhood for an individual,  randomly selected student at a particular time instance, where  first and second degree connections are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In addi-  tion to capturing detailed link-formation behavior evaluated  later in this study, evaluating a visual representation from  the perspective of a single student provides an intuition for  individual student contributions and demonstrates the presence  of “hub” students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The lack of multiple paths between students  highlights the underlying sparse nature of G(i), requiring users  to traverse one long path rather than choose from several short  connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Additionally, the relative small false positive rate  (denoted by blue links in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 3) demonstrates our framework’s  efficacy for link prediction, as we will describe further in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Two particular subsets of G(i) are of interest in the link  prediction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We define  Ω = (u, v) : u, v ∈ N(G), u ̸= v,  (1)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', all possible learner pairs in the SLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We then define  two subsets of Ω : G(L), which is the set of formed links  at the final time i = L (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', with yuv(L) = 1), and  Gc(L) = Ω \\ G(L), the complement graph of un-formed links  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', yuv(L) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Note that |Gc(L)| ≫ |G(L)| for each  dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', most learners are never linked).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This large class  imbalance between formed and unformed links informs our  link prediction framework in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' SLN Feature Engineering  We now define our features, computed for each learner pair  (u, v), u ̸= v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' These quantities serve as the inputs to our  prediction algorithms in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Neighborhood-based Features: These features, as well as  path-based features discussed next, are extracted from the  topology of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Letting N(G) be the set of nodes in  the SLN G and Γu(i) ⊆ N(G) denote the set of neighbors  of u at time i, the neighborhood-based features qualitatively  measure the “similarity” of u and v’s neighborhoods [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' They  are quantified as follows:  1) Jaccard coefficient:  Jauv = |Γu(i) ∩ Γv(i)|/|Γu(i) ∪ Γv(i)|  2) Adamic-Adar index:  Aduv =  �  n∈Γu(i)∩Γv(i)  1/log|Γn(i)|  3) Resource allocation index:  Reuv =  �  n∈Γu(i)∩Γv(i)  1/|Γn(i)|  4) Preferential attachment score:  Pruv = |Γu(i)| · |Γv(i)|  We let buv denote the vector of these features for pair (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Note that a larger value of each of these features, roughly  speaking, indicates that u and v share more common, low  degree neighbors than they do with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Path-based Features: These features measure the proximity  of u and v in the SLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' They are as follows:  5) Shortest path length (Lpuv): The length of the shortest  path between u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  6) Number of paths (Npuv): The number of shortest paths  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', of length Lp) between u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  We let auv denote the vector of these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Note that as  Lp decreases, u and v become more closely connected, while  a larger Np indicates more redundancy in these paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Post-based Features: Besides topology-based attributes,  learners’ interests in different course topics will also influence  their probability of forming links in an SLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In particular,  we would expect those with similar topic interests to be more  likely to post in the same thread, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', form links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We thus  compare the topics of different learners’ posts to compute  another feature that shows the learners’ similarity in interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  To do this, we apply the Latent Dirichlet Allocation (LDA)  algorithm [39] on the dictionary of all course words (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', all  unique words used in all the considered posts of a course) to  extract a set, K, of latent topics across posts, and a model of  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Coursera  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Piazza  u1  Post 1  u1  Question  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='5)  4  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='3)  u2  Reply 1  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='3)  u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' u5  Answer  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='3)  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='3)  4  u3  Reply 2  u3  Follow Up 1  i1  u4  Follow Up 2  u4  Reply 3  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='4)  u2  Follow Up 3  i2  u2  Reply 4  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='Course Title  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='4955  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='TABLE I: Descriptive metrics on our six considered forum datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The title, beginning date (m/dd/yy), duration (weeks), number of users,  threads, learner pairs, and posts by the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' All courses were broken into 20 time instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 3: A snapshot of the SLN graph model for a single user  (represented by a unique ID string) and their close neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  The visual demonstrates the lack of multiple paths between users,  underlying the sparse nature of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  posts as a probability vector of these topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In our application,  we view each post as a separate “document,” since learners  are likely to discuss many distinct topics over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' For each  learner, u, we obtain the latent topic vector of their posts  through time i as the average of their post vectors through  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We denote the set of topics for learner u that exceed a  minimum threshold of coverage across their posts through time  i as Ku(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' With this, we define the last feature which captures  the number of common topics between learners u and v:  7) Number of common topics (To): |Ku(i) ∩ Kv(i)|  We use cuv as the time-series version of To, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', the number  of common topics discussed by u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Link Prediction Methodology  As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II-B, the features extracted from the  graph topology contain spatially and temporally correlated  patterns between learner pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Therefore, we employ pre-  diction models that are capable of exploiting these patterns  for accurate link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In this capacity, we consider the  efficacy of four distinct deep learning architectures for our  proposed framework: (i) the fully connected neural network  (FCNN), which offers effective latent space prediction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' (ii) the  convolutional neural network (CNN), which is highly effective  for processing spatially correlated patterns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' (iii) the long-  short-term memory (LSTM) based recurrent neural network  (RNN), which is desirable for time-series modeling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' (iv)  the convolutional recurrent neural network (CRNN), which  extracts both spatial and temporal correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' As baselines to  these methods, and to demonstrate the necessity of the afore-  mentioned classifiers and their corresponding architectures, we  compare our proposed deep learning prediction framework to  five traditional prediction models: an unsupervised predictor,  two linear prediction models (support vector machines and  linear discriminant analysis), a graph neural network [45], and  a Bayesian neural network [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  For a given pair of users (u, v), the input feature vector into  each of the following models is given by euv = [buv, auv, cuv]  while the target output is the link state yuv(i) ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In the  following, we describe the latent state of each model as well  as their corresponding training procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  1) Unsupervised Predictor: We begin by using a simple  prediction algorithm as a benchmark for the parameter-based  models described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Choosing the feature most associated  with link formation, we follow [16] and turn the resource  allocation index (Re) feature into an unsupervised predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  To do this, we compute Re for each (u, v) ∈ Ω, normalize the  vector of values to [0, 1], and use this as ˆyuv(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  2) Linear Classifiers: Next, we consider two relatively sim-  ple linear models for SLN link prediction: linear discriminant  analysis (LinDA) and support vector machines (SVMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Both  models attempt to find a separating linear hyper-plane between  learners who did and did not form links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' However, both models  are learned using different methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Specifically, LinDA  uses every sample during training and assumes samples in  each class follow the same distribution and have the same  covariance matrix whereas SVM makes no prior assumptions  on the data’s distribution and aims to find a decision boundary  using the points that result in the highest error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  3) Graph Neural Networks (GNN): GNNs are a class of  neural networks for learning over datasets expressed as graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  They have been employed to perform link prediction on a  variety of graph topologies [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' A potential advantage  of GNNs in our setting would be obviating much of the  feature engineering in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II-B, as they can learn directly  from the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Thus, we compare the efficacy of  GNNs to our proposed method for predicting link formation  in SLNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Specifically, we adopt a two-layer convolutional  GraphSAGE model [47], where node attributes of the SLN  are self-generated during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Here, the adjacency matrix  of the SLN is used as input into the GraphSAGE model at a  given time in order to predict future links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  4) Deep Learning Classifiers: One potential limitation of  linear classifiers is their small parameter space, which pre-  vents learning intricate non-linear relationships between input  features extracted from an SLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' GraphSAGE GNNs aim to  address this challenge, but they lose the ability to model  yourID  1stdegreeconnections  2nddegreeconnections  Peersyouareencouragedtotalkto  FormedConnections  EncouragedFutureConnections6  Features  SNR  Mean  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=') – for the network features of the two link groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The top row  for each feature corresponds to formed links (yuv(L) = 1), and the bottom to non-formed links (yuv(L) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Taken individually, the  neighborhood-based features Re and Ad have the strongest correlations with link formation, while the topic-based To tends to have the least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='0615  point report fine  (f) s20  TABLE III: Summary of the top five topics extracted by LDA for  each online discussion forum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' For each course, the topics tend to be  reasonably disjoint, with the exception of common words  explicit features between node pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' To mitigate each of these  shortcomings, we propose a deep learning approach on specif-  ically engineered features in which various characteristics of  (u, v) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', spatial and time-varying properties) are expected  to be learned for stronger prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Specifically, we propose five deep architectures for link  prediction: the Bayesian neural network (BNN), the fully  connected neural network (FCNN), the convolutional neural  network (CNN), the recurrent neural network (RNN), and the  convolutional recurrent neural network (CRNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Each model  (excluding the Bayesian Neural Network) applies the Rectified  Linear Unit (ReLU) activation function, given by σ(a) =  max{0, a}, in its hidden layers followed by a two-unit output  layer, which applies the softmax activation function, which  allows for a probabilistic interpretation of link formation for  a learner pair (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The model architecture for each of our  considered models are discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The hyper-parameter  selection of each model was empirically determined to best fit  the diverse datasets utilized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Bayesian Neural Network (BNN): The Bayesian Network  (BNet) model [40] defines the probability density of latent  variable zuv as a Gaussian:  P(zuv|euv) = N(wT euv, σ2),  (2)  where w is the weight vector and σ2 is the variance, both  to be estimated when the model is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' From this, yuv is  estimated according to  P(yuv = 1|zuv) = σ(φφφT zuv + b),  (3)  where φφφ and b are a vector and scalar, respectively, to be  estimated during training, and σ(·) is the logistic sigmoid  function given by σ(·) = 1/(1 + e−(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Our BNN architecture is composed of a hidden layer  encoding the latent variable zuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This hidden layer has 10  7  (a) Ja  (b) Ad  (c) Re  (d) Pr  (e) Np  (f) Lp  (g) To  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 4: Cumulative distribution functions (CDFs) for each of the seven feature vectors from s20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' CDFs of non-formed links are marked in  blue, and CDFs of formed links are shown in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' These demonstrate that there is (a) an observable difference in distribution between  the two populations for each feature and (b) an inverse relationship between number of shortest paths and shortest path length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  units, each represents a normal distribution with weight wi  and variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Following this hidden layer is a dense output  layer with softmax activation function given in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Fully Connected Neural Network (FCNN): FCNNs are  considered a higher dimensional non-linear extension of link  classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Such models can potentially represent more so-  phisticated non-linear relationships for better link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Our fully connected multi-layer artificial neural network is  composed of two hidden layers each containing 128 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Convolutional Neural Network (CNN): In addition to  FCNN models, we also consider deep convolutional neural  networks (CNNs), which in addition to providing a large  parameter space for learning, capture spatial characteristics  between features for each learning pair (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In the domain of  link prediction, capturing spatial correlations between signal  features is especially important since the majority of features  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', buv and auv) are extracted from the topology of the SLN  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our proposed CNN for link prediction is composed of  two convolutional layers with 64 3 × 1 feature maps and 32  2 × 1 feature maps, respectively, followed by a 32-unit fully  connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Recurrent Neural Network (RNN): BNNs, FCNNs and  CNNs, as well as linear classifiers, do not explicitly model  the evolution of latent space variables over time based on euv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  This could potentially provide useful information for modeling  an SLN, particularly so that the predictor could respond to  sudden changes in the input relative to the prior state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This  may occur, for example, when the topic of the course shifts,  which could be reflected in a sudden change in cuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  To address this challenge, we consider a long-short-term  memory (LSTM) based RNN with input duv = [euv, huv(i −  1)]T , where huv(0) = 0 and huv(i − 1) is the output vector  from the previous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We then define the interaction gate,  relationship gain gate, and relationship fading gate vectors at  each time interval, i, as  guv(i) = ψ(Wgduv(i) + bg),  (4)  iuv(i) = σ(Widuv(i) + bi),  (5)  fuv(i) = σ(Wfduv(i) + bf),  (6)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Here, ψ(·) and σ(·) are the tanh and sigmoid  functions, respectively, and the matrices Wg, Wi, and Wf  as well as the vectors bg, bi, and bf contain parameters that  are estimated during the model training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Formally,  the latent cell state, zuv(i), is updated as  zuv = guv(i) ⊙ iuv(i) + zuv(i − 1) ⊙ fuv(i),  (7)  where ⊙ denotes element-wise matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' An out-  put gate, ouv(i), is then used to determine the factor to which  each element of zuv(i) should be used in the definition of  huv(i):  ouv(i) = σ(woduv(i) + bo), huv(i) = σ(zuv(i) ⊙ ouv(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  (8)  With this, yuv(i) is estimated as  P(yuv(i) = 1|zuv(i)) = σ(h1(i)),  (9)  where h1(i) is the first element of h(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our implemented  RNN is composed of 64-cell LSTM layer followed by 128-  unit fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Convolutional Recurrent Neural Network (CRNN): Con-  volutional recurrent neural networks contain both convolu-  tional layers and recurrent LSTM layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Although such mod-  els are typically computationally costly to train, they capture  both spatial and time-varying correlations between learner  pair feature vectors, thus providing the advantages of high  parameter deep learning models with CNNs and RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our  proposed CRNN architecture consists of two convolutional  layers, containing 64 3 × 1 and 32 2 × 1 feature maps  respectively, followed by a 32-cell LSTM layer, and a 32 unit  fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='0  1  2  3  4  5  Feature  eValue8  5) Deep Learning Parameter Training: We train each deep  learning algorithm using the Adam optimizer as well as the  categorical cross entropy loss function, which for our link  prediction setup is given by  L = − 1  N  N  �  n=1  2  �  j=1  yjlog( ˆyj),  (10)  where N is the total number of samples being used to calculate  the loss and ˆy is the probability of link formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Each model  uses a batch size of 64 as well as a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Finally, each model is trained using 300 epochs, which is  sufficient for convergence on each dataset but simultaneously  allows for convergence at slightly different optima, resulting  in robust and reliable evaluation when used with k-fold cross  validation as further discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' LINK PREDICTION EVALUATION  In this section, we begin by describing our considered  courses along with their corresponding datasets (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III-A)  as well as our model evaluation procedure (Sec III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We  then evaluate our framework’s performance for predicting link  formation (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III-C) and examine the time-accuracy of our  prediction model (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Datasets  We consider the SLNs formed in six courses: four Coursera-  based MOOC courses and two traditional courses offered  at Purdue University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The four MOOC courses – “Machine  Learning” (ml), “Algorithms: Design and Analysis, Part 1”  (algo), “English Composition I” (comp), and “Shakespeare in  Community” (shake) – were selected to represent a diverse  set of subjects: two quantitative in nature and two in the  humanities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In addition, we also consider the course “Python  for Data Science” hosted through Purdue University over two  semesters: “Fall 2019” (f19) and “Spring 2020” (s20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The  availability of data from two offerings of a single course  provides a unique opportunity to evaluate behavior in a single  course over multiple semesters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The s20 dataset is of particular  interest because of its relation with the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Specifically, this course was held in-person from January -  March, allowing students to begin forming in-person links,  which carried into their relationship in the course’s SLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  However, with the pandemic forcing a transition to fully online  learning, link formation between students became completely  dependent on discussion forum communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The inclusion  of the f19 and s20 datasets, which differ both in size and  in format, demonstrate our framework’s broad applicability  to different online course formats in dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Table I shows detailed metrics of the six considered datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 5 summarizes the graph topology at the termination of  each course under evaluation in terms of five social network  metrics: number of nodes, number of edges, shortest path  lengths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', the Lpuv feature), degree per node, and user  clustering coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The diverse nature of each course is  evident from each of the shown metrics and particularly from  the varying number of edges and nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We observe the largest  differences between the Purdue f19 and s20 courses versus  the MOOC courses: the f19 and s20 courses are significantly  smaller in nodes/edges and also have significantly larger  degree per node and clustering coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We also observe  the difference in both the number of edges and the average  degree per node between the f19 and s20 courses, which  demonstrates the increase in student utilization of discussion  forums in the absence of in-person instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Next, we describe the SLNs in terms of the features in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We make several observations on associations with  link formation within and across datasets before evaluating the  link-prediction portion of our proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  1) Data Preparation: To obtain a representative set of  student behavior from a course, and to ensure that data  gathered from each source is uniformly formatted, we filter  each considered dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Specifically, we remove the instruc-  tors from the list of learners and remove all links formed  between learners and instructors, since we are interested in  developing models targeted towards peer-to-peer interaction,  with the goal of requiring less direct instructor intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Furthermore, interactions before the beginning of a course are  removed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' only links formed during a course are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Both course-hosting sites offer an option for full anonymity  to learners – posts made with anonymity are ignored, as we  cannot make meaningful connections with unknown users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Enrolled learners who did not access the forum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', an empty  adjacency matrix), are not considered to remove confusion –  a lack of behavior excludes a helpful metric for predicting  future behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Such students would likely benefit from more  traditional intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' After filtering, less than 2% of the  learner pairs in each dataset demonstrated a formed link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  This underscores an extreme sparsity of learner pairs for link  prediction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' the methodology applied to avoid overfitting will  be discussed further in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  2) Topic extraction: To obtain the post similarities cuv(i),  we must first extract the topics, K, and distributions for  each post according to the LDA algorithm discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Prior to building the dictionary of topics, all URLs,  punctuations, and stopwords are removed from each post’s  text and all words are stemmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Table III summarizes the  topic extraction results for each dataset using |K| = 20 topics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  the top three words shown are from the five topics that have  the highest supports across posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We find that |K| = 20  produces a set of topics that have reasonably large supports  across posts while retaining granular information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', able  to convey differences between student posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In our manual  inspection, larger values of |K| lacked the support to generate  informative features, while smaller values of |K| resulted in  too much intersection between topics for a good understanding  of content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Model Evaluation Procedure  To evaluate the models proposed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II, we use the  following metrics, training procedures, and evaluation criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  1) Metrics: We use three metrics to evaluate prediction  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' First, we compute the overall Accuracy (ACC),  9  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 5: Social network graph metrics on our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We see the largest distinction in characteristics between the four MOOC courses and  the two Purdue courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Model  ml  algo  shake  comp  f19  s20  Re  AUC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='5005 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='5188 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0322  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='5061 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0034  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='5167 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0266  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='5689 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0401  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='5238 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0121  ACC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='5995 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0054  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8338 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8296 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0073  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8349 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0082  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9524 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0057  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9599 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0020  BNet  AUC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9053 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9488 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0058  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8603 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0095  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8684 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0116  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='7413 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0546  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='7495 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0269  ACC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9175 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0066  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9805 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0019  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9472 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9492 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0026  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9600 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0053  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9672 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0013  FCNN  AUC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9766 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0033  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9706 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0039  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9670 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0059  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9714 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0084  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8991 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0367  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8844 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0330  ACC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9782 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0027  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9871 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0029  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9853 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0019  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9850 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0022  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9688 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0037  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9729 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0022  SVM  AUC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9122 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0027  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9523 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8982 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0071  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8618 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0071  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8437 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0343  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='9137 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='9755 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9608 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='9462 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0022  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9670 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0040  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9700 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='8486 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='8674 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='9713 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='9969 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='0011  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9955 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='7395 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='5628 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='9967 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='9988 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='9965 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='9958 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0019  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='6557 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0751  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='5500 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0997  TABLE IV: Performance of each considered link prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The CNN model is among the best performing model across all six  datasets with respect to the AUC and ACC metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' All results in bold highlight the best performing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We see that the GNN results in  strong performance on the four MOOC courses while performing poorly on the two Purdue courses, indicating that GNNs are effective for  link prediction in large courses whereas our method delivers strong performance in small courses as well as large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  or the fraction of predictions over all time that are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' For  iteration k, it is obtained as:  1  |Ωke| · L  �  (u,v)∈Ωk  e  L  �  i=1  1{yuv(i) = ¯yuv(i)},  (11)  where yuv(i) ∈ {0, 1} is the binary prediction made based on  ˜yuv(i) and 1 is the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Second, we compute  the Area Under the ROC Curve (AUC), which assesses the  tradeoff between true and false positive rates for a classifier  [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Third, we define a metric called Time Accuracy (TAC)  to be the fraction of links that are predicted to form within  a fixed window w of when they actually form (among those  that eventually form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Letting nuv = mini{yuv(i) = 1} be  the actual time at which link (u, v) ∈ Ωf  k forms and ˜nuv =  mini{˜yuv(i) = 1} the predicted time, the TAC is defined as  1  |Ωf  k|  �  (u,v)∈Ωf  k  1{|˜nuv − nuv| ≤ w}  (12)  for iteration k, where Ωk  f ⊂ Ωk  e is the set of correctly predicted  links in the test set that will eventually form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We compute  the mean and standard deviation of each metric across three  evaluation iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  2) Training and Testing: k-fold cross validation is used to  evaluate each predictor with k = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Following Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III-A, we  again consider the link sets G(L) and Gc(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our objective  is to train models capable of accurate link prediction despite  the large class imbalance between G(L) and Gc(L) that will  be observed during training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' To achieve this,  we take an equal proportion of samples from both G(L) and  Gc(L) to form each training fold, which, in turn, retains the  overall class imbalance in the training set during each training  iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The corresponding testing set of each training fold  contains the same class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' After each training fold, we  calculate the metrics of interest on the respective testing set of  the validation run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This sampling, along with the utilization of  the AUC measurement, allows us to quantify the false alarm  versus true positive rate, since the prediction accuracies on a  poorly trained model could be very high due to the large class  imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  In each of the k iterations, we consider a set of time intervals  from which the model parameters are estimated considering  each pair (u, v) ∈ Ωr  k, using the procedures in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III-B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Then, for each (u, v) ∈ Ωe  k, the inputs are used to make a  prediction ˜yuv(i) ∈ [0, 1] of the link state yuv(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Link Prediction Evaluation  Table IV gives the overall performance of the baseline,  linear, GNN, and deep learning models in terms of the AUC  and ACC metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Overall, we see that the CNN consistently  outperforms the other predictors for each considered dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  In addition, the GNN achieves strong (comparable to the CNN)  User Clustering Coefficients  Shotest Path Length  Number of Edges  Number of Nodes  Degree per Node  30000  9  200  10000  2  2910  prediction performance on the four MOOC datasets, but it  performs poorly on both f19 and s20, achieving AUCs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='74  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='56 in f19 and s20, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This behavior is con-  sistent with observations in prior work [46] that GNNs require  large datasets for effective generalization – a characteristic that  MOOCs are able to provide (with at least 1,000 users in each  case, see Table I) whereas the Purdue courses, f19 and s20,  are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our explicit feature engineered methodology paired  with a CNN classifier, on the other hand, is more robust against  variations in SLN course size and type in comparison to the  GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Of particular interest is the s20 dataset and its performance  relative to the other five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Because s20 was held  partially in-person prior to the COVID-19 outbreak in March  2020, the behavior represented includes both in-person and  online interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Furthermore, it contains a rapid change  in behavior midway through the semester that models must  account for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' It follows from the high accuracies and AUCs  demonstrated by each deep-learning model on this dataset that  our prediction model can be applied to hybrid-online courses  with a similar level of accuracy to fully online courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' It also  suggests that our proposed model is responsive to large-scale  shifts in student behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' From Table IV, we see neither the  GNN nor the other baseline models are capable of capturing  either of these desirable characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' As a result, we find  that our proposed framework is capable of increasing both  course quality and learner interactions during the pandemic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  an attribute that can be leveraged to improve instruction in a  post-pandemic course offering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Considering all courses, the CNN model has slightly higher  performance across the metrics and datasets, reaching average  AUCs between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='92 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='99 and average ACCs between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='97  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The AUC of Re is nearly random, but demonstrates  a high accuracy in all cases because of the large class im-  balance present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Similarly, the linear classifiers demonstrate  high ACC values because of the large class imbalance as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Although the Bayesian model consistently outperforms  the baseline models, the lower accuracy and AUC relative  to the CNN and CRNN models confirms our hypothesis from  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II that capturing spatial and temporal variance leads to  improvement in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' More specifically, the evolution  of the state of an SLN between different time periods, both  temporally and spatially, is important to predicting learner  interactions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' this aspect is effectively included in the LSTM-  based CRNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We further observe that the CNN model, capturing  spatial variance, and the RNN capturing temporal variance,  each perform similarly to the CRNN model for several datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  This suggests that while spatial and temporal variance both  individually assist in prediction, their combined usage may  not result in significant performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Although an accurate prediction is most informative on the  efficacy of a connection between learners, recommendations  may also be supported by false predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' If a high-accuracy  model falsely predicts that two users will connect, we may  infer that the formation of a link between these two users  would be beneficial based on model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Conversely,  there is a strong correlation between false negative predictions  and weak links between learners, implying that the benefits of  forming a connection between two such users would be trivial  compared to other, more highly-weighted connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Early Detection of Link Formation  The models proposed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III-C consider the ability to  predict link formation in subsequent time intervals up until  the end of the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' However, it does not consider links  that will form at an earlier or later interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' These occurrences  of a delay between link formation and prediction can lend  additional information of importance to learners: if we can  predict in advance which learners may form connections, we  may encourage them to connect sooner, potentially resulting in  a stronger connection or faster replies from learners expected  to have delayed responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' On the other hand, if we find that  a link forms much sooner than predicted by our model, this  may indicate that learners would benefit from re-connecting  on the current topic later in the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  To study these cases, we evaluate the TAC metric from  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III-B for our RNN, CNN, FCNN, and CRNN models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=',  we measure whether links form within a given window w of  when they are predicted to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Note that the TAC metric was  only calculated for the deep learning models, since they were  consistently the best performing link formation predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  The granular value of 20 time intervals used to generate the  SLN graph model gives the predictive model access to more  frequently updated features, and allows the model to respond  quickly to changes in SLN behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 6 shows the TAC  values as w is increased from 0 to 20 for several of our  proposed deep learning prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The sharp increase  of each TAC curve for small w of each model – with the  exception of the RNN – indicates that many links form close to  when they are predicted to form, reinforcing our observations  of model quality from other performance metrics in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  A window of w = 2, for example, is already sufficient for all  six forums to reach a TAC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='5 or above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Observing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 6e, which represents the TAC curve of the  f19 dataset, it is clear that our TAC metric demonstrates  a lower accuracy for small datasets but the performance of  individual models has more variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This is largely attributed  to the smaller number of learner pairs contained in the f19  dataset with which to train the model compared to a MOOC  forum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' However, with the exception of the RNN, we can  observe the same curve shape and sharp initial increase present  for larger datasets, indicating that TAC is both a consistent and  useful evaluation metric of model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We can further  observe in the ml, f19, and s20 datasets that the RNN model  fails to correctly predict links consistently across datasets  within a small interval of when they actually occur, further  suggesting that spatial features play a more important role in  the problem of link prediction, which is further discussed in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Furthermore, there are very few links with large w, once  again reinforcing the results of other performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The  small quantity of links with large w in each forum present a  significant opportunity to recommend early formation of links  (when predictions are early) and potential times for learners  to reconnect (when predictions are late).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Though there is less  11  (a) ml  (b) algo  (c) shake  (d) comp  (e) f19  (f) s20  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 6: TAC with different windows w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The TAC curves all exhibit sharp increases initially, indicating many links form around the time they  are predicted to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The links at higher w, on the other hand, indicate potential for recommending early link formation and future reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  room for change on links with smaller w, learners may be  more willing to act on recommendations in these cases since  they induce less modification to actual behavior [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' after all, a  learner may be reluctant to reach out to others on the basis of  outdated threads or on the assumption that they will eventually  collaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' LINK FORMATION ANALYTICS  In this section, we consider several descriptive analytic  tools and visualizations for instructors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We first describe the  evolution of model parameters during prediction (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' IV-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  We then examine the correlations between features (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  IV-B) and analyze their individual and collective impact on  prediction (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' IV-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Finally, we analyze the importance of  the predictor’s architecture in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Time-Series Variable Evolution  Because the hidden layers of deep-learning models cannot  be understood intuitively, we provide an alternate form of  visualizing their behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' It is possible to observe the de-  cisions made by the deep learning model during prediction by  investigating changes in state for each model gate over time,  and making inferences about the final prediction from these  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The stability exhibited by the gates over time  supports the viability of early link formation prediction from  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' To demonstrate this, we consider an example of  how the CRNN LSTM layer parameters specified in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II-C  for deep learning prediction models evolve over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  By examining the relationship fading gate, f, in particular,  we are able to demonstrate how the inputs from time interval  i−1 affect the model output at time interval i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', how much  information is carried over from interval to interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' To do so,  we choose a link (u, v) ∈ G(L) at random from algo, and  feed euv(i) into the trained model for L = 20 to generate the  predictions ˜yuv(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The prediction has high accuracy on the  chosen link, which forms within one time interval of when it  is predicted to form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  The neuron activation values for the gates g, i, f, o and the  state z and output h are additionally considered and shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The vertical axis is the vector dimension (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', neuron  number), and the horizontal is the time instance i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' A few of the  input gate dimensions, g, change at about the time the link is  formed (around i = 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' These changes propagate through the  network, causing the output, h, as well as some dimensions  of the intermediate gates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', f, i, and o) to change around  i = 17 as well, thus forming an accurate prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The fact  that i and f in particular tend to take extreme values indicates  that the input, g, and prior state, z, are either fully passed or  blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  We also observe that several dimensions in z evolve gradu-  ally over time, with several non-zero dimensions in f passing  information across multiple time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This result helps  explain why models using an LSTM layer in conjunction with  other methods perform better than the Bayesian model: passing  information from one time interval to another increases the  prediction quality compared to only updating the input features  at each time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Feature Correlations  Investigating the relationship between individual features  provides insights into the shape of an SLN in a different  capacity than the predictions made by our deep-learning mod-  els, and it provides an analytical tool with which instructors  can monitor an online classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Table II summarizes the  distributions of G(L) (top row) and Gc(L) (bottom row), with  the top 5% of outliers removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We show the means and  standard deviations (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=') of each feature for both groups, as  well as the signal-to-noise ratio (SNR) for each feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The  large difference in magnitude for both mean and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' between  formed and unformed links indicates a clear difference in  behavior between these two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The large gap in values  reinforces the results of our predictive algorithms discussed in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The SNR measures how effectively a feature can  distinguish between the two groups, with a higher magnitude  indicating more efficacy [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We make a few impactful  observations for link prediction from these statistics:  (i) Infrequent short paths: The length and number of shortest  paths between learners are both negatively associated with link  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The former is consistent with the intuition that  learners who are closer together (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', smaller shortest path  lengths) are more likely to form links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The latter, however,  indicates that links are more likely to form when fewer such  shortest paths exist, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', the paths should be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' An  interesting analogy can be drawn here to the small world  phenomenon, where users can discover short paths in a social  network even when only one or a few exist [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' in other  words, the presence of fewer short paths makes each of those  neighboring connections more important and more likely to  foster link creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  (ii) Low-degreed shared neighbors: In order of increasing  SNR, Ja, Re and Ad are each positively associated with link  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Each of these measures the common neighborhood  of two learners, with increasing penalty placed on the degrees  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='4  crnn  cnn  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='6  A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='4  crnn  cnn  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='2  fcnn  rnn  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0  0  2  4  6  8  10 12 14 16 18 20  W12  (a) fuv(i)  (b) guv(i)  (c) huv(i)  (d) iuv(i)  (e) ouv(i)  (f) zuv(i)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 7: Neuron activations of each gate fuv(i), guv(i), huv(i), iuv(i), ouv(i), and zuv(i) over time of the LSTM layer inside the CRNN  model for two particular links (u, v) in algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The fact that several gate dimensions are non-zero indicates that information is propagating  across multiple time periods for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The top row demonstrates activations for a link formed late in the course, and the bottom row  demonstrates activations for an early-formed link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  of these neighbors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', Ja does not include degree at all,  while Re is inversely proportional to it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The fact that Ad  has the highest SNR, then, implies that shared neighbors with  fewer links are more prone to facilitate link formation, which  is consistent with the the point above on unique paths being  more predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  (iii) Low ceiling feature values: Taking the statistics present  in Table II in conjunction with each feature’s cumulative  distribution function (CDF), shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 4, it is evident for  several features including To and Pr that no learner pairs reach  the maximum possible value for the feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Most notably  with respect to To, the maximum number of shared topics  between two connected users is always less than 15 of the  20 extracted topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Given the highly connected nature of  “hub” students that possess a large number of shortest path  connections, it would be expected that the maximum number  of shared topics would be 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This discrepancy in number  of shared topics suggests that hub students connect frequently  with less-engaged students, but rarely interact with each other,  creating smaller student ecosystems within the course centered  around their knowledge dissemination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Another possibility  is a difference in student knowledge state/engagement on  particular topics, indicating that learners are more motivated to  post about topics they are confident in or interested in learning  and avoid topics they are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  (iv) Topology vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' post properties: Pr and To are both  positively associated with link formation, as one would expect:  those with higher degrees (Pr) and focusing on similar topics  (To) should be more likely to interact in the discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Surprisingly, though, these features have lower SNRs than  the other neighborhood-based features, indicating that the  network topology drives link formation in an SLN more than  individual learner properties like a learner’s tendency to post,  for example, or topic interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Furthermore, the SNR of To is  higher in the less densely populated courses (f19 and s20),  indicating that clearer signals may emerge around topics when  there is less overall volume of discussion in the forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This  is consistent with the performance differential of the GNN  model in link prediction on the large vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' small datasets, since  it does not learn from topic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  (v) Quantitative vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' humanities courses: Among the four  MOOC courses, Pr is higher in comp and shake (particularly  shake) than in ml and algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This is consistent with human-  ities courses tending to invite more open-ended discussions,  whereas quantitative courses have questions requiring explicit  answers [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' More learners would then be motivated to post in  the forums of humanities courses – in fact, such participation  may be a course requirement – leading to more links forming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Table I confirms the intuition that even with a smaller class  size, comp and shake have a higher ratio of learner pairs to  learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The distinction between quantitative and humanities  courses also helps explain which settings temporal behavior is  helpful for link prediction, as we will discuss in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Feature Importance Analysis  Recall in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' II-B that we define three groups of fea-  tures: (i) Nei, which quantify the overlap between learner  neighborhoods, (ii) Path, which are the length and number  of shortest paths, and (iii) Post, or the similarity in what  learners discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' To complement the correlation analysis in  Table II that was done for each feature individually, we now  analyze the contribution of each feature type to the prediction  quality of our CRNN model, by evaluating it using different  input feature combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  To evaluate smaller groups of features using our CNN  and CRNN models, a modification in model architecture  is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our implementation of the CRNN model for  computing links with all features contained both a 3 × 1  kernel layer and a 2 × 1 kernel layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' To classify samples  using a subset of less than five of the seven features, the  second convolutional layer using a 2 × 1 kernel was removed,  leaving a single convolutional layer with a 3×1 kernel before  the fully connected and output layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='8541 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0024  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8922 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0078  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='6735 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0519  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='6346 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0118  ACC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9446 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9753 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9314 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9482 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0029  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9538 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9627 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0011  Path + Post  AUC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9332 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0034  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9455 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+page_content='9255 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0096  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9444 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0078  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8832 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0358  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='8848 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0175  ACC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9418 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0031  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9659 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9650 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0038  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9736 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0039  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9679 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0051  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='9736 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='0022  TABLE V: Performance of the CRNN Model with selected input feature groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The top two highest performing groups for each course metric  are bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The combinations of Nei + Path and Path + Post outperform Nei + Post consistently, indicating that while neighborhood-  based features are most important for prediction, the other feature types contribute significantly to link prediction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Table V shows the results when each course is broken  into 20 time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' None of the combinations reach the  performance of the original model with all input variables  in Table IV, indicating that each feature group contributes  to the prediction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' The Nei + Path and Path + Post  combinations show the highest overall performance across all  six forums, indicating that the combination of Nei + Path has  a confounding effect on the model – we would expect both  Nei-based groups to share a higher AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Combining these  values with the SNRs in Table II indicates that the Nei features  contribute the most to model accuracy, followed by Post and  then Path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  If we compare the individual feature groups, we generally  find that the Nei features perform the best, followed by Path,  and then Post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This is consistent with the behavior of these  features within groups as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This ordering of Post and Path  is opposite of the SNR magnitudes from Table II: here, the  single feature To outperforms the combined impact of Path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Given that Table II is concerned with the eventual formation of  links but not the time at which they form, we conjecture that  in the absence of Nei, Post is more important to pinpointing  the time of link formation while Path is more important to  whether they form at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' After all, the timing of particular  topic coverage should influence when learners interested in  those topics connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Model Architecture Analysis  Here, we first analyze the importance of spatial pattern  preserving convolutional layers and temporal pattern preserv-  ing recurrent layers for link prediction in SLNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' We find  that, in general, classification models that incorporate only  spatial pattern dependencies (CNN) outperform models that  only incorporate time dependencies (RNN), as shown in Table  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This is consistent with Table V, where we find that SLN  topology features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', neighborhood and path-based features),  which explain spatial relationships between links, are the  most important for accurate link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' However, we also  find that incorporating time dependencies into link prediction  models (e,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', RNN and CRNN) obtains strong performance in  large courses such as MOOCs, whereas these models become  less accurate on small courses such as f19 and s20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Interest-  ingly, although the RNN accurately predicts whether links will  form (as shown in Table IV), they do not accurately predict  when the links will form as shown from the TAC curves  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 6, particularly on ml, f19, and s20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This behavior  is consistent with such quantitative courses requiring short  answers in fast time intervals whereas the humanities courses  typically involve threads of discussion that persist over longer  periods of time [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 7, we further explored the efficacy  of recurrent layers by visualizing the various gates of the CRNN  in the algo course, where we saw that information propagates  from multiple time periods to aid link prediction after spatial  patterns have been identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This reinforces that recurrent  layers may carry long-term information for link prediction,  but convolutional layers are more robust for in SLNs on both  large and small courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  In addition, as shown in Table IV, convolutional GNNs  achieve strong link prediction performance on each of the  MOOC datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Rather than employing our explicitly de-  fined model features, GraphSAGE embeds features across the  SLN topology that exploit spatial patterns, hence resulting in  strong performance for these datasets captured by the GNN’s  convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' However, for the GNN to learn such  discriminative features, it may require a large graph to train on  [46], thus making the model less effective for smaller courses  such as f19 and s20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Our proposed framework, in which we  explicitly model features between node pairs, on the other  hand, is better able to learn and generalize on the smaller  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' More generally, these results indicate that in the  SLN domain, informed feature engineering (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', using spatial  features) paired with corresponding layers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', convolutional)  results in better trained models with less data than that required  by GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' This is useful for generating analytics in the early  stages of courses before a significant amount of links have  formed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', before interaction data has been observed) on the  forums [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' CONCLUSION  In this work, we developed a link prediction framework  specifically tailored to operate in social learning networks  (SLNs) based on neighborhood-based, path-based, and post-  based modeling features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Through evaluation in six different  courses, we demonstrated our framework’s ability to perform  accurate link prediction in a variety of learning environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  In particular, we examined the efficacy of our framework on a  course forced online after approximately eight weeks of tradi-  tional instruction due to the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' In addition,  we considered the SLNs formed in four Massive Open Online  Courses (MOOCs) as well as one traditional undergraduate  course, with a heavy reliance on student participation in an  online discussion forum, offered through Purdue University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  While our work establishes an initial framework and re-  sults for link prediction in SLNs, many avenues remain for  exploring the challenges of link prediction in this new type of  online social network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' One is additional feature engineering:  other features that we did not consider – such as learners’  14  background knowledge, level of education, and personal goals  – may also be associated with link formation, and may  allow further improvements in link prediction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' As  demonstrated here, our proposed framework is applicable  across multiple datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' thus, additional evaluation variants  on forums or classes with different structures, such as those  present in K-12 education, may be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  REFERENCES  [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Brinton and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Joe-Wong, ”Predicting Learner Interactions  in Social Learning Networks,” IEEE INFOCOM, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1322-1330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Brinton and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Chiang, “Social Learning Networks: A Brief  Survey,” IEEE CISS, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Miller, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Soh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Samal, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Kupzyk, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Nugent, ”A Compar-  ison of Educational Statistics and Data Mining Approaches to Identify  Characteristics that Impact Online Learning,” in Journal of Educational  Data Mining, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 117-150, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Pendry and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Salvatore, “Individual and Social Benefits of On-  line Discussion Forums,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Learning Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 50, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  211–220, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Brinton and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Chiang, “MOOC Performance Prediction via  Clickstream Data and Social Learning Networks,” IEEE INFOCOM,  2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2299–2307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Brinton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Buccapatnam, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Wong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Chiang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Poor, “Social learning networks: Efficiency optimization for mooc  forums,” IEEE INFOCOM, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Liben-Nowell and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Kleinberg, “The Link-Prediction Problem for  Social Networks,” Journal of the Association for Information Science  and Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1019–1031, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Brinton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Buccapatnam, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Zheng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Cao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Lan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Wong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Ha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Chiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Poor, ”On the Efficiency of Online  Social Learning Networks,” in IEEE/ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 5,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2076-2089, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [9] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Zhang, ”Socially-Driven Learning-Based  Prefetching in Mobile Online Social Networks,” in IEEE/ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2320-2333, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Wong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Chiang, ”On the Efficiency of Social  Recommender Networks,” in IEEE/ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 4,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2512–2524, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Divakaran and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Mohan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' “Temporal Link Prediction: A Survey.”  New Generation Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 213–258, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Lorenzen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Hjuler, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Alstrup, ”Tracking Behavioral Patterns  Among Students in an Online Education System,” EDM, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 280-  285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Aghababaei and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Makrehchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' “Interpolative Self-Training Approach  for Link Prediction,” Intelligent Data Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  1379–1395, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Muniz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Goldschmidt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Choren, ”Combining contextual,  temporal and topological information for unsupervised link prediction  in social networks,” Knowledge-Based Systems, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 129–137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [15] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Jie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Li and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Liu, ”Social Network Group Identification based  on Local Attribute Community Detection,” 3rd IEEE ITNEC, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  443-447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Backstrom and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Leskovec, ”Supervised Random Walks: Predicting  and Recommending Links in Social Networks,” Web Search and Data  Mining, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [17] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Aghabozorgi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Khayyambashi, “A New Study of Using  Temporality and Weights to Improve Similarity Measures for Link  Prediction of Social Networks,” Journal of Intelligent & Fuzzy Systems,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2667–2678, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [18] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' ”A supervised link prediction method  for dynamic networks,” Journal of Intelligent & Fuzzy Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 31,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 291-299, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Amershi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Conati, ”Combining Unsupervised and Supervised Clas-  sification to Build User Models for Exploratory Learning Environments,”  in Journal of Educational Data Mining, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 18-71, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [20] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Gurjar, “Leveraging Social Networks for Authentic Learning in  Distance Learning Teacher Education,” TechTrends: Linking Research &  Practice to Improve Learning, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 64, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 666–677, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Lynch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Barnes, ”How Many Friends Can You Make in  a Week?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Evolving Social Relationships in MOOCs over Time,” EDM,  2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 97-103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Cui, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Yang , C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Homan ”Modeling Information Sharing Behav-  ior on Q&A Forums,” Pacific-Asia Conference on Knowledge Discovery  and Data Mining, Lecture Notes in Computer Science, 2017, vol 10235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [23] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Koprulu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Kim and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Shroff, ”Battle of Opinions Over Evolving  Social Networks,” in IEEE/ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 532-545,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Yang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Kraut, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Rose, “Exploring the Effect of Student Confusion  in Massive Open Online Courses,” in Journal of Educational Data  Mining, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 52-83, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Liang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Wei and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Zhang, ”On Structural Features, User  Social Behavior, and Kinship Discrimination in Communication Social  Networks,” in IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Computat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Social Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 425-  436, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [26] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Xiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Neville, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Rogati, “Modeling Relationship Strength  in Online Social Networks,” in WWW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' ACM, 2010, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 981–990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [27] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Dalipi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Imran and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Kastrati, ”MOOC dropout prediction  using machine learning techniques: Review and research challenges,”  2018 IEEE EDUCON, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1007-1014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Tsiakmaki, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Kostopoulos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Kotsiantis, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='Ragos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' ”Transfer  Learning from Deep Neural Networks for Predicting Student Perfor-  mance.” Applied Sciences, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [29] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Jiang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Dai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Jia and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Hirota, ”Student Eye  Gaze Tracking During MOOC Teaching,” SCIS, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 875-880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [30] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Papamitsiou and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Economides, ”Motivating Students in Col-  laborative Activities With Game-Theoretic Group Recommendations,” in  IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 374-386, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [31] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Almatrafi and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Johri, ”Systematic Review of Discussion Forums  in Massive Open Online Courses (MOOCs),” in IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 413-428, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Joksimovi´c and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Jovanovi´c and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Kovanovi´c and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Gaˇsevi´c and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  Miliki´c and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Zouaq and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' van Staalduinen, ”Comprehensive Analysis  of Discussion Forum Participation: From Speech Acts to Discussion  Dynamics and Course Outcomes,” in IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 38-51, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [33] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Moreno-Marcos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Alario-Hoyos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Mu˜noz-Merino, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Est´evez-  Ayres and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Kloos, ”A Learning Analytics Methodology for Under-  standing Social Interactions in MOOCs,” in IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=',  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 442-455, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [34] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Moreno-Marcos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Alario-Hoyos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Mu˜noz-Merino and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Kloos, ”Prediction in MOOCs: A Review and Future Research  Directions,” in IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 384-401,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Pigeau, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Aubert, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Prie, ”Success Prediction in MOOCs,”  EDM, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 390-395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [36] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Calefato, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Lanubile and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Novielli, ”Moving to Stack Overflow:  Best-Answer Prediction in Legacy Developer Forums,” ACM/IEEE Inter-  national Symposium, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Rezvanian and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Meybodi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' “A New Learning Automata-Based  Sampling Algorithm for Social Networks.” International Journal of  Communication Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 30, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 5, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [38] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Cheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Guo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Cui, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Shan, ”Dynamical Modeling, Analysis,  and Control of Information Diffusion over Social Networks: A Deep  Learning-Based Recommendation Algorithm in Social Network,” Dis-  crete Dynamics in Nature & Society, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Blei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Ng, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Jordan, “Latent Dirichlet Allocation,”  JMLR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 993–1022, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [40] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Varshney, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Kumar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Gupta, ”Predicting information diffusion  probabilities in social networks: A Bayesian networks based approach,”  Knowledge-Based Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 133, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 66-76, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Kung, Kernel Methods and Machine Learning, Cambridge Uni-  versity Press, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Fire, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Tenenboim, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Lesser, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Puzis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Rokach, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Elovici,  “Link Prediction in Social Networks using Computationally Efficient  Topological Features,” in SocialCom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 73–80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Aswathy Divakaran and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Mohan, “Temporal Link Prediction: A  Survey,” New Gener.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 38, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 213–258, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [44] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Yu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Cheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Aggarwal, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Chen, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Wang, ”Link Prediction  with Spatial and Temporal Consistency in Dynamic Networks,” IJCAI,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [45] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Zhang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Chen, ”Link prediction based on graph neural net-  works,” NeurIPS, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Zhou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Cui, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Hu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Li, and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Sun, “Graph neural networks: A review of methods and applications,”  AI Open, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 57-81, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [47] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Hamilton, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Ying, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' Leskovec, “Inductive Representation  Learning on Large Graphs,” NeurIPS, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content='  [48] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
+page_content=' He, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfo_2T/content/2301.01606v1.pdf'}
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+Diffusion-based Data Augmentation for Skin
+Disease Classification: Impact Across Original
+Medical Datasets to Fully Synthetic Images
+Mohamed Akrout1∗, B´alint Gyepesi1∗, P´eter Holl´o2, Adrienn Po´or2,
+Bl´aga Kincs˝o2, Stephen Solis1, Katrina Cirone1, Jeremy Kawahara1,
+Dekker Slade1, Latif Abid1, M´at´e Kov´acs1, and Istv´an Fazekas1
+1 AIP Labs, Budapest, Hungary
+2 Semmelweis University, Faculty of Medicine, Department of Dermatology,
+Venereology and Dermatooncology, Budapest, Hungary
+Abstract. Despite continued advancement in recent years, deep neural
+networks still rely on large amounts of training data to avoid overfitting.
+However, labeled training data for real-world applications such as health-
+care is limited and difficult to access given longstanding privacy, and
+strict data sharing policies. By manipulating image datasets in the pixel
+or feature space, existing data augmentation techniques represent one of
+the effective ways to improve the quantity and diversity of training data.
+Here, we look to advance augmentation techniques by building upon the
+emerging success of text-to-image diffusion probabilistic models in aug-
+menting the training samples of our macroscopic skin disease dataset.
+We do so by enabling fine-grained control of the image generation pro-
+cess via input text prompts. We demonstrate that this generative data
+augmentation approach successfully maintains a similar classification ac-
+curacy of the visual classifier even when trained on a fully synthetic skin
+disease dataset. Similar to recent applications of generative models, our
+study suggests that diffusion models are indeed effective in generating
+high-quality skin images that do not sacrifice the classifier performance,
+and can improve the augmentation of training datasets after curation.
+Keywords: Data augmentation · Skin condition classification · AI for
+dermatology · Diffusion models · Synthetic medical datasets
+Fig. 1: Synthetic melanoma images generated by the stable diffusion model after
+fine-tuning it with melanoma images using the input text prompt “melanoma”.
+* equal contribution
+arXiv:2301.04802v1  [cs.LG]  12 Jan 2023
+
+2
+M. Akrout, B. Gyepesi et al.
+1
+Introduction
+The last months have witnessed the emergence of diffusion probabilistic models
+(DPM) [10] as a powerful generator of high-fidelity synthetic datasets, leading
+to record-breaking performances in various applications such as image synthesis
+[21], natural language processing [4], and computational chemistry [3], to name
+a few. When compared to other types of generative models, such as generative
+adversarial networks (GANs) and variational autoencoders, DPMs are easier to
+train and offer state-of-the-art image generation quality [7].
+Given that synthetic images play a crucial role in privacy-preserving gener-
+ation and small dataset augmentation, DPMs attracted significant attention in
+the medical imaging field. Table 1 provides an overview of the prior studies of
+DPMs, including their medical applications and dataset domains. At first glance,
+the reader can identify that the study in [23] is the closest one to this work where
+synthetic images were generated from seed images in the Fitzpatrick 17k dataset
+using the OpenAI’s DALL·E 2 model [19].
+Table 1: Summary of existing applications of diffusion models in medical imaging.
+Medical applications
+Dataset domain
+Papers
+Image generation
+lungs X-Ray, CT, MRI
+[2, 5, 16, 17]
+Image segmentation
+MRI, CT, ultrasound
+[9, 13, 30]
+Image inpainting
+MRI
+[22]
+Image denoising
+MRI, CT, retinal OCT
+[6, 11, 32]
+Lesion detection
+MRI
+[24, 29, 31]
+Image translation
+MRI, CT
+[13, 15]
+Seed-image based augmentation
+Dermatology
+[23]
+Skin disease classification
+Dermatology
+This work
+using large synthetic datasets
+Inspired by the recent early success of DPMs, we propose to use diffusion models
+for image augmentation as part of supervised machine learning pipelines. More
+specifically, we study how diffusion models can i) increase the classification met-
+rics for skin diseases, and ii) augment skin condition datasets by effectively ma-
+nipulating the generated images’ features conditioned on the input text prompts.
+This paper makes the following contributions:
+– We study the potential of DPMs for skin disease classifications by fine-tuning
+them on six different disease conditions: basal cell carcinoma, melanoma,
+actinic keratosis, atypical melanocytic nevus, lentigo, seborrheic keratosis.
+We do so by learning the embeddings of each disease using text inversion.
+
+Diffusion-based Image Augmentation for Skin Disease Classification
+3
+– We demonstrate that the classification accuracies of skin disease classifiers
+trained on generated synthetic images is similar to training on real images,
+where the performance is maintained when using half the number of real
+images, and only slightly deteriorates when using a fully synthetic dataset.
+This result suggests that the recent success of generative models can help
+minimize the barriers of sharing labeled medical datasets, with minimal per-
+formance deterioration.
+– We illustrate how DPMs are powerful tools to add visual aspects of skin
+images guided by domain experts in complementing training datasets.
+2
+Diffusion-based data augmentation
+In this section, we begin by describing the methods used for training the embed-
+dings of the aforementioned six skin diseases on our macroscopic skin images.
+Then, we present the datasets associated with the two DPM training scenarios:
+a hybrid dataset compromising 50% synthetic and 50% real images, and a 100%
+fully synthetic dataset generated by the trained embeddings.
+2.1
+Stable diffusion
+The stable diffusion model proposed in [21] is not a monolithic model, but rather
+a pipeline of three components, as depicted in Fig. 2:
+1) Text encoding, based on the CLIP model [18], which transforms each token
+of the input text prompt into an embedding vector.
+2) Latent space U-Net generator, which takes all the token embeddings and a
+random noise array (a.k.a., latent array) and sequentially generates multiple
+arrays that better resemble the input text and the visual images on which
+the U-Net has been trained.
+3) Image decoder, based on a variational autoencoder (VAE) to transform the
+obtained latent array into the pixel space.
+In this pipeline, the embedding vectors of the text encoding control both the
+generation of the U-Net latent space representations as well as the VAE decoding.
+2.2
+Training dataset for synthetic image generation
+The limited number of available labeled images is one of the leading limita-
+tions faced by medical classification applications. Our internal macroscopic im-
+age dataset consists of thousands of skin condition images curated and classified
+by dermatologists to cover more than 700 different diseases. Here, we choose six
+widely spread classes across three distinct categories:
+– Malignant classes: basal cell carcinoma and melanoma;
+– Pre-malignant classes: actinic keratosis and atypical melanocytic nevus;
+– Benign classes: lentigo and seborrheic keratosis.
+
+4
+M. Akrout, B. Gyepesi et al.
+Fig. 2: The diffusion model pipeline for synthetic skin image generation.
+Table 2 provides an overview of the number of images used for each disease in
+training the text embedding with the stable diffusion model.
+In order to train the text embeddings associated to each skin disease, we use the
+stable diffusion architecture [20] based on latent diffusion models [21]. Using a
+model of the latter pretrained on multiple LAION datasets [1], we fine-tune each
+Table 2: The number of real training images for the considered skin diseases.
+Category
+Skin disease
+Data source
+Benign
+Seborrheic keratosis
+2134
+Lentigo
+680
+Pre-malignant
+Actinic keratosis
+3298
+Atypical melanocytic nevus
+623
+Malignant
+Basal cell carcinoma
+7081
+Melanoma
+3381
+
+Text Prompt
+"a photo of melanoma with irregular edges"
+Text
+Encoder
+Random
+Noise
+Conditioning
+U-net
+Denoising
+Variational
+Autoencoder
+Synthetic Skin ImageDiffusion-based Image Augmentation for Skin Disease Classification
+5
+embedding on our real-world image skin condition dataset for two million steps
+using the default hyperparameters proposed in [25]. We use PyTorch for both
+training and inference. Each embedding is trained on three NVIDIA GeForce
+RTX 3090 GPUs.
+2.3
+Curation of generated images
+While most of the generated skin disease images are of high quality, it is not
+unusual to obtain generated images of medium or low quality. To isolate high-
+quality images from lower qualities, Fig.3 depicts the full pipeline for augmenting
+our skin disease dataset composed of the following four steps:
+Fig. 3: Summary of the four steps of the generation pipeline for skin disease data
+augmentation.
+1) Synthetic data generation: Using the stable diffusion model described in Sec-
+tion 2.1, we generate 30.000 images for each one of the considered six skin
+diseases to get a synthetic dataset.
+2) Non-skin image filtering: We run the obtained synthetic dataset in 1) through
+a pretrained binary EfficientNet classifier [26] to filter out any non-skin images.
+The binary classifier has been trained on the skin images of the macroscopic
+dataset presented in Table 2 and non-skin images from ImageNet. The ac-
+cepted images as skin images by the binary classifier represent more than 99%
+of the generated images and constitute the filtered synthetic dataset.
+3) Skin disease image filtering: We use the filtered synthetic dataset to pre-
+dict the skin disease label using a pretrained ensemble model composed of
+two CNN models (EfficientNetV2 [27], RegNet [8]) and a visual transformer
+(Swin-Transformer [14]). This ensemble model has been pretrained on the
+macroscopic dataset presented in Table 2.
+4) Data augmentation: We use the correctly labeled images by the pretrained
+ensemble classifier as the data source for augmenting our initial dataset.
+
+Synthetic
+Filtered Synthetic
+Dataset
+Dataset
+Stable
+EfficientNet
+Diffusion
+non-skin filter
+Skin Disease Dataset
+RegNet
+(real, hybrid or fully synthetic)
+EfficientNetV2
+Swim Transformer
+Correctly Labeled
+Authentic Dataset
+Pretrained Ensemble
+Classifier
+Data
+Augmentation6
+M. Akrout, B. Gyepesi et al.
+3
+Experiments and Results
+3.1
+Dataset scenarios for synthetic image generation
+Based on the filtered images whose labels were correctly predicted by the pre-
+trained ensemble classifier, we build a fully synthetic dataset consisting of 500
+images per skin disease. For the real images, we randomly sample 500 images
+per class from our macroscopic skin image dataset. To examine the impact of the
+synthetic dataset on classification metrics, we consider the following datasets:
+– a small real dataset (real-small) containing 250 real images only,
+– a real dataset containing 500 real images only,
+– a hybrid dataset consisting of 250 real images and 250 synthetic images,
+– a synthetic dataset containing 500 synthetic images only.
+Note that the four datasets are balanced across skin diseases with varying pro-
+portions of real and synthetic images. This allows us to assess the efficiency of
+substituting real data with synthetic ones.
+3.2
+Medical synthetic data samples using text prompt inputs
+Here, we demonstrate the quality of the synthetic skin disease images stemming
+from the generation pipeline in Fig. 3 by providing four synthetic images for
+each disease. Similar to the synthetic melanoma images in Fig. 1, we present
+synthetic images of seborrheic keratosis, lentigo, atypical melanocytic nevus,
+basal cell carcinoma and actinic keratosis in Figs. 4, 5, 6, 7, and 8, respectively.
+Fig. 4: Synthetic seborrheic keratosis images generated by the stable diffusion
+model after fine-tuning it with seborrheic keratosis images using the input text
+prompt “seborrheic keratosis”.
+Fig. 5: Synthetic lentigo images generated by the stable diffusion model after
+fine-tuning it with lentigo images using the input text prompt “lentigo”.
+
+Diffusion-based Image Augmentation for Skin Disease Classification
+7
+Fig. 6: Synthetic synthetic atypical melanocytic nevus images generated by the
+stable diffusion model after fine-tuning it with atypical melanocytic nevus images
+using the input text prompt “atypical melanocytic nevus”.
+Fig. 7: Synthetic basal cell carcinoma images generated by the stable diffusion
+model after fine-tuning it with basal cell carcinoma images using the input text
+prompt “basal cell carcinoma”.
+Fig. 8: Synthetic actinic keratosis images generated by the stable diffusion model
+after fine-tuning it with actinic keratosis images using the input text prompt
+“actinic keratosis”.
+While the impressive generative capabilities of AI models have already been es-
+tablished for normal and glaucomatous eyes in [12], our generated macroscopic
+images for different skin diseases similarly establishes the effectiveness for der-
+matology using larger synthetic datasets. This is to be opposed to seed-image
+based augmentation in [23] where synthetic datasets where not used to fine-tune
+the generative model.
+3.3
+Classification of Skin Conditions
+In this section, we first describe the training and inference procedures of the skin
+disease ensemble classifier on the four datasets described in Section 3.1.
+The Training Step We start by training three networks of the ensemble clas-
+sifier (i.e., Swin-Transformer [14], EfficientNetV2 [27], and RegNetZ [8]) on each
+one of the datasets (i.e., real, hybrid, and synthetic). We do so using the PyTorch
+Image Models library [28]. We make use of the default training hyperparameters
+
+8
+M. Akrout, B. Gyepesi et al.
+and set the number of training epochs and batch size to 100 and 8, respectively.
+We also use early stopping* by monitoring the validation loss, and opt for the
+stochastic gradient descent (SGD) optimizer. We also use a data split of 80%
+and 20% for training and validation dataset sizes, respectively.
+For every dataset, we calculate the mean and standard deviation for each
+one of the RBG image channels. They are accustomed to preprocessing the input
+images to normalize the images fed to all the networks. It is worth noting that the
+early stopping criterion occurs when we train the models on the fully synthetic
+dataset only. This is as opposed to training on real or hybrid datasets, where
+early stopping does not occur because the validation accuracy stagnates with
+very little increase, and peaks at 89% only. This observation suggests that the
+fully synthetic dataset generated with stable diffusion exhibits non-perceptible
+differentiating features that is allowing for faster training and convergence.
+The Inference Step We evaluate the trained ensemble model by running in-
+ference on our test dataset consisting of 3582 real images. Table 3 shows their
+distribution across the skin disease categories and classes.
+Table 3: The number of test images for the six considered skin diseases
+Category
+Skin disease
+Number of images
+Benign
+Seborrheic keratosis
+1597
+Lentigo
+293
+Pre-malignant
+Actinic keratosis
+282
+Atypical melanocytic nevus
+885
+Malignant
+Basal cell carcinoma
+345
+Melanoma
+180
+We do not carry out any preprocessing to the test images other than the same
+normalization applied to the training images.
+3.4
+Classification results
+We now evaluate three ensemble classifiers where each classifier is separately
+trained on one of the real-small, real, hybrid and synthetic datasets, as described
+in Section 3.1. We run inference on our test dataset and report in Table 4 the
+associated top-k classification accuracy. The latter computes the number of times
+where the correct skin disease is among the top-k predicted diseases (ranked from
+highest to lowest predicted scores).
+* Here, early stopping occurs as soon as the validation accuracy does not improve over
+10 consecutive epochs.
+
+Diffusion-based Image Augmentation for Skin Disease Classification
+9
+Table 4: Top-1 to top-5 skin disease classification accuracy on real-small, real,
+hybrid and fully synthetic datasets.
+Dataset
+# of images
+Accuracy
+Real
+Synthetic
+Top-1
+Top-2
+Top-3
+Top-4
+Top-5
+Real-small
+250
+0
+53.41%
+73.51%
+83.22%
+89.75 %
+95.45%
+Real
+500
+0
+54.05%
+73.95%
+84.84%
+91.49 %
+96.96%
+Hybrid
+250
+250
+54.13%
+73.23%
+85.01%
+92.16%
+96.65%
+Synthetic
+0
+500
+47.29%
+70.71%
+84.09%
+92.16%
+96.85%
+From Table 4, it can be seen that the top-k accuracies of the four classifiers are
+very comparable. More importantly, we observe how the use of synthetic images
+improves the overall accuracy of skin classifiers. Indeed, their performances on
+the real and hybrid datasets have been improved. As ascertained by our clinical
+partners at Semmelweis University, this result confirms that beyond their im-
+pressive visual quality across thousands of images, diffusion models also provide
+significant benefit as synthetic images for real-world medical applications.
+4
+Conclusion
+In this paper, we demonstrate the impressive generative capabilities of proba-
+bilistic diffusion models in generating macroscopic skin disease images. We show
+how it is possible to condition the probabilistic diffusion-based generation on
+text prompt inputs in obtaining fine-grained synthetic images. Furthermore, we
+propose a closed loop data augmentation pipeline to automatically curate the
+generated images while complementing real-world skin disease datasets. Finally,
+our classification task of six skin diseases highlights how synthetic images are
+reliable data sources given that they have been demonstrated beneficial for skin
+disease classification. This result underlines the importance of the recent gen-
+erative modelling success for medical applications as an effective means of data
+sharing without infringing confidentiality issues. Several exciting avenues for
+further investigation remain open such as conditioning the image generation in
+relation to skin tone, with skin tone diversification in datasets being another
+leading limitation, or the use of input images in addition to the text prompt.
+References
+[1] Large-scale Artificial Intelligence Open Network. https://laion.ai, accessed:
+2023-01-11
+[2] Ali, H., Murad, S., Shah, Z.: Spot the fake lungs: Generating synthetic med-
+ical images using neural diffusion models. arXiv preprint arXiv:2211.00902
+(2022)
+
+10
+M. Akrout, B. Gyepesi et al.
+[3] Anand,
+N.,
+Achim,
+T.:
+Protein
+structure
+and
+sequence
+generation
+with equivariant denoising diffusion probabilistic models. arXiv preprint
+arXiv:2205.15019 (2022)
+[4] Austin, J., Johnson, D.D., Ho, J., Tarlow, D., van den Berg, R.: Structured
+denoising diffusion models in discrete state-spaces. Advances in Neural In-
+formation Processing Systems 34, 17981–17993 (2021)
+[5] Chambon, P., Bluethgen, C., Langlotz, C.P., Chaudhari, A.: Adapting pre-
+trained vision-language foundational models to medical imaging domains.
+arXiv preprint arXiv:2210.04133 (2022)
+[6] Chung, H., Lee, E.S., Ye, J.C.: Mr image denoising and super-resolution
+using regularized reverse diffusion. arXiv preprint arXiv:2203.12621 (2022)
+[7] Croitoru, F.A., Hondru, V., Ionescu, R.T., Shah, M.: Diffusion models in
+vision: A survey. arXiv preprint arXiv:2209.04747 (2022)
+[8] Doll´ar, P., Singh, M., Girshick, R.: Fast and accurate model scaling. FAIR
+(2021)
+[9] Guo, X., Yang, Y., Ye, C., Lu, S., Xiang, Y., Ma, T.: Accelerating diffusion
+models via pre-segmentation diffusion sampling for medical image segmen-
+tation. arXiv preprint arXiv:2210.17408 (2022)
+[10] Ho, J., Jain, A., Abbeel, P.: Denoising diffusion probabilistic models. Ad-
+vances in Neural Information Processing Systems 33, 6840–6851 (2020)
+[11] Hu, D., Tao, Y.K., Oguz, I.: Unsupervised denoising of retinal oct with
+diffusion probabilistic model. In: Medical Imaging 2022: Image Processing.
+vol. 12032, pp. 25–34. SPIE (2022)
+[12] Kumar, A.J.S., Chong, R.S., Crowston, J.G., Chua, J., Bujor, I., Husain, R.,
+Vithana, E.N., Girard, M.J., Ting, D.S., Cheng, C.Y., et al.: Evaluation of
+generative adversarial networks for high-resolution synthetic image genera-
+tion of circumpapillary optical coherence tomography images for glaucoma.
+JAMA ophthalmology 140(10), 974–981 (2022)
+[13] La Barbera, G., Boussaid, H., Maso, F., Sarnacki, S., Rouet, L., Gori, P.,
+Bloch, I.: Anatomically constrained ct image translation for heterogeneous
+blood vessel segmentation. arXiv preprint arXiv:2210.01713 (2022)
+[14] Liu, Z., Lin, Y., Cao, Y., Hu, H., Wei, Y., Zhang, Z., Lin, S., Guo, B.:
+Swin transformer: Hierarchical vision transformer using shifted windows.
+Microsoft Research Asia (2021)
+[15] ¨Ozbey, M., Dar, S.U., Bedel, H.A., Dalmaz, O., ¨Ozturk, S¸., G¨ung¨or, A.,
+C¸ukur, T.: Unsupervised medical image translation with adversarial diffu-
+sion models. arXiv preprint arXiv:2207.08208 (2022)
+[16] Packh¨auser, K., Folle, L., Thamm, F., Maier, A.: Generation of anonymous
+chest radiographs using latent diffusion models for training thoracic abnor-
+mality classification systems. arXiv preprint arXiv:2211.01323 (2022)
+[17] Pinaya, W.H., Tudosiu, P.D., Dafflon, J., Da Costa, P.F., Fernandez, V.,
+Nachev, P., Ourselin, S., Cardoso, M.J.: Brain imaging generation with
+latent diffusion models. In: MICCAI Workshop on Deep Generative Models.
+pp. 117–126. Springer (2022)
+[18] Radford, A., Kim, J.W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sas-
+try, G., Askell, A., Mishkin, P., Clark, J., et al.: Learning transferable visual
+
+Diffusion-based Image Augmentation for Skin Disease Classification
+11
+models from natural language supervision. In: International Conference on
+Machine Learning. pp. 8748–8763. PMLR (2021)
+[19] Ramesh, A., Dhariwal, P., Nichol, A., Chu, C., Chen, M.: Hierarchi-
+cal text-conditional image generation with clip latents. arXiv preprint
+arXiv:2204.06125 (2022)
+[20] Rombach, R., Blattman, A., Lorenz, D., Esser, P., Ommer, B.: Stable dif-
+fusion. https://github.com/CompVis/stable-diffusion (2022)
+[21] Rombach, R., Blattmann, A., Lorenz, D., Esser, P., Ommer, B.: High-
+resolution image synthesis with latent diffusion models. In: Proceedings of
+the IEEE/CVF Conference on Computer Vision and Pattern Recognition.
+pp. 10684–10695 (2022)
+[22] Rouzrokh, P., Khosravi, B., Faghani, S., Moassefi, M., Vahdati, S., Erickson,
+B.J.: Multitask brain tumor inpainting with diffusion models: A method-
+ological report. arXiv preprint arXiv:2210.12113 (2022)
+[23] Sagers, L.W., Diao, J.A., Groh, M., Rajpurkar, P., Adamson, A.S., Manrai,
+A.K.: Improving dermatology classifiers across populations using images
+generated by large diffusion models. arXiv preprint arXiv:2211.13352 (2022)
+[24] Sanchez, P., Kascenas, A., Liu, X., O’Neil, A.Q., Tsaftaris, S.A.: What
+is healthy? generative counterfactual diffusion for lesion localization. In:
+MICCAI Workshop on Deep Generative Models. pp. 34–44. Springer (2022)
+[25] Stein, L.: Invoke ai. https://github.com/invoke-ai/InvokeAI (2022)
+[26] Tan, M., V.Le, Q.: Efficientnet: Rethinking model scaling for convolutional
+neural networks (2020)
+[27] Tan, M., V.Le, Q.: Efficientnetv2: Smaller models and faster training (2021)
+[28] Wightmann, R.: Pytorch image models. https://github.com/rwightman/
+pytorch-image-models (2022)
+[29] Wolleb, J., Bieder, F., Sandk¨uhler, R., Cattin, P.C.: Diffusion models for
+medical anomaly detection. arXiv preprint arXiv:2203.04306 (2022)
+[30] Wu, J., Fang, H., Zhang, Y., Yang, Y., Xu, Y.: Medsegdiff: Medical
+image segmentation with diffusion probabilistic model. arXiv preprint
+arXiv:2211.00611 (2022)
+[31] Wyatt, J., Leach, A., Schmon, S.M., Willcocks, C.G.: Anoddpm: Anomaly
+detection with denoising diffusion probabilistic models using simplex noise.
+In: Proceedings of the IEEE/CVF Conference on Computer Vision and
+Pattern Recognition. pp. 650–656 (2022)
+[32] Xia, W., Lyu, Q., Wang, G.: Low-dose ct using denoising diffusion proba-
+bilistic model for 20x speedup. arXiv preprint arXiv:2209.15136 (2022)
+
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new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf,len=477
+page_content='Diffusion-based Data Augmentation for Skin  Disease Classification: Impact Across Original  Medical Datasets to Fully Synthetic Images  Mohamed Akrout1∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' B´alint Gyepesi1∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' P´eter Holl´o2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Adrienn Po´or2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Bl´aga Kincs˝o2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Stephen Solis1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Katrina Cirone1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Jeremy Kawahara1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Dekker Slade1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Latif Abid1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' M´at´e Kov´acs1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' and Istv´an Fazekas1  1 AIP Labs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Hungary  2 Semmelweis University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Faculty of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Department of Dermatology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Venereology and Dermatooncology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Hungary  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Despite continued advancement in recent years, deep neural  networks still rely on large amounts of training data to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  However, labeled training data for real-world applications such as health-  care is limited and difficult to access given longstanding privacy, and  strict data sharing policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' By manipulating image datasets in the pixel  or feature space, existing data augmentation techniques represent one of  the effective ways to improve the quantity and diversity of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Here, we look to advance augmentation techniques by building upon the  emerging success of text-to-image diffusion probabilistic models in aug-  menting the training samples of our macroscopic skin disease dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  We do so by enabling fine-grained control of the image generation pro-  cess via input text prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' We demonstrate that this generative data  augmentation approach successfully maintains a similar classification ac-  curacy of the visual classifier even when trained on a fully synthetic skin  disease dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Similar to recent applications of generative models, our  study suggests that diffusion models are indeed effective in generating  high-quality skin images that do not sacrifice the classifier performance,  and can improve the augmentation of training datasets after curation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Keywords: Data augmentation · Skin condition classification · AI for  dermatology · Diffusion models · Synthetic medical datasets  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 1: Synthetic melanoma images generated by the stable diffusion model after  fine-tuning it with melanoma images using the input text prompt “melanoma”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  equal contribution  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='04802v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='LG]  12 Jan 2023  2  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Akrout, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Gyepesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  1  Introduction  The last months have witnessed the emergence of diffusion probabilistic models  (DPM) [10] as a powerful generator of high-fidelity synthetic datasets, leading  to record-breaking performances in various applications such as image synthesis  [21], natural language processing [4], and computational chemistry [3], to name  a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' When compared to other types of generative models, such as generative  adversarial networks (GANs) and variational autoencoders, DPMs are easier to  train and offer state-of-the-art image generation quality [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Given that synthetic images play a crucial role in privacy-preserving gener-  ation and small dataset augmentation, DPMs attracted significant attention in  the medical imaging field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Table 1 provides an overview of the prior studies of  DPMs, including their medical applications and dataset domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' At first glance,  the reader can identify that the study in [23] is the closest one to this work where  synthetic images were generated from seed images in the Fitzpatrick 17k dataset  using the OpenAI’s DALL·E 2 model [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Table 1: Summary of existing applications of diffusion models in medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Medical applications  Dataset domain  Papers  Image generation  lungs X-Ray,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' MRI  [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 17]  Image segmentation  MRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' ultrasound  [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 30]  Image inpainting  MRI  [22]  Image denoising  MRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' retinal OCT  [6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 32]  Lesion detection  MRI  [24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 31]  Image translation  MRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' CT  [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 15]  Seed-image based augmentation  Dermatology  [23]  Skin disease classification  Dermatology  This work  using large synthetic datasets  Inspired by the recent early success of DPMs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' we propose to use diffusion models  for image augmentation as part of supervised machine learning pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' More  specifically, we study how diffusion models can i) increase the classification met-  rics for skin diseases, and ii) augment skin condition datasets by effectively ma-  nipulating the generated images’ features conditioned on the input text prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  This paper makes the following contributions:  – We study the potential of DPMs for skin disease classifications by fine-tuning  them on six different disease conditions: basal cell carcinoma, melanoma,  actinic keratosis, atypical melanocytic nevus, lentigo, seborrheic keratosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  We do so by learning the embeddings of each disease using text inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Diffusion-based Image Augmentation for Skin Disease Classification  3  – We demonstrate that the classification accuracies of skin disease classifiers  trained on generated synthetic images is similar to training on real images,  where the performance is maintained when using half the number of real  images, and only slightly deteriorates when using a fully synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  This result suggests that the recent success of generative models can help  minimize the barriers of sharing labeled medical datasets, with minimal per-  formance deterioration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  – We illustrate how DPMs are powerful tools to add visual aspects of skin  images guided by domain experts in complementing training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  2  Diffusion-based data augmentation  In this section, we begin by describing the methods used for training the embed-  dings of the aforementioned six skin diseases on our macroscopic skin images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Then, we present the datasets associated with the two DPM training scenarios:  a hybrid dataset compromising 50% synthetic and 50% real images, and a 100%  fully synthetic dataset generated by the trained embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='1  Stable diffusion  The stable diffusion model proposed in [21] is not a monolithic model, but rather  a pipeline of three components, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 2:  1) Text encoding, based on the CLIP model [18], which transforms each token  of the input text prompt into an embedding vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  2) Latent space U-Net generator, which takes all the token embeddings and a  random noise array (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', latent array) and sequentially generates multiple  arrays that better resemble the input text and the visual images on which  the U-Net has been trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  3) Image decoder, based on a variational autoencoder (VAE) to transform the  obtained latent array into the pixel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  In this pipeline, the embedding vectors of the text encoding control both the  generation of the U-Net latent space representations as well as the VAE decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='2  Training dataset for synthetic image generation  The limited number of available labeled images is one of the leading limita-  tions faced by medical classification applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Our internal macroscopic im-  age dataset consists of thousands of skin condition images curated and classified  by dermatologists to cover more than 700 different diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Here, we choose six  widely spread classes across three distinct categories:  – Malignant classes: basal cell carcinoma and melanoma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  – Pre-malignant classes: actinic keratosis and atypical melanocytic nevus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  – Benign classes: lentigo and seborrheic keratosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  4  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Akrout, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Gyepesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 2: The diffusion model pipeline for synthetic skin image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Table 2 provides an overview of the number of images used for each disease in  training the text embedding with the stable diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  In order to train the text embeddings associated to each skin disease, we use the  stable diffusion architecture [20] based on latent diffusion models [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Using a  model of the latter pretrained on multiple LAION datasets [1], we fine-tune each  Table 2: The number of real training images for the considered skin diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Category  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Skin disease  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Data source  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Benign  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Seborrheic keratosis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='2134  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Lentigo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='680  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Pre-malignant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Actinic keratosis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='3298  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Atypical melanocytic nevus  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='623  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Malignant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Basal cell carcinoma  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='7081  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Melanoma  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='3381  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Text Prompt  "' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='a photo of melanoma with irregular edges"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Text  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Random  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Noise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Conditioning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='U-net  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Denoising  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Variational  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Autoencoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Synthetic Skin ImageDiffusion-based Image Augmentation for Skin Disease Classification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='embedding on our real-world image skin condition dataset for two million steps  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='using the default hyperparameters proposed in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' We use PyTorch for both  training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Each embedding is trained on three NVIDIA GeForce  RTX 3090 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='3  Curation of generated images  While most of the generated skin disease images are of high quality, it is not  unusual to obtain generated images of medium or low quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' To isolate high-  quality images from lower qualities, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='3 depicts the full pipeline for augmenting  our skin disease dataset composed of the following four steps:  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 3: Summary of the four steps of the generation pipeline for skin disease data  augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  1) Synthetic data generation: Using the stable diffusion model described in Sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='1, we generate 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='000 images for each one of the considered six skin  diseases to get a synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  2) Non-skin image filtering: We run the obtained synthetic dataset in 1) through  a pretrained binary EfficientNet classifier [26] to filter out any non-skin images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  The binary classifier has been trained on the skin images of the macroscopic  dataset presented in Table 2 and non-skin images from ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' The ac-  cepted images as skin images by the binary classifier represent more than 99%  of the generated images and constitute the filtered synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  3) Skin disease image filtering: We use the filtered synthetic dataset to pre-  dict the skin disease label using a pretrained ensemble model composed of  two CNN models (EfficientNetV2 [27], RegNet [8]) and a visual transformer  (Swin-Transformer [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' This ensemble model has been pretrained on the  macroscopic dataset presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  4) Data augmentation: We use the correctly labeled images by the pretrained  ensemble classifier as the data source for augmenting our initial dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Synthetic  Filtered Synthetic  Dataset  Dataset  Stable  EfficientNet  Diffusion  non-skin filter  Skin Disease Dataset  RegNet  (real, hybrid or fully synthetic)  EfficientNetV2  Swim Transformer  Correctly Labeled  Authentic Dataset  Pretrained Ensemble  Classifier  Data  Augmentation6  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Akrout, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Gyepesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  3  Experiments and Results  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='1  Dataset scenarios for synthetic image generation  Based on the filtered images whose labels were correctly predicted by the pre-  trained ensemble classifier, we build a fully synthetic dataset consisting of 500  images per skin disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' For the real images, we randomly sample 500 images  per class from our macroscopic skin image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' To examine the impact of the  synthetic dataset on classification metrics, we consider the following datasets:  – a small real dataset (real-small) containing 250 real images only,  – a real dataset containing 500 real images only,  – a hybrid dataset consisting of 250 real images and 250 synthetic images,  – a synthetic dataset containing 500 synthetic images only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Note that the four datasets are balanced across skin diseases with varying pro-  portions of real and synthetic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' This allows us to assess the efficiency of  substituting real data with synthetic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='2  Medical synthetic data samples using text prompt inputs  Here, we demonstrate the quality of the synthetic skin disease images stemming  from the generation pipeline in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 3 by providing four synthetic images for  each disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Similar to the synthetic melanoma images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 1, we present  synthetic images of seborrheic keratosis, lentigo, atypical melanocytic nevus,  basal cell carcinoma and actinic keratosis in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 4, 5, 6, 7, and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 4: Synthetic seborrheic keratosis images generated by the stable diffusion  model after fine-tuning it with seborrheic keratosis images using the input text  prompt “seborrheic keratosis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 5: Synthetic lentigo images generated by the stable diffusion model after  fine-tuning it with lentigo images using the input text prompt “lentigo”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Diffusion-based Image Augmentation for Skin Disease Classification  7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 6: Synthetic synthetic atypical melanocytic nevus images generated by the  stable diffusion model after fine-tuning it with atypical melanocytic nevus images  using the input text prompt “atypical melanocytic nevus”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 7: Synthetic basal cell carcinoma images generated by the stable diffusion  model after fine-tuning it with basal cell carcinoma images using the input text  prompt “basal cell carcinoma”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 8: Synthetic actinic keratosis images generated by the stable diffusion model  after fine-tuning it with actinic keratosis images using the input text prompt  “actinic keratosis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  While the impressive generative capabilities of AI models have already been es-  tablished for normal and glaucomatous eyes in [12], our generated macroscopic  images for different skin diseases similarly establishes the effectiveness for der-  matology using larger synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' This is to be opposed to seed-image  based augmentation in [23] where synthetic datasets where not used to fine-tune  the generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='3  Classification of Skin Conditions  In this section, we first describe the training and inference procedures of the skin  disease ensemble classifier on the four datasets described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  The Training Step We start by training three networks of the ensemble clas-  sifier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Swin-Transformer [14], EfficientNetV2 [27], and RegNetZ [8]) on each  one of the datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', real, hybrid, and synthetic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' We do so using the PyTorch  Image Models library [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' We make use of the default training hyperparameters  8  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Akrout, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Gyepesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  and set the number of training epochs and batch size to 100 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  We also use early stopping* by monitoring the validation loss, and opt for the  stochastic gradient descent (SGD) optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' We also use a data split of 80%  and 20% for training and validation dataset sizes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  For every dataset, we calculate the mean and standard deviation for each  one of the RBG image channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' They are accustomed to preprocessing the input  images to normalize the images fed to all the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' It is worth noting that the  early stopping criterion occurs when we train the models on the fully synthetic  dataset only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' This is as opposed to training on real or hybrid datasets, where  early stopping does not occur because the validation accuracy stagnates with  very little increase, and peaks at 89% only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' This observation suggests that the  fully synthetic dataset generated with stable diffusion exhibits non-perceptible  differentiating features that is allowing for faster training and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  The Inference Step We evaluate the trained ensemble model by running in-  ference on our test dataset consisting of 3582 real images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Table 3 shows their  distribution across the skin disease categories and classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Table 3: The number of test images for the six considered skin diseases  Category  Skin disease  Number of images  Benign  Seborrheic keratosis  1597  Lentigo  293  Pre-malignant  Actinic keratosis  282  Atypical melanocytic nevus  885  Malignant  Basal cell carcinoma  345  Melanoma  180  We do not carry out any preprocessing to the test images other than the same  normalization applied to the training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='4  Classification results  We now evaluate three ensemble classifiers where each classifier is separately  trained on one of the real-small, real, hybrid and synthetic datasets, as described  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' We run inference on our test dataset and report in Table 4 the  associated top-k classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' The latter computes the number of times  where the correct skin disease is among the top-k predicted diseases (ranked from  highest to lowest predicted scores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Here, early stopping occurs as soon as the validation accuracy does not improve over  10 consecutive epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Diffusion-based Image Augmentation for Skin Disease Classification  9  Table 4: Top-1 to top-5 skin disease classification accuracy on real-small, real,  hybrid and fully synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Dataset  # of images  Accuracy  Real  Synthetic  Top-1  Top-2  Top-3  Top-4  Top-5  Real-small  250  0  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='41%  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='51%  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='22%  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='75 %  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='45%  Real  500  0  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='05%  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='95%  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='84%  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='49 %  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='96%  Hybrid  250  250  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='13%  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='23%  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='01%  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='16%  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='65%  Synthetic  0  500  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='29%  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='71%  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='09%  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='16%  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='85%  From Table 4, it can be seen that the top-k accuracies of the four classifiers are  very comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' More importantly, we observe how the use of synthetic images  improves the overall accuracy of skin classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Indeed, their performances on  the real and hybrid datasets have been improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' As ascertained by our clinical  partners at Semmelweis University, this result confirms that beyond their im-  pressive visual quality across thousands of images, diffusion models also provide  significant benefit as synthetic images for real-world medical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  4  Conclusion  In this paper, we demonstrate the impressive generative capabilities of proba-  bilistic diffusion models in generating macroscopic skin disease images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' We show  how it is possible to condition the probabilistic diffusion-based generation on  text prompt inputs in obtaining fine-grained synthetic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Furthermore, we  propose a closed loop data augmentation pipeline to automatically curate the  generated images while complementing real-world skin disease datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Finally,  our classification task of six skin diseases highlights how synthetic images are  reliable data sources given that they have been demonstrated beneficial for skin  disease classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' This result underlines the importance of the recent gen-  erative modelling success for medical applications as an effective means of data  sharing without infringing confidentiality issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Several exciting avenues for  further investigation remain open such as conditioning the image generation in  relation to skin tone, with skin tone diversification in datasets being another  leading limitation, or the use of input images in addition to the text prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  References  [1] Large-scale Artificial Intelligence Open Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' https://laion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='ai, accessed:  2023-01-11  [2] Ali, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Murad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Shah, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Spot the fake lungs: Generating synthetic med-  ical images using neural diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='00902  (2022)  10  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Akrout, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Gyepesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  [3] Anand,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=',  Achim,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=':  Protein  structure  and  sequence  generation  with equivariant denoising diffusion probabilistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' arXiv preprint  arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='15019 (2022)  [4] Austin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Johnson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Ho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Tarlow, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', van den Berg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Structured  denoising diffusion models in discrete state-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Advances in Neural In-  formation Processing Systems 34, 17981–17993 (2021)  [5] Chambon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Bluethgen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Langlotz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Chaudhari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Adapting pre-  trained vision-language foundational models to medical imaging domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='04133 (2022)  [6] Chung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Lee, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Ye, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Mr image denoising and super-resolution  using regularized reverse diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='12621 (2022)  [7] Croitoru, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Hondru, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Ionescu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Shah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Diffusion models in  vision: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='04747 (2022)  [8] Doll´ar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Singh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Girshick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Fast and accurate model scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' FAIR  (2021)  [9] Guo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Ye, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Lu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Xiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Ma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Accelerating diffusion  models via pre-segmentation diffusion sampling for medical image segmen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='17408 (2022)  [10] Ho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Jain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Abbeel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Denoising diffusion probabilistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Ad-  vances in Neural Information Processing Systems 33, 6840–6851 (2020)  [11] Hu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Tao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Oguz, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Unsupervised denoising of retinal oct with  diffusion probabilistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' In: Medical Imaging 2022: Image Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 12032, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 25–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' SPIE (2022)  [12] Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Chong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Crowston, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Chua, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Bujor, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Husain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=',  Vithana, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Girard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Ting, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Cheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' : Evaluation of  generative adversarial networks for high-resolution synthetic image genera-  tion of circumpapillary optical coherence tomography images for glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  JAMA ophthalmology 140(10), 974–981 (2022)  [13] La Barbera, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Boussaid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Maso, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Sarnacki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Rouet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Gori, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=',  Bloch, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Anatomically constrained ct image translation for heterogeneous  blood vessel segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='01713 (2022)  [14] Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Wei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Guo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=':  Swin transformer: Hierarchical vision transformer using shifted windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  Microsoft Research Asia (2021)  [15] ¨Ozbey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Dar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Bedel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Dalmaz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', ¨Ozturk, S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', G¨ung¨or, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=',  C¸ukur, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Unsupervised medical image translation with adversarial diffu-  sion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='08208 (2022)  [16] Packh¨auser, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Folle, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Thamm, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Maier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Generation of anonymous  chest radiographs using latent diffusion models for training thoracic abnor-  mality classification systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='01323 (2022)  [17] Pinaya, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Tudosiu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Dafflon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Da Costa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Fernandez, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=',  Nachev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Ourselin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Cardoso, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Brain imaging generation with  latent diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' In: MICCAI Workshop on Deep Generative Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 117–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Springer (2022)  [18] Radford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Hallacy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Ramesh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Goh, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Agarwal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Sas-  try, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Askell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Mishkin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Clark, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' : Learning transferable visual  Diffusion-based Image Augmentation for Skin Disease Classification  11  models from natural language supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' In: International Conference on  Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 8748–8763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' PMLR (2021)  [19] Ramesh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Dhariwal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Nichol, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Chu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Hierarchi-  cal text-conditional image generation with clip latents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' arXiv preprint  arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='06125 (2022)  [20] Rombach, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
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+page_content=', Lorenz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
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+page_content=', Ommer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Stable dif-  fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
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+page_content=' 10684–10695 (2022)  [22] Rouzrokh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Khosravi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Faghani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Moassefi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Vahdati, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Erickson,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
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+page_content=' : Multitask brain tumor inpainting with diffusion models: A method-  ological report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
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+page_content='12113 (2022)  [23] Sagers, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Diao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Groh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Rajpurkar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Adamson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Manrai,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' : Improving dermatology classifiers across populations using images  generated by large diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
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+page_content='13352 (2022)  [24] Sanchez, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Kascenas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', O’Neil, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Tsaftaris, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': What  is healthy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' generative counterfactual diffusion for lesion localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' In:  MICCAI Workshop on Deep Generative Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 34–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' Springer (2022)  [25] Stein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Invoke ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='com/invoke-ai/InvokeAI (2022)  [26] Tan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Le, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Efficientnet: Rethinking model scaling for convolutional  neural networks (2020)  [27] Tan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='Le, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Efficientnetv2: Smaller models and faster training (2021)  [28] Wightmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Pytorch image models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='com/rwightman/  pytorch-image-models (2022)  [29] Wolleb, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Bieder, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Sandk¨uhler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Cattin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' : Diffusion models for  medical anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='04306 (2022)  [30] Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Fang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Medsegdiff: Medical  image segmentation with diffusion probabilistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' arXiv preprint  arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='00611 (2022)  [31] Wyatt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Leach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Schmon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Willcocks, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' : Anoddpm: Anomaly  detection with denoising diffusion probabilistic models using simplex noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content='  In: Proceedings of the IEEE/CVF Conference on Computer Vision and  Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' 650–656 (2022)  [32] Xia, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Lyu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=', Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=': Low-dose ct using denoising diffusion proba-  bilistic model for 20x speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
+page_content=' arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQf7wvD/content/2301.04802v1.pdf'}
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@@ -0,0 +1,1388 @@
+Cavity Quantum Electrodynamics with Hyperbolic van der Waals Materials
+Yuto Ashida,1, 2, ∗ Atac¸ ˙Imamo˘glu,3 and Eugene Demler4
+1Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
+2Institute for Physics of Intelligence, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan
+3Institute of Quantum Electronics, ETH Zurich, CH-8093 Z¨urich, Switzerland
+4Institute for Theoretical Physics, ETH Zurich, 8093 Z¨urich, Switzerland
+The ground-state properties and excitation energies of a quantum emitter can be modified in the ultrastrong
+coupling regime of cavity quantum electrodynamics (QED) where the light-matter interaction strength becomes
+comparable to the cavity resonance frequency. Recent studies have started to explore the possibility to control
+an electronic material by embedding it in a cavity that confines electromagnetic fields in deep subwavelength
+scales. Currently, there is a strong motivation to realize ultrastrong-coupling cavity QED in the terahertz (THz)
+range, since most of the elementary excitations of quantum materials are in this frequency window. We propose
+and analyze an ideal platform to achieve this aim where a two-dimensional electronic material is encapsulated
+by a planar cavity consisting of ultrathin polar van der Waals crystals. As a concrete setup, we show that
+nanometer-thick hexagonal boron nitride layers allow for reaching the ultrastrong coupling regime for single-
+electron cyclotron resonance in a bilayer graphene. The proposed cavity setting can be realized by a wide
+variety of thin dielectric materials with hyperbolic dispersions. Consequently, van der Waals heterostructures
+could provide an ideal playground for exploring the ultrastrong-coupling physics of cavity QED materials.
+Strong coupling regime of cavity quantum electrodynam-
+ics (QED), where the emitter-cavity coupling strength exceeds
+decay rates, has played a central role in quantum informa-
+tion science. For instance, cavity-mediated coherent interac-
+tion between two distant qubits has allowed for implementing
+two-qubit gates with high fidelity [1, 2]. Moreover, cavity-
+mediated Raman transitions have the potential to realize all-
+to-all coupling with a tunable range and strength, providing an
+indispensable tool for quantum simulators [3, 4]. Recent stud-
+ies have started to explore the possibility of further increasing
+the coupling strength so that it becomes comparable to the
+elementary excitation energies. Many of the common simpli-
+fications in cavity QED fail in this regime, rendering the theo-
+retical analysis challenging [5–8]. A remarkable feature here
+is that ultrastrong light-matter coupling can alter the ground-
+state electronic properties due to virtual processes where both
+emitters and cavity photons are excited [9–11]. Consequently,
+a natural question to address is if and when the ultrastrong
+coupling regime can be attained and used to control material
+properties simply by cavity confinement in the absence of an
+external driving.
+A quick calculation for cavities supporting purely photonic
+excitations suggests that the ultrastrong coupling regime is out
+of reach due to the smallness of the fine structure constant
+[12]. However, this limit can be overcome in structured elec-
+tromagnetic environments because hybridization with matter
+excitations enables one to control cavity frequencies indepen-
+dently of wavelengths. For instance, in superconducting mi-
+crowave circuits, a large kinetic inductance allows for high-
+impedance electromagnetic excitations, leading to the single-
+quantum ultrastrong coupling in the microwave range [13–
+16].
+Meanwhile, many of the elementary excitations in quan-
+tum materials are in the terahertz (THz) regime. Recent ex-
+perimental and theoretical studies have shown the potential of
+utilizing the cavity confinement as a means to modify such ex-
+citations [17–27], which shows great promise for controlling
+quantum many body states and endowing them with new func-
+tionalities, including superconductivity [28–31], ferroelectric-
+ity [32–34], magnetotransport [35–37] and topological prop-
+erties [38, 39]. In view of the prospects of observing these in-
+triguing phenomena in cavity QED materials, there is a strong
+motivation to examine the feasibility of attaining the single-
+quantum ultrastrong coupling at THz frequencies.
+The aim of this Letter is to propose and analyze a planar
+cavity setup consisting of polar van der Waals (vdW) crys-
+tals as an ideal platform to explore the ultrastrong-coupling
+physics of cavity QED materials (Fig. 1a). We point out that
+one can attain a single-quantum ultrastrong coupling by em-
+bedding a 2D material with electronic transitions in the THz
+regime into ultrathin hexagonal boron nitride (h-BN) layers.
+A special feature here is that electrons in the 2D material
+couple to the electric field component of tightly confined hy-
+perbolic phonon polaritons. The strong photon-phonon hy-
+d
+L
+ω
+Ω
+L
+x
+z
+ε
+ε
+εt
+z
+z
+Ω t
+(a)
+(b)
+0
+FIG. 1.
+Schematic figure illustrating the proposed planar cavity
+setup consisting of thin polar van der Waals crystals (green shaded)
+whose optical axis is along z direction. In the narrow air gap be-
+tween the two slabs, a 2D material (red shaded) is inserted and the
+electron there can ultrastrongly couple to electromagnetic fields of
+tightly confined hyperbolic polaritons (blue solid curve). The thick-
+ness (lateral extension) of surrounding materials is d (L). (b) Out-
+of-plane (solid curve) and in-plane (dashed curve) permittivities ϵz,t
+of hyperbolic materials. The Restrahlen bands (red shaded) appear
+above each of the out-of-plane and in-plane phonon resonances Ωz,t.
+arXiv:2301.03712v1  [cond-mat.mes-hall]  9 Jan 2023
+
+2
+bridization together with the low frequencies of polaritons
+then results in the significantly enhanced coupling strength
+over a broad range of momenta. This should be contrasted
+to conventional polar dielectrics with an isotropic dispersion,
+where sizable light-matter hybridization takes place in a lim-
+ited range of momenta since light and matter are almost de-
+coupled at large momenta due to the fast speed of light.
+As a proof-of-concept demonstration, we consider the case
+of a 2D electron with parabolic dispersion under the static
+magnetic field. We analyze the hybridization between hyper-
+bolic phonon polaritons in h-BN and the cyclotron motion of
+the parabolic electrons. We show that the single-electron ul-
+trastrong coupling regime is within the reach provided that the
+thicknesses of h-BN layers are chosen to be nanometer-scale.
+This consideration is motivated by recent advances demon-
+strating ultrasmall mode volumes of hyperbolic phonon po-
+laritons in h-BN nanostructures [40–45]. While we present
+quantitative estimates for h-BN nanocavities for the sake of
+concreteness, the proposed cavity scheme is applicable to the
+majority of polar vdW crystals [46], which exhibit hyper-
+bolic polaritons originating from distinct in- and out-of-plane
+infrared-active phonons.
+Hyperbolic phonon polaritons.— We begin our analysis by
+reviewing the properties of a planar cavity made out of layered
+thin polar vdW materials. Due to the weakness of interlayer
+coupling, such materials naturally possess two types of opti-
+cal phonons corresponding to in-plane and out-of-plane ionic
+oscillations in the THz or mid-infrared regimes. This leads to
+the uniaxial anisotropy characterized by the out-of-plane (in-
+plane) dielectric permittivities ϵz (ϵt). In the frequency win-
+dows above each of the phonon resonances, the two dielectric
+responses can have opposite signs (Fig. 1b). This unique fea-
+ture leads to the hyperbolic isofrequency surfaces defined by
+the dispersion relation of the transverse magnetic modes,
+|q|2
+ϵz
++ |κ|2
+ϵt
+= ω2
+ϵ0c2 ,
+(1)
+where q (κ) is the in-plane (out-of-plane) wavevector, ϵ0 is
+the vacuum permittivity, and c is the speed of light. The op-
+posite signs of ϵz,t allow for excitations to have low frequen-
+cies even at large momenta. The resulting hybridized elemen-
+tary excitations of photons and optical phonons are known
+as hyperbolic phonon polaritons. The corresponding disper-
+sions in this range of frequencies are called the Reststrahlen
+bands of either Type I when ϵz < 0, ϵt > 0 or Type II when
+ϵz > 0, ϵt < 0.
+As a representative hyperbolic material, h-BN has a strong
+crystalline anisotropy leading to two spectrally distinct Rest-
+strahlen bands with ultralow loss [40–45]. Below we focus on
+the electromagnetic couplings with Type I h-BN hyperbolic
+modes ωqn lying in the narrow frequency window above the
+out-of-plane phonon frequency Ωz = 41.1 THz (cf. top panel
+in Fig. 2). As we discuss below, these modes exhibit several
+advantages for the purpose of attaining the ultrastrong cou-
+plings thanks to their relatively low frequencies and sizable
+photon components over a broad range of momenta q.
+Single-electron
+ultrastrong
+coupling
+with
+hyperbolic
+materials.— We consider the planar cavity setup where a 2D
+material (e.g., a bilayer graphene) is inserted into the narrow
+air gap between two thin h-BN slabs whose thickness and lat-
+eral extensions are denoted by d and L, respectively (Fig. 1a).
+Our starting point is the following cavity QED Hamiltonian of
+the 2D parabolic electron interacting with hyperbolic phonon
+polaritons:
+ˆH =
+�
+ˆp + eAs(ˆr) + e ˆA(ˆr)
+�2
+2m
++ ˆHpol,
+(2)
+which can be derived from an effective theory of uniaxial po-
+lar dielectrics coupled to quantized electromagnetic fields in
+the Coulomb gauge [47, 48]. Here, e is the elementary charge,
+m is the electron mass, ˆr and ˆp are the electron position and
+momentum operators in the 2D lateral directions, respectively,
+As represents an arbitrary static field, and ˆHpol is the free po-
+lariton Hamiltonian,
+ˆHpol =
+�
+qn
+ℏωqnˆγ†
+qnˆγqn,
+(3)
+where ˆγqn (ˆγ†
+qn) annihilates (creates) a hyperbolic polariton
+with the in-plane wavevector q in the branch n ∈ N. The 2D
+vector field ˆA(r) is obtained by projecting the vector potential
+onto the 2D plane where the electron is placed, and can be
+n=1
+qd
+qn
+z
+ω
+/Ω
+0
+0
+0.05
+0.1
+0.15
+5
+10
+15
+20
+n=2
+n=3
+1
+1.02
+1.04
+1.06
+1.08
+n=1
+n=2
+n=3
+√d/dqn
+z
+t
+E
+n=1
+n=2
+n=3
+q d
+＊
+FIG. 2.
+(Top) Dispersions ωqn of the hyperbolic phonon polari-
+tons in the planar cavity setting. The inset shows the spatial pro-
+files of the in-plane electric fields along z direction for each mode at
+qd = 4. For the sake of visibility, only the three lowest modes are
+plotted. (Bottom) Inverse of the square root of the effective dimen-
+sionless confinement length,
+�
+d/dqn, characterizing the dimension-
+less single-electron coupling strength for each mode. The maximum
+value in the principal branch n = 1 is reached at q = q∗.
+
+3
+expanded in terms of the polariton operators by
+ˆA(ˆr) =
+�
+qn
+Aqneq
+�
+ˆγqneiq·ˆr + ˆγ†
+qne−iq·ˆr�
+,
+(4)
+where Aqn ≃
+�
+ℏ/(2L2ϵ0ωqndqn) is the amplitude with the
+effective confinement length dqn whose value is characterized
+by the polariton mode function. The effective polarization
+vector becomes eq ≡ q/|q| because, for the symmetric ar-
+rangement we assumed, the electric fields of the polaritons
+only have in-plane components along the propagation direc-
+tion in the 2D plane where the electronic material is located.
+In general, while the vector potential is originally a 3D trans-
+verse vector field in the Coulomb gauge, it can effectively ac-
+quire longitudinal components when projected onto the 2D
+tangential plane.
+Figure 2 shows the results for Type I h-BN hyperbolic
+modes. The dispersions (top panel) start from the longitudinal
+phonon frequency and saturate to Ωz at high q ≡ |q|. Natu-
+rally, these hybridized modes are almost purely longitudinal
+(transverse) phonon excitations in the limit q → 0 (q → ∞),
+which do not allow for a strong coupling to electrons in the air
+gap. Crucially, however, except for these two limits, there still
+exist the nonvanishing photon contributions leading to the siz-
+able electric-dipole couplings at intermediate momenta (bot-
+tom panel). This is because the in-plane component, which
+couples to the 2D electron, has almost equal photonic and
+phononic contents while the out-of-plane component is pre-
+dominantly phonon-like [48].
+To further proceed, we use the unitary transformation ˆU =
+e−iˆr· ˆP b/ℏ with ˆP b = �
+qn ℏqˆγ†
+qnˆγqn [49], resulting in the
+Hamiltonian ˆHU ≡ ˆU † ˆH ˆU given by
+ˆHU =
+�
+ˆp+eAs(ˆr)+e ˆA(0)− ˆP b
+�2
+2m
++ ˆHpol.
+(5)
+In this way, the electron-coordinate dependence in the quan-
+tized vector potential can be eliminated at the expense of gen-
+erating the polariton momentum ˆP b, leading to the nonlocal
+interaction among polaritons mediated by the electron. The
+dimensionful coupling strength between the electron and each
+of the dynamical quantized electromagnetic modes is given
+by
+gqn = eAqn
+�ωqn
+mℏ .
+(6)
+Since gqn characterizes the single-electron coupling strength
+rather than the collective one, it depends on the lateral size L
+through gqn ∝ L−1. Consequently, a natural measure for the
+effective coupling strength between the 2D electron and the
+continuum of polariton modes is given by geff ≡
+��
+qn g2qn,
+which scales as geff ∝ O(L0) [50].
+Previously, the deep strong coupling regime has been ex-
+perimentally realized in superconducting circuits, where geff
+becomes comparable to the microwave photon frequency.
+In the present setting, the use of ultrathin h-BN slabs with
+nanometer-scale thicknesses enables one to reach the deep or
+extremely strong coupling regimes, where geff becomes com-
+parable to or even exceeds a THz cavity frequency. For in-
+stance, using the h-BN parameters [40], we estimate coupling
+strengths of order gqn/Ωz = 1.7 ×
+�
+(10 nm)3/(L2dqn),
+which, together with the results of the effective confinement
+length dqn in Fig. 2b, can lead to geff ∼ Ωz in nanoscale het-
+erostructures. More specifically, one can attain geff ≃ 2.0 Ωz
+for the n = 1 principal branch when the cavity thickness (in-
+plane momentum cutoff) is set to be d = 5 nm (Λ = 2 nm−1).
+As demonstrated below, one important consequence of attain-
+ing geff ∼ Ωz is that the electron mixes the otherwise indepen-
+dent cavity modes and creates a localized state at a frequency
+below the cavity resonance. We emphasize that this key fea-
+ture is largely insensitive to a choice of the lateral size L and
+thus remains even in the 2D thermodynamic limit L/d → ∞.
+Application to a 2D electron under the magnetic field.— As
+a proof-of-concept demonstration, we now focus on a proto-
+typical setting of ultrastrong-coupling physics, namely, a 2D
+parabolic electron subject to a static perpendicular magnetic
+field. From now on, we consider the n = 1 principal hyper-
+bolic mode that most strongly couples to the cyclotron motion
+of the electron, and abbreviate the subscript n for the sake of
+notational simplicity. To simplify the Hamiltonian, we choose
+the symmetric gauge, As(r) = (−By/2, Bx/2)T, with the
+magnetic field B and introduce the annihilation operator of
+Landau levels by
+ˆa =
+1
+√
+2
+�lB
+ℏ (ˆpx − iˆpy) −
+i
+2lB
+(ˆx − iˆy)
+�
+,
+(7)
+where lB =
+�
+ℏ/(eB) is the magnetic length. Using the
+cyclotron frequency ωc = eB/m and the operator ˆπ =
+1
+√
+2
+�
+ˆa + ˆa†, i(ˆa − ˆa†)
+�T, we obtain
+ˆHU = ℏωc
+2
+�
+ˆπ+
+�
+q
+eq
+�
+cq
+�
+ˆγq+ˆγ†
+q
+�
+−qlBˆγ†
+qˆγq
+�
+�2
++ ˆHpol,
+(8)
+where cq = gq/√ωqωc is the dimensionless coefficient char-
+acterizing the coupling strength of the cyclotron motion to
+the hyperbolic polaritons with momentum q.
+In the long-
+wavelength limit qlB → 0, the polariton-polariton interac-
+tion in Eq. (8) disappears and the problem reduces to the
+quadratic one. As demonstrated below, however, this interac-
+tion term can in general contribute to the dynamics and affect
+the absorption spectrum especially in the ultrastrong coupling
+regimes. This is because the electron couples most strongly
+with the electromagnetic modes at a finite momentum around
+q ∼ q∗ (cf. Fig. 2b) whose corresponding length scale 1/q∗ is
+comparable to the magnetic length lB near the cyclotron reso-
+nance. More generally, such spatial dependence of the vector
+potential must be taken into account when 1/q∗ is comparable
+to the characteristic length scale of the electron motion.
+The low-energy excitations can be studied by analyzing the
+
+4
+c
+ω /Ω
+0
+0
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+c
+ω /Ω
+0
+1
+2
+3
+4
+5
+c
+ω /Ω
+0
+1
+2
+3
+4
+5
+ω/Ω
+d=3nm
+d=6nm
+d=10nm
+z
+z
+z
+z
+FIG. 3.
+Magnetoabsorption spectra A(ω) of the 2D electron in the cavity plotted against a cyclotron resonance ωc = eB/m at different
+cavity thicknesses d. The blue solid curve in each panel corresponds to the bound-state energy of the Landau polariton. We impose the periodic
+boundary conditions on the lateral directions and fix the aspect ratio L/d = π. We use the h-BN parameters in Ref. [40] and set the in-plane
+momentum cutoff Λ = 2 nm−1.
+magnetoabsorption spectrum
+A(ω) = Re
+�� ∞
+0
+eiωt⟨GS|ˆae−i ˆ
+Htˆa†|GS⟩
+�
+,
+(9)
+where |GS⟩ is the ground state. To reveal its qualitative fea-
+tures, we perform a simple variational analysis as follows.
+Specifically, we first determine the variational ground state of
+Eq. (8) in the form of a product of coherent states. We then ex-
+pand the Hamiltonian around this state, obtain the fluctuations
+up to the quadratic terms, and determine the excitation spec-
+trum via the exact diagonalization of the effective quadratic
+Hamiltonian [48].
+Figure 3 shows the obtained magnetoabsorption spectra,
+where the cavity thickness d is varied while the aspect ratio
+L/d is kept constant. The blue solid curve in each panel shows
+the bound-state energy of the Landau polariton. As the thick-
+ness is decreased, the spectrum starts to exhibit the anticross-
+ings around the cyclotron resonance. In particular, when the
+cavity length becomes a few nanometers, the large separations
+between the branches that are comparable to the elementary
+excitation energies themselves emerge; this is a hallmark of
+the ultrastrong coupling regime.
+As discussed before, a key feature here is the formation of
+the dressed bound state consisting of the electron and the lo-
+calized phonon polaritons, which manifests itself as the anti-
+crossed lower branch in the spectrum (cf. the blue curve in
+Fig. 3). Importantly, this feature remains independently of the
+lateral size L, while the effect of increasing L/d is the appear-
+ance of a continuum of cavity modes above the lower branch
+[48]. It is worthwhile to note that the positions of the anti-
+crossings are shifted above the bare resonances ωc/Ωz ≃ 1.
+This upward shift originates from the renormalization of the
+effective polariton energies due to the repulsive polariton-
+polariton interaction. Since the latter comes from the spatial
+dependence of the vector potential, this effect is absent in the
+long-wavelength limit. We also remark that the appearance
+of the multiple anticrossed branches in Fig. 3 are due to the
+discretized in-plane momentum q, whose value is set by the
+periodic boundary conditions in the lateral directions.
+Discussions.— The proposed platform for ultrastrong-
+coupling cavity QED materials can be realized by various po-
+lar vdW materials exhibiting hyperbolic phonon polaritons,
+including Bi2Se3, Bi2Te3, MoS2 and MoO3 as well as h-
+BN [46, 51–55]. Thus, by confining materials in the cavity
+consisting of these vdW structures, one can strongly couple
+electronic excitations to quantized electromagnetic modes in a
+wide spectral range from mid- or far-infrared to THz regimes.
+Moreover, the layered nature of vdW crystals should readily
+allow one to tune the cavity coupling strengths by controlling
+the thickness of the surrounding crystals.
+While polaritons in these materials are known to exhibit ul-
+tralow loss [40–45, 53–55], it merits further study to examine
+a role of dissipation in the present cavity setup. In particular, it
+is intriguing to explore if and when a confinement of quantum
+materials in the cavity with dissipative hyperbolic polaritons
+could lead to phase transitions or dynamics unique to open
+systems [56, 57]. We also note that in general the contribu-
+tion from the instantaneous Coulomb energy could be impor-
+tant when the 2D material is placed near the surfaces. In the
+present planar setup, this contribution might lead to sizable ef-
+fects in many-body regimes due to dielectric screening of the
+electron-electron Coulomb interaction [58].
+In summary, we showed that the planar cavity consisting of
+vdW heterostructures provides a promising platform to attain
+the single-electron ultrastrong coupling in the THz or mid-
+infrared regions. As a proof-of-concept demonstration, we
+presented the analysis of the magnetoabsorption spectrum for
+the 2D electron confined in the h-BN cavity, where the cou-
+pling can be ultrastrong provided that the thicknesses are ju-
+diciously controlled. We expect that our results open a way
+to study wealth of recent predictions in the emerging field of
+cavity QED materials. We hope that our work simulates fur-
+ther studies in these directions.
+We are grateful to Iliya Esin, Ilya Esterlis, Tim Kaxiras,
+Igor Khanonkin, Frank Koppens, Kanta Masuki, Gil Refael,
+Tao Shi, and Kenji Yasuda for fruitful discussions. Y.A. ac-
+knowledges support from the Japan Society for the Promotion
+of Science through Grant No. JP19K23424. A.I. was sup-
+
+5
+ported by the Swiss National Science Foundation (SNSF) un-
+der Grant Number 200020 207520. E.D. acknowledges sup-
+port from the ARO grant “Control of Many-Body States Us-
+ing Strong Coherent Light-Matter Coupling in Terahertz Cav-
+ities” and the Swiss National Science Foundation under Divi-
+sion II.
+∗ ashida@phys.s.u-tokyo.ac.jp
+[1] L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R.
+Johnson, D. Schuster, J. Majer, A. Blais, L. Frunzio, S. Girvin,
+and R. J. Schoelkopf, Nature 460, 240 (2009).
+[2] A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Rev.
+Mod. Phys. 93, 025005 (2021).
+[3] A. Stute, B. Casabone, P. Schindler, T. Monz, P. Schmidt,
+B. Brandst¨atter, T. Northup,
+and R. Blatt, Nature 485, 482
+(2012).
+[4] V. D. Vaidya, Y. Guo, R. M. Kroeze, K. E. Ballantine, A. J.
+Koll´ar, J. Keeling,
+and B. L. Lev, Phys. Rev. X 8, 011002
+(2018).
+[5] D. Hagenm¨uller, S. De Liberato, and C. Ciuti, Phys. Rev. B 81,
+235303 (2010).
+[6] D. De Bernardis, P. Pilar, T. Jaako, S. De Liberato, and P. Rabl,
+Phys. Rev. A 98, 053819 (2018).
+[7] A. Le Boite, Adv. Quant. Tech. 3, 1900140 (2020).
+[8] Y. Ashida, A. ˙Imamo˘glu, and E. Demler, Phys. Rev. Lett. 126,
+153603 (2021).
+[9] F. J. Garcia-Vidal, C. Ciuti, and T. W. Ebbesen, Science 373,
+eabd0336 (2021).
+[10] F. Schlawin, D. M. Kennes, and M. A. Sentef, Appl. Phys. Rev.
+9, 011312 (2022).
+[11] J. Bloch, A. Cavalleri, V. Galitski, M. Hafezi, and A. Rubio,
+Nature 606, 41 (2022).
+[12] M. Devoret, S. Girvin, and R. Schoelkopf, Ann. Phys. 519, 767
+(2007).
+[13] P. Forn-D´ıaz, J. J. Garc´ıa-Ripoll, B. Peropadre, J.-L. Orgiazzi,
+M. Yurtalan, R. Belyansky, C. M. Wilson, and A. Lupascu, Nat.
+Phys. 13, 39 (2017).
+[14] F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and
+K. Semba, Nat. Phys. 13, 44 (2017).
+[15] P. Forn-D´ıaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Rev.
+Mod. Phys. 91, 025005 (2019).
+[16] A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta,
+and F. Nori, Nat. Rev. Phys. 1, 19 (2019).
+[17] G. Scalari, C. Maissen, S. Cibella, R. Leoni, P. Carelli, F. Val-
+morra, M. Beck, and J. Faist, New J. Phys. 16, 033005 (2014).
+[18] S. Smolka, W. Wuester, F. Haupt, S. Faelt, W. Wegscheider,
+and A. Imamoglu, Science 346, 332 (2014).
+[19] C. Maissen, G. Scalari, F. Valmorra, M. Beck, J. Faist,
+S. Cibella,
+R. Leoni,
+C. Reichl,
+C. Charpentier,
+and
+W. Wegscheider, Phys. Rev. B 90, 205309 (2014).
+[20] Q. Zhang, M. Lou, X. Li, J. L. Reno, W. Pan, J. D. Watson,
+M. J. Manfra, and J. Kono, Nature Phys 12, 1005 (2016).
+[21] A. Bayer, M. Pozimski, S. Schambeck, D. Schuh, R. Huber,
+D. Bougeard, and C. Lange, Nano Lett. 17, 6340 (2017).
+[22] J. Keller, G. Scalari, F. Appugliese, S. Rajabali, M. Beck,
+J. Haase, C. A. Lehner, W. Wegscheider, M. Failla, M. My-
+ronov, D. R. Leadley, J. Lloyd-Hughes, P. Nataf, and J. Faist,
+Phys. Rev. B 101, 075301 (2020).
+[23] A. Thomas, E. Devaux, K. Nagarajan, G. Rogez, M. Seidel,
+F. Richard, C. Genet, M. Drillon, and T. W. Ebbesen, Nano
+Lett. 21, 4365 (2021).
+[24] J. Flick, M. Ruggenthaler, H. Appel, and A. Rubio, Proc. Natl.
+Acad. Sci. U.S.A. 114, 3026 (2017).
+[25] V. Rokaj, M. Penz, M. A. Sentef, M. Ruggenthaler, and A. Ru-
+bio, Phys. Rev. Lett. 123, 047202 (2019).
+[26] D. M. Juraschek, T. Neuman, J. Flick, and P. Narang, Phys.
+Rev. Research 3, L032046 (2021).
+[27] J. Rom´an-Roche and D. Zueco, SciPost Phys. Lect. Notes , 50
+(2022).
+[28] F. Schlawin, A. Cavalleri, and D. Jaksch, Phys. Rev. Lett. 122,
+133602 (2019).
+[29] J. B. Curtis, Z. M. Raines, A. A. Allocca, M. Hafezi, and V. M.
+Galitski, Phys. Rev. Lett. 122, 167002 (2019).
+[30] M. A. Sentef, M. Ruggenthaler, and A. Rubio, Sci. Adv. 4,
+eaau6969 (2018).
+[31] H. Gao, F. Schlawin, and D. Jaksch, Phys. Rev. B 104, L140503
+(2021).
+[32] Y. Ashida, A. ˙Imamo˘glu, J. Faist, D. Jaksch, A. Cavalleri, and
+E. Demler, Phys. Rev. X 10, 041027 (2020).
+[33] S. Latini, D. Shin, S. A. Sato, C. Sch¨afer, U. De Giovannini,
+H. H¨ubener, and A. Rubio, Proc. Natl. Acad. Sci. U.S.A. 118,
+e2105618118 (2021).
+[34] K. Lenk, J. Li, P. Werner, and M. Eckstein, arXiv:2205.05559
+(2022).
+[35] G. L. Paravicini-Bagliani, F. Appugliese, E. Richter, F. Val-
+morra, J. Keller, M. Beck, N. Bartolo, C. R¨ossler, T. Ihn, K. En-
+sslin, C. Ciuti, G. Scalari,
+and J. Faist, Nat. Phys. 15, 186
+(2019).
+[36] F. Appugliese, J. Enkner, G. L. Paravicini-Bagliani, M. Beck,
+C. Reichl, W. Wegscheider, G. Scalari, C. Ciuti, and J. Faist,
+Science 375, 1030 (2022).
+[37] V. Rokaj, M. Penz, M. A. Sentef, M. Ruggenthaler, and A. Ru-
+bio, Phys. Rev. B 105, 205424 (2022).
+[38] X. Wang, E. Ronca, and M. A. Sentef, Phys. Rev. B 99, 235156
+(2019).
+[39] K. Masuki and Y. Ashida, arXiv:2209.01363 (2022).
+[40] J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M.
+Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods,
+A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier,
+and K. S. Novoselov, Nat. Commun. 5, 1 (2014).
+[41] S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod,
+M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi,
+M. Thiemens, G. Dominguez, A. H. C. Neto, A. Zettl, F. Keil-
+mann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, Sci-
+ence 343, 1125 (2014).
+[42] A. J. Giles, S. Dai, I. Vurgaftman, T. Hoffman, S. Liu, L. Lind-
+say, C. T. Ellis, N. Assefa, I. Chatzakis, T. L. Reinecke, J. G.
+Tischler, M. M. Fogler, J. H. Edgar, D. N. Basov, and J. D.
+Caldwell, Nat. Mater. 17, 134 (2018).
+[43] S. Dai, W. Fang, N. Rivera, Y. Stehle, B.-Y. Jiang, J. Shen,
+R. Y. Tay, C. J. Ciccarino, Q. Ma, D. Rodan-Legrain, P. Jarillo-
+Herrero, E. H. T. Teo, M. M. Fogler, P. Narang, J. Kong, and
+D. N. Basov, Adv. Mater. 31, 1806603 (2019).
+[44] E. Y. Ma, J. Hu, L. Waldecker, K. Watanabe, T. Taniguchi,
+F. Liu, and T. F. Heinz, Nano Lett. 22, 8389 (2022).
+[45] H. H. Sheinfux, L. Orsini, M. Jung, I. Torre, M. Ceccanti,
+R. Maniyara, D. B. Ruiz, A. H¨otger, R. Bertini, S. Castilla,
+N. C. H. Hesp, E. Janzen, A. Holleitner, V. Pruneri, J. H. Edgar,
+G. Shvets, and F. H. L. Koppens, arXiv:2202.08611 (2022).
+[46] A. K. Geim and I. V. Grigorieva, Nature 499, 419 (2013).
+[47] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Pho-
+tons and Atoms (Wiley, New York, 1989).
+[48] See Supplemental Materials for further details on the statements
+and the derivations.
+
+6
+[49] T. D. Lee, F. E. Low, and D. Pines, Phys. Rev. 90, 297 (1953).
+[50] Y. Ashida, T. Yokota, A. ˙Imamo˘glu, and E. Demler, Phys. Rev.
+Research 4, 023194 (2022).
+[51] M. Esslinger,
+R. Vogelgesang,
+N. Talebi,
+W. Khunsin,
+P. Gehring, S. De Zuani, B. Gompf, and K. Kern, ACS Pho-
+tonics 1, 1285 (2014).
+[52] J. Sun, N. M. Litchinitser, and J. Zhou, ACS Photon. 1, 293
+(2014).
+[53] W. Ma, P. Alonso-Gonz´alez, S. Li, A. Y. Nikitin, J. Yuan,
+J. Mart´ın-S´anchez, J. Taboada-Guti´errez, I. Amenabar, P. Li,
+S. V´elez, et al., Nature 562, 557 (2018).
+[54] Z. Zheng, J. Chen, Y. Wang, X. Wang, X. Chen, P. Liu, J. Xu,
+W. Xie, H. Chen, S. Deng, and N. Xu, Adv. Mater. 30, 1705318
+(2018).
+[55] Z. Zheng, N. Xu, S. L. Oscurato, M. Tamagnone, F. Sun,
+Y. Jiang, Y. Ke, J. Chen, W. Huang, W. L. Wilson, A. Ambrosio,
+S. Deng, and H. Chen, Sci. Adv. 5, eaav8690 (2019).
+[56] Y. Ashida, Z. Gong, and M. Ueda, Adv. Phys. 69, 249 (2020).
+[57] F. Mivehvar, F. Piazza, T. Donner, and H. Ritsch, Adv. Phys.
+70, 1 (2021).
+[58] L. H. Li, E. J. Santos, T. Xing, E. Cappelluti, R. Rold´an,
+Y. Chen, K. Watanabe, and T. Taniguchi, Nano Lett. 15, 218
+(2015).
+
+7
+Supplementary Materials
+Derivation of the effective cavity QED Hamiltonian
+We here derive the effective Hamiltonian of the proposed cavity QED system discussed in the main text. To this end, we start
+from a general microscopic model of the 2D electron coupled to quantized electromagnetic fields in uniaxial polar dielectrics in
+the Coulomb gauge [32, 47]:
+ˆH =
+�
+ˆp + eAs(ˆx, ˆy) + e ˆA(ˆx, ˆy, z = 0)
+�2
+2m
++ ˆH0,
+ˆp =
+� ˆpx
+ˆpy
+�
+,
+(S1)
+where m is the electron mass, e is the elementary charge, and As is an arbitrary static external field. We note that the 2D vector
+field ˆA(x, y, z = 0) is defined by the projection of the following 3D quantized vector potential onto the 2D tangential plane at
+z = 0:
+ˆA(r) =
+�
+kλ
+�
+ℏ
+2V ϵ0ck
+�
+ˆakλϵkλeik·r + H.c.
+�
+,
+(S2)
+where V is the volume of the entire space, and ˆakλ (ˆa†
+kλ) is the annihilation (creation) operator of photons with wavevector k
+and polarization λ; note that these excitations correspond to bare photons defined in the absence of dielectric materials. In the
+Coulomb gauge, the orthonormal transverse polarization vectors ϵkλ satisfy k · ϵkλ = 0 and ϵ†
+kλϵkν = δλν. The term ˆH0 is the
+quadratic Hamiltonian consisting of the energies of the quantized electromagnetic fields and the polar phonons in dielectrics:
+ˆH0 =
+�
+d3r
+� ˆΠ
+2
+2ϵ(z) + ϵ0c2
+2
+�
+∇ × ˆA
+�2
+�
++
+�
+D
+d3r
+v
+�
+i∈{x,y,z}
+�
+��
+�
+ˆπi − Z∗e ˆAi
+�2
+2Mi
++ 1
+2MiΩ2
+i ˆφ2
+i +
+�
+Z∗eˆφ∥
+i
+�2
+2ϵi(z)v
+�
+�� ,
+(S3)
+where ˆΠ(r) is the canonically conjugate vector field of ˆA(r), v is the unit-cell volume of the dielectrics, ˆφ(r) is a vector field
+representing polar optical phonons, ˆπ(r) is its canonically conjugate vector field, Z∗e is the effective charge of polar phonons,
+and
+�
+D d3r represents the integration over the dielectric regions,
+�
+D d3r =
+�
+d2r
+�� −δ/2
+−d/2 +
+� d/2
+δ/2
+�
+dz, with δ being the thickness
+of the narrow air gap that is assumed to be much shorter than the cavity thickness d. We also define the permittivity tensor ϵ(z)
+by
+ϵ(z) = diag (ϵt∞, ϵt∞, ϵz∞) ϑ
+�d
+2 − |z|
+�
+ϑ
+�
+|z| − δ
+2
+�
++ ϵ0I
+�
+ϑ
+�
+|z| − d
+2
+�
++ ϑ
+�δ
+2 − |z|
+��
+,
+(S4)
+where ϵt(z)∞ is the in-plane (out-of-plane) permittivity in the infinite-frequency limit and I is the identity matrix. We denote the
+in-plane masses and phonon frequencies by Mx,y = Mt and Ωx,y = Ωt, respectively. We remark that the permittivity in the air
+gap region can in general take a value different from the vacuum value depending on a specific choice of the cavity geometry,
+while we take it to be ϵ0 in the present work for the sake of simplicity.
+To derive the effective Hamiltonian, we first need to identify a polariton eigenmode that diagonalizes the quadratic part ˆH0.
+This can be done by solving the macroscopic Maxwell’s equations derived from the above model. We remark that the influence
+of the static external field can be neglected when constructing such polariton modes. Without loss of generality, we consider
+eigenmodes propagating along the x direction with in-plane momentum q = (q, 0, 0)T. Also, we recall that we are interested in
+the hyperbolic eigenmodes that strongly couple to the 2D electron moving on the z = 0 plane. Such modes correspond to the
+P-polarized (or equivalently, the transverse magnetic) modes with the even parity for the tangential component, which satisfy the
+following conditions: [B]y ̸= 0, [B]x,z = 0, [E]y = 0, [E]x,z ̸= 0, and Ex(z) = Ex(−z), Ez(z) = −Ez(−z). The resulting
+eigenequations that characterize these modes are
+q2
+ϵz(ω) +
+κ2
+ϵt(ω) = ω2
+ϵ0c2 ,
+(S5)
+tan
+�κd
+2
+�
+= −
+κ
+ϵt
+�
+q2 − ω2
+c2
+,
+(S6)
+where κ is the out-of-plane momentum and the in-plane and out-of-plane permittivities ϵt,z(ω) are given by
+ϵt(ω) = ϵt∞
+�
+1 +
+g2
+t
+Ω2
+t − ω2
+�
+, ϵz(ω) = ϵz∞
+�
+1 +
+g2
+z
+Ω2z − ω2
+�
+.
+(S7)
+
+8
+n=1,
+n=2,
+n=3,
+Mode amplitude
+qd=4
+z/d
+z/d
+z/d
+qd=8
+qd=12
+FIG. S1. Mode profiles of the h-BN Type I hyperbolic phonon polaritons. The top, middle, and bottom panels plot the spatial dependences of
+the mode functions along the thickness direction for their in-plane components, out-of-plane components, and root mean squares, respectively.
+The red solid (blue dotted) curves correspond to the electric-field (phonon) mode functions uqn (uφ
+qn).
+Here, we denote the coupling strengths of the dielectrics by
+gt =
+�
+(Z∗e)2
+ϵt∞Mtv ,
+gz =
+�
+(Z∗e)2
+ϵz∞Mzv .
+(S8)
+For the h-BN crystals considered in the main text, we use the parameters in Ref. [40] where the out-of-plane and in-plane phonon
+frequencies are Ωz = 23.3 THz and Ωt = 41.1 THz, the corresponding couplings are gz = 0.37Ωz and gt = 0.61Ωt, and the
+out-of-plane and in-plane infinite-frequency permittivities are ϵz∞ = 2.95 and ϵt∞ = 4.87, respectively.
+Solving the eigenequations (S5) and (S6) with respect to κ and ω for a given q, we obtain a hyperbolic dispersion ωqn
+associated with the out-of-plane momentum κqn, which are labeled by the discrete number n. The corresponding mode function,
+uqn, characterizes the spatial profile of the electric field and is given by
+uqn = fqn
+�
+cos(κqnz)ex − iϵtq
+ϵzκqn
+sin(κqnz)ez
+�
+eiqxϑ
+�d
+2 − |z|
+�
+ϑ
+�
+|z| − δ
+2
+�
++ f ′
+qn
+�
+s=±1
+[ex + is(q/νqn)ez] eiqx−sνqnzϑ
+�
+|z| − d
+2
+�
++ f ′′
+qn [cosh(νqnz)ex − i(q/νqn) sinh(νqnz)ez] eiqxϑ
+�δ
+2 − |z|
+�
+,
+(S9)
+where νqn =
+�
+q2 − ω2qn/c2 and the amplitude coefficients satisfy the following conditions:
+fqn/f ′
+qn = e−νqnd/2/ cos(κqnd/2),
+(S10)
+fqn/f ′′
+qn = cosh (νqnδ/2) / cos(κqnδ/2).
+(S11)
+Without loss of generality, we choose fqn > 0. The corresponding phonon mode function is given by
+uφ
+qn = fqn
+�
+gtΩt
+Ω2
+t − ω2qn
+cos(κqnz)ex − iϵtq
+ϵzκqn
+gzΩz
+Ω2z − ω2qn
+sin(κqnz)ez
+�
+eiqxϑ
+�d
+2 − |z|
+�
+ϑ
+�
+|z| − δ
+2
+�
+.
+(S12)
+We then impose the normalization condition,
+� ∞
+−∞
+dz
+�
+|uqn(z)|2 +
+��uφ
+qn(z)
+��2�
+= 1,
+(S13)
+
+1.5
+1.0
+0.5
+0.0
+-1.0
+-0.5
+0.0
+0.5
+1.01.0
+TT
+0.5
+0.0
+-0.5
+-1.00.2
+0.1
+0.0
+-0.1
+-0.21.5
+1.0
+0.5
+0.0
+-1.0
+-0.5
+0.0
+0.5
+1.01.0
+0.5
+0.0
+-0.5
+-1.00.2
+0.1
+0.0
+-0.1
+-0.21.5
+1.0
+0.5
+0.0
+-1.0
+-0.5
+0.0
+0.5
+1.01.0
+TT
+0.5
+0.0
+-0.5
+TT
+-1.00.2
+0.1
+0.0
+0.1
+-0.29
+which fixes the amplitude coefficients.
+The results for the three lowest branches of the h-BN Type I hyperbolic modes are shown in Fig. S1. We remark that, on
+the z = 0 plane where the 2D material is inserted, the electric fields have only the in-plane components along the propagation
+direction. We also note that, in the air gap region, the electric field is written as the curl of the magnetic field and thus it is purely
+transverse, i.e., uqn ∝ ∇ ×
+�
+sinh(νqnz)eiqxey
+�
+for |z| < δ
+2. Since we assume the narrow gap ν, κ ≪ δ−1, we can introduce
+the effective confinement length dqn through the relation |uqn(z = 0)| ≃ fqn ≡ 1/
+�
+dqn, which characterizes the coupling
+strength gqn of the 2D electron on the z = 0 plane as discussed in the main text.
+Details about the mean-field analysis of the magnetoabsorption spectrum
+We here provide technical details about the mean-field variational analysis of the magnetoabsorption spectrum discussed in
+the main text. To this end, we first perform an additional unitary transformation,
+ˆV = e
+�
+qn
+cqn
+lB (ˆγ†
+qn−ˆγqn),
+ˆV †ˆγqn ˆV = ˆγqn + cqn
+lB
+,
+(S14)
+which displaces the polariton operators. Using the relation �
+qn qc2
+qn = 0, the Hamiltonian in Eq. (8) in the main text is
+transformed to
+ˆHV U = ˆV † ˆHU ˆV = ℏωc
+2
+�
+ˆπ −
+�
+qn
+qlBˆγ†
+qnˆγqn
+�2
++
+�
+qn
+ℏωqn
+�
+ˆγ†
+qnˆγqn + cqn
+lB
+�
+ˆγqn + ˆγ†
+qn
+��
+.
+(S15)
+It is useful to rewrite it in terms of the canonically conjugate variables as follows:
+ˆHV U
+ℏωc
+=
+ˆX2 + ˆP 2
+2
++
+�
+qn
+ϵqn
+ˆX2
+qn + ˆP 2
+qn
+2
+− ˆπ · ˆΞb + 1
+2 : ˆΞ
+2
+b : +
+�
+qn
+ξqn ˆXqn,
+(S16)
+where we denote the conjugate variables associated with the cyclotron motion and the polaritons by
+ˆπ =
+�ˆa + ˆa†
+√
+2
+, i(ˆa − ˆa†)
+√
+2
+�T
+=
+�
+ˆX, ˆP
+�T
+,
+�
+ˆγqn + ˆγ†
+qn
+√
+2
+, i(ˆγqn − ˆγ†
+qn)
+√
+2
+�T
+=
+�
+ˆXqn, ˆPqn
+�T
+,
+(S17)
+and introduce
+ˆΞb =
+�
+qn
+lBq
+2
+�
+ˆX2
+qn + ˆP 2
+qn
+�
+, ϵqn = ωqn
+ωc
++ l2
+Bq2
+2
+, ξqn =
+�
+2ωqn
+ω3c
+gqn
+qlB
+,
+(S18)
+which correspond to the dimensionless total polariton momentum, the normalized effective polariton energies, and the dimen-
+sionless coupling strengths, respectively. The mean-field energy E is given by taking the expectation value of Eq. (S16) with
+respect to the product of coherent states for these conjugate variables. The mean-field ground state is then determined by finding
+the zero of the derivatives of E, leading to
+π = Ξb =
+�
+qn
+lBq
+2
+�
+X
+2
+qn + P
+2
+qn
+�
+, Xqn = −ξqn
+ϵqn
+, P qn = 0.
+(S19)
+c
+ω /Ω
+0
+0
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+c
+ω /Ω
+0
+1
+2
+3
+4
+5
+c
+ω /Ω
+0
+1
+2
+3
+4
+5
+ω/Ω
+L/d=2π
+z
+z
+z
+z
+L/d=3π
+L/d=4π
+FIG. S2.
+Magnetoabsorption spectra A(ω) at different aspect ratios L/d. The cavity thickness d is fixed to be d = 3 nm. The in-plane
+momentum q is discretized as in Eq. (S27). The results are plotted for the h-BN parameters and the in-plane momentum cutoff Λ = 2 nm−1.
+
+T
+7
+T
+T
+F10
+q
+[1/nm]
+0
+0
+0.1
+0.2
+0.3
+0.5
+0.4
+1
+2
+3
+q
+z
+g   /Ω
+d=3nm
+d=6nm
+d=10nm
+q
+[1/nm]
+0
+0
+0.05
+0.1
+0.15
+0.25
+0.2
+1
+2
+3
+q
+z
+g   /Ω
+d=3nm
+d=6nm
+d=10nm
+L/d=2π
+L/d=3π
+L/d=4π
+(a)
+(b)
+FIG. S3.
+Coupling strengths gq at (a) different cavity thicknesses d with L/d = π and (b) different aspect ratios L/d with d = 3 nm. The
+values corresponding to each of the discrete in-plane momenta q of hyperbolic modes are indicated by the filled red circles. The coupling
+strengths indicated in panel (a) and (b) correspond to Fig. 3 in the main text and Fig. S2, respectively. The results are plotted for the h-BN
+parameters and the in-plane momentum cutoff Λ = 2 nm−1.
+To obtain the low-energy excitation spectrum, we perform the projection ˆP onto the quadratic fluctuations on top of this
+mean-field ground state. Specifically, we describe the low-energy excitations in terms of the effective quadratic Hamiltonian
+given by
+ˆHeff = ˆP ˆD† ˆHV U ˆD ˆP,
+(S20)
+where ˆD induces the mean-field displacements
+ˆD† ˆπ ˆD = ˆπ + π,
+ˆD† ˆXqn ˆD = ˆXqn + Xqn,
+ˆD† ˆPqn ˆD = ˆPqn.
+(S21)
+The result is
+ˆHeff
+ℏωc
+=
+ˆX2 + ˆP 2
+2
++
+�
+qn
+˜ϵqn
+ˆX2
+qn + ˆP 2
+qn
+2
++ 1
+2
+��
+qn
+˜ξqneq ˆXqn
+�2
+− ˆπ ·
+�
+qn
+˜ξqneq ˆXqn
+≡ 1
+2
+ˆφ
+TM ˆφ,
+(S22)
+where we introduce the modified effective polariton energies and the effective coupling strengths by
+˜ϵqn ≡ ωqn
+ωc
++ l2
+Bq2
+2
+− 1
+2lBq · Ξb,
+˜ξqn ≡ gqn
+ωc
+�
+2ωqn
+ωc
+1
+ωqn
+ωc + l2
+Bq2
+2
+,
+(S23)
+respectively. In the last line, we rewrite the Hamiltonian by using the matrix M and the vector of the conjugate variables
+ˆφ =
+�
+ˆX, . . . , ˆXqn, . . . , ˆP, . . . , ˆPqn, . . .
+�T
+.
+(S24)
+The low-energy excitation spectrum {ωλ} can then be obtained from the Williamson eigenvalues of M as follows:
+STMS = diag({ωλ/ωc} , {ωλ/ωc}),
+(S25)
+where S is a symplectic matrix. We note that this treatment is exact in the long-wavelength limit qlB → 0 in which Eq. (S16)
+simplifies to the quadratic Hamiltonian.
+Finally, the absorption spectrum can be obtained from
+A(ω) ≃ Re
+�� ∞
+0
+eiωt⟨0| ˆD†ˆae−i ˆ
+HV Utˆa† ˆD|0⟩
+�
+≃ Re
+�� ∞
+0
+eiωt⟨0|
+�
+ˆa + πx − iπy
+√
+2
+�
+e−i ˆ
+Hefft
+�
+ˆa† + πx + iπy
+√
+2
+�
+|0⟩
+�
+.
+(S26)
+The last line can easily be evaluated by using the diagonalized basis in Eq. (S25). We note that the results in the main text are
+obtained for the discretized in-plane wavenumbers as follows:
+qi ∈
+�
+−Λ, −N − 1
+N
+Λ, . . . , − Λ
+N , 0, Λ
+N , . . . , N − 1
+N
+Λ, Λ
+�
+, i ∈ {x, y}.
+(S27)
+
+000
+810
+800880
+00
+0011
+Here, the integer number N is related to the lateral system size L via
+� ΛL
+2π
+�
+= N. The results obtained for the cavity consisting
+of ultrathin h-BN materials with different aspect ratios L/d are plotted in Fig. S2. The coupling strengths to each hyperbolic
+phonon polariton mode with discretized in-plane momentum are plotted for different thicknesses and aspect ratios in Fig. S3.
+When the lateral size L is increased, the coupling strength gq to each mode decreases uniformly as gq ∝ L−1 while there appear
+a larger number of cavity modes. Consequently, a key feature of the ultrastrong coupling regime in the spectrum, that is, the
+formation of the localized Landau-polariton mode remains almost the same independently of the lateral size L. Meanwhile, the
+increase of L/d leads to the appearance of dense anticrossed branches originating from the hybridization with the continuum
+cavity modes above the lower Landau-polariton mode.
+
diff --git a/sNE2T4oBgHgl3EQfLQam/content/tmp_files/load_file.txt b/sNE2T4oBgHgl3EQfLQam/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..aadaff32b84f171df7d1646de921e62f091b8dde
--- /dev/null
+++ b/sNE2T4oBgHgl3EQfLQam/content/tmp_files/load_file.txt
@@ -0,0 +1,1010 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf,len=1009
+page_content='Cavity Quantum Electrodynamics with Hyperbolic van der Waals Materials  Yuto Ashida,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ∗ Atac¸ ˙Imamo˘glu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='3 and Eugene Demler4  1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 7-3-1 Hongo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Bunkyo-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Tokyo 113-0033,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Japan  2Institute for Physics of Intelligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 7-3-1 Hongo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Tokyo 113-0033,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Japan  3Institute of Quantum Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ETH Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' CH-8093 Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Switzerland  4Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ETH Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 8093 Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Switzerland  The ground-state properties and excitation energies of a quantum emitter can be modified in the ultrastrong  coupling regime of cavity quantum electrodynamics (QED) where the light-matter interaction strength becomes  comparable to the cavity resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Recent studies have started to explore the possibility to control  an electronic material by embedding it in a cavity that confines electromagnetic fields in deep subwavelength  scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Currently, there is a strong motivation to realize ultrastrong-coupling cavity QED in the terahertz (THz)  range, since most of the elementary excitations of quantum materials are in this frequency window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We propose  and analyze an ideal platform to achieve this aim where a two-dimensional electronic material is encapsulated  by a planar cavity consisting of ultrathin polar van der Waals crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' As a concrete setup, we show that  nanometer-thick hexagonal boron nitride layers allow for reaching the ultrastrong coupling regime for single-  electron cyclotron resonance in a bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The proposed cavity setting can be realized by a wide  variety of thin dielectric materials with hyperbolic dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Consequently, van der Waals heterostructures  could provide an ideal playground for exploring the ultrastrong-coupling physics of cavity QED materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Strong coupling regime of cavity quantum electrodynam-  ics (QED), where the emitter-cavity coupling strength exceeds  decay rates, has played a central role in quantum informa-  tion science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' For instance, cavity-mediated coherent interac-  tion between two distant qubits has allowed for implementing  two-qubit gates with high fidelity [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Moreover, cavity-  mediated Raman transitions have the potential to realize all-  to-all coupling with a tunable range and strength, providing an  indispensable tool for quantum simulators [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Recent stud-  ies have started to explore the possibility of further increasing  the coupling strength so that it becomes comparable to the  elementary excitation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Many of the common simpli-  fications in cavity QED fail in this regime, rendering the theo-  retical analysis challenging [5–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' A remarkable feature here  is that ultrastrong light-matter coupling can alter the ground-  state electronic properties due to virtual processes where both  emitters and cavity photons are excited [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Consequently,  a natural question to address is if and when the ultrastrong  coupling regime can be attained and used to control material  properties simply by cavity confinement in the absence of an  external driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  A quick calculation for cavities supporting purely photonic  excitations suggests that the ultrastrong coupling regime is out  of reach due to the smallness of the fine structure constant  [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' However, this limit can be overcome in structured elec-  tromagnetic environments because hybridization with matter  excitations enables one to control cavity frequencies indepen-  dently of wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' For instance, in superconducting mi-  crowave circuits, a large kinetic inductance allows for high-  impedance electromagnetic excitations, leading to the single-  quantum ultrastrong coupling in the microwave range [13–  16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Meanwhile, many of the elementary excitations in quan-  tum materials are in the terahertz (THz) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Recent ex-  perimental and theoretical studies have shown the potential of  utilizing the cavity confinement as a means to modify such ex-  citations [17–27], which shows great promise for controlling  quantum many body states and endowing them with new func-  tionalities, including superconductivity [28–31], ferroelectric-  ity [32–34], magnetotransport [35–37] and topological prop-  erties [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' In view of the prospects of observing these in-  triguing phenomena in cavity QED materials, there is a strong  motivation to examine the feasibility of attaining the single-  quantum ultrastrong coupling at THz frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  The aim of this Letter is to propose and analyze a planar  cavity setup consisting of polar van der Waals (vdW) crys-  tals as an ideal platform to explore the ultrastrong-coupling  physics of cavity QED materials (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We point out that  one can attain a single-quantum ultrastrong coupling by em-  bedding a 2D material with electronic transitions in the THz  regime into ultrathin hexagonal boron nitride (h-BN) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  A special feature here is that electrons in the 2D material  couple to the electric field component of tightly confined hy-  perbolic phonon polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The strong photon-phonon hy-  d  L  ω  Ω  L  x  z  ε  ε  εt  z  z  Ω t  (a)  (b)  0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Schematic figure illustrating the proposed planar cavity  setup consisting of thin polar van der Waals crystals (green shaded)  whose optical axis is along z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' In the narrow air gap be-  tween the two slabs, a 2D material (red shaded) is inserted and the  electron there can ultrastrongly couple to electromagnetic fields of  tightly confined hyperbolic polaritons (blue solid curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The thick-  ness (lateral extension) of surrounding materials is d (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' (b) Out-  of-plane (solid curve) and in-plane (dashed curve) permittivities ϵz,t  of hyperbolic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The Restrahlen bands (red shaded) appear  above each of the out-of-plane and in-plane phonon resonances Ωz,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='03712v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='mes-hall]  9 Jan 2023  2  bridization together with the low frequencies of polaritons  then results in the significantly enhanced coupling strength  over a broad range of momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' This should be contrasted  to conventional polar dielectrics with an isotropic dispersion,  where sizable light-matter hybridization takes place in a lim-  ited range of momenta since light and matter are almost de-  coupled at large momenta due to the fast speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  As a proof-of-concept demonstration, we consider the case  of a 2D electron with parabolic dispersion under the static  magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We analyze the hybridization between hyper-  bolic phonon polaritons in h-BN and the cyclotron motion of  the parabolic electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We show that the single-electron ul-  trastrong coupling regime is within the reach provided that the  thicknesses of h-BN layers are chosen to be nanometer-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  This consideration is motivated by recent advances demon-  strating ultrasmall mode volumes of hyperbolic phonon po-  laritons in h-BN nanostructures [40–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' While we present  quantitative estimates for h-BN nanocavities for the sake of  concreteness, the proposed cavity scheme is applicable to the  majority of polar vdW crystals [46], which exhibit hyper-  bolic polaritons originating from distinct in- and out-of-plane  infrared-active phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Hyperbolic phonon polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='— We begin our analysis by  reviewing the properties of a planar cavity made out of layered  thin polar vdW materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Due to the weakness of interlayer  coupling, such materials naturally possess two types of opti-  cal phonons corresponding to in-plane and out-of-plane ionic  oscillations in the THz or mid-infrared regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' This leads to  the uniaxial anisotropy characterized by the out-of-plane (in-  plane) dielectric permittivities ϵz (ϵt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' In the frequency win-  dows above each of the phonon resonances, the two dielectric  responses can have opposite signs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' This unique fea-  ture leads to the hyperbolic isofrequency surfaces defined by  the dispersion relation of the transverse magnetic modes,  |q|2  ϵz  + |κ|2  ϵt  = ω2  ϵ0c2 ,  (1)  where q (κ) is the in-plane (out-of-plane) wavevector, ϵ0 is  the vacuum permittivity, and c is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The op-  posite signs of ϵz,t allow for excitations to have low frequen-  cies even at large momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The resulting hybridized elemen-  tary excitations of photons and optical phonons are known  as hyperbolic phonon polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The corresponding disper-  sions in this range of frequencies are called the Reststrahlen  bands of either Type I when ϵz < 0, ϵt > 0 or Type II when  ϵz > 0, ϵt < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  As a representative hyperbolic material, h-BN has a strong  crystalline anisotropy leading to two spectrally distinct Rest-  strahlen bands with ultralow loss [40–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Below we focus on  the electromagnetic couplings with Type I h-BN hyperbolic  modes ωqn lying in the narrow frequency window above the  out-of-plane phonon frequency Ωz = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='1 THz (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' top panel  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' As we discuss below, these modes exhibit several  advantages for the purpose of attaining the ultrastrong cou-  plings thanks to their relatively low frequencies and sizable  photon components over a broad range of momenta q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Single-electron  ultrastrong  coupling  with  hyperbolic  materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='— We consider the planar cavity setup where a 2D  material (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=', a bilayer graphene) is inserted into the narrow  air gap between two thin h-BN slabs whose thickness and lat-  eral extensions are denoted by d and L, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Our starting point is the following cavity QED Hamiltonian of  the 2D parabolic electron interacting with hyperbolic phonon  polaritons:  ˆH =  �  ˆp + eAs(ˆr) + e ˆA(ˆr)  �2  2m  + ˆHpol,  (2)  which can be derived from an effective theory of uniaxial po-  lar dielectrics coupled to quantized electromagnetic fields in  the Coulomb gauge [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Here, e is the elementary charge,  m is the electron mass, ˆr and ˆp are the electron position and  momentum operators in the 2D lateral directions, respectively,  As represents an arbitrary static field, and ˆHpol is the free po-  lariton Hamiltonian,  ˆHpol =  �  qn  ℏωqnˆγ†  qnˆγqn,  (3)  where ˆγqn (ˆγ†  qn) annihilates (creates) a hyperbolic polariton  with the in-plane wavevector q in the branch n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The 2D  vector field ˆA(r) is obtained by projecting the vector potential  onto the 2D plane where the electron is placed, and can be  n=1  qd  qn  z  ω  /Ω  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='15  5  10  15  20  n=2  n=3  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='08  n=1  n=2  n=3  √d/dqn  z  t  E  n=1  n=2  n=3  q d  ＊  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (Top) Dispersions ωqn of the hyperbolic phonon polari-  tons in the planar cavity setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The inset shows the spatial pro-  files of the in-plane electric fields along z direction for each mode at  qd = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' For the sake of visibility, only the three lowest modes are  plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' (Bottom) Inverse of the square root of the effective dimen-  sionless confinement length,  �  d/dqn, characterizing the dimension-  less single-electron coupling strength for each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The maximum  value in the principal branch n = 1 is reached at q = q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  3  expanded in terms of the polariton operators by  ˆA(ˆr) =  �  qn  Aqneq  �  ˆγqneiq·ˆr + ˆγ†  qne−iq·ˆr�  ,  (4)  where Aqn ≃  �  ℏ/(2L2ϵ0ωqndqn) is the amplitude with the  effective confinement length dqn whose value is characterized  by the polariton mode function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The effective polarization  vector becomes eq ≡ q/|q| because, for the symmetric ar-  rangement we assumed, the electric fields of the polaritons  only have in-plane components along the propagation direc-  tion in the 2D plane where the electronic material is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  In general, while the vector potential is originally a 3D trans-  verse vector field in the Coulomb gauge, it can effectively ac-  quire longitudinal components when projected onto the 2D  tangential plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Figure 2 shows the results for Type I h-BN hyperbolic  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The dispersions (top panel) start from the longitudinal  phonon frequency and saturate to Ωz at high q ≡ |q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Natu-  rally, these hybridized modes are almost purely longitudinal  (transverse) phonon excitations in the limit q → 0 (q → ∞),  which do not allow for a strong coupling to electrons in the air  gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Crucially, however, except for these two limits, there still  exist the nonvanishing photon contributions leading to the siz-  able electric-dipole couplings at intermediate momenta (bot-  tom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' This is because the in-plane component, which  couples to the 2D electron, has almost equal photonic and  phononic contents while the out-of-plane component is pre-  dominantly phonon-like [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  To further proceed, we use the unitary transformation ˆU =  e−iˆr· ˆP b/ℏ with ˆP b = �  qn ℏqˆγ†  qnˆγqn [49], resulting in the  Hamiltonian ˆHU ≡ ˆU † ˆH ˆU given by  ˆHU =  �  ˆp+eAs(ˆr)+e ˆA(0)− ˆP b  �2  2m  + ˆHpol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (5)  In this way, the electron-coordinate dependence in the quan-  tized vector potential can be eliminated at the expense of gen-  erating the polariton momentum ˆP b, leading to the nonlocal  interaction among polaritons mediated by the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The  dimensionful coupling strength between the electron and each  of the dynamical quantized electromagnetic modes is given  by  gqn = eAqn  �ωqn  mℏ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (6)  Since gqn characterizes the single-electron coupling strength  rather than the collective one, it depends on the lateral size L  through gqn ∝ L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Consequently, a natural measure for the  effective coupling strength between the 2D electron and the  continuum of polariton modes is given by geff ≡  ��  qn g2qn,  which scales as geff ∝ O(L0) [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Previously, the deep strong coupling regime has been ex-  perimentally realized in superconducting circuits, where geff  becomes comparable to the microwave photon frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  In the present setting, the use of ultrathin h-BN slabs with  nanometer-scale thicknesses enables one to reach the deep or  extremely strong coupling regimes, where geff becomes com-  parable to or even exceeds a THz cavity frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' For in-  stance, using the h-BN parameters [40], we estimate coupling  strengths of order gqn/Ωz = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='7 ×  �  (10 nm)3/(L2dqn),  which, together with the results of the effective confinement  length dqn in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 2b, can lead to geff ∼ Ωz in nanoscale het-  erostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' More specifically, one can attain geff ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='0 Ωz  for the n = 1 principal branch when the cavity thickness (in-  plane momentum cutoff) is set to be d = 5 nm (Λ = 2 nm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  As demonstrated below, one important consequence of attain-  ing geff ∼ Ωz is that the electron mixes the otherwise indepen-  dent cavity modes and creates a localized state at a frequency  below the cavity resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We emphasize that this key fea-  ture is largely insensitive to a choice of the lateral size L and  thus remains even in the 2D thermodynamic limit L/d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Application to a 2D electron under the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='— As  a proof-of-concept demonstration, we now focus on a proto-  typical setting of ultrastrong-coupling physics, namely, a 2D  parabolic electron subject to a static perpendicular magnetic  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' From now on, we consider the n = 1 principal hyper-  bolic mode that most strongly couples to the cyclotron motion  of the electron, and abbreviate the subscript n for the sake of  notational simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' To simplify the Hamiltonian, we choose  the symmetric gauge, As(r) = (−By/2, Bx/2)T, with the  magnetic field B and introduce the annihilation operator of  Landau levels by  ˆa =  1  √  2  �lB  ℏ (ˆpx − iˆpy) −  i  2lB  (ˆx − iˆy)  �  ,  (7)  where lB =  �  ℏ/(eB) is the magnetic length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Using the  cyclotron frequency ωc = eB/m and the operator ˆπ =  1  √  2  �  ˆa + ˆa†, i(ˆa − ˆa†)  �T, we obtain  ˆHU = ℏωc  2  �  ˆπ+  �  q  eq  �  cq  �  ˆγq+ˆγ†  q  �  −qlBˆγ†  qˆγq  �  �2  + ˆHpol,  (8)  where cq = gq/√ωqωc is the dimensionless coefficient char-  acterizing the coupling strength of the cyclotron motion to  the hyperbolic polaritons with momentum q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  In the long-  wavelength limit qlB → 0, the polariton-polariton interac-  tion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' (8) disappears and the problem reduces to the  quadratic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' As demonstrated below, however, this interac-  tion term can in general contribute to the dynamics and affect  the absorption spectrum especially in the ultrastrong coupling  regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' This is because the electron couples most strongly  with the electromagnetic modes at a finite momentum around  q ∼ q∗ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 2b) whose corresponding length scale 1/q∗ is  comparable to the magnetic length lB near the cyclotron reso-  nance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' More generally, such spatial dependence of the vector  potential must be taken into account when 1/q∗ is comparable  to the characteristic length scale of the electron motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  The low-energy excitations can be studied by analyzing the  4  c  ω /Ω  0  0  1  2  3  4  5  1  2  3  4  5  c  ω /Ω  0  1  2  3  4  5  c  ω /Ω  0  1  2  3  4  5  ω/Ω  d=3nm  d=6nm  d=10nm  z  z  z  z  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Magnetoabsorption spectra A(ω) of the 2D electron in the cavity plotted against a cyclotron resonance ωc = eB/m at different  cavity thicknesses d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The blue solid curve in each panel corresponds to the bound-state energy of the Landau polariton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We impose the periodic  boundary conditions on the lateral directions and fix the aspect ratio L/d = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We use the h-BN parameters in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' [40] and set the in-plane  momentum cutoff Λ = 2 nm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  magnetoabsorption spectrum  A(ω) = Re  �� ∞  0  eiωt⟨GS|ˆae−i ˆ  Htˆa†|GS⟩  �  ,  (9)  where |GS⟩ is the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' To reveal its qualitative fea-  tures, we perform a simple variational analysis as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Specifically, we first determine the variational ground state of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' (8) in the form of a product of coherent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We then ex-  pand the Hamiltonian around this state, obtain the fluctuations  up to the quadratic terms, and determine the excitation spec-  trum via the exact diagonalization of the effective quadratic  Hamiltonian [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Figure 3 shows the obtained magnetoabsorption spectra,  where the cavity thickness d is varied while the aspect ratio  L/d is kept constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The blue solid curve in each panel shows  the bound-state energy of the Landau polariton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' As the thick-  ness is decreased, the spectrum starts to exhibit the anticross-  ings around the cyclotron resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' In particular, when the  cavity length becomes a few nanometers, the large separations  between the branches that are comparable to the elementary  excitation energies themselves emerge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' this is a hallmark of  the ultrastrong coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  As discussed before, a key feature here is the formation of  the dressed bound state consisting of the electron and the lo-  calized phonon polaritons, which manifests itself as the anti-  crossed lower branch in the spectrum (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' the blue curve in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Importantly, this feature remains independently of the  lateral size L, while the effect of increasing L/d is the appear-  ance of a continuum of cavity modes above the lower branch  [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' It is worthwhile to note that the positions of the anti-  crossings are shifted above the bare resonances ωc/Ωz ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  This upward shift originates from the renormalization of the  effective polariton energies due to the repulsive polariton-  polariton interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Since the latter comes from the spatial  dependence of the vector potential, this effect is absent in the  long-wavelength limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We also remark that the appearance  of the multiple anticrossed branches in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 3 are due to the  discretized in-plane momentum q, whose value is set by the  periodic boundary conditions in the lateral directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='— The proposed platform for ultrastrong-  coupling cavity QED materials can be realized by various po-  lar vdW materials exhibiting hyperbolic phonon polaritons,  including Bi2Se3, Bi2Te3, MoS2 and MoO3 as well as h-  BN [46, 51–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Thus, by confining materials in the cavity  consisting of these vdW structures, one can strongly couple  electronic excitations to quantized electromagnetic modes in a  wide spectral range from mid- or far-infrared to THz regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Moreover, the layered nature of vdW crystals should readily  allow one to tune the cavity coupling strengths by controlling  the thickness of the surrounding crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  While polaritons in these materials are known to exhibit ul-  tralow loss [40–45, 53–55], it merits further study to examine  a role of dissipation in the present cavity setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' In particular, it  is intriguing to explore if and when a confinement of quantum  materials in the cavity with dissipative hyperbolic polaritons  could lead to phase transitions or dynamics unique to open  systems [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We also note that in general the contribu-  tion from the instantaneous Coulomb energy could be impor-  tant when the 2D material is placed near the surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' In the  present planar setup, this contribution might lead to sizable ef-  fects in many-body regimes due to dielectric screening of the  electron-electron Coulomb interaction [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  In summary, we showed that the planar cavity consisting of  vdW heterostructures provides a promising platform to attain  the single-electron ultrastrong coupling in the THz or mid-  infrared regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' As a proof-of-concept demonstration, we  presented the analysis of the magnetoabsorption spectrum for  the 2D electron confined in the h-BN cavity, where the cou-  pling can be ultrastrong provided that the thicknesses are ju-  diciously controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We expect that our results open a way  to study wealth of recent predictions in the emerging field of  cavity QED materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We hope that our work simulates fur-  ther studies in these directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  We are grateful to Iliya Esin, Ilya Esterlis, Tim Kaxiras,  Igor Khanonkin, Frank Koppens, Kanta Masuki, Gil Refael,  Tao Shi, and Kenji Yasuda for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ac-  knowledges support from the Japan Society for the Promotion  of Science through Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' JP19K23424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' was sup-  5  ported by the Swiss National Science Foundation (SNSF) un-  der Grant Number 200020 207520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' acknowledges sup-  port from the ARO grant “Control of Many-Body States Us-  ing Strong Coherent Light-Matter Coupling in Terahertz Cav-  ities” and the Swiss National Science Foundation under Divi-  sion II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  ∗ ashida@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='jp  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' DiCarlo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Chow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Gambetta, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Bishop, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Johnson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Schuster, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Majer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Blais, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Frunzio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Girvin,  and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Schoelkopf, Nature 460, 240 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Blais, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Grimsmo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Girvin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Wallraff, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 93, 025005 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Stute, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Casabone, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Schindler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Monz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Schmidt,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Brandst¨atter, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Northup,  and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Blatt, Nature 485, 482  (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Vaidya, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Guo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Kroeze, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ballantine, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Koll´ar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Keeling,  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' X 8, 011002  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Hagenm¨uller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' De Liberato, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ciuti, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' B 81,  235303 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' De Bernardis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Pilar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Jaako, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' De Liberato, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rabl,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' A 98, 053819 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Le Boite, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 3, 1900140 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ashida, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ˙Imamo˘glu, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Demler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 126,  153603 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Garcia-Vidal, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ciuti, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ebbesen, Science 373,  eabd0336 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Schlawin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Kennes, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Sentef, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  9, 011312 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Bloch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Cavalleri, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Galitski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Hafezi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rubio,  Nature 606, 41 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Devoret, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Girvin, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Schoelkopf, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' 519, 767  (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Forn-D´ıaz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Garc´ıa-Ripoll, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Peropadre, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Orgiazzi,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Yurtalan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Belyansky, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Wilson, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lupascu, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 13, 39 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [14] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Yoshihara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Fuse, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ashhab, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Kakuyanagi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Saito, and  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Semba, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 13, 44 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Forn-D´ıaz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lamata, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rico, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Kono, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Solano, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 91, 025005 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Frisk Kockum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Miranowicz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' De Liberato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Savasta,  and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Nori, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 1, 19 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [17] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Scalari, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Maissen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Cibella, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Leoni, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Carelli, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Val-  morra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Beck, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Faist, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 16, 033005 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Smolka, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Wuester, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Haupt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Faelt, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Wegscheider,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Imamoglu, Science 346, 332 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Maissen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Scalari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Valmorra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Beck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Faist,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Cibella,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Leoni,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Reichl,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Charpentier,  and  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Wegscheider, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' B 90, 205309 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [20] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Reno, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Pan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Watson,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Manfra, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Kono, Nature Phys 12, 1005 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Bayer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Pozimski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Schambeck, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Schuh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Huber,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Bougeard, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lange, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 17, 6340 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Keller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Scalari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Appugliese, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rajabali, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Beck,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Haase, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lehner, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Wegscheider, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Failla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' My-  ronov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Leadley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lloyd-Hughes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Nataf, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Faist,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' B 101, 075301 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Thomas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Devaux, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Nagarajan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rogez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Seidel,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Richard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Genet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Drillon, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ebbesen, Nano  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 21, 4365 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Flick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ruggenthaler, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Appel, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rubio, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 114, 3026 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [25] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rokaj, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Penz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Sentef, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ruggenthaler, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ru-  bio, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 123, 047202 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [26] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Juraschek, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Neuman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Flick, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Narang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Research 3, L032046 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rom´an-Roche and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Zueco, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Notes , 50  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [28] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Schlawin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Cavalleri, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Jaksch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 122,  133602 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Curtis, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Raines, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Allocca, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Hafezi, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Galitski, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 122, 167002 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Sentef, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ruggenthaler, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rubio, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 4,  eaau6969 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [31] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Gao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Schlawin, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Jaksch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' B 104, L140503  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [32] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ashida, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ˙Imamo˘glu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Faist, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Jaksch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Cavalleri, and  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Demler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' X 10, 041027 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Latini, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Shin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Sato, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Sch¨afer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' De Giovannini,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' H¨ubener, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rubio, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 118,  e2105618118 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [34] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lenk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Li, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Werner, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Eckstein, arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='05559  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [35] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Paravicini-Bagliani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Appugliese, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Richter, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Val-  morra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Keller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Beck, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Bartolo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' R¨ossler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ihn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' En-  sslin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ciuti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Scalari,  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Faist, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 15, 186  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [36] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Appugliese, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Enkner, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' B 105, 205424 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [38] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' Ronca, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Sentef, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' B 99, 235156  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' Ellis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Maier,  and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' 5, 1 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Dai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' Ma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' McLeod,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Gannett, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Neto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Fogler, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' Ellis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' Edgar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Basov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' 17, 134 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [43] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content=' Teo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Fogler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Narang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Kong, and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Basov, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 31, 1806603 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [44] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Hu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Waldecker, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Taniguchi,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Liu, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Heinz, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 22, 8389 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [45] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Sheinfux, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Orsini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Jung, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Torre, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ceccanti,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Maniyara, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ruiz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' H¨otger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Bertini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Castilla,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Hesp, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Janzen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Holleitner, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Pruneri, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Edgar,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Shvets, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Koppens, arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='08611 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [46] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Geim and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Grigorieva, Nature 499, 419 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [47] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Cohen-Tannoudji, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Dupont-Roc, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Grynberg, Pho-  tons and Atoms (Wiley, New York, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [48] See Supplemental Materials for further details on the statements  and the derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  6  [49] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Lee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Low, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Pines, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 90, 297 (1953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [50] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ashida, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Yokota, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ˙Imamo˘glu, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Demler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Research 4, 023194 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [51] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Esslinger,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Vogelgesang,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Talebi,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Khunsin,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Gehring, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' De Zuani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Gompf, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Kern, ACS Pho-  tonics 1, 1285 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [52] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Sun, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Litchinitser, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Zhou, ACS Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 1, 293  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [53] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Alonso-Gonz´alez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Nikitin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Yuan,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Mart´ın-S´anchez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Taboada-Guti´errez, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Amenabar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Li,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' V´elez, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=', Nature 562, 557 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [54] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Xu,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Xie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Deng, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Xu, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 30, 1705318  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [55] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Zheng, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Oscurato, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Tamagnone, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Sun,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Huang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Wilson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ambrosio,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Deng, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Chen, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 5, eaav8690 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [56] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ashida, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Gong, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ueda, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 69, 249 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [57] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Mivehvar, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Piazza, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Donner, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Ritsch, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  70, 1 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  [58] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Li, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Santos, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Xing, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Cappelluti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Rold´an,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Watanabe, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Taniguchi, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 15, 218  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  7  Supplementary Materials  Derivation of the effective cavity QED Hamiltonian  We here derive the effective Hamiltonian of the proposed cavity QED system discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' To this end, we start  from a general microscopic model of the 2D electron coupled to quantized electromagnetic fields in uniaxial polar dielectrics in  the Coulomb gauge [32, 47]:  ˆH =  �  ˆp + eAs(ˆx, ˆy) + e ˆA(ˆx, ˆy, z = 0)  �2  2m  + ˆH0,  ˆp =  � ˆpx  ˆpy  �  ,  (S1)  where m is the electron mass, e is the elementary charge, and As is an arbitrary static external field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We note that the 2D vector  field ˆA(x, y, z = 0) is defined by the projection of the following 3D quantized vector potential onto the 2D tangential plane at  z = 0:  ˆA(r) =  �  kλ  �  ℏ  2V ϵ0ck  �  ˆakλϵkλeik·r + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  �  ,  (S2)  where V is the volume of the entire space, and ˆakλ (ˆa†  kλ) is the annihilation (creation) operator of photons with wavevector k  and polarization λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' note that these excitations correspond to bare photons defined in the absence of dielectric materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' In the  Coulomb gauge, the orthonormal transverse polarization vectors ϵkλ satisfy k · ϵkλ = 0 and ϵ†  kλϵkν = δλν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The term ˆH0 is the  quadratic Hamiltonian consisting of the energies of the quantized electromagnetic fields and the polar phonons in dielectrics:  ˆH0 =  �  d3r  � ˆΠ  2  2ϵ(z) + ϵ0c2  2  �  ∇ × ˆA  �2  �  +  �  D  d3r  v  �  i∈{x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='z}  �  ��  �  ˆπi − Z∗e ˆAi  �2  2Mi  + 1  2MiΩ2  i ˆφ2  i +  �  Z∗eˆφ∥  i  �2  2ϵi(z)v  �  �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S3)  where ˆΠ(r) is the canonically conjugate vector field of ˆA(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' v is the unit-cell volume of the dielectrics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ˆφ(r) is a vector field  representing polar optical phonons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ˆπ(r) is its canonically conjugate vector field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Z∗e is the effective charge of polar phonons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  and  �  D d3r represents the integration over the dielectric regions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  �  D d3r =  �  d2r  �� −δ/2  −d/2 +  � d/2  δ/2  �  dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' with δ being the thickness  of the narrow air gap that is assumed to be much shorter than the cavity thickness d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We also define the permittivity tensor ϵ(z)  by  ϵ(z) = diag (ϵt∞, ϵt∞, ϵz∞) ϑ  �d  2 − |z|  �  ϑ  �  |z| − δ  2  �  + ϵ0I  �  ϑ  �  |z| − d  2  �  + ϑ  �δ  2 − |z|  ��  ,  (S4)  where ϵt(z)∞ is the in-plane (out-of-plane) permittivity in the infinite-frequency limit and I is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We denote the  in-plane masses and phonon frequencies by Mx,y = Mt and Ωx,y = Ωt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We remark that the permittivity in the air  gap region can in general take a value different from the vacuum value depending on a specific choice of the cavity geometry,  while we take it to be ϵ0 in the present work for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  To derive the effective Hamiltonian, we first need to identify a polariton eigenmode that diagonalizes the quadratic part ˆH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  This can be done by solving the macroscopic Maxwell’s equations derived from the above model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We remark that the influence  of the static external field can be neglected when constructing such polariton modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Without loss of generality, we consider  eigenmodes propagating along the x direction with in-plane momentum q = (q, 0, 0)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Also, we recall that we are interested in  the hyperbolic eigenmodes that strongly couple to the 2D electron moving on the z = 0 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Such modes correspond to the  P-polarized (or equivalently, the transverse magnetic) modes with the even parity for the tangential component, which satisfy the  following conditions: [B]y ̸= 0, [B]x,z = 0, [E]y = 0, [E]x,z ̸= 0, and Ex(z) = Ex(−z), Ez(z) = −Ez(−z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The resulting  eigenequations that characterize these modes are  q2  ϵz(ω) +  κ2  ϵt(ω) = ω2  ϵ0c2 ,  (S5)  tan  �κd  2  �  = −  κ  ϵt  �  q2 − ω2  c2  ,  (S6)  where κ is the out-of-plane momentum and the in-plane and out-of-plane permittivities ϵt,z(ω) are given by  ϵt(ω) = ϵt∞  �  1 +  g2  t  Ω2  t − ω2  �  , ϵz(ω) = ϵz∞  �  1 +  g2  z  Ω2z − ω2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S7)  8  n=1,  n=2,  n=3,  Mode amplitude  qd=4  z/d  z/d  z/d  qd=8  qd=12  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Mode profiles of the h-BN Type I hyperbolic phonon polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The top, middle, and bottom panels plot the spatial dependences of  the mode functions along the thickness direction for their in-plane components, out-of-plane components, and root mean squares, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  The red solid (blue dotted) curves correspond to the electric-field (phonon) mode functions uqn (uφ  qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Here, we denote the coupling strengths of the dielectrics by  gt =  �  (Z∗e)2  ϵt∞Mtv ,  gz =  �  (Z∗e)2  ϵz∞Mzv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S8)  For the h-BN crystals considered in the main text, we use the parameters in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' [40] where the out-of-plane and in-plane phonon  frequencies are Ωz = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='3 THz and Ωt = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='1 THz, the corresponding couplings are gz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='37Ωz and gt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='61Ωt, and the  out-of-plane and in-plane infinite-frequency permittivities are ϵz∞ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='95 and ϵt∞ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='87, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Solving the eigenequations (S5) and (S6) with respect to κ and ω for a given q, we obtain a hyperbolic dispersion ωqn  associated with the out-of-plane momentum κqn, which are labeled by the discrete number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The corresponding mode function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  uqn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' characterizes the spatial profile of the electric field and is given by  uqn = fqn  �  cos(κqnz)ex − iϵtq  ϵzκqn  sin(κqnz)ez  �  eiqxϑ  �d  2 − |z|  �  ϑ  �  |z| − δ  2  �  + f ′  qn  �  s=±1  [ex + is(q/νqn)ez] eiqx−sνqnzϑ  �  |z| − d  2  �  + f ′′  qn [cosh(νqnz)ex − i(q/νqn) sinh(νqnz)ez] eiqxϑ  �δ  2 − |z|  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S9)  where νqn =  �  q2 − ω2qn/c2 and the amplitude coefficients satisfy the following conditions:  fqn/f ′  qn = e−νqnd/2/ cos(κqnd/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S10)  fqn/f ′′  qn = cosh (νqnδ/2) / cos(κqnδ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S11)  Without loss of generality, we choose fqn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The corresponding phonon mode function is given by  uφ  qn = fqn  �  gtΩt  Ω2  t − ω2qn  cos(κqnz)ex − iϵtq  ϵzκqn  gzΩz  Ω2z − ω2qn  sin(κqnz)ez  �  eiqxϑ  �d  2 − |z|  �  ϑ  �  |z| − δ  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S12)  We then impose the normalization condition,  � ∞  −∞  dz  �  |uqn(z)|2 +  ��uφ  qn(z)  ��2�  = 1,  (S13)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='0  TT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+page_content='29  which fixes the amplitude coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  The results for the three lowest branches of the h-BN Type I hyperbolic modes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We remark that, on  the z = 0 plane where the 2D material is inserted, the electric fields have only the in-plane components along the propagation  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We also note that, in the air gap region, the electric field is written as the curl of the magnetic field and thus it is purely  transverse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=', uqn ∝ ∇ ×  �  sinh(νqnz)eiqxey  �  for |z| < δ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Since we assume the narrow gap ν, κ ≪ δ−1, we can introduce  the effective confinement length dqn through the relation |uqn(z = 0)| ≃ fqn ≡ 1/  �  dqn, which characterizes the coupling  strength gqn of the 2D electron on the z = 0 plane as discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Details about the mean-field analysis of the magnetoabsorption spectrum  We here provide technical details about the mean-field variational analysis of the magnetoabsorption spectrum discussed in  the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' To this end, we first perform an additional unitary transformation,  ˆV = e  �  qn  cqn  lB (ˆγ†  qn−ˆγqn),  ˆV †ˆγqn ˆV = ˆγqn + cqn  lB  ,  (S14)  which displaces the polariton operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Using the relation �  qn qc2  qn = 0, the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' (8) in the main text is  transformed to  ˆHV U = ˆV † ˆHU ˆV = ℏωc  2  �  ˆπ −  �  qn  qlBˆγ†  qnˆγqn  �2  +  �  qn  ℏωqn  �  ˆγ†  qnˆγqn + cqn  lB  �  ˆγqn + ˆγ†  qn  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S15)  It is useful to rewrite it in terms of the canonically conjugate variables as follows:  ˆHV U  ℏωc  =  ˆX2 + ˆP 2  2  +  �  qn  ϵqn  ˆX2  qn + ˆP 2  qn  2  − ˆπ · ˆΞb + 1  2 : ˆΞ  2  b : +  �  qn  ξqn ˆXqn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S16)  where we denote the conjugate variables associated with the cyclotron motion and the polaritons by  ˆπ =  �ˆa + ˆa†  √  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' i(ˆa − ˆa†)  √  2  �T  =  �  ˆX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ˆP  �T  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  �  ˆγqn + ˆγ†  qn  √  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' i(ˆγqn − ˆγ†  qn)  √  2  �T  =  �  ˆXqn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ˆPqn  �T  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S17)  and introduce  ˆΞb =  �  qn  lBq  2  �  ˆX2  qn + ˆP 2  qn  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ϵqn = ωqn  ωc  + l2  Bq2  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' ξqn =  �  2ωqn  ω3c  gqn  qlB  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S18)  which correspond to the dimensionless total polariton momentum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' the normalized effective polariton energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' and the dimen-  sionless coupling strengths,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The mean-field energy E is given by taking the expectation value of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' (S16) with  respect to the product of coherent states for these conjugate variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The mean-field ground state is then determined by finding  the zero of the derivatives of E, leading to  π = Ξb =  �  qn  lBq  2  �  X  2  qn + P  2  qn  �  , Xqn = −ξqn  ϵqn  , P qn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S19)  c  ω /Ω  0  0  1  2  3  4  5  1  2  3  4  5  c  ω /Ω  0  1  2  3  4  5  c  ω /Ω  0  1  2  3  4  5  ω/Ω  L/d=2π  z  z  z  z  L/d=3π  L/d=4π  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Magnetoabsorption spectra A(ω) at different aspect ratios L/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The cavity thickness d is fixed to be d = 3 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The in-plane  momentum q is discretized as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' (S27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The results are plotted for the h-BN parameters and the in-plane momentum cutoff Λ = 2 nm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  T  7  T  T  F10  q  [1/nm]  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='4  1  2  3  q  z  g   /Ω  d=3nm  d=6nm  d=10nm  q  [1/nm]  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='2  1  2  3  q  z  g   /Ω  d=3nm  d=6nm  d=10nm  L/d=2π  L/d=3π  L/d=4π  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Coupling strengths gq at (a) different cavity thicknesses d with L/d = π and (b) different aspect ratios L/d with d = 3 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The  values corresponding to each of the discrete in-plane momenta q of hyperbolic modes are indicated by the filled red circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The coupling  strengths indicated in panel (a) and (b) correspond to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' 3 in the main text and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' S2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The results are plotted for the h-BN  parameters and the in-plane momentum cutoff Λ = 2 nm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  To obtain the low-energy excitation spectrum, we perform the projection ˆP onto the quadratic fluctuations on top of this  mean-field ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Specifically, we describe the low-energy excitations in terms of the effective quadratic Hamiltonian  given by  ˆHeff = ˆP ˆD† ˆHV U ˆD ˆP,  (S20)  where ˆD induces the mean-field displacements  ˆD† ˆπ ˆD = ˆπ + π,  ˆD† ˆXqn ˆD = ˆXqn + Xqn,  ˆD† ˆPqn ˆD = ˆPqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S21)  The result is  ˆHeff  ℏωc  =  ˆX2 + ˆP 2  2  +  �  qn  ˜ϵqn  ˆX2  qn + ˆP 2  qn  2  + 1  2  ��  qn  ˜ξqneq ˆXqn  �2  − ˆπ ·  �  qn  ˜ξqneq ˆXqn  ≡ 1  2  ˆφ  TM ˆφ,  (S22)  where we introduce the modified effective polariton energies and the effective coupling strengths by  ˜ϵqn ≡ ωqn  ωc  + l2  Bq2  2  − 1  2lBq · Ξb,  ˜ξqn ≡ gqn  ωc  �  2ωqn  ωc  1  ωqn  ωc + l2  Bq2  2  ,  (S23)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' In the last line, we rewrite the Hamiltonian by using the matrix M and the vector of the conjugate variables  ˆφ =  �  ˆX, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' , ˆXqn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' , ˆP, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' , ˆPqn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  �T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S24)  The low-energy excitation spectrum {ωλ} can then be obtained from the Williamson eigenvalues of M as follows:  STMS = diag({ωλ/ωc} , {ωλ/ωc}),  (S25)  where S is a symplectic matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We note that this treatment is exact in the long-wavelength limit qlB → 0 in which Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' (S16)  simplifies to the quadratic Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  Finally, the absorption spectrum can be obtained from  A(ω) ≃ Re  �� ∞  0  eiωt⟨0| ˆD†ˆae−i ˆ  HV Utˆa† ˆD|0⟩  �  ≃ Re  �� ∞  0  eiωt⟨0|  �  ˆa + πx − iπy  √  2  �  e−i ˆ  Hefft  �  ˆa† + πx + iπy  √  2  �  |0⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S26)  The last line can easily be evaluated by using the diagonalized basis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' (S25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' We note that the results in the main text are  obtained for the discretized in-plane wavenumbers as follows:  qi ∈  �  −Λ, −N − 1  N  Λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' , − Λ  N , 0, Λ  N , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' , N − 1  N  Λ, Λ  �  , i ∈ {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  (S27)  000  810  800880  00  0011  Here, the integer number N is related to the lateral system size L via  � ΛL  2π  �  = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The results obtained for the cavity consisting  of ultrathin h-BN materials with different aspect ratios L/d are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' The coupling strengths to each hyperbolic  phonon polariton mode with discretized in-plane momentum are plotted for different thicknesses and aspect ratios in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content='  When the lateral size L is increased, the coupling strength gq to each mode decreases uniformly as gq ∝ L−1 while there appear  a larger number of cavity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Consequently, a key feature of the ultrastrong coupling regime in the spectrum, that is, the  formation of the localized Landau-polariton mode remains almost the same independently of the lateral size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
+page_content=' Meanwhile, the  increase of L/d leads to the appearance of dense anticrossed branches originating from the hybridization with the continuum  cavity modes above the lower Landau-polariton mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQfLQam/content/2301.03712v1.pdf'}
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+1 
+ 
+Domain Wall-Magnetic Tunnel Junction Analog 
+Content Addressable Memory Using Current and 
+Projected Data 
+ 
+Harrison Jin, Hanqing Zhu, Keren Zhu, Thomas Leonard, Jaesuk Kwon, Mahshid Alamdar, Kwangseok Kim, 
+Jungsik Park, Naoki Hase, David Z. Pan, and Jean Anne C. Incorvia 
+ 
+     Abstract—With the rise in in-memory computing architectures 
+to reduce the compute-memory bottleneck, a new bottleneck is 
+present between analog and digital conversion. Analog content-
+addressable memories (ACAM) are being recently studied for in-
+memory computing to efficiently convert between analog and 
+digital signals. Magnetic memory elements such as magnetic 
+tunnel junctions (MTJs) could be useful for ACAM due to their 
+low read/write energy and high endurance, but MTJs are usually 
+restricted to digital values. The spin orbit torque-driven domain 
+wall-magnetic tunnel junction (DW-MTJ) has been recently 
+shown to have multi-bit function. Here, an ACAM circuit is 
+studied that uses two domain wall-magnetic tunnel junctions (DW-
+MTJs) as the analog storage elements. Prototype DW-MTJ data is 
+input into the magnetic ACAM (MACAM) circuit simulation, 
+showing ternary CAM function. Device-circuit co-design is carried 
+out, showing that 8-10 weight bits are achievable, and that 
+designing asymmetrical spacing of the available DW positions in 
+the device leads to evenly spaced ACAM search bounds. Analyzing 
+available spin orbit torque materials shows platinum provides the 
+largest MACAM search bound while still allowing spin orbit 
+torque domain wall motion, and that the circuit is optimized with 
+minimized MTJ resistance, minimized spin orbit torque material 
+resistance, and maximized tunnel magnetoresistance. These 
+results show the feasibility of using DW-MTJs for MACAM and 
+provide design parameters. 
+ 
+Index Terms—analog circuits, associative memory, content 
+addressable memory, magnetic domain walls, magnetic tunnel 
+junctions, memory, neural networks, spin electronics, spintronics 
+I. INTRODUCTION 
+S data size increases and Moore’s law slows down, 
+alternative in-memory computing (IMC) architectures 
+are being studied and used to efficiently process 
+information. Conventional analog IMC often uses 
+crossbar array architectures and nonvolatile memory (NVM) 
+such as resistive random-access memory (RRAM) and has 
+shown many applications in neural network acceleration [1]–
+[3], neuromorphic computing [4], statistical learning [5], signal 
+processing [6], [7], scientific computing [8], [9], and other 
+fields. However, there is a challenge extending these 
+ 
+This work was supported by the Samsung Global Research Outreach (GRO) 
+Program. The authors also acknowledge support from the Department of 
+Energy Office of Science Microelectronics Co-Design project COINFLIPS (J. 
+Kwon) and the National Science Foundation Graduate Research Fellowship 
+under Grant No. 2020307514 (T. Leonard). (Corresponding author: Jean Anne 
+C. Incorvia). 
+architectures to operations other than matrix dot multiplication. 
+Existing IMC architectures rely on extensive analog to digital 
+conversion (ADC) and digital to analog conversion (DAC) to 
+switch between matrix multiplication and other computations 
+such as activation functions [10]. The consequent conversion 
+cost greatly dilutes efficiency in both power and speed; e.g., 
+data converters can consume around 85% of the total energy in 
+a typical RRAM-based neural network accelerator [11]. This 
+energy overhead is especially critical for edge computing. 
+To address this challenge, analog content-addressable      
+memories (ACAM) are being recently studied for IMC [12], 
+[13]. ACAM cells are designed to detect whether the value 
+corresponding to input search data is located within a range. 
+Unlike conventional CAM, the input data of ACAM are 
+allowed to be analog values or multi-bit. ACAM can be used to 
+fuse computation and data conversion for time- and energy-
+efficient conversion between analog and digital signals. For 
+example, a recently proposed RRAM-based  ACAM shows 
+delays of 350 ps and >1 fJ energy consumption, and a recently 
+shown ferroelectric-based ACAM shows up to 3-bit precision 
+to perform in-memory nearest neighbor searching to perform 
+few-shot learning [14], [15]. The ideal requirements of NVM 
+for ACAM include low switching energy, low read energy, high 
+endurance, and controllability of setting the memory element to 
+a given analog value. Thus, magnetic tunnel junctions (MTJs) 
+are a natural choice for ACAM, due to their theoretically 
+unlimited endurance, modest switching voltage, and back-end-
+of-the-line compatibility for integration into the ACAM cell 
+[16]. The domain wall-magnetic tunnel junction (DW-MTJ) 
+allows 
+for 
+analog-like 
+programming 
+of 
+tunnel 
+magnetoresistance (TMR) through modulation of the position 
+of a DW underneath an MTJ, using either spin transfer torque 
+(STT) or spin orbit torque (SOT). Recent work has 
+demonstrated the use of STT-based MTJs in ternary CAM 
+applications, but ternary CAMs are associated with 
+considerably greater area consumption costs in order to achieve 
+the same density of bits as their contemporary analog and multi-
+bit counterparts [15], [17], [18].  STT magnetic random-access 
+H. Jin, H. Zhu, K. Zhu, T. Leonard, J. Kwon, M. Alamdar, D. Z. Pan, and J. 
+A. C. Incorvia are with the Department of Electrical and Computer Engineering, 
+University of Texas at Austin, Austin, TX 78712 USA (e-mail: 
+incorvia@austin.utexas.edu). M. Alamdar is now with Samsung Austin 
+Semiconductor, Austin, TX, 78754 USA. 
+K. Kim, J. Park, and N. Hase are with Samsung Advanced Institute of 
+Technology (SAIT), Samsung Electronics Co., Suwon, 16678, South Korea (e-
+mail: stone99.kim@samsung.com). 
+A 
+
+2 
+ 
+memory (STT-MRAM) based on the MTJ has also been shown 
+in crossbar arrays to perform analog multiply-and-accumulate 
+operations [19]. But, much of the previous works of spintronics 
+for CAM has been focused on binary and ternary functionality. 
+This is because achieving controllable multiple resistance levels 
+in an MTJ is a challenge.  
+Here, we propose and verify the performance of a DW-MTJ 
+ACAM prototype for high-throughput, high-speed searches. 
+The DW-MTJ ACAM cell compares an analog input to a range 
+of stored values, which is set using the programmable resistance 
+of the DW-MTJ through the modulation of the DW. Circuit 
+simulations using prototype results from DW-MTJ device 
+cycling data verify that MTJs can be effectively integrated into 
+ACAMs as programmable elements. Additionally, projected 
+data using optimized magnetic stack parameters demonstrate 
+the feasibility of performing analog multi-bit search operation 
+for implementation in high-throughput computing. Our 
+proposed DW-MTJ ACAM benefits from up to a 44 × decrease 
+in energy consumption per search, and 2.86 × faster search 
+time, per bit compared to existing MTJ ternary CAM circuits. 
+Additionally, the high programmability in linearity through 
+different DW-MTJ geometries, combined with low variation 
+within each weight, allows our proposed ACAM to circumvent 
+time-costly weight programming that is necessary in other 
+emergent ACAM designs; thus, potentially reducing write 
+times by up to 3 orders of magnitude. These results show the 
+potential for DWs and MTJs to be used in these energy-efficient 
+circuits. 
+ 
+II. CELL DESIGN AND METHODS 
+Fig. 1a shows the DW-MTJ multi-weight NVM used for the 
+magnetic ACAM (MACAM) design. A top-pinned MTJ stack 
+has its bottom heavy metal and magnetic free layer extended 
+into a magnetic wire that hosts a magnetic DW. SOT current 
+applied from IN to CLK sets the DW position at one of the 
+notches, which in turn sets the resistance between the CLK and 
+OUT terminals. We have previously shown DW-MTJ 
+prototypes with 3-5 stable resistance levels at room 
+temperature; due to the physical setting of the resistance by the 
+DW position, highly controllable weights are achievable as long 
+as the DW is set to the desired notch [20]. 
+Two DW-MTJs are integrated into the 8-transistor MACAM 
+cell shown in Fig. 1b [18]. Minimum and maximum voltage 
+bounds are set using the search lines (𝑆𝐿!"#$, 	𝑆𝐿%&'), and input 
+search voltage is applied through the data line, 𝑉(%. The search 
+result is reflected on the match line (𝑀𝐿) behavior. Fig. 1c 
+depicts how the DW position in the DW-MTJ determines a 
+match. Programmable resistance states demonstrate how 
+different voltage states on both MTJs can be used to write 
+different voltage bounds. A match can be yielded anywhere 
+between the upper and lower bounds. 
+To understand the cause, the drain-to-source voltage of the 
+input transistor 𝑇)*+ (see Fig. 1b), 𝑉(,, and its drain current, 𝐼(, 
+can be described using the form: 
+ 
+𝐼( =	
+,%!"#!-.$%
+(0&"'(10)*+), 
+(1) 
+where 𝑅'"34 is the resistance of the heavy metal plus free layer 
+patterned wire and 𝑅567 is the read-out resistance of the DW-
+MTJ. Subsequently, 
+ 
+𝑉(, = 𝑆𝐿$"#$ − 𝐼(2𝑅'"34 + 𝑅5674.  
+(2) 
+ 
+The saturation region of the 𝑇)*+ in which lower bounds will 
+remain matched can then be defined as: 
+\ 
+(a) 
+ 
+(b) 
+ 
+(c) 
+ 
+Fig. 1. 
+(a) Diagram of the three-terminal DW-MTJ. Resistance states are 
+programmed using voltage pulses from IN and CLK, and read from IN to OUT. 
+Notches are shown that assist repeatable setting of the DW. (b) Circuit 
+schematic of proposed MACAM circuit. Minimum and maximum voltage 
+bounds are set using 𝑆𝐿  lines, and input search voltage is applied through the 
+data line, 𝑉(%. The search result is reflected on the match line (𝑀𝐿) behavior. 
+(c) Programmable resistance states demonstrate how different voltage states on 
+both MTJs can be used to write different voltage bounds. A match can be 
+yielded anywhere between the upper and lower bounds. Different notch 
+locations are shown on a 5-weight DW-MTJ synapse. “N4” corresponds to 
+antiparallel and “N0” corresponds to parallel resistance relative to the reference 
+magnetic layer. 
+
+CLK
+MTJ
+Z
+OUT
+X
+DW
+INLower
+Upper
+Bound
+Bound
+ML
+SLHigh
+W.
+N
+IN2
+V
+SL
+VDL
+Z
+Pinned Layer
+Domain Wall
+OUT
+X
+IN
+CLK
+Free
+Layer
+Heavy MetalLower Bounds
+Upper Bounds
+NO
+N4
+NO
+N4
+SL
+SL
+high
+low3 
+ 
+ 
+𝑉(% > 6
+,%!"#!-.*!,-$
+80&"'(10)*+9:. + 𝑉6!,)*, 
+(3) 
+ 
+Where 𝑉6!,<( and 𝑉6!,)* (in Figs. 3b, e) are the threshold 
+voltages of the pulldown (𝑇<() and input (𝑇)*) transistors, 
+respectively. 𝑘= = µ=𝐶&>
+?
+% , the large signal MOSFET 
+transconductance parameter, where µ= is the mobility of 
+electrons on the channel surface, 𝐶&> is the oxide capacitance, 
+and 
+?
+%  is the ratio of channel width to channel length. 
+Meanwhile, the linear region where the lower bounds will 
+remain matched can be defined using the approximation: 
+ 
+𝑉𝐷𝐿 <
+𝑉𝑇ℎ,𝑃𝐷
+𝑔𝑚 !
+1
+𝑟𝑜 +
+1
+𝑅𝑤𝑖𝑟𝑒+𝑅𝑀𝑇𝐽". 
+(4) 
+ 
+Here, 𝑔𝑚 is the small signal transconductance and 𝑟𝑜 is the 
+output resistance, both of which are intrinsic constants to the n-
+type MOSFET, 𝑇)*+. The opposite can be said about the upper 
+bound, as the gate voltage of the pulldown transistor is inverted.  
+    As the input search voltage decreases to 0, so does the drain 
+current, 𝐼(, of the input transistor, as is expected by the 𝑉J,-𝐼( 
+relationship of a n-type MOSFET. In the ACAM circuit, 𝑉J,, 
+the gate-to-source voltage of the input transistor, is equal to the 
+analog search input, 𝑉(%. To maintain the matching condition, 
+it is crucial that 𝑉(, remains less than that of 𝑉6$,<( in order to 
+maintain the match line voltage. To counteract this, the MTJ 
+resistance must increase accordingly to maintain the adequate 
+voltage drop across the MTJ to keep 𝑇<( in its OFF state.  
+    Cadence Virtuoso and Spectre are used to characterize the 
+functionality of the MACAM circuit. The circuit is constructed 
+using 40 nm gate processes technology, and the MTJ is modeled 
+as two resistances in series,	𝑅'"34 + 𝑅567. To verify 
+performance, a two-dimensional parametric sweep of search 
+voltages at different 𝑅567 is run. The input search voltage is 
+swept from the full range established by the search lines (𝑆𝐿%&' 
+and 𝑆𝐿!"#$	set to 0 V and 1 V respectively) at a preprogrammed 
+𝑅567. This is repeated at different 𝑅567 to determine the 𝑀𝐿 
+behavior as a function of the search inputs, to understand the 
+limits of both the DW-MTJ device and CMOS circuitry.  
+ 
+III. TCAM FUNCTIONALITY WITH PROTOTYPE DATA 
+    To start, experimental data from DW-MTJ prototypes is 
+input into the constructed circuit model, to study their function 
+for CAM. Fig. 2a shows the device data with device SEM 
+shown in the inset. A 50 ns voltage pulse is applied from IN to 
+CLK, followed by measurement of 𝑅567. The voltage pulse 
+amplitude is increased from 2 to 4 V in 0.1 V steps, showing 
+three distinct resistance values as the DW eventually de-pins 
+and moves to another notch. This is repeated for 10 cycles; see 
+Ref [20] for details. Nominal TMR of the magnetic stack was 
+measured using current in-plane TMR = 170%. The resistance-
+area product for parallel MTJ resistance, RA, was measure RA 
+= Ω × µ𝑚K, with a heavy metal layer of tantalum. The 
+trapezoidal synapse device used an MTJ with top-down area of 
+1.575 µ𝑚K. 
+    From this data, we extract total resistances of 𝑅567 = 67 Ω, 
+75 Ω, and 93 Ω, with cycle-to-cycle resistance variation = 2.5%. 
+Inputting these values into the MACAM circuit, ternary CAM 
+behavior is seen, shown in Fig 2b, which plots the ML voltage 
+vs. search voltage for the different relative MTJ weights. The 
+search voltage is the analog search input from 𝑉(%. From these 
+curves, the lower bound 𝐵% (V) and upper bound 𝐵L (V) are 
+defined as the search voltage values that set 𝑀𝐿 = 0.5. The 
+storage range is defined as 𝑆𝑅	 =	𝐵L − 𝐵%. The resulting 
+maximum 𝑆𝑅 = 0.109	V that can be achieved using these 
+measured resistance weights is between 𝐵% = 0.854	V and 
+𝐵M = 0.963	V, which can be seen as the don’t care or X, state 
+that includes all values within this range; alternatively, by 
+setting the upper and lower bounds DW-MTJs to the same 
+weight, we can also achieve a cell which will always result in a 
+mismatch. Because there are 3 resistance weights, it is also 
+possible to achieve two smaller resistance states as well, which 
+can be used in binary implementation, depicted as the 0 and 1 
+states in Fig. 2b. The combination of the X bit with the smaller 
+0 and 1 bits can then be used to implement ternary CAM 
+functions. Ternary CAM application in memory-augmented 
+neural networks have previously been demonstrated for one-
+shot learning in Ref. [21]. 
+ 
+(a) 
+ 
+(b) 
+ 
+(c) 
+ 
+Fig. 2.  
+(a) Data of 𝑅567 vs. applied voltage pulse amplitude of device 
+shown in inset, showing 3 distinct weights. Each color is another cycle of the 
+same device showing repeatable cycle-to-cycle behavior. (b) Calculated 
+ternary CAM performance of device from (a). The dotted line at 0.5 V on the 
+ML shows the minimum ML voltage necessary to yield a match. The lower 
+and upper bound of each discrete level (0, 1, or X) is marked by the two points 
+of intersection with the dotted line. (c) Top-down design of 9 weight DW-MTJ 
+wire, with lithographically patterned magnetic wire shown in grey with 9 
+notches, and the MTJ for read-out is shown in teal. 
+ 
+ 
+
+100
+1.0
+Notch 0-2 (X)
+Notch 0-1 (1)
+0.8
+90
+Notch 1-2 (0)
+80
+70
+0.2
+500 nm
+60
+0.0
+2.0
+2.5
+3.0
+3.5
+4.0
+0.7
+0.8
+0.9
+1.0
+50 ns write pulse amplitude (V)
+Search Voltage (V)y
+NO
+N1 N2 N3 N4 N5 N6 N7
+N84 
+ 
+IV. ACAM FUNCTION AND DESIGN FOR 9-RESISTANCE 
+WEIGHT DW-MTJ 
+With existing prototypes showing 3-5 resistance levels, it is 
+feasible to extend to 8 resistance levels by extending the length 
+of the DW track and including 9 notches, depicted in Fig. 2c.  
+Assuming 𝑅'"34	 = 40	Ω, and 𝑅567 = 	70	Ω, with TMR = 
+170%, more analog and multi-bit capabilities of the cell can be 
+demonstrated.  
+Under these assumptions, we can simulate the performance 
+of 𝐵% and 𝐵L’s circuit components. The specific voltage value 
+of the bound associated to the 𝐵% circuit can be programmed 
+using the MTJ-transistor voltage divider circuit, which is 
+demonstrated in Figs. 3a, b. Furthermore, it can be seen that the 
+relationship between the DW-MTJ resistance weights and their 
+associated bounds is such that lower resistances are necessary 
+to write higher bounds, while larger resistances are required to 
+achieve the lower bounds, shown in Figs. 3a, d where the 
+parallel resistance is in notch “0” and increases with each notch 
+up to notch “8”, which is the anti-parallel state. 𝐵L (see Figs. 
+3d, e) experiences a similar matching condition, but conversely 
+requires an input voltage smaller than the threshold at 𝑇<(K due 
+to the CMOS inverter prior to the pulldown transistor. Fig. 3c 
+depicts the match line current (blue) and match line voltage 
+(black) during an input voltage sweep from 0 to 1 V, depicting 
+a mismatch-match-mismatch event. To assess the 𝑀𝐿 threshold 
+voltage to yield a match, a buffer is placed at the end of the 𝑀𝐿, 
+and its transfer function is shown in Fig. 3f, revealing a 
+minimum 𝑀𝐿 voltage at 0.5 V to be considered a match. 
+    When the 𝑀𝐿 is plotted against the search voltage for the 9 
+evenly spaced notches in the DW-MTJ, the resulting 
+ 
+Fig. 3. 
+(a), (b) 𝐵! match performance based off different programmed resistance weights. An input signal less than the 𝐵! threshold yields a mismatch by 
+forcing the transistor PD1 in (b) to pull down the ML voltage to ground. (c), (d) Conversely, an input signal greater than the 𝐵" threshold in (d) forces the transistor 
+PD2 to pull down ML voltage to ground using similarly to (c) but with an inverting CMOS prior to the pull-down transistor PD2. (e) Simulation of voltage and 
+current measured through the match line during an input voltage sweep. (f) Transfer characteristics of voltage buffer at the match line output indicating a matching 
+threshold at 500 mV. 
+ 
+Fig. 4. 
+(a) Linear notch spacing vs. (b) nonlinear notch spacing effects on 
+9-weight multi-bit performance. 
+
+ML D
+1.0
+1.0
+3E-5
+SL
+0.8
+(A)
+0.8
+2E-5
+M0.6
+M0.6
+MTJ1
+Notch 8 
+-Notch 0
+W
+PD1
+MTJ,1
+0.4
+1E-5 
+ML
+0.2
+0.2
+SL
+0E+0
+LOW
+0.0
+0.0
+VDL
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Search Voltage (V)
+Search Voltage (V)
+(a)
+(b)
+(c)
+1.0
+1.0
+M0.8
+High
+0.8
+o.6
+MTJ2
+PD2
+Notch 8
+Notch 0
+0.4
+0.4
+IN2
+B
+0.2
+Low
+0.0
+VDL
+0.0
+0.0
+0.2
+0.4
+0.5
+0.6
+0.7
+0.8
+¥0.9
+1.0
+0.6
+0.8
+1.0
+Search Voltage (V)
+ML (V)
+(d)
+(e)
+(f)1.0
+0.8
+s7
+so
+M
+0.4
+0.2
+0.0
+0.4
+0.6
+0.8
+1.0
+Search Voltage (V)
+1 μm
+s0 s1 s2 s3s4 s5 s6 s7
+MTJ
+(a)
+1.0
+s0
+0.8
+s7
+0.2
+0.0
+0.4
+0.6
+0.8
+1.0
+Search Voltage (V)
+1 μm
+s0-s4
+s5
+s6
+s7
+MTJ
+(b)5 
+ 
+relationship is 9 unevenly spaced discrete levels, as shown in 
+Fig. 4a. While ACAM behavior is still achieved, it is shown that 
+linearly spaced notches produce uneven widths of the distinct 
+levels in the multi-bit circuit operation. With these matching 
+characteristics, it would not be suitable to implement multi-bit 
+performance, such as that shown in Ref. [15]. Thus, designing 
+the DW-MTJ with unevenly spaced notches, to achieve 
+approximately exponentially increasing MTJ resistance vs. 
+notch, alleviates this issue. When considering notch spacing, 
+the minimum pitch between notches is no less than the width of 
+the notch itself to avoid stochastic movement of the DW 
+between adjacent notches from factors like variations in 
+magnetic wire geometry and thermal effects (i.e., a notch with 
+a 100 nm width is restricted to have a minimum spacing of 100 
+nm from the next notch). Fig. 4b shows the re-designed 9 
+notches, and the resulting ACAM function of ML vs. search 
+voltage, which shows evenly spaced discrete levels. These 
+results show a useful benefit of DW-MTJs for these circuits 
+because device behavior can be tuned to the circuit by adjusting 
+device geometry.  
+ 
+V. MACAM USING MTJ WAFER DATA 
+    While the previous section focused on the impact of 𝑅567 on 
+the ACAM function, the resistance of the heavy metal plus 
+magnetic track, 𝑅'"34, will also impact the circuit performance, 
+since the read current of the DW-MTJ runs through the DW 
+racetrack wire and out the MTJ. For SOT-driven DW motion, 
+𝑅'"34 is dominated by the resistivity of the heavy metal 
+underneath the DW track. Here, we consider 3 common heavy 
+metals used in SOT-MRAM: platinum, 𝛼-tungsten, and 
+tantalum; 𝛽-tungsten and other similar large spin Hall angle 
+materials were not included due to their known high resistivities 
+[22]–[24]. 
+ 
+A. Magnetic Stack Material and Device Characteristics 
+    SOT-MRAM thin film stacks were grown with heavy metal 
+layer of 7 nm-thick α-tungsten and measured using CIPT, 
+showing average TMR = 170%, and average RA product = 35 
+Ω × µ𝑚K. Using these measured stack characteristics, we then 
+evaluated the impact of heavy metal resistivity on device 
+performance. 𝑅'"34 was calculated using the excess length of 
+wire outside the area of the MTJ, with 𝑙 × 𝑤 dimensions of 
+0.75	µ𝑚	 × 400	𝑛𝑚 on all 3 stacks. The resistivities of the 
+heavy metal thin films used were assumed from literature to be 
+15 µΩ × 𝑐𝑚 for platinum, 21	µΩ × 𝑐𝑚 for 𝛼-tungsten, and 25 
+µΩ × 𝑐𝑚 for tantalum [25]–[27]; thus, resulting in a projected 
+wire resistance of 40 Ω, 56 Ω, and 67 Ω, respectively. The top-
+down geometry of the MTJ has a 𝑙 × 𝑤 dimensions of 
+3	µ𝑚	 × 100	𝑛𝑚, resulting in a parallel resistance of ~117 Ω. 
+ 
+B. Simulated Results 
+    Fig. 5 shows the performance of the circuit simulated using 
+device parameters extrapolated from each of the 3 stacks. Fig. 
+5a shows the maximum storage range of all 3 of these devices 
+plotted against each other. The storage range, 𝑆𝑅, is the 
+maximum distance that can be achieved between 𝐵L and 𝐵%. 
+Achieving larger 𝑆𝑅 allows for greater density of discrete levels 
+in analog multi-bit applications. Additionally, maximizing the 
+accessible storage range within the minimum and maximum 
+possible bounds, established by 𝑆𝐿$"#$ and 𝑆𝐿O&', reduces the 
+energy cost of peripheral circuitry used to scale down that of 
+the two DW-MTJ-CMOS subcircuits. This is important because 
+the limited TMR ratios available in current MTJ devices allows 
+programming bounds to only a fraction of the total available 
+range. For the three SOT material types, Fig. 6a shows the 
+MACAM bound (V) associated with different values of 𝑅567. 
+The presence of wire resistance results in unwanted static 
+voltage drops within the subcircuits shown in Figs 3b, e. Due to 
+the low resistivity of platinum thin films, platinum has the 
+highest 𝐵L overall due to having the lowest wire resistance, 
+seeing as both 𝐵% and 𝐵L increase with decreasing MTJ 
+resistance. Consequently, the maximum 𝑆𝑅 of the ACAM cell 
+utilizing the platinum stack is 245 mV, which is 16% greater 
+than α-tungsten and 28% greater than tantalum. Another 
+demonstration showing the ability to achieve 5 discrete levels 
+can be seen in Figs. 5b-d. Within the range of voltages available 
+to all three stacks, they are all capable of comfortably fitting 5 
+discrete levels; that is, the minimum pitch between notches is 
+reliably spaced as described in Section IV. 
+ 
+VI. DW-MTJ MATERIALS PARAMETERS OPTIMIZATION FOR 
+ACAM 
+   The results so far show that the MACAM circuit can achieve 
+both ternary and multi-bit-like functionality, using prototype 
+data, measured MTJ stack data with often-used heavy metal 
+materials, and feasible extension from the measured 3-5 notches 
+to 9 notches. Here, we inspect design considerations of the DW-
+MTJ to further optimize the ACAM cell’s performance, to 
+predict what the ideal properties of the DW-MTJ should be for 
+this application.  
+  
+Fig. 5. 
+(a) Simulation of three different SOT heavy metals showing total 
+range, as well as individual states assuming 5 notches in the (b) platinum stack, 
+(c) 𝛼-tungsten stack, and (d) tantalum stack. 
+ 
+
+1.0
+Pt
+0.8
+αW
+0.8
+Ta
+0.6
+M
+M 0.4
+0.2
+0.2
+0.0
+0.0
+0.5
+0.6
+0.7
+0.8
+0.9
+1.
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Search Voltage (V)
+Search Voltage (V)
+(a)
+(b)
+0.8
+0.8
+0.6
+0.6
+M
+M 0.4
+0.2
+0.2
+0.0
+0.0
+0.5
+0.6
+0.7
+0.8
+0.9
+1.
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Search Voltage (V)
+Search Voltage (V)
+(c)
+(d)6 
+ 
+ A. Stack Characteristics and DW-MTJ Geometry 
+Considerations in Design Optimization 
+When considering the factors important to optimizing the 
+DW-MTJ for greater storage range in the ACAM cell, stack 
+characteristics (RA, TMR, and resistivity) and device geometry 
+(heavy metal thickness and MTJ top-down area) play large roles 
+in minimizing resistance losses and increasing total storage 
+range. Figs. 6b, c show the storage range decreases with 
+increasing RA and increasing 𝑅'"34 ; Fig 6d shows storage 
+range improves with TMR with decreasing benefits for high 
+TMRs. The area of the MTJ with respect to stack RA cannot be 
+neglected, since proportionally scaling down a device can have 
+unintended consequences on the total storage range of said 
+device. Fig. 6b shows proportionally scaling down a device 
+with feature node size of 50 nm down to 25 nm results in a 
+subsequent 4 × increase of parallel resistance, as both the 
+length and width of the MTJ are each proportionally scaled 
+down by 
++
+K ×. The resistance vs. feature node can be accounted 
+for in the circuit design. 
+ 
+B. Design and Simulation of Prototype with Projected Data 
+    Thus, we design a theoretical MTJ stack with ideal 
+parameters to demonstrate projected prototype performance. 
+Platinum is chosen as the heavy metal due to its low resistivity 
+while still having a good spin Hall angle for energy efficient 
+DW motion [21]. The Pt thin film layer is assumed to be 15 nm 
+thick, and the RA is assumed to be 5 Ω × µ𝑚K, on the low end 
+of what is feasible with today’s MgO-based MTJs. We first 
+consider a reasonable TMR = 200%, which is currently 
+achievable. The device is designed to accommodate an MTJ 
+with dimensions of 1.5	µ𝑚 × 50	𝑛𝑚 to be able to make use of 
+25 nm pitch between notches. With this, the parallel resistance 
+of the device is 67 Ω and the anti-parallel resistance is 200 Ω. 
+Fig. 7 shows the simulation results, where the ACAM is 
+demonstrating 8 and 10 discrete levels by choosing 9 or 11 
+notches respectively, or also a minimum resolution of 3-bits. 
+The trends observed in Fig. 7  reveal that large TMR and low 
+RA work to improve the maximum storage range of the ACAM 
+cell: the storage range increases from 245 mV, projected from 
+the realistic magnetic stacks in the previous section, up to 300 
+mV in the ideal stack. The increase in heavy metal layer 
+thickness from 7 nm to 15 nm constitutes a decrease in wire 
+resistance from 40 Ω to 19 Ω. This, in combination with the 
+30% increase of TMR, extends the projected storage range by 
+~18%. Given the nonlinear behavior of wire resistance, shown 
+in Figs. 4 & 5, the increased range of programmable resistances, 
+and their associated voltage bounds, allow for larger density of 
+notch spacing necessary in the lower discrete levels for multi-
+bit implementations. Fig. 7 shows 8 and 10 discrete levels, with 
+devices designed such that they meet the design considerations 
+for minimum notch spacing described in Section IV.  
+C. Simulation of Prototype with 1000% TMR Ratio Projected 
+Data 
+    In Fig. 8, the same parameters are assumed except TMR = 
+1000%, not yet commercially achievable today. The increase in 
+storage ranges from 200% TMR to 1000% TMR is 300 mV to 
+480 mV. If this high on/off ratio could be achieved, the cell 
+would be capable of greater multi-bit precision, as much as 16 
+discrete levels, or 4-bits, as shown in Fig. 7d.  
+VII. ANALYSIS AND DISCUSSION 
+    To evaluate the energy consumption of the MACAM, we 
+simulate the average DC current through the 𝑀𝐿 and integrated 
+it over several inference passes. Using this method, the 
+estimated energy consumption during one search period in our 
+cell is roughly 0.92 fJ per search operation. It should be noted, 
+however, that the energy consumption from periphery circuits 
+(match line pre-charging, search line drivers, DAC, etc.) is 
+estimated to consume up to an additional 0.52 fJ [18]. To 
+estimate the total area consumption of the CMOS components, 
+the total sum of all transistors was taken and assumed to be 
+~90% of the total area consumption; thus, giving a top-down 
+circuit area consumption of  ~36 µ𝑚K using a 40 nm CMOS 
+technology node. The largest dimension of MTJ devices used 
+in previous simulations does not exceed 5.54 µ𝑚K; thus, DW-
+MTJ placement back-end-of-the-line on the CMOS circuit 
+would not affect the overall top-down area of the circuit. 
+ 
+Fig. 6. 
+(a) Relation of ML threshold bound vs. 𝑅567 for the three heavy 
+metal types. (b) Performance changes from proportional geometric scaling of 
+device with 50 nm and 25 nm pitch between notches. (c) Change in maximum 
+writable 𝑆𝑅 of platinum-based MACAM device with wire resistance. (d) 
+Change in maximum writable storage range with TMR. 
+  
+ 
+(a) 
+ 
+(b) 
+ 
+Fig. 7 
+(a) DW-MTJ utilizing platinum heavy metal layer with ideally 
+optimized RA, scaled geometry, 200% TMR, and minimal wire resistance to 
+demonstrate multi-bit performance of 9 notches and 8 distinct levels. (b) 
+Identical parameters assumed for 11 notches and 10 distinct levels. 
+ 
+
+0.18
+25 nm notch pitch
+1.0 Pt
+50 nm notch pitch
+aw
+0.16
+Ta
+M0.9
+ound
+0
+0.8
+S
+B
+0.10
+0.7
+0.08
+50
+100
+150
+200
+20
+30
+40
+50
+60
+70
+RMTJ (2)
+RA (Q·μum?)
+(a)
+(b)
+0.5
+0.5
+0.4
+0.4
+M
+M
+0.3
+SR
+0.2
+0.2
+0.1
+0
+50
+100
+150
+200
+0
+200
+400
+600
+800 1000
+Rwire (Q2)
+TMR (%)
+(c)
+(d)0.8
+0.8
+0.6
+0.6
+M
+0.4
+0.4
+ML
+M
+0.2
+0.2
+0.0
+0.0
+0.6
+0.7
+0.8
+0.9
+1.0
+0.6
+0.7
+0.8
+0.9
+1.0
+Search Voltage (V)
+Search Voltage (V)7 
+ 
+ 
+TABLE I 
+COMPARISON OF EMERGENT ACAM ARCHITECTURES* 
+ 
+PARAMETER 
+FEFET 
+[28] 
+RRAM 
+[18][29] 
+SRAM 
+[18] 
+DW-MTJ 
+[30] 
+AREA 
+CONSUMPTION 
+49 F2 
+48,828 F2 
+918,750 
+F2 
+22,875 F2 
+TECHNOLOGY 
+NODE 
+45 nm 
+16 nm 
+16 nm 
+40 nm 
+NON-
+VOLATILITY 
+Yes 
+Yes 
+No 
+Yes 
+ON/OFF RATIO 
+104-106 
+106 
+106 
+1.5-6 
+VARIATION 
+High 
+High 
+Low 
+Low 
+LINEARITY 
+Low 
+Low 
+High 
+High 
+SEARCH 
+LATENCY 
+~10 ns 
+~50 ps 
+N/A 
+350 ps 
+ENDURANCE 
+105 
+1012 
+>1015 
+>1015 
+ENERGY (PER 
+SEARCH) 
+0.07 fJ 
+0.52 fJ 
+0.165 fJ 
+0.92 fJ 
+*Values from individual cell 
+ 
+    Some additional energy costs can be found in the necessary 
+circuits to perform read and write operations within the ACAM 
+cell. The energy required to update the DW-MTJ by a single 
+weight is on average ~0.1 pJ in few-100 nm prototypes [20], 
+which can be scaled to ~2 fJ for 15 nm feature sizes [31]. This 
+energy can be reduced through scaling and device engineering. 
+The match line output also requires a sensing circuit based on a 
+transimpedance amplifier (TIA), which has an associated 
+energy dissipation of about 2.5 pJ over the course of one search 
+operation [32]. This relatively high energy can be effectively 
+reduced using large ACAM arrays to amplify integrated 
+currents. 
+    The modest TMR of MTJs are considerably smaller than that 
+of the relatively large on/off conductance ratios of FeFETs and 
+RRAM. The operation in the relatively small range of 
+conductivities leads to reduced noise robustness and potential 
+energy costs to scale voltage inputs to a range that can be 
+accommodated by the MACAM. However, FeFETs and 
+modern memristor technology continue to suffer from high 
+non-linearity as well as inconsistent cycle-to-cycle weight 
+variation without the assistance of external circuitry, which in 
+the case of FeFETs can results in a verification period that is 
+microseconds in length [15]. Additionally, the physical 
+robustness of SOT switching MTJs introduces a considerably 
+larger endurance than 2-terminal devices, The ability to tune the 
+change in resistance through the device geometry also provides 
+unique ways to adapt MTJs for the circuit. 
+    Furthermore, at the system level, the DW-MTJ-informed 
+ACAM demonstrates the ability to perform a “fusion” of 
+nonlinear activation and ADC. The search operation of ACAM 
+can be used to binarize an analog input signal, while also 
+introducing an in-situ nonlinearity characteristic. Thus, this 
+eliminates the need for costly A/D converters for non-linear 
+activation 
+in 
+analog 
+computing 
+applications. 
+The 
+approximation of the ReLU-alpha activation function using this 
+concept is verified in our work, Ref. 14. There, the cost of the 
+MACAM search operation is 0.92 fJ with an associated 0.52 
+fJ/search cost from the peripheral circuitry, as compared to 
+ADC configuration of ~10.1 pJ [adc-1] and ~18.6 pJ [adc-2] in 
+Ref. [14].  
+ 
+VIII. CONCLUSIONS 
+    Our 
+10-transistor, 
+2-DW-MTJ 
+circuit 
+utilizes 
+the 
+programmable behavior of shape-depended multi-weight DW-
+MTJ synapses to perform analog CAM operation. We examined 
+the many trade-offs and design considerations in magnetic stack 
+characteristics and device geometry in the process of designing 
+DW-MTJs to optimize performance in ACAMs. With this, we 
+were able to demonstrate 5 discrete multi-bit levels with 
+realistic magnetic stack parameters, and up to 16 discrete levels 
+using 
+ideal 
+projected 
+stack 
+parameters. 
+The 
+analog 
+programmability made available by the introduction of DW-
+MTJs eliminates the need to interface with ADCs, which are 
+heavily energy intensive. Additionally, the digital output 
+enables the ACAM to also act as an alternative to ADCs. The 
+programmable 
+weights 
+in 
+ACAM 
+makes 
+for 
+ideal 
+implementation in high-throughput computing, such as one-
+shot/few-shot learning using decision trees.  
+    We did not account for ACAM’s intended use in large arrays 
+to be able to handle input word lengths. This type of system-
+level application of ACAM is associated with changes to both 
+average latency per search per cell, as well as energy 
+consumption per cell. These are important considerations to be 
+addressed in future work. 
+REFERENCES 
+[1] 
+M. Hu, C. E. Graves, C. Li, Y. Li, N. Ge, E. Montgomery, N. Davila, H. 
+Jiang, R. S. Williams, J. J. Yang, Q. Xia & J. P. Strachan. Memristor-
+Based Analog Computation and Neural Network Classification with a Dot 
+Product Engine. Advanced Materials 30, 1–10 (2018). 
+[2] 
+C. Li, D. Belkin, Y. Li, P. Yan, M. Hu, N. Ge, H. Jiang, E. Montgomery, 
+P. Lin, Z. Wang, W. Song, J. P. Strachan, M. Barnell, Q. Wu, R. S. 
+Williams, J. J. YZang & Q. Xia. Efficient and self-adaptive in-situ 
+ 
+Fig. 8. 
+(a) DW-MTJ utilizing platinum heavy metal layer with ideally 
+optimized RA, scaled geometry, 1000% TMR, and minimized wire resistance 
+to demonstrate multi-bit performance with 3 weights. (b-d) Identical 
+parameters are used to extend the resolution up to 5, 8, and 16 states. 
+
+1.0
+1.0
+0.8
+0.8
+≥ 0.4
+0.2
+0.2
+0.0
+0.0
+0.4
+0.6
+0.8
+1.0
+0.4
+0.6
+0.8
+1.0
+Search Voltage (V)
+Search Voltage (V)
+(a)
+(b)
+1.0
+1.0
+0.8
+0.8
+≥ 0.4
+0.2
+0.2
+0.0
+0.0
+0.4
+0.6
+0.8
+1.0
+0.4
+0.6
+0.8
+1.0
+Search Voltage (V)
+Search Voltage (V)
+(c)
+(d)8 
+ 
+learning in multilayer memristor neural networks. Nat Commun 9, 7–14 
+(2018). 
+[3] 
+S. Ambrogio, P. Narayanan, H. Tsai, R. M. Shelby, I. Boybat, C. Di Nolfo, 
+S. Sidler, M. Giordano, M. Bodini, N. C. P. Farinha, B. Killeen, C. Cheng, 
+Y. Jaoudi & G. W. Burr. Equivalent-accuracy accelerated neural-network 
+training using analogue memory. Nature 558, 60–67 (2018). 
+[4] 
+P. Yao, H. Wu, B. Gao, S. B. Eryilmaz, X. Huang, W. Zhang, Q. Zhang, 
+N. Deng, L. Shi, H. S. P. Wong & H. Qian. Face classification using 
+electronic synapses. Nat Commun 8, 1–8 (2017). 
+[5] 
+B. Wu, D. Feng, W. Tong, J. Liu, C. Wang, W. Zhao & M. Peng. ReRAM 
+crossbar-based analog computing architecture for naive bayesian engine. 
+Proceedings - 2019 IEEE International Conference on Computer Design, 
+ICCD 2019 147–155 (2019) 
+[6] 
+C. Li, M. Hu, Y. Li, H. Jiang, N. Ge, E. Montgomery, J. Zhang, W. Song, 
+N. Dávila, C. E. Graves, Z. Li, J. P. Strachan, P. Lin, Z. Wang, M. Barnell, 
+Q. Wu, R. S. Williams, J. J. Yang & Q. Xia. Analogue signal and image 
+processing with large memristor crossbars. Nat Electron 1, 52–59 (2018). 
+[7] 
+P. M. Sheridan, F. Cai, C. Du, W. Ma, Z. Zhang & W. D. Lu. Sparse 
+coding with memristor networks. Nat Nanotechnol 12, 784–789 (2017). 
+[8] 
+M. A. Zidan, Y. J. Jeong, J. Lee, B. Chen, S. Huang, M. J. Kushner & W. 
+D. Lu. A general memristor-based partial differential equation solver. Nat 
+Electron 1, 411–420 (2018). 
+[9] 
+Z. Sun, G. Pedretti, E. Ambrosi, A. Bricalli, W. Wang & D. Ielmini. 
+Solving matrix equations in one step with cross-point resistive arrays. 
+Proc Natl Acad Sci U S A 116, 4123–4128 (2019). 
+[10] W. Haensch, T. Gokmen & R. Puri. The Next Generation of Deep 
+Learning Hardware: Analog Computing. Proceedings of the IEEE 107, 
+108–122 (2019). 
+[11] Q. Liu, B. Gao, P. Yao, D. Wu, J. Chen, Y. Pang, W. Zhang, Y. Liao, C. 
+Xue, W. Chen, J. Tang, Y. Wang, M. Chang, H. Qian & H. Wu. A fully 
+integrated analog reram based 78.4tops/w compute-in-memory chip with 
+fully parallel mac computing. 2020 IEEE International Solid- State 
+Circuits Conference - (ISSCC) 500–502 (2020). 
+[12] C. Li, C. E. Graves, X. Sheng, D. Miller, M. Foltin, G. Pedretti & J. P. 
+Strachan. Analog content-addressable memories with memristors. Nat 
+Commun 11, 1–8 (2020). 
+[13] X. Yin, C. Li, Q. Huang, L. Zhang, M. Niemier, X. S. Hu, C. Zhuo & K. 
+Ni. FeCAM: A Universal Compact Digital and Analog Content 
+Addressable Memory Using Ferroelectric. IEEE Trans Electron Devices 
+67, 2785–2792 (2020). 
+[14] H. Zhu, K. Zhu, J. Gu, H. Jin, R. Chen, J. A. Incorvia & D. Z. Pan. Fuse 
+and Mix: MACAM-Enabled Analog Activation for Energy-Efficient 
+Neural Acceleration. arXiv preprint arXiv:2208.08099 (2022). 
+[15] A. Kazemi, M. M. Sharifi, A. F. Laguna, F. Müller, X. Yin, T. Kämpfe, 
+M. Niemier & X. S. Hu. FeFET Multi-Bit Content-Addressable Memories 
+for In-Memory Nearest Neighbor Search. IEEE Transactions on 
+Computers 71, 2565–2576 (2022). 
+[16] T. Leonard, S. Liu, M. Alamdar, H. Jin, C. Cui, O. G. Akinola, L. Xue, T. 
+P. Xiao, J. S. Friedman, M. J. Marinella, C. H. Bennett & J. A. C. Incorvia. 
+Shape-Dependent Multi-Weight Magnetic Artificial Synapses for 
+Neuromorphic Computing. Adv Electron Mater n/a, 2200563 (2022). 
+[17] B. Song, T. Na, J. P. Kim, S. H. Kang & S.-O. Jung. A 10T-4MTJ 
+Nonvolatile Ternary CAM Cell for Reliable Search Operation and a 
+Compact Area. IEEE Transactions on Circuits and Systems II: Express 
+Briefs 64, 700–704 (2017). 
+[18] C. Li, C. E. Graves, X. Sheng, D. Miller, M. Foltin, G. Pedretti & J. P. 
+Strachan. Analog content-addressable memories with memristors. Nat 
+Commun 11, 1638 (2020). 
+[19] S. Jung, H. Lee, S. Myung, H. Kim, S. K. Yoon, S.-W. Kwon, Y. Ju, M. 
+Kim, W. Yi, S. Han, B. Kwon, B. Seo, K. Lee, G.-H. Koh, K. Lee, Y. 
+Song, C. Choi, D. Ham & S. J. Kim. A crossbar array of magnetoresistive 
+memory devices for in-memory computing. Nature 601, 211–216 (2022). 
+[20] T. Leonard, S. Liu, M. Alamdar, C. Cui, O. G. Akinola, L. Xue, T. P. 
+Xiao, J. S. Friedman, M. J. Marinella, C. H. Bennett & J. A. C. Incorvia. 
+Shape-Dependent Multi-Weight Magnetic Artificial Synapses for 
+Neuromorphic Computing. Adv Electron Mater 2200563, 1–11 (2022). 
+[21] K. Ni, X. Yin, A. F. Laguna, S. Joshi, S. Dünkel, M. Trentzsch, J. Müller, 
+S. Beyer, M. Niemier, X. S. Hu & S. Datta. Ferroelectric ternary content-
+addressable memory for one-shot learning. Nat Electron 2, 521–529 
+(2019). 
+[22] M. Han, P. Yuan, J. Liu, S. Si, X. Zhao, Y. Yue, X. Wang & X. Xiao. 
+Interface Energy Coupling between β-tungsten Nanofilm and Few-
+layered Graphene. Sci Rep 7, 12213 (2017). 
+[23] B. Lenoir, M. Cassart, J.-P. Michenaud, H. Scherrer & S. Scherrer. 
+Transport properties of Bi-RICH Bi-Sb alloys. Journal of Physics and 
+Chemistry of Solids 57, 89–99 (1996). 
+[24] J. Sasaki, H. H. Huy, N. H. D. Khang, P. N. Hai, Q. Le, B. York, X. Liu, 
+S. Le, C. Hwang, M. Ho & H. Takano. Improvement of the Effective Spin 
+Hall Angle by Inserting an Interfacial Layer in Sputtered BiSb 
+Topological 
+Insulator 
+(Bottom)/Ferromagnet 
+With 
+In-Plane 
+Magnetization. IEEE Trans Magn 58, 1–4 (2022). 
+[25] W. Zhang, W. Han, X. Jiang, S.-H. Yang & S. S. P. Parkin. Role of 
+transparency of platinum–ferromagnet interfaces in determining the 
+intrinsic magnitude of the spin Hall effect. Nat Phys 11, 496–502 (2015). 
+[26] P. Catania, J. P. Doyle & J. J. Cuomo. Low resistivity body‐centered cubic 
+tantalum thin films as diffusion barriers between copper and silicon. 
+Journal of Vacuum Science & Technology A 10, 3318–3321 (1992). 
+[27] C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph & R. A. Buhrman. Spin 
+transfer torque devices utilizing the giant spin Hall effect of tungsten. Appl 
+Phys Lett 101, 122404 (2012). 
+[28] X. Yin, C. Li, Q. Huang, L. Zhang, M. Niemier, X. S. Hu, C. Zhuo & K. 
+Ni. FeCAM: A Universal Compact Digital and Analog Content 
+Addressable Memory Using Ferroelectric. IEEE Trans Electron Devices 
+67, 2785–2792 (2020). 
+[29] S.-G. Ren, R. Ni, X.-D. Huang, Y. Li & X.-S. Miao. Large On/Off and 
+Rectification 
+Ratios, 
+Self-Compliance, 
+High-Uniformity 
+in 
+Pt/Al2O3/TaOx/Ta 
+Self-Rectifying 
+Memristors. 
+in 
+2021 
+IEEE 
+International Conference on Integrated Circuits, Technologies and 
+Applications (ICTA) 29–30 (2021).  
+[30] Krizakova, V., Perumkunnil, M., Couet, S., Gambardella, P., & Garello, 
+K. Spin-orbit torque switching of magnetic tunnel junctions for memory 
+applications. Journal of Magnetism and Magnetic Materials, 562, 
+169692 (2022).  
+[31] M. Alamdar, T. Leonard, C. Cui, B. P. Rimal, L. Xue, O. G. Akinola, T. 
+Patrick Xiao, J. S. Friedman, C. H. Bennett, M. J. Marinella & J. A. C. 
+Incorvia. Domain wall-magnetic tunnel junction spin-orbit torque devices 
+and circuits for in-memory computing. Appl Phys Lett 118, (2021). 
+[32] F. Sunny, A. Mirza, M. Nikdast & S. Pasricha. CrossLight: A Cross-Layer 
+Optimized Silicon Photonic Neural Network Accelerator. in 2021 58th 
+ACM/IEEE Design Automation Conference (DAC) 1069–1074 (2021).  
+ 
+ 
+ 
+ 
+
diff --git a/u9E3T4oBgHgl3EQfkgqO/content/tmp_files/load_file.txt b/u9E3T4oBgHgl3EQfkgqO/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8c2c8b7d64eb3924dd57c0000aa031eaab1a6aa5
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@@ -0,0 +1,1002 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf,len=1001
+page_content='1  Domain Wall-Magnetic Tunnel Junction Analog  Content Addressable Memory Using Current and  Projected Data  Harrison Jin, Hanqing Zhu, Keren Zhu, Thomas Leonard, Jaesuk Kwon, Mahshid Alamdar, Kwangseok Kim,  Jungsik Park, Naoki Hase, David Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Pan, and Jean Anne C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Incorvia  Abstract—With the rise in in-memory computing architectures  to reduce the compute-memory bottleneck, a new bottleneck is  present between analog and digital conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Analog content-  addressable memories (ACAM) are being recently studied for in-  memory computing to efficiently convert between analog and  digital signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Magnetic memory elements such as magnetic  tunnel junctions (MTJs) could be useful for ACAM due to their  low read/write energy and high endurance, but MTJs are usually  restricted to digital values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The spin orbit torque-driven domain  wall-magnetic tunnel junction (DW-MTJ) has been recently  shown to have multi-bit function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Here, an ACAM circuit is  studied that uses two domain wall-magnetic tunnel junctions (DW-  MTJs) as the analog storage elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Prototype DW-MTJ data is  input into the magnetic ACAM (MACAM) circuit simulation,  showing ternary CAM function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Device-circuit co-design is carried  out, showing that 8-10 weight bits are achievable, and that  designing asymmetrical spacing of the available DW positions in  the device leads to evenly spaced ACAM search bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Analyzing  available spin orbit torque materials shows platinum provides the  largest MACAM search bound while still allowing spin orbit  torque domain wall motion, and that the circuit is optimized with  minimized MTJ resistance, minimized spin orbit torque material  resistance, and maximized tunnel magnetoresistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' These  results show the feasibility of using DW-MTJs for MACAM and  provide design parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Index Terms—analog circuits, associative memory, content  addressable memory, magnetic domain walls, magnetic tunnel  junctions, memory, neural networks, spin electronics, spintronics  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' INTRODUCTION  S data size increases and Moore’s law slows down,  alternative in-memory computing (IMC) architectures  are being studied and used to efficiently process  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Conventional analog IMC often uses  crossbar array architectures and nonvolatile memory (NVM)  such as resistive random-access memory (RRAM) and has  shown many applications in neural network acceleration [1]–  [3], neuromorphic computing [4], statistical learning [5], signal  processing [6], [7], scientific computing [8], [9], and other  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' However, there is a challenge extending these  This work was supported by the Samsung Global Research Outreach (GRO)  Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The authors also acknowledge support from the Department of  Energy Office of Science Microelectronics Co-Design project COINFLIPS (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Kwon) and the National Science Foundation Graduate Research Fellowship  under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 2020307514 (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Leonard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (Corresponding author: Jean Anne  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Incorvia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  architectures to operations other than matrix dot multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Existing IMC architectures rely on extensive analog to digital  conversion (ADC) and digital to analog conversion (DAC) to  switch between matrix multiplication and other computations  such as activation functions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The consequent conversion  cost greatly dilutes efficiency in both power and speed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=',  data converters can consume around 85% of the total energy in  a typical RRAM-based neural network accelerator [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' This  energy overhead is especially critical for edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  To address this challenge, analog content-addressable  memories (ACAM) are being recently studied for IMC [12],  [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' ACAM cells are designed to detect whether the value  corresponding to input search data is located within a range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Unlike conventional CAM, the input data of ACAM are  allowed to be analog values or multi-bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' ACAM can be used to  fuse computation and data conversion for time- and energy-  efficient conversion between analog and digital signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' For  example, a recently proposed RRAM-based  ACAM shows  delays of 350 ps and >1 fJ energy consumption, and a recently  shown ferroelectric-based ACAM shows up to 3-bit precision  to perform in-memory nearest neighbor searching to perform  few-shot learning [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The ideal requirements of NVM  for ACAM include low switching energy, low read energy, high  endurance, and controllability of setting the memory element to  a given analog value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Thus, magnetic tunnel junctions (MTJs)  are a natural choice for ACAM, due to their theoretically  unlimited endurance, modest switching voltage, and back-end-  of-the-line compatibility for integration into the ACAM cell  [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The domain wall-magnetic tunnel junction (DW-MTJ)  allows  for  analog-like  programming  of  tunnel  magnetoresistance (TMR) through modulation of the position  of a DW underneath an MTJ, using either spin transfer torque  (STT) or spin orbit torque (SOT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Recent work has  demonstrated the use of STT-based MTJs in ternary CAM  applications, but ternary CAMs are associated with  considerably greater area consumption costs in order to achieve  the same density of bits as their contemporary analog and multi-  bit counterparts [15], [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' STT magnetic random-access  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Jin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zhu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zhu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Leonard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Kwon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Alamdar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Pan, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Incorvia are with the Department of Electrical and Computer Engineering,  University of Texas at Austin, Austin, TX 78712 USA (e-mail:  incorvia@austin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Alamdar is now with Samsung Austin  Semiconductor, Austin, TX, 78754 USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Park, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Hase are with Samsung Advanced Institute of  Technology (SAIT), Samsung Electronics Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=', Suwon, 16678, South Korea (e-  mail: stone99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='kim@samsung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  A  2  memory (STT-MRAM) based on the MTJ has also been shown  in crossbar arrays to perform analog multiply-and-accumulate  operations [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' But, much of the previous works of spintronics  for CAM has been focused on binary and ternary functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  This is because achieving controllable multiple resistance levels  in an MTJ is a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Here, we propose and verify the performance of a DW-MTJ  ACAM prototype for high-throughput, high-speed searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  The DW-MTJ ACAM cell compares an analog input to a range  of stored values, which is set using the programmable resistance  of the DW-MTJ through the modulation of the DW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Circuit  simulations using prototype results from DW-MTJ device  cycling data verify that MTJs can be effectively integrated into  ACAMs as programmable elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Additionally, projected  data using optimized magnetic stack parameters demonstrate  the feasibility of performing analog multi-bit search operation  for implementation in high-throughput computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Our  proposed DW-MTJ ACAM benefits from up to a 44 × decrease  in energy consumption per search, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='86 × faster search  time, per bit compared to existing MTJ ternary CAM circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Additionally, the high programmability in linearity through  different DW-MTJ geometries, combined with low variation  within each weight, allows our proposed ACAM to circumvent  time-costly weight programming that is necessary in other  emergent ACAM designs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' thus, potentially reducing write  times by up to 3 orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' These results show the  potential for DWs and MTJs to be used in these energy-efficient  circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' CELL DESIGN AND METHODS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 1a shows the DW-MTJ multi-weight NVM used for the  magnetic ACAM (MACAM) design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A top-pinned MTJ stack  has its bottom heavy metal and magnetic free layer extended  into a magnetic wire that hosts a magnetic DW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' SOT current  applied from IN to CLK sets the DW position at one of the  notches, which in turn sets the resistance between the CLK and  OUT terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' We have previously shown DW-MTJ  prototypes with 3-5 stable resistance levels at room  temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' due to the physical setting of the resistance by the  DW position, highly controllable weights are achievable as long  as the DW is set to the desired notch [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Two DW-MTJs are integrated into the 8-transistor MACAM  cell shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 1b [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Minimum and maximum voltage  bounds are set using the search lines (𝑆𝐿!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' "#$,  𝑆𝐿%&\'), and input  search voltage is applied through the data line, 𝑉(%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The search  result is reflected on the match line (𝑀𝐿) behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 1c  depicts how the DW position in the DW-MTJ determines a  match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Programmable resistance states demonstrate how  different voltage states on both MTJs can be used to write  different voltage bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A match can be yielded anywhere  between the upper and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  To understand the cause, the drain-to-source voltage of the  input transistor 𝑇)*+ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 1b), 𝑉(,, and its drain current, 𝐼(,  can be described using the form:  𝐼( =  ,%!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' "#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='$%  (0&"\'(10)*+),  (1)  where 𝑅\'"34 is the resistance of the heavy metal plus free layer  patterned wire and 𝑅567 is the read-out resistance of the DW-  MTJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Subsequently,  𝑉(, = 𝑆𝐿$"#$ − 𝐼(2𝑅\'"34 + 𝑅5674.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  (2)  The saturation region of the 𝑇)*+ in which lower bounds will  remain matched can then be defined as:  \\  (a)  (b)  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  (a) Diagram of the three-terminal DW-MTJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Resistance states are  programmed using voltage pulses from IN and CLK, and read from IN to OUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Notches are shown that assist repeatable setting of the DW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (b) Circuit  schematic of proposed MACAM circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Minimum and maximum voltage  bounds are set using 𝑆𝐿  lines, and input search voltage is applied through the  data line, 𝑉(%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The search result is reflected on the match line (𝑀𝐿) behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  (c) Programmable resistance states demonstrate how different voltage states on  both MTJs can be used to write different voltage bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A match can be  yielded anywhere between the upper and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Different notch  locations are shown on a 5-weight DW-MTJ synapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' “N4” corresponds to  antiparallel and “N0” corresponds to parallel resistance relative to the reference  magnetic layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  CLK  MTJ  Z  OUT  X  DW  INLower  Upper  Bound  Bound  ML  SLHigh  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  N  IN2  V  SL  VDL  Z  Pinned Layer  Domain Wall  OUT  X  IN  CLK  Free  Layer  Heavy MetalLower Bounds  Upper Bounds  NO  N4  NO  N4  SL  SL  high  low3  𝑉(% > 6  ,%!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='"#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' *!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=',-$  80&"\'(10)*+9:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' + 𝑉6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=',)*,  (3)  Where 𝑉6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=',<( and 𝑉6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=',)* (in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 3b, e) are the threshold  voltages of the pulldown (𝑇<() and input (𝑇)*) transistors,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 𝑘= = µ=𝐶&>  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  % , the large signal MOSFET  transconductance parameter, where µ= is the mobility of  electrons on the channel surface, 𝐶&> is the oxide capacitance,  and  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  %  is the ratio of channel width to channel length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Meanwhile, the linear region where the lower bounds will  remain matched can be defined using the approximation:  𝑉𝐷𝐿 <  𝑉𝑇ℎ,𝑃𝐷  𝑔𝑚 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  1  𝑟𝑜 +  1  𝑅𝑤𝑖𝑟𝑒+𝑅𝑀𝑇𝐽".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  (4)  Here, 𝑔𝑚 is the small signal transconductance and 𝑟𝑜 is the  output resistance, both of which are intrinsic constants to the n-  type MOSFET, 𝑇)*+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The opposite can be said about the upper  bound, as the gate voltage of the pulldown transistor is inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  As the input search voltage decreases to 0, so does the drain  current, 𝐼(, of the input transistor, as is expected by the 𝑉J,-𝐼(  relationship of a n-type MOSFET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' In the ACAM circuit, 𝑉J,,  the gate-to-source voltage of the input transistor, is equal to the  analog search input, 𝑉(%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' To maintain the matching condition,  it is crucial that 𝑉(, remains less than that of 𝑉6$,<( in order to  maintain the match line voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' To counteract this, the MTJ  resistance must increase accordingly to maintain the adequate  voltage drop across the MTJ to keep 𝑇<( in its OFF state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Cadence Virtuoso and Spectre are used to characterize the  functionality of the MACAM circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The circuit is constructed  using 40 nm gate processes technology, and the MTJ is modeled  as two resistances in series, 𝑅\'"34 + 𝑅567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' To verify  performance, a two-dimensional parametric sweep of search  voltages at different 𝑅567 is run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=" The input search voltage is  swept from the full range established by the search lines (𝑆𝐿%&'  and 𝑆𝐿!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' "#$ set to 0 V and 1 V respectively) at a preprogrammed  𝑅567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' This is repeated at different 𝑅567 to determine the 𝑀𝐿  behavior as a function of the search inputs, to understand the  limits of both the DW-MTJ device and CMOS circuitry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' TCAM FUNCTIONALITY WITH PROTOTYPE DATA  To start, experimental data from DW-MTJ prototypes is  input into the constructed circuit model, to study their function  for CAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 2a shows the device data with device SEM  shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A 50 ns voltage pulse is applied from IN to  CLK, followed by measurement of 𝑅567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The voltage pulse  amplitude is increased from 2 to 4 V in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='1 V steps, showing  three distinct resistance values as the DW eventually de-pins  and moves to another notch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' This is repeated for 10 cycles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' see  Ref [20] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Nominal TMR of the magnetic stack was  measured using current in-plane TMR = 170%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The resistance-  area product for parallel MTJ resistance, RA, was measure RA  = Ω × µ𝑚K, with a heavy metal layer of tantalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The  trapezoidal synapse device used an MTJ with top-down area of  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='575 µ𝑚K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  From this data, we extract total resistances of 𝑅567 = 67 Ω,  75 Ω, and 93 Ω, with cycle-to-cycle resistance variation = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Inputting these values into the MACAM circuit, ternary CAM  behavior is seen, shown in Fig 2b, which plots the ML voltage  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' search voltage for the different relative MTJ weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The  search voltage is the analog search input from 𝑉(%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' From these  curves, the lower bound 𝐵% (V) and upper bound 𝐵L (V) are  defined as the search voltage values that set 𝑀𝐿 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The  storage range is defined as 𝑆𝑅  = 𝐵L − 𝐵%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The resulting  maximum 𝑆𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='109 V that can be achieved using these  measured resistance weights is between 𝐵% = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='854 V and  𝐵M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='963 V, which can be seen as the don’t care or X, state  that includes all values within this range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' alternatively, by  setting the upper and lower bounds DW-MTJs to the same  weight, we can also achieve a cell which will always result in a  mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Because there are 3 resistance weights, it is also  possible to achieve two smaller resistance states as well, which  can be used in binary implementation, depicted as the 0 and 1  states in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The combination of the X bit with the smaller  0 and 1 bits can then be used to implement ternary CAM  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ternary CAM application in memory-augmented  neural networks have previously been demonstrated for one-  shot learning in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  (a)  (b)  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  (a) Data of 𝑅567 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' applied voltage pulse amplitude of device  shown in inset, showing 3 distinct weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Each color is another cycle of the  same device showing repeatable cycle-to-cycle behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (b) Calculated  ternary CAM performance of device from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The dotted line at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='5 V on the  ML shows the minimum ML voltage necessary to yield a match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The lower  and upper bound of each discrete level (0, 1, or X) is marked by the two points  of intersection with the dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (c) Top-down design of 9 weight DW-MTJ  wire, with lithographically patterned magnetic wire shown in grey with 9  notches, and the MTJ for read-out is shown in teal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  Notch 0-2 (X)  Notch 0-1 (1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='8  90  Notch 1-2 (0)  80  70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='2  500 nm  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  50 ns write pulse amplitude (V)  Search Voltage (V)y  NO  N1 N2 N3 N4 N5 N6 N7  N84  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' ACAM FUNCTION AND DESIGN FOR 9-RESISTANCE  WEIGHT DW-MTJ  With existing prototypes showing 3-5 resistance levels, it is  feasible to extend to 8 resistance levels by extending the length  of the DW track and including 9 notches, depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Assuming 𝑅\'"34  = 40 Ω, and 𝑅567 =  70 Ω, with TMR =  170%, more analog and multi-bit capabilities of the cell can be  demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Under these assumptions, we can simulate the performance  of 𝐵% and 𝐵L’s circuit components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The specific voltage value  of the bound associated to the 𝐵% circuit can be programmed  using the MTJ-transistor voltage divider circuit, which is  demonstrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 3a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Furthermore, it can be seen that the  relationship between the DW-MTJ resistance weights and their  associated bounds is such that lower resistances are necessary  to write higher bounds, while larger resistances are required to  achieve the lower bounds, shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 3a, d where the  parallel resistance is in notch “0” and increases with each notch  up to notch “8”, which is the anti-parallel state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 𝐵L (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  3d, e) experiences a similar matching condition, but conversely  requires an input voltage smaller than the threshold at 𝑇<(K due  to the CMOS inverter prior to the pulldown transistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 3c  depicts the match line current (blue) and match line voltage  (black) during an input voltage sweep from 0 to 1 V, depicting  a mismatch-match-mismatch event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' To assess the 𝑀𝐿 threshold  voltage to yield a match, a buffer is placed at the end of the 𝑀𝐿,  and its transfer function is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 3f, revealing a  minimum 𝑀𝐿 voltage at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='5 V to be considered a match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  When the 𝑀𝐿 is plotted against the search voltage for the 9  evenly spaced notches in the DW-MTJ, the resulting  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  (a), (b) 𝐵!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' match performance based off different programmed resistance weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' An input signal less than the 𝐵!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' threshold yields a mismatch by  forcing the transistor PD1 in (b) to pull down the ML voltage to ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (c), (d) Conversely, an input signal greater than the 𝐵" threshold in (d) forces the transistor  PD2 to pull down ML voltage to ground using similarly to (c) but with an inverting CMOS prior to the pull-down transistor PD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (e) Simulation of voltage and  current measured through the match line during an input voltage sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (f) Transfer characteristics of voltage buffer at the match line output indicating a matching  threshold at 500 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  (a) Linear notch spacing vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (b) nonlinear notch spacing effects on  9-weight multi-bit performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  ML D  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  3E-5  SL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='8  (A)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='8  2E-5  M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='6  M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='6  MTJ1  Notch 8  Notch 0  W  PD1  MTJ,1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='4  1E-5  ML  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='2  SL  0E+0  LOW  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  VDL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  Search Voltage (V)  Search Voltage (V)  (a)  (b)  (c)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='8  High  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='8  o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='0  Search Voltage (V)  ML (V)  (d)  (e)  (f)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='8  s7  so  M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='0  Search Voltage (V)  1 μm  s0 s1 s2 s3s4 s5 s6 s7  MTJ  (a)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  s0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  Search Voltage (V)  1 μm  s0-s4  s5  s6  s7  MTJ  (b)5  relationship is 9 unevenly spaced discrete levels, as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' While ACAM behavior is still achieved, it is shown that  linearly spaced notches produce uneven widths of the distinct  levels in the multi-bit circuit operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' With these matching  characteristics, it would not be suitable to implement multi-bit  performance, such as that shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Thus, designing  the DW-MTJ with unevenly spaced notches, to achieve  approximately exponentially increasing MTJ resistance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  notch, alleviates this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' When considering notch spacing,  the minimum pitch between notches is no less than the width of  the notch itself to avoid stochastic movement of the DW  between adjacent notches from factors like variations in  magnetic wire geometry and thermal effects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=', a notch with  a 100 nm width is restricted to have a minimum spacing of 100  nm from the next notch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 4b shows the re-designed 9  notches, and the resulting ACAM function of ML vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' search  voltage, which shows evenly spaced discrete levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' These  results show a useful benefit of DW-MTJs for these circuits  because device behavior can be tuned to the circuit by adjusting  device geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' MACAM USING MTJ WAFER DATA  While the previous section focused on the impact of 𝑅567 on  the ACAM function, the resistance of the heavy metal plus  magnetic track, 𝑅\'"34, will also impact the circuit performance,  since the read current of the DW-MTJ runs through the DW  racetrack wire and out the MTJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' For SOT-driven DW motion,  𝑅\'"34 is dominated by the resistivity of the heavy metal  underneath the DW track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Here, we consider 3 common heavy  metals used in SOT-MRAM: platinum, 𝛼-tungsten, and  tantalum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 𝛽-tungsten and other similar large spin Hall angle  materials were not included due to their known high resistivities  [22]–[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Magnetic Stack Material and Device Characteristics  SOT-MRAM thin film stacks were grown with heavy metal  layer of 7 nm-thick α-tungsten and measured using CIPT,  showing average TMR = 170%, and average RA product = 35  Ω × µ𝑚K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Using these measured stack characteristics, we then  evaluated the impact of heavy metal resistivity on device  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 𝑅\'"34 was calculated using the excess length of  wire outside the area of the MTJ, with 𝑙 × 𝑤 dimensions of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='75 µ𝑚  × 400 𝑛𝑚 on all 3 stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The resistivities of the  heavy metal thin films used were assumed from literature to be  15 µΩ × 𝑐𝑚 for platinum, 21 µΩ × 𝑐𝑚 for 𝛼-tungsten, and 25  µΩ × 𝑐𝑚 for tantalum [25]–[27];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' thus, resulting in a projected  wire resistance of 40 Ω, 56 Ω, and 67 Ω, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The top-  down geometry of the MTJ has a 𝑙 × 𝑤 dimensions of  3 µ𝑚  × 100 𝑛𝑚, resulting in a parallel resistance of ~117 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Simulated Results  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 5 shows the performance of the circuit simulated using  device parameters extrapolated from each of the 3 stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  5a shows the maximum storage range of all 3 of these devices  plotted against each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The storage range, 𝑆𝑅, is the  maximum distance that can be achieved between 𝐵L and 𝐵%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Achieving larger 𝑆𝑅 allows for greater density of discrete levels  in analog multi-bit applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Additionally, maximizing the  accessible storage range within the minimum and maximum  possible bounds, established by 𝑆𝐿$"#$ and 𝑆𝐿O&\', reduces the  energy cost of peripheral circuitry used to scale down that of  the two DW-MTJ-CMOS subcircuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' This is important because  the limited TMR ratios available in current MTJ devices allows  programming bounds to only a fraction of the total available  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' For the three SOT material types, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 6a shows the  MACAM bound (V) associated with different values of 𝑅567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  The presence of wire resistance results in unwanted static  voltage drops within the subcircuits shown in Figs 3b, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Due to  the low resistivity of platinum thin films, platinum has the  highest 𝐵L overall due to having the lowest wire resistance,  seeing as both 𝐵% and 𝐵L increase with decreasing MTJ  resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Consequently, the maximum 𝑆𝑅 of the ACAM cell  utilizing the platinum stack is 245 mV, which is 16% greater  than α-tungsten and 28% greater than tantalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Another  demonstration showing the ability to achieve 5 discrete levels  can be seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 5b-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Within the range of voltages available  to all three stacks, they are all capable of comfortably fitting 5  discrete levels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' that is, the minimum pitch between notches is  reliably spaced as described in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' DW-MTJ MATERIALS PARAMETERS OPTIMIZATION FOR  ACAM  The results so far show that the MACAM circuit can achieve  both ternary and multi-bit-like functionality, using prototype  data, measured MTJ stack data with often-used heavy metal  materials, and feasible extension from the measured 3-5 notches  to 9 notches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Here, we inspect design considerations of the DW-  MTJ to further optimize the ACAM cell’s performance, to  predict what the ideal properties of the DW-MTJ should be for  this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  (a) Simulation of three different SOT heavy metals showing total  range, as well as individual states assuming 5 notches in the (b) platinum stack,  (c) 𝛼-tungsten stack, and (d) tantalum stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='8  αW  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='6  M  M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='6  M  M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  Search Voltage (V)  Search Voltage (V)  (c)  (d)6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Stack Characteristics and DW-MTJ Geometry  Considerations in Design Optimization  When considering the factors important to optimizing the  DW-MTJ for greater storage range in the ACAM cell, stack  characteristics (RA, TMR, and resistivity) and device geometry  (heavy metal thickness and MTJ top-down area) play large roles  in minimizing resistance losses and increasing total storage  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 6b, c show the storage range decreases with  increasing RA and increasing 𝑅\'"34 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Fig 6d shows storage  range improves with TMR with decreasing benefits for high  TMRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The area of the MTJ with respect to stack RA cannot be  neglected, since proportionally scaling down a device can have  unintended consequences on the total storage range of said  device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 6b shows proportionally scaling down a device  with feature node size of 50 nm down to 25 nm results in a  subsequent 4 × increase of parallel resistance, as both the  length and width of the MTJ are each proportionally scaled  down by  +  K ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The resistance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' feature node can be accounted  for in the circuit design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Design and Simulation of Prototype with Projected Data  Thus, we design a theoretical MTJ stack with ideal  parameters to demonstrate projected prototype performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Platinum is chosen as the heavy metal due to its low resistivity  while still having a good spin Hall angle for energy efficient  DW motion [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The Pt thin film layer is assumed to be 15 nm  thick, and the RA is assumed to be 5 Ω × µ𝑚K, on the low end  of what is feasible with today’s MgO-based MTJs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' We first  consider a reasonable TMR = 200%, which is currently  achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The device is designed to accommodate an MTJ  with dimensions of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='5 µ𝑚 × 50 𝑛𝑚 to be able to make use of  25 nm pitch between notches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' With this, the parallel resistance  of the device is 67 Ω and the anti-parallel resistance is 200 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 7 shows the simulation results, where the ACAM is  demonstrating 8 and 10 discrete levels by choosing 9 or 11  notches respectively, or also a minimum resolution of 3-bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  The trends observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 7  reveal that large TMR and low  RA work to improve the maximum storage range of the ACAM  cell: the storage range increases from 245 mV, projected from  the realistic magnetic stacks in the previous section, up to 300  mV in the ideal stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The increase in heavy metal layer  thickness from 7 nm to 15 nm constitutes a decrease in wire  resistance from 40 Ω to 19 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' This, in combination with the  30% increase of TMR, extends the projected storage range by  ~18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Given the nonlinear behavior of wire resistance, shown  in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 4 & 5, the increased range of programmable resistances,  and their associated voltage bounds, allow for larger density of  notch spacing necessary in the lower discrete levels for multi-  bit implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 7 shows 8 and 10 discrete levels, with  devices designed such that they meet the design considerations  for minimum notch spacing described in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Simulation of Prototype with 1000% TMR Ratio Projected  Data  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 8, the same parameters are assumed except TMR =  1000%, not yet commercially achievable today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The increase in  storage ranges from 200% TMR to 1000% TMR is 300 mV to  480 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' If this high on/off ratio could be achieved, the cell  would be capable of greater multi-bit precision, as much as 16  discrete levels, or 4-bits, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 7d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' ANALYSIS AND DISCUSSION  To evaluate the energy consumption of the MACAM, we  simulate the average DC current through the 𝑀𝐿 and integrated  it over several inference passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Using this method, the  estimated energy consumption during one search period in our  cell is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='92 fJ per search operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' It should be noted,  however, that the energy consumption from periphery circuits  (match line pre-charging, search line drivers, DAC, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=') is  estimated to consume up to an additional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='52 fJ [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' To  estimate the total area consumption of the CMOS components,  the total sum of all transistors was taken and assumed to be  ~90% of the total area consumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' thus, giving a top-down  circuit area consumption of  ~36 µ𝑚K using a 40 nm CMOS  technology node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The largest dimension of MTJ devices used  in previous simulations does not exceed 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='54 µ𝑚K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' thus, DW-  MTJ placement back-end-of-the-line on the CMOS circuit  would not affect the overall top-down area of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  (a) Relation of ML threshold bound vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 𝑅567 for the three heavy  metal types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (b) Performance changes from proportional geometric scaling of  device with 50 nm and 25 nm pitch between notches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (c) Change in maximum  writable 𝑆𝑅 of platinum-based MACAM device with wire resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (d)  Change in maximum writable storage range with TMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  (a)  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 7  (a) DW-MTJ utilizing platinum heavy metal layer with ideally  optimized RA, scaled geometry, 200% TMR, and minimal wire resistance to  demonstrate multi-bit performance of 9 notches and 8 distinct levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (b)  Identical parameters assumed for 11 notches and 10 distinct levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='18  25 nm notch pitch  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0 Pt  50 nm notch pitch  aw  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='16  Ta  M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='9  ound  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='8  S  B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='08  50  100  150  200  20  30  40  50  60  70  RMTJ (2)  RA (Q·μum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=')  (a)  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='1  0  50  100  150  200  0  200  400  600  800 1000  Rwire (Q2)  TMR (%)  (c)  (d)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='4  ML  M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='0  Search Voltage (V)  Search Voltage (V)7  TABLE I  COMPARISON OF EMERGENT ACAM ARCHITECTURES*  PARAMETER  FEFET  [28]  RRAM  [18][29]  SRAM  [18]  DW-MTJ  [30]  AREA  CONSUMPTION  49 F2  48,828 F2  918,750  F2  22,875 F2  TECHNOLOGY  NODE  45 nm  16 nm  16 nm  40 nm  NON-  VOLATILITY  Yes  Yes  No  Yes  ON/OFF RATIO  104-106  106  106  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='5-6  VARIATION  High  High  Low  Low  LINEARITY  Low  Low  High  High  SEARCH  LATENCY  ~10 ns  ~50 ps  N/A  350 ps  ENDURANCE  105  1012  >1015  >1015  ENERGY (PER  SEARCH)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='07 fJ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='52 fJ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='165 fJ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='92 fJ  Values from individual cell  Some additional energy costs can be found in the necessary  circuits to perform read and write operations within the ACAM  cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The energy required to update the DW-MTJ by a single  weight is on average ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='1 pJ in few-100 nm prototypes [20],  which can be scaled to ~2 fJ for 15 nm feature sizes [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' This  energy can be reduced through scaling and device engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  The match line output also requires a sensing circuit based on a  transimpedance amplifier (TIA), which has an associated  energy dissipation of about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='5 pJ over the course of one search  operation [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' This relatively high energy can be effectively  reduced using large ACAM arrays to amplify integrated  currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  The modest TMR of MTJs are considerably smaller than that  of the relatively large on/off conductance ratios of FeFETs and  RRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The operation in the relatively small range of  conductivities leads to reduced noise robustness and potential  energy costs to scale voltage inputs to a range that can be  accommodated by the MACAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' However, FeFETs and  modern memristor technology continue to suffer from high  non-linearity as well as inconsistent cycle-to-cycle weight  variation without the assistance of external circuitry, which in  the case of FeFETs can results in a verification period that is  microseconds in length [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Additionally, the physical  robustness of SOT switching MTJs introduces a considerably  larger endurance than 2-terminal devices, The ability to tune the  change in resistance through the device geometry also provides  unique ways to adapt MTJs for the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Furthermore, at the system level, the DW-MTJ-informed  ACAM demonstrates the ability to perform a “fusion” of  nonlinear activation and ADC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The search operation of ACAM  can be used to binarize an analog input signal, while also  introducing an in-situ nonlinearity characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Thus, this  eliminates the need for costly A/D converters for non-linear  activation  in  analog  computing  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  The  approximation of the ReLU-alpha activation function using this  concept is verified in our work, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' There, the cost of the  MACAM search operation is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='92 fJ with an associated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='52  fJ/search cost from the peripheral circuitry, as compared to  ADC configuration of ~10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='1 pJ [adc-1] and ~18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='6 pJ [adc-2] in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' CONCLUSIONS  Our  10-transistor,  2-DW-MTJ  circuit  utilizes  the  programmable behavior of shape-depended multi-weight DW-  MTJ synapses to perform analog CAM operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' We examined  the many trade-offs and design considerations in magnetic stack  characteristics and device geometry in the process of designing  DW-MTJs to optimize performance in ACAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' With this, we  were able to demonstrate 5 discrete multi-bit levels with  realistic magnetic stack parameters, and up to 16 discrete levels  using  ideal  projected  stack  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  The  analog  programmability made available by the introduction of DW-  MTJs eliminates the need to interface with ADCs, which are  heavily energy intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Additionally, the digital output  enables the ACAM to also act as an alternative to ADCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The  programmable  weights  in  ACAM  makes  for  ideal  implementation in high-throughput computing, such as one-  shot/few-shot learning using decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  We did not account for ACAM’s intended use in large arrays  to be able to handle input word lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' This type of system-  level application of ACAM is associated with changes to both  average latency per search per cell, as well as energy  consumption per cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' These are important considerations to be  addressed in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  REFERENCES  [1]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Hu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Graves, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Li, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ge, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Montgomery, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Davila, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Jiang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Yang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Xia & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Strachan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Memristor-  Based Analog Computation and Neural Network Classification with a Dot  Product Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Advanced Materials 30, 1–10 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [2]  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Belkin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Li, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Yan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Hu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ge, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Jiang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Montgomery,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Song, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Strachan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Barnell, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Wu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' YZang & Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Efficient and self-adaptive in-situ  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  (a) DW-MTJ utilizing platinum heavy metal layer with ideally  optimized RA, scaled geometry, 1000% TMR, and minimized wire resistance  to demonstrate multi-bit performance with 3 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' (b-d) Identical  parameters are used to extend the resolution up to 5, 8, and 16 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='8  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content=' Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content=' Yang & Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content='  [7]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Sheridan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Cai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Du, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zhang & W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+page_content=' Nat Nanotechnol 12, 784–789 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [8]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zidan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Jeong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Lee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Kushner & W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A general memristor-based partial differential equation solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Nat  Electron 1, 411–420 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [9]  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Sun, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Pedretti, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ambrosi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Bricalli, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Wang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ielmini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Solving matrix equations in one step with cross-point resistive arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Proc Natl Acad Sci U S A 116, 4123–4128 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Haensch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Gokmen & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Puri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' The Next Generation of Deep  Learning Hardware: Analog Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Proceedings of the IEEE 107,  108–122 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [11] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Gao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Yao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Pang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Liao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Xue, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Tang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Chang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Qian & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A fully  integrated analog reram based 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='4tops/w compute-in-memory chip with  fully parallel mac computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' 2020 IEEE International Solid- State  Circuits Conference - (ISSCC) 500–502 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Graves, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Sheng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Miller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Foltin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Pedretti & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Strachan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Analog content-addressable memories with memristors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Nat  Commun 11, 1–8 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [13] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Yin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Niemier, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Hu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zhuo & K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' FeCAM: A Universal Compact Digital and Analog Content  Addressable Memory Using Ferroelectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' IEEE Trans Electron Devices  67, 2785–2792 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [14] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zhu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Gu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Jin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Incorvia & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Pan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Fuse  and Mix: MACAM-Enabled Analog Activation for Energy-Efficient  Neural Acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='08099 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Kazemi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Sharifi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Laguna, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Müller, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Yin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Kämpfe,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Niemier & X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' FeFET Multi-Bit Content-Addressable Memories  for In-Memory Nearest Neighbor Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' IEEE Transactions on  Computers 71, 2565–2576 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Leonard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Alamdar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Jin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Cui, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Akinola, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Xue, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Xiao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Friedman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Marinella, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Bennett & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Incorvia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Shape-Dependent Multi-Weight Magnetic Artificial Synapses for  Neuromorphic Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Adv Electron Mater n/a, 2200563 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [17] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Song, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Na, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Kang & S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Jung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A 10T-4MTJ  Nonvolatile Ternary CAM Cell for Reliable Search Operation and a  Compact Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' IEEE Transactions on Circuits and Systems II: Express  Briefs 64, 700–704 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Graves, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Sheng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Miller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Foltin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Pedretti & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Strachan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Analog content-addressable memories with memristors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Nat  Commun 11, 1638 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Jung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Myung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Yoon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Kwon, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ju, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Kim, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Yi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Han, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Kwon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Seo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Lee, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Koh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Lee, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Song, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Choi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ham & S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A crossbar array of magnetoresistive  memory devices for in-memory computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Nature 601, 211–216 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [20] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Leonard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Alamdar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Cui, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Akinola, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Xue, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Xiao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Friedman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Marinella, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Bennett & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Incorvia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Shape-Dependent Multi-Weight Magnetic Artificial Synapses for  Neuromorphic Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Adv Electron Mater 2200563, 1–11 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [21] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ni, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Yin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Laguna, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Joshi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Dünkel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Trentzsch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Müller,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Beyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Niemier, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Hu & S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Datta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ferroelectric ternary content-  addressable memory for one-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Nat Electron 2, 521–529  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Han, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Si, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Yue, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Wang & X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Xiao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Interface Energy Coupling between β-tungsten Nanofilm and Few-  layered Graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Sci Rep 7, 12213 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [23] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Lenoir, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Cassart, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Michenaud, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Scherrer & S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Scherrer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Transport properties of Bi-RICH Bi-Sb alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Journal of Physics and  Chemistry of Solids 57, 89–99 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Sasaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Huy, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Khang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Hai, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Le, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' York, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Liu,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Le, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Hwang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ho & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Takano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Improvement of the Effective Spin  Hall Angle by Inserting an Interfacial Layer in Sputtered BiSb  Topological  Insulator  (Bottom)/Ferromagnet  With  In-Plane  Magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' IEEE Trans Magn 58, 1–4 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [25] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Han, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Jiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Yang & S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Parkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Role of  transparency of platinum–ferromagnet interfaces in determining the  intrinsic magnitude of the spin Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Nat Phys 11, 496–502 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [26] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Catania, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Doyle & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Cuomo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Low resistivity body‐centered cubic  tantalum thin films as diffusion barriers between copper and silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Journal of Vacuum Science & Technology A 10, 3318–3321 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [27] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Pai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Tseng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ralph & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Buhrman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Spin  transfer torque devices utilizing the giant spin Hall effect of tungsten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Appl  Phys Lett 101, 122404 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [28] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Yin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Niemier, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Hu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Zhuo & K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' FeCAM: A Universal Compact Digital and Analog Content  Addressable Memory Using Ferroelectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' IEEE Trans Electron Devices  67, 2785–2792 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ren, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Ni, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Li & X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Miao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Large On/Off and  Rectification  Ratios,  Self-Compliance,  High-Uniformity  in  Pt/Al2O3/TaOx/Ta  Self-Rectifying  Memristors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  in  2021  IEEE  International Conference on Integrated Circuits, Technologies and  Applications (ICTA) 29–30 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [30] Krizakova, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=', Perumkunnil, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=', Couet, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=', Gambardella, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=', & Garello,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Spin-orbit torque switching of magnetic tunnel junctions for memory  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Journal of Magnetism and Magnetic Materials, 562,  169692 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Alamdar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Leonard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Cui, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Rimal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Xue, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Akinola, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Patrick Xiao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Friedman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Bennett, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Marinella & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  Incorvia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Domain wall-magnetic tunnel junction spin-orbit torque devices  and circuits for in-memory computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Appl Phys Lett 118, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content='  [32] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Sunny, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Mirza, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Nikdast & S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' Pasricha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' CrossLight: A Cross-Layer  Optimized Silicon Photonic Neural Network Accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
+page_content=' in 2021 58th  ACM/IEEE Design Automation Conference (DAC) 1069–1074 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQfkgqO/content/2301.04598v1.pdf'}
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+arXiv:2301.11567v1  [math.AP]  27 Jan 2023
+Threshold dynamics of a nonlocal dispersal SIS epidemic
+model with free boundaries ∗
+Yachun Tonga, Inkyung Ahnb and Zhigui Lina†
+a School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
+b Department of Mathematics, Korea University, Sejong 339-700, South Korea
+Abstract.
+To study the influence of the moving front of the infected interval and the
+spatial movement of individuals on the spreading or vanishing of infectious disease, we
+consider a nonlocal SIS (susceptible-infected-susceptible) reaction-diffusion model with
+media coverage, hospital bed numbers and free boundaries. The principal eigenvalue of
+the integral operator is defined, and the impacts of the diffusion rate of infected individuals
+and interval length on the principal eigenvalue are analyzed. Furthermore, the sufficient
+conditions for spreading and vanishing of the disease are derived. Our results show that
+large media coverage and hospital bed numbers are beneficial to the prevention and control
+of disease. The difference between the model with nonlocal diffusion and that with local
+diffusion is also discussed and nonlocal diffusion leads more possibilities.
+MSC: 35K57, 92D30; secondary: 35R35.
+Keywords: SIS model; Free boundary; Nonlocal diffusion; Spreading and vanishing
+1
+Introduction
+With the emergence and outbreak of COVID-19 [3,30] in recent years, infectious disease models
+have become one of the most popular research topics. To study the spread and dynamics of
+COVID-19, most scholars use the SIR (susceptible-infected-recovered) [3,28], SEIR (susceptible-
+exposed-infected-recovered) [25,30] and SEAIR (susceptible-exposed-asymptomatic-infectious-
+removed) [2, 46] models to describe the spread of COVID-19. Meanwhile, the classical SIS
+model has received great attention in mathematical epidemiology.
+Considering the impact of the spatial heterogeneity of the environment and the movement
+of individuals on infectious diseases, Allen et al. in [1] proposed and discussed an SIS reaction-
+diffusion system
+
+
+
+
+
+
+
+St − dS∆S = −β(x)SI
+S+I + γ(x)I,
+t > 0, x ∈ Ω,
+It − dI∆I = β(x)SI
+S+I − γ(x)I,
+t > 0, x ∈ Ω,
+∂S
+∂η = ∂I
+∂η = 0,
+t > 0, x ∈ ∂Ω.
+(1.1)
+∗The first author is supported by the Postgraduate Research & Practice Innovation Program of Jiangsu
+Province (KYCX21-3188), the second author is supported under the framework of international cooperation
+program managed by the National Research Foundation of Korea (NRF-2019K2A9A2A06025237) and the third
+author is supported by the National Natural Science Foundation of China (Grant No. 12271470).
+†Corresponding author. Email: zglin@yzu.edu.cn (Z. Lin).
+1
+
+Here, Ω ⊂ Rn (n ≥ 1) is a bounded domain; S(t, x) and I(t, x) indicate the density of susceptible
+and infected individuals at location x and time t, respectively; dS and dI are positive constants
+that account for the diffusion rate of susceptible and infected individuals, respectively; and
+the positive bounded H¨older continuous functions β(x) and γ(x) can be interpreted as rates of
+disease transmission and recovery for x ∈ Ω, respectively. The authors in [1] mainly discussed
+the existence, uniqueness and stability of DFE (disease-free equilibrium) and EE (endemic
+equilibrium) and used the basic reproduction number R0 to characterize the risk of the region.
+Afterwards, Peng and Liu [34] confirmed the conjecture proposed by Allen et al. in [1] that
+a unique EE is globally asymptotically stable in some special cases. Further results that the
+effect of individual movement (large or small) on the existence and disappearance of disease were
+obtained in [33]. For more results of the SIS reaction-diffusion model, one can see [24, 35, 42]
+and the references therein.
+It is easy to find that the above articles are devoted to the study of SIS models on a fixed
+domain. In real life, the movement of species leads to changes in biological habitats, and in
+mathematics, the free boundary can be used to described this phenomenon, such as the healing
+of wounds [10] and the expansion of new species or invasive species [6,14,27,38]. Free boundary
+problems can also be used to describe the transmission of disease, such as the SIRS model [7],
+SIS model [22], SIR model [23,47] and references therein.
+To explore the moving front of the infected individual, Wang and Guo [40] introduced the
+free boundary and studied the dynamics of the following SIS reaction-diffusion model:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+St − d∆S = σ − µS − β(x)SI + γ(x)I,
+t > 0, x ∈ R,
+It − d∆I = β(x)SI − µI − γ(x)I,
+t > 0, x ∈ (g(t), h(t)),
+I(t, x) = 0,
+t > 0, x ∈ R\(g(t), h(t)),
+g′(t) = −kIx(t, g(t)), g(0) = −h0,
+t ≥ 0,
+h′(t) = −kIx(t, h(t)), h(0) = h0,
+t ≥ 0,
+S(0, x) = S0(x), I(0, x) = I0(x),
+x ∈ R.
+(1.2)
+The basic reproduction number was given, and the spreading-vanishing dichotomy was estab-
+lished. Some conditions for disease spreading or vanishing were presented by investigating the
+effect of the diffusion rate (d), initial value (I0) and expanding capability (k) on the asymptotic
+behavior of the infected individuals.
+It is widely known that random dispersal or local diffusion describes the local behavior
+of the movements of organisms between adjacent spatial locations [26]. Briefly, the classical
+Laplace diffusion operator is used to describe that the movement of the infectious agent and
+infected population only occurs between adjacent spatial positions [43]. However, Murray [32]
+noted that a local or short-range diffusive flux proportional to the gradient is not suitable to
+characterize some biological phenomena. In the real world, the movements and interactions of
+some organisms occur at nonadjacent spatial positions, and such dispersal are called nonlocal
+diffusion [13].
+Recently, nonlocal diffusion equations have attracted extensive attention and have been used
+to characterize long-range dispersal in population ecology [6,26]. In addition, scholars have also
+extensively investigated infectious disease models with nonlocal diffusion, such as the West Nile
+virus model [15], SIS epidemic model [17, 44], and SIR reaction-diffusion model [18, 45]. For
+other epidemic models with nonlocal diffusion, see references [9,39,41] and references therein.
+In addition, there are many factors that affect the spread of infectious disease, such as the
+contact transmission rate and the recovery rate. Educating the public about the disease through
+mass media (such as television, radio, newspapers, billboards, internet, magazines etc), is one
+2
+
+of the important precautions. Therefore, media coverage can indirectly reduce the contact rate
+between people and infectious diseases, thus reducing the contact transmission rate of infectious
+diseases [36]. In general, the main factor impacting the recovery rate is the availability of health
+care (such as the number of physicians, nurses, hospital beds and isolation places). In fact,
+health and medical institutions use the hospital bed-population ratio (HBPR) (the number of
+hospital beds per 10000 people) as a method of reckoning available resources to the public [31].
+Taking into account of nonlocal diffusion, media coverage and hospital bed numbers, we
+consider the following nonlocal dispersal SIS epidemic model with a free boundary
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+St = dL1[S] + σ − µ1S − β(m(x), I, x)SI + γ(b(x), I, x)I,
+t > 0, x ∈ R,
+It = dL2[I; g, h] − µ2I + β(m(x), I, x)SI − γ(b(x), I, x)I,
+t > 0, x ∈ (g(t), h(t)),
+I(t, x) = 0,
+t ≥ 0, x ∈ R\(g(t), h(t)),
+h′(t) = k
+� h(t)
+g(t)
+� +∞
+h(t) J(x − y)I(t, x)dydx,
+t > 0,
+g′(t) = −k
+� h(t)
+g(t)
+� g(t)
+−∞ J(x − y)I(t, x)dydx,
+t > 0,
+S(0, x) = S0(x), g(0) = −h0, h(0) = h0,
+x ∈ R,
+I(0, x) = I0(x),
+x ∈ (−h0, h0),
+(1.3)
+where
+L1[S] =
+�
+R
+J(x − y)S(t, y)dy − S(t, x),
+L2[I; g, h] =
+� h(t)
+g(t)
+J(x − y)I(t, y)dy − I(t, x),
+and d, S(t, x) and I(t, x) have the same epidemiological interpretation as in (1.1). The con-
+stants σ, µ1 and µ2 are positive, where σ accounts for the environment carrying capability; the
+natural mortality rate of the susceptible individuals is expressed by µ1, and µ2 denotes the sum
+of the natural mortality and disease-caused death rates of the infected individuals. The func-
+tions β(m(x), I, x), γ(b(x), I, x), m(x), b(x) are nonnegative, where m(x) represents the media
+coverage, and b(x) stands for the number of hospital beds. In this paper, we assume that
+(1) the contact infectious rate β(m(x), I, x) is Lipschitz continuous and monotonically de-
+creasing in m(x) and increasing in I;
+(2) the recovery rate γ(b(x), I, x) is Lipschitz continuous and increasing in b(x) and mono-
+tonically decreasing in I;
+(3) βI(m(x), I, x) and γI(b(x), I, x) are continuous and bounded for m(x) ∈ [0, ∞), I ∈
+[0, ∞) and x ∈ (−∞, ∞).
+For instance, Cui and Zhu [12] used the function β(I) = βemI to model the impact of media
+coverage on the transmission rate; and Shan and Zhu [37] used the function γ(b, I, x) = γ0 +
+(γ1 − γ0)
+b
+b+I to describe the hospital resource impact factors.
+Recalling that S(t, x) denotes the density at point x and time t, the kernel function J(x−y)
+is regarded as the probability distribution of jumping from place y to place x, then the integral
+operator
+�
+R J(x − y)S(t, y)dy accounts for the rate at which the individuals are gathering at
+point x from all other places, and −S(t, x) is the rate at which the individuals are leaving
+at point x to other places. In addition, the infected individuals stay in the infected interval
+(g(t), h(t)). We further suppose that the initial function S0(x) satisfies
+S0(x) ∈ C(R) ∩ L∞(R) and S0(x) > 0 in R,
+(1.4)
+and I0(x) satisfies
+I0(x) ∈ C([−h0, h0]), I0(±h0) = 0, I0(x) > 0 in (−h0, h0).
+(1.5)
+3
+
+For system (1.3), we assume that the kernel function J : R → R is continuous and nonneg-
+ative, and has the properties
+(J) : J ∈ C(R) ∩ L∞(R) is symmetric, J(0) > 0,
+�
+R
+J(x)dx = 1.
+The free boundary conditions h′(t) = k
+� h(t)
+g(t)
+� +∞
+h(t) J(x−y)I(t, x)dydx and g′(t) = −k
+� h(t)
+g(t)
+� g(t)
+−∞ J(x−
+y)I(t, x)dydx in (1.3) imply that the expanding rate of the interval (g(t), h(t)) is determined
+by the infected individuals and is proportional to the outward flux of the infected individuals
+across the interval (g(t), h(t)) [5].
+It is worth mentioning that there are links and differences between local diffusion and
+nonlocal diffusion. Local diffusion, expressed by the Laplace operator ∆u (the Laplace in Rn,
+n ≥ 2) or uxx (in one-dimensional space), is used to describe the influence between adjacent
+positions, and nonlocal diffusion, expressed by the integral operator (is given by
+�
+R J(x −
+y)u(t, y)dy−u(t, x)), is used to describe long-distance dispersal. However, the Laplace operator
+can be regarded as a local approximation of a nonlocal diffusion operator. In fact, when J(·) is
+symmetric and has compact supports, such as J(x) = (1/ǫ)K(x/ǫ) with 0 < ǫ ≪ 1 and K(x) is
+a general mollification function with support x ∈ [−1, 1], we can transform nonlocal operators
+into local operators by using the Taylor formula [29].
+This article is organized as follows: the existence and uniqueness of the global solution
+are given in Section 2. Section 3 is devoted to defining and studying the properties of the
+principal eigenvalue. Section 4 gives some sufficient conditions for the disease to spread or
+vanish. Finally, a brief discussion is presented in Section 5.
+2
+Global existence and uniqueness
+In this section, we assume that h0 > 0, S0(x) and I0(x) satisfy (1.4) and (1.5). For any given
+T > 0, we first introduce the notations as follows:
+HT := {h ∈ C([0, T]) : h(0) = h0,
+inf
+0≤t1<t2≤T
+h(t2)−h(t1)
+t2−t1
+> 0},
+GT := {g ∈ C([0, T]) : −g ∈ HT},
+Dg,h
+T
+:= {(t, x) ∈ R2 : 0 < t ≤ T, g(t) < x < h(t)},
+Dh0
+T := {(t, x) ∈ R2 : 0 < t ≤ T, −h0 < x < h0},
+D∞
+T := {(t, x) ∈ R2 : 0 < t ≤ T, x ∈ R},
+XS0
+T := {φ(t, x) ∈ C(D∞
+T ) ∩ L∞(D∞
+T ) : φ(0, x) = S0(x) in R, φ(t, x) ≥ 0 in D∞
+T },
+XI0
+T := {ψ(t, x) ∈ C(D∞
+T ) : ψ(0, x) = I0(x) in [−h0, h0], ψ(t, x) ≥ 0 in Dg,h
+T ,
+ψ(t, x) = 0 for t ∈ (0, T), x ∈ R\(g(t), h(t))}.
+To prove the existence and uniqueness of the global solution of problem (1.3), we first give
+the following result for problem (1.3) without a free boundary.
+4
+
+Lemma 2.1 For any given T > 0 and (g, h) ∈ HT × GT , the problem
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+St = dL1[S] + σ − µ1S − β(m(x), I, x)SI + γ(b(x), I, x)I,
+0 < t ≤ T, x ∈ R,
+It = dL2[I; g, h] − µ2I + β(m(x), I, x)SI − γ(b(x), I, x)I,
+0 < t ≤ T, x ∈ (g(t), h(t)),
+I(t, x) = 0,
+0 ≤ t ≤ T, x ∈ R\(g(t), h(t)),
+S(0, x) = S0(x),
+x ∈ R,
+I(0, x) = I0(x),
+x ∈ (−h0, h0)
+(2.1)
+admits a unique solution (Sg,h, Ig,h) ∈ C(D
+∞
+T ) × C(D
+g,h
+T ). Moreover,
+0 < Sg,h(t, x) ≤ A
+for any (t, x) ∈ D∞
+T ,
+(2.2)
+0 < Ig,h(t, x) ≤ A
+for any (t, x) ∈ Dg,h
+T ,
+(2.3)
+where A = max{ σ
+µ1, ∥S0∥∞ + ∥I0∥∞}.
+Proof. The main idea of this proof comes from [45]. We divide the proof into three steps.
+Step 1. The parameterized ODE problem.
+For any given x ∈ R, s ∈ (0, T], denote
+tx =
+
+
+
+
+
+
+
+
+
+
+
+tg
+x,
+x ∈ (g(s), −h0) and x = g(tg
+x),
+0,
+x ∈ [−h0, h0],
+th
+x,
+x ∈ (h0, h(s)) and x = h(th
+x),
+s,
+x ∈ R\(g(s), h(s)).
+Clearly, tx > 0 for x ∈ R\[−h0, h0], tx < s for x ∈ (g(s), h(s)). For any given (φ, ψ) ∈ XS0
+s ×XI0
+s ,
+define
+A1 = max{A, σ + ∥ψ∥∞ sup γ
+µ1
+, ∥φ∥∞},
+A2 = max{A, (d + A1 sup β)∥ψ∥∞
+d + µ2
+}.
+We discuss it in the following two cases:
+Case 1: x ∈ R\[−h0, h0], t ∈ [0, tx].
+Clearly, I(t, x) = 0 for (t, x) ∈ [0, tx] × R\[−h0, h0]. Consider the ODE problem
+�
+St = d
+�
+R J(x − y)φ(t, y)dy − dS + σ − µ1S,
+0 < t ≤ tx,
+S(0, x) = S0(x),
+x ∈ R\[−h0, h0].
+(2.4)
+For any S1, S2 ∈ [0, A1],
+|d
+�
+R J(x − y)φ(t, y)dy − dS1 + σ − µ1S1 − d
+�
+R J(x − y)φ(t, y)dy + dS2 − σ + µ1S2|
+=
+(d + µ1)|S1 − S2|.
+Therefore, F := d
+�
+R J(x−y)φ(t, y)dy−dS+σ−µ1S is Lipschitz continuous in S for S ∈ [0, A1].
+By the fundamental theory of ODEs, problem (2.4) has a unique solution Sφ(t, x) defined in
+t ∈ [0, �tx), and Sφ(t, x) is continuous in both t and x. To see that t → S(·, x) can be uniquely
+extended to [0, tx], we need to prove that if Sφ(t, x) is uniquely defined for t ∈ [0, �tx] with
+�tx ∈ (0, tx], then
+0 ≤ Sφ(t, x) ≤ A1, for t ∈ [0, �tx] and x ∈ R\[−h0, h0].
+5
+
+Obviously,
+d
+�
+R J(x − y)φ(t, y)dy − dA1 + σ − µ1A1
+≤
+d∥φ∥∞ − dA1 + σ − µ1A1
+≤
+0,
+and ∥S0∥∞ ≤ A1. Thanks to the direct comparison argument, one can derive Sφ(t, x) ≤ A1
+for t ∈ [0, �tx] and x ∈ R\[−h0, h0]. We use similar method to prove that Sφ(t, x) ≥ 0 for
+t ∈ [0, �tx], x ∈ R\[−h0, h0].
+Case 2: x ∈ (g(s), h(s)), t ∈ [tx, s].
+Define
+�Sφ(x) =
+�
+S0(x),
+x ∈ [−h0, h0]
+Sφ(tx, x),
+x /∈ [−h0, h0]
+and
+�I(x) =
+�
+I0(x),
+x ∈ [−h0, h0]
+0,
+x /∈ [−h0, h0].
+Consider the ODE problem
+
+
+
+
+
+
+
+St = F1(t, x, S, I),
+tx < t ≤ s,
+It = F2(t, x, S, I),
+tx < t ≤ s,
+S(tx, x) = �Sφ(x), I(tx, x) = �I(x),
+x ∈ (g(s), h(s))
+(2.5)
+with
+F1 = d
+�
+R
+J(x − y)φ(t, y)dy − dS + σ − µ1S + γ(b, I, x)ψ − β(m, I, x)SI,
+F2 = d
+� h(t)
+g(t)
+J(x − y)ψ(t, y)dy − dI − µ2I − γ(b, I, x)I + β(m, I, x)Sψ.
+For any (Si, Ii) ∈ [0, A1] × [0, A2](i = 1, 2), obviously, Fi(t, x, S, I) is Lipschitz continuous in
+(S, I) for (Si, Ii) ∈ [0, A1] × [0, A2] by the continuity and monotonicity of β(m(x), I, x) and
+γ(b(x), I, x), and it is uniformly continuous for x ∈ (g(s), h(s)) and t ∈ [tx, s]. In addition,
+Fi(t, x, S, I) is continuous in all its variables in this range. Problem (2.5) has a unique solution
+(Sφ,ψ(t, x), Iφ,ψ(t, x)) for t ∈ [tx, sx), and (Sφ,ψ(t, x), Iφ,ψ(t, x)) is continuous in both t and x by
+the fundamental theorem of ODEs.
+To show that (Sφ,ψ(t, x), Iφ,ψ(t, x)) can be uniquely extended to [tx, s], it suffices to prove
+that if (Sφ,ψ(t, x), Iφ,ψ(t, x)) is uniquely defined for t ∈ [tx, �t] with �t ∈ (tx, s], then
+0 ≤ Sφ,ψ(t, x) ≤ A1, 0 ≤ Iφ,ψ(t, x) ≤ A2 for t ∈ [tx, �t].
+(2.6)
+In fact, it is easy to see that
+F1(t, x, A1, A2)
+=
+d
+�
+R J(x − y)φ(t, y)dy − dA1 + σ − µ1A1 + γ(b, A2, x)ψ − β(m, A2, x)A1A2
+≤
+d∥φ∥∞ − dA1 + σ − µ1A1 + γ(b, A2, x)∥ψ∥ − β(m, A2, x)A1A2
+<
+d∥φ∥∞ − dA1 + σ − µ1A1 + ∥ψ∥∞ sup γ
+≤
+0
+and
+F2(t, x, A1, A2)
+=
+d
+� h(t)
+g(t) J(x − y)ψ(t, y)dy − dA2 − µ2A2 − γ(b, A2, x)A2 + β(m, A2, x)A1ψ
+≤
+d
+� h(t)
+g(t) J(x − y)ψ(t, y)dy − dA2 − µ2A2 + A1∥ψ∥∞ sup β
+≤
+0.
+6
+
+Since A1 ≥ ∥S0∥∞, A2 ≥ ∥I0∥∞, we have Sφ,ψ(t, x) ≤ A1 and Iφ,ψ(t, x) ≤ A2 in t ∈ [tx, �t] by the
+comparison argument. The left part of (2.6) can be obtained similarly by using Fi(t, x, 0, 0) ≥ 0
+(i = 1, 2).
+Step 2. A fixed point theorem.
+For any s ∈ (0, T), we note
+XS0
+s
+:= {φ|D
+∞
+s : φ ∈ XS0
+T }, XI0
+s := {ψ|D
+g,h
+s
+: ψ ∈ XI0
+T }.
+Denote
+(�S(t, x), �I(t, x)) =
+�
+(Sφ(t,x), 0),
+x ∈ R\[−h0, h0], t = [0, tx],
+(Sφ,ψ(t, x), Iφ,ψ(t, x)),
+x ∈ (g(s), h(s)), t ∈ [tx, s],
+where Sφ(t, x), Sφ,ψ(t, x) and Iφ,ψ(t, x)) are given in Step 1. By Step 1, for any (φ, ψ), we have
+a unique solution (�S, �I) for t ∈ [0, s]. It is easy to check that �S(t, x) is continuous in D
+∞
+s ,
+and �I(t, x) is continuous in D
+g,h
+s
+due to the continuous dependence of the ODE solution on the
+parameters. Therefore, (�S, �I) ∈ XS0
+s ×XI0
+s . Note that XS0
+s
+and XI0
+s are complete metric spaces,
+respectively, with the norms
+d1(φ1, φ2) = ∥φ1 − φ2∥C(D
+∞
+s ), d2(ψ1, ψ2) = ∥ψ1 − ψ2∥C(Dg,h
+s
+).
+Hence, we find a mapping Γ : XS0
+s × XI0
+s → XS0
+s × XI0
+s by Γ(φ, ψ) = (�S, �I).
+Setting
+M1 = max{A, 4∥S0∥∞, 4σ
+µ1
+, 4(σ + M2)
+µ1 + d
+}, M2 = max{A, 2∥I0∥∞}.
+Define
+XM1
+s
+= {φ| φ ∈ XS0
+s , ∥φ∥C(D∞
+s ) ≤ M1},
+XM2
+s
+= {ψ| ψ ∈ XI0
+s , ∥ψ∥C(Dg,h
+s
+) ≤ M2}.
+Using the same arguments as Lemma 2.1 in [45], we can deduce that Γ is a contraction map and
+has a unique fixed point (S∗, I∗) ∈ XM1
+s
+× XM2
+s
+for any s ∈ (0, �s] by the contraction mapping
+theorem, where �s relies on d, M1, β, γ and M2.
+To prove that (S∗, I∗) is the unique solution (2.1) for t ∈ [0, s] with s ∈ (0, �s], it suffices to
+discuss that any nonnegative solution (S, I) of (2.1) for t ∈ [0, s] belongs to XM1
+s
+× XM2
+s
+.
+We claim that
+S + I ≤ A for t ∈ [0, s] and x ∈ R,
+(2.7)
+which implies that
+0 ≤ S(t, x) ≤ A,
+(t, x) ∈ [0, s] × R,
+0 ≤ I(t, x) ≤ A,
+(t, x) ∈ [0, s] × [g(t), h(t)].
+Consequently, we obtain that for any s ∈ (0, �s], (2.1) admits a unique solution for t ∈ [0, s]. To
+complete the proof, it only needs to prove that the claim (2.7) is true.
+Let N = S + I, then for t ∈ [0, s] and x ∈ (g(t), h(t)),
+Nt
+≤
+d
+�
+R J(x − y)N(t, y)dy − dN(t, x) − d(
+� g(t)
+−∞ +
+� ∞
+h(t))J(x − y)I(t, y)dy + σ − µ1N
+≤
+d
+�
+R J(x − y)N(t, y)dy − dN(t, x) − µ1N + σ.
+Since ∥S0∥∞ + ∥I0∥∞ ≤ A, we have N(t, x) ≤ A for t ∈ [0, s] and x ∈ (g(t), h(t)) by using the
+comparison principle.
+7
+
+While t ∈ [0, s] and x ∈ R\(g(t), h(t)), then I(t, x) = 0, which implies
+Nt = d
+�
+R
+J(x − y)N(t, y)dy − dN(t, x) + σ − µ1N.
+It is clear that N ≤ A for t ∈ [0, s] and x ∈ R\(g(t), h(t)). Next, we prove that (2.7) holds. We
+argue by contradiction and suppose that
+max
+(t,x)∈[0, s]×R N(t, x) > A, there exists a point (t0, x0) ∈
+[0, s] × R such that max N = N(t0, x0) > A. According to the above analysis, we can obtain
+that x0 = g(t0) or x0 = h(t0). Without loss of generality, we assume that x0 = g(t0). Since
+I(t0, g(t0)) = 0, S(t0, x0) satisfies
+St(t0, x0) = d
+�
+R
+J(x0 − y)S(t0, y)dy − dS(t0, x0) + σ − µ1S(t0, x0).
+Obviously, St(t0, x0) ≥ 0 and S(t0, x0) ≤ A, which contradicts the assumption that
+max
+(t,x)∈[0, s]×R N(t, x) >
+A.
+Step 3. Extension of the solution.
+We now prove that the unique solution of (2.1) for 0 < t ≤ s can be extended to 0 < t ≤ T.
+In Step 2, �s depends only on d, γ, β, and A. With the help of the iterative method, we obtain
+that problem (2.1) has a unique solution for t ∈ [0, T]. We omit it here; see Step 3 of the proof
+in Lemma 2.1 in [45] for more details.
+□
+Theorem 2.2 Assume that (J) holds.
+For any S0 satisfying (1.4) and I0 satisfying (1.5),
+problem (1.3) admits a unique positive solution (S(t, x), I(t, x); g(t), h(t)) defined for all t > 0.
+Proof.
+We will prove this result by using Lemma 2.1 and the fixed point theorem. For any
+given T > 0 and (g∗, h∗) ∈ HT × GT , we know that (2.1) with (g, h) = (g∗, h∗) has a unique
+solution (S∗, I∗). Define
+
+
+
+�g = −h0 − k
+� t
+0
+� h∗(τ)
+g∗(τ)
+� g∗(τ)
+−∞
+J(x − y)I∗(τ, x)dydxdτ,
+�h = h0 + k
+� t
+0
+� h∗(τ)
+g∗(τ)
+� +∞
+h∗(τ) J(x − y)I∗(τ, x)dydxdτ.
+(2.8)
+In view of (J) and J(0) > 0, there exist constants ǫ0 ∈ (0, h0/4) and δ0 > 0 such that
+J(x) ≥ δ0 if |x| ≤ ǫ0.
+By virtue of the above inequality and proof of Theorem 2.1 in [5], there exists
+T0 = T0(k, A, h0, ǫ0, I0, J) > 0,
+such that, for any T ∈ (0, T0],
+sup
+0≤t1<t2≤T
+�g(t2) − �g(t1)
+t2 − t1
+≤ −kη1,
+inf
+0≤t1<t2≤T
+�h(t2) − �h(t1)
+t2 − t1
+≥ kη2,
+�h(t) − �g(t) ≤ 2h0 + ǫ0/4 for t ∈ [0, T],
+where
+η1 = 1
+4ǫ0δ0e−(d+µ2+sup γ)T0
+� −h0+ ǫ0
+4
+−h0
+I0(x)dx, η2 = 1
+4ǫ0δ0e−(d+µ2+sup γ)T0
+� h0
+h0− ǫ0
+4
+I0(x)dx.
+8
+
+Let
+ΣT := {(g, h) ∈ Hh0
+T × Gh0
+T :
+sup
+0≤t1<t2≤T
+g(t2)−g(t1)
+t2−t1
+≤ −kη1,
+inf
+0≤t1<t2≤T
+h(t2)−h(t1)
+t2−t1
+≥ kη2, h(t) − g(t) ≤ 2h0 + ǫ0
+4 for t ∈ [0, T]},
+and define the mapping F(g∗, h∗) = (�g,�h). Clearly, the above analysis implies that
+F(ΣT) ⊂ ΣT for T ∈ (0, T0].
+In the following, similar to the proof of Theorem 1.1 in [45], we first prove that F is a contraction
+mapping, and then F admits a unique fixed point in ΣT by the contraction mapping theorem.
+Next, we can derive that (g, h) ∈ ΣT holds for any solution (S, I; g, h) of (1.3) for t ∈ [0, T],
+that is, (S, I; g, h) is a unique solution of (1.3) for t ∈ [0, T]. Finally, we can show that the
+solution (S, I; g, h) of (1.3) is uniquely extended to t ∈ (0, +∞). Here, we omit the proof here;
+see Step 3 in the proof of Theorem 1.1 in [45] for more details.
+□
+3
+The eigenvalue problem
+For any −∞ < L1 < L2 < +∞ and d > 0, denote
+L{(L1, L2), d}[φ](x) = d
+� L2
+L1
+J(x − y)φ(y)dy − dφ(x).
+Now we introduce some results on the principal eigenvalue of the linear operator L{(L1, L2), d} +
+a(x) : C([L1, L2]) �→ C([L1, L2]) defined by
+(L{(L1, L2), d} + a(x))[φ](x) = d
+� L2
+L1
+J(x − y)φ(y)dy − dφ(x) + a(x)φ(x),
+where a(x) = σβ(m(x),0,x)
+µ1
+− µ2 − γ(b(x), 0, x) ∈ C([L1, L2]) and J satisfies (J). Furthermore, we
+assume
+(H) : a(x) is Lipschitz continuous and achieves its maximum in [L1, L2] at some point x0 ∈ (L1, L2).
+Define the generalized principal eigenvalue as
+λp(L{(L1, L2), d} + a(x))
+:= inf{λ ∈ R | ∃φ ∈ C([L1, L2]), φ > 0 s. t. (L{(L1, L2), d} + a(x))[φ] ≤ λφ in (L1, L2)}.
+(3.1)
+Furthermore, we call it a principal eigenvalue if λp(L{(L1, L2), d} + a(x)) is an eigenvalue of the
+operator L{(L1, L2), d} + a(x) with a continuous and positive eigenfunction. Recalling that a(x)
+is Lipschitz continuous and achieves a global maximum in (L1, L2), thus a(x) automatically
+satisfies the condition
+1
+(supx∈(L1, L2) a(x))−a(x) ̸∈ L1. Therefore, it follows from Theorem 1.1 or
+Theorem 1.2 in [11] that the generalized principal eigenvalue λp(L{(L1, L2), d}+a(x)) is a principal
+eigenvalue.
+In this section, we are interested in the properties of the generalized principal eigenvalue
+λp(L{(L1, L2), d}+a(x)) with media coverage m(x) and hospital bed number b(x), and asymptotic
+behavior of the principal eigenvalue in large and small interval lengths (L1, L2) or diffusion rate
+d.
+Before stating our primary result, we recall a useful proposition from [11].
+9
+
+Proposition 3.1 ( [11]) The following assertions hold:
+(i) Assume (L1, L2) ⊂ (L3, L4). Then,
+λp(L{(L1, L2), d} + a(x)) ≤ λp(L{(L3, L4), d} + a(x)).
+(ii) Fix L1, L2 and suppose that a1(x) ≤ a2(x). Then,
+λp(L{(L1, L2), d} + a1(x)) ≤ λp(L{(L1, L2), d} + a2(x)).
+Moreover, if a1(x) + δ < a2(x) for some δ > 0, then
+λp(L{(L1, L2), d} + a1(x)) < λp(L{(L1, L2), d} + a2(x)).
+(iii) λp(L{(L1, L2), d} + a(x)) is Lipschitz continuous in a(x). More precisely,
+|λp(L{(L1, L2), d} + a1(x)) − λp(L{(L1, L2), d} + a2(x))| ≤ ∥a1(x) − a2(x)∥∞.
+Let us now analyze the impact of media coverage m(x) and hospital bed number b(x) on
+the generalized principal eigenvalue. Obviously, the following result holds by Proposition 3.1
+(ii).
+Theorem 3.2 Suppose that (J) and (H) hold. Then,
+(i) λp(L{(L1, L2), d} + a(x)) is a strictly monotone decreasing in m(x).
+(ii) λp(L{(L1, L2), d} + a(x)) is a strictly monotone decreasing in b(x).
+From now on, we discuss the effect of interval length on the principal eigenvalue λp(L{(L1,L2), d}+
+a(x)).
+Theorem 3.3 Suppose that (J) and (H) hold, then the following three conclusions hold:
+(i) λp(L{(L1, L2), d} + a(x)) is continuous for L1, L2 ∈ (−∞, +∞).
+(ii)
+lim
+L1, L2→0 λp(L{(L1, L2), d} + a(x)) = a(0) − d.
+(iii)
+lim
+−L1,L2→+∞ λp(L{(L1, L2), d} + a(x)) = sup
+x∈R
+a(x).
+Proof. The proof of (i) is similar to Proposition 3.4 in [5], and we omit it here.
+(ii) Due to the continuity of a(x), for any given ǫ > 0, there exists h > 0 small enough such
+that
+|a(x) − a(0)| < ǫ, x ∈ [−h, h].
+Since λp(L{(−h, h), d} + a(x)) is a principal eigenvalue, there exists a positive function φ(x) ∈
+C([−h, h]) such that
+d
+� h
+−h
+J(x − y)φ(y)dy − dφ(x) + a(x)φ(x) = λp(L{(−h, h), d} + a(x))φ(x), x ∈ [−h, h],
+which gives by integrating,
+|λp(L{(−h, h), d} + a(x)) − a(0) + d|
+=
+|
+d
+� h
+−h
+� h
+−h J(x−y)φ(y)φ(x)dydx
+� h
+−h φ2(x)dx
++
+� h
+−h a(x)φ2(x)dx
+� h
+−h φ2(x)dx
+− a(0)|
+=
+|
+d
+� h
+−h
+� h
+−h J(x−y)φ(y)φ(x)dydx
+� h
+−h φ2(x)dx
++
+� h
+−h(a(x)−a(0))φ2(x)dx
+� h
+−h φ2(x)dx
+|
+≤
+d∥J∥∞(� h
+−h φ(x)dx)2
+� h
+−h φ2(x)dx
++ |
+� h
+−h(a(x)−a(0))φ2(x)dx
+� h
+−h φ2(x)dx
+|
+≤
+2d∥J∥∞h + ǫ → ǫ as h → 0+.
+10
+
+From the arbitrariness of ǫ, we have
+|λp(L{(−h, h), d} + a(x)) − a(0) + d| → 0 as h → 0+,
+which together with the continuity of λp(L{(L1, L2), d} + a(x)) about L1, L2 give
+lim
+L1, L2→0 λp(L{(L1, L2), d} + a(x)) = a(0) − d.
+(iii) According to the monotonicity of λp(L{(L1, L2), d}+a(x)) with respect to interval (L1, L2)
+and function a(x) yields
+λp(L{(L1, L2), d} + a(x)) ≤ λp(L{(L1, L2), d} + sup
+x∈R
+a(x)) ≤ λp(L{(−∞, +∞), d} + sup
+x∈R
+a(x)).
+Consider the following eigenvalue problem
+d
+� ∞
+−∞
+J(x − y)φ(y)dy − dφ(x) + φ(x) sup
+x∈R
+a(x) = λp(L{(−∞, +∞), d} + sup
+x∈R
+a(x))φ, x ∈ R. (3.2)
+It is easily seen that λp(L{(−∞, +∞), d} + sup
+x∈R
+a(x)) = sup
+x∈R
+a(x).
+So the principal eigenvalue
+λp(L{(L1, L2), d} + a(x)) ≤ sup
+x∈R
+a(x). Therefore,
+lim sup
+−L1, L2→+∞
+λp(L{(L1, L2), d} + a(x)) ≤ sup
+x∈R
+a(x).
+To prove (iii), it suffices to prove that
+lim inf
+−L1, L2→+∞ λp(L{(L1, L2), d} + a(x)) ≥ sup
+x∈R
+a(x) holds.
+In fact, by the continuity of a(x) and the definition of sup, for given ǫ > 0, there exists some
+x0 ∈ R such that
+sup
+x∈R
+a(x) − ǫ ≤ a(x0).
+By (J), for given ǫ > 0, there exist L1 < x0 − 1 and L2 > x0 + 1 such that
+� L2
+L1
+J(z)dz > 1 − ǫ.
+Now take
+δn(x − x0) =
+�
+k1e−1/(1−n2(x−x0)2) > 0,
+x ∈ (x0 − 1/n, x0 + 1/n),
+= 0,
+x /∈ (x0 − 1/n, x0 + 1/n),
+where k1 is positive and satisfies
+�
+R δn(x − x0)dx = 1.
+It is easy to check that the se-
+quence {δn(x − x0)} weakly converges to some δ(x − x0) in L1((L1, L2)). By the definition
+of λp(L{(L1, L2), d} + a(x)), one can easily obtain
+d
+� L2
+L1 J(x − y)δn(y − x0)dy − dδn(x − x0) + a(x)δn(x − x0)
+≤
+λp(L{(L1, L2), d} + a(x))δn(x − x0), x ∈ (L1, L2).
+(3.3)
+Integrating the equation of (3.3) over (L1, L2) yields
+λp(L{(L1, L2), d} + a(x))
+≥
+d
+� L2
+L1
+� L2
+L1 J(x−y)δn(y−x0)dydx−d
+� L2
+L1 δn(x−x0)dx+
+� L2
+L1 a(x)δn(x−x0)dx
+� L2
+L1 δn(x−x0)dx
+≥
+d
+� L2
+L1
+� x0+1
+x0−1 J(x−y)δn(y−x0)dydx−d
+� L2
+L1 δn(x−x0)dx+
+� L2
+L1 a(x)δn(x−x0)dx
+� L2
+L1 δn(x−x0)dx
+≥
+d � x0+1
+x0−1 δn(y−x0)[� L2−x0−1
+L1−x0+1 J(z)dz]dy−d � L2
+L1 δn(x−x0)dx+� L2
+L1 a(x)δn(x−x0)dx
+� L2
+L1 δn(x−x0)dx
+≥
+(d(1−ǫ)−d) � L2
+L1 δn(x−x0)dx+� L2
+L1 a(x)δn(x−x0)dx
+� L2
+L1 δn(x−x0)dx
+=
+−dǫ +
+� L2
+L1 a(x)δn(x−x0)dx
+� L2
+L1 δn(x−x0)dx ,
+11
+
+where we have used that
+� x0+1
+x0−1 δn(x−x0)dx =
+� L2
+L1 δn(x−x0)dx for sufficiently large n. Therefore,
+by taking n → +∞,
+lim inf
+−L1, L2→+∞ λp(L{(L1,L2), d} + a(x))
+≥
+−dǫ +
+� +∞
+−∞ a(x)δ(x − x0)dx
+=
+−dǫ + a(x0)
+≥
+−dǫ + sup
+x∈R
+a(x) − ǫ.
+It follows from the arbitrarily of ǫ that
+lim inf
+−L1,L2→+∞ λp(L{(L1, L2), d} + a(x)) ≥ sup
+x∈R
+a(x).
+□
+In the following, we discuss the monotonicity of the principal eigenvalue with respect to d
+and its limiting behaviors as d → 0 or d → +∞.
+Theorem 3.4 Suppose that (J) and (H) hold. Then, the following statements hold:
+(i) λp(L{(L1, L2), d} + a(x)) is a strictly monotone decreasing function of d.
+(ii) lim
+d→0 λp(L{(L1, L2), d} + a(x)) =
+max
+x∈[L1, L2] a(x).
+(iii) If
+� L1−L2
+−∞
+J(z)dz > 0(symmetrically,
+� +∞
+L2−L1 J(z)dz > 0) holds, then lim
+d→+∞ λp(L{(L1, L2), d}+
+a(x)) = −∞.
+Proof.
+(i) Assume that λp(d1) := λp(L{(L1,L2), d} + a(x)) is the principal eigenvalue and
+φ(x) is the corresponding positive eigenfunction with ∥φ∥L2 = 1, we have
+λp(d1)φ(x) = d1
+� L2
+L1
+J(x − y)φ(y)dy − d1φ(x) + a(x)φ(x), x ∈ (L1, L2).
+Suppose that d < d1, then
+λp(d1)
+=
+d1
+� L2
+L1
+� L2
+L1 J(x − y)φ(y)φ(x)dydx − d1 +
+� L2
+L1 a(x)φ2(x)dx
+=
+d
+� L2
+L1
+� L2
+L1 J(x − y)φ(y)φ(x)dydx − d1 +
+� L2
+L1 a(x)φ2(x)dx
++(d1 − d)
+� L2
+L1
+� L2
+L1 J(x − y)φ(y)φ(x)dydx
+<
+d
+� L2
+L1
+� L2
+L1 J(x − y)φ(y)φ(x)dydx − d1 +
+� L2
+L1 a(x)φ2(x)dx + d1 − d
+=
+d
+� L2
+L1
+� L2
+L1 J(x − y)φ(y)φ(x)dydx − d +
+� L2
+L1 a(x)φ2(x)dx
+≤
+λp(d).
+Therefore λp(d1) < λp(d).
+(ii) The idea of this proof is from Theorem 2.8 in [44]. For the eigenvalue problem
+d
+� L2
+L1
+J(x − y)ϕ(y)dy − dϕ(x) + ( max
+x∈[L1, L2] a(x)) ϕ(x) = λ∗
+pϕ(x), x ∈ (L1, L2).
+(3.4)
+It follows from [19] that λ∗
+p ≤
+max
+x∈[L1, L2] a(x).
+So we have λp(L{(L1, L2), d} + a(x)) ≤ λ∗
+p ≤
+max
+x∈[L1, L2] a(x) by (ii) of Proposition 3.1.
+12
+
+Next, we prove that lim inf
+d→0
+λp(L{(L1, L2), d} + a(x)) ≥
+max
+x∈[L1, L2] a(x). Assume for the contrary
+that lim inf
+d→0
+λp(L{(L1, L2), d} + a(x)) ≤
+max
+x∈[L1, L2] a(x) − ǫ for some ǫ > 0 . By the definition of
+lim inf, there exists some �d > 0 such that if d ≤ �d, then
+λp(L{(L1, L2), d} + a(x)) ≤
+max
+x∈[L1, L2] a(x) − ǫ
+2.
+On the other hand, by the continuity of a(x), there exist x0 ∈ (L1, L2) and r > 0 such that
+max
+x∈[L1, L2] a(x) ≤ a(x) + ǫ
+4, x ∈ Ur(x0) ⊂ (L1, L2).
+Therefore,
+λp(L{(L1, L2), d} + a(x)) ≤ a(x) − ǫ
+4
+for 0 < d < �d and x ∈ Ur(x0).
+Let (λp(L{(L1, L2), d} + a(x)), ψ(x)) be the eigenpair of the
+following eigenvalue problem:
+d
+� L2
+L1
+J(x − y)ψ(y)dy − dψ(x) + a(x)ψ(x) = λp(L{(L1, L2), d} + a(x))ψ(x), x ∈ (L1, L2).
+Then,
+� L2
+L1
+J(x − y)ψ(y)dy − ψ(x) = λp(L{(L1, L2), d} + a(x)) − a(x)
+d
+ψ(x) ≤ − ǫ
+4dψ(x) in Ur(x0).
+Let �λ be the principal eigenvalue of the linear problem
+� �
+R J(x − y)u(y)dy − u(x) = λu(x)
+in Ur(x0),
+u(x) = 0,
+in R\Ur(x0).
+(3.5)
+It is well known that −1 < �λ < 0 by Theorem 2.1 in [20]. Let Ψ(x) be the eigenfunction
+corresponding to �λ and ∥Ψ(x)∥L∞ = 1. Take
+ψ(x) =
+ψ(x)
+infUr(x0) ψ(x), ψ(x) = Ψ(x).
+We consider the following problem
+� �
+R J(x − y)u(y)dy − u(x) = − ǫ
+4du(x)
+in Ur(x0),
+u(x) = 0,
+in R\Ur(x0).
+(3.6)
+Direct calculation yields
+�
+Ur(x0) J(x − y)ψ(y)dy − ψ(x) +
+ǫ
+4dψ(x)
+=
+1
+inf ψ[
+�
+Ur(x0) J(x − y)ψ(y)dy − ψ(x)] +
+ǫ
+4dψ(x)
+≤
+1
+inf ψ(− ǫ
+4dψ(x) +
+ǫ
+4dψ(x))
+=
+0,
+13
+
+and
+�
+Ur(x0) J(x − y)ψ(y)dy − ψ(x) +
+ǫ
+4dψ(x)
+=
+�
+Ur(x0) J(x − y)Ψ(x)dy − Ψ(x)] +
+ǫ
+4dΨ(x)
+=
+�λΨ(x) +
+ǫ
+4dΨ(x)
+≥
+0
+provided d < min{�d, − ǫ
+4�λ}. Hence, by the super-sub solution method in [21], one can yield (3.6)
+has a positive solution between ψ(x) and ψ , which implies that �λ = − ǫ
+4d. This contradicts to
+the independence of �λ about d. Therefore, lim
+d→0 λp(L{(L1, L2), d} + a(x)) =
+max
+x∈[L1, L2] a(x).
+(iii) We claim that if
+� L1−L2
+−∞
+J(z)dz > 0 (or
+� +∞
+L2−L1 J(z)dz > 0), then lim
+d→+∞ λp(L{(L1, L2), d} +
+a(x)) := λ∞ = −∞.
+We argue by contradiction and suppose that λ∞ > −∞.
+Now let
+ϕ(x) = C1 (positive constant), without loss of generality, taking C1 = 1, then for x ∈ (L1, L2),
+d
+� L2
+L1 J(x − y)ϕ(y)dy − dϕ(x) + a(x)ϕ(x)
+<
+d
+� L2
+L1 J(x − y)dy − d +
+max
+x∈[L1,L2] a(x)
+=
+−d
+�
+R\[L1,L2] J(x − y)dy +
+max
+x∈[L1,L2] a(x)
+=
+−d(
+� x−L2
+−∞
++
+� +∞
+x−L1)J(z)dz +
+max
+x∈[L1,L2] a(x)
+≤
+−d(
+� L1−L2
+−∞
++
+� +∞
+L2−L1)J(z)dz +
+max
+x∈[L1,L2] a(x)
+as
+�
+R J(x)dx = 1. Owing to the assumption that
+� L1−L2
+−∞
+J(z)dz > 0 (or
+� +∞
+L2−L1 J(z)dz > 0),
+there exists a d adequately large such that
+d
+� L2
+L1
+J(x − y)ϕ(y)dy − dϕ(x) + a(x)ϕ(x) ≤ (λ∞ − 1)ϕ(x).
+Thus, by definition of λp(L{(L1, L2), d} + a(x)), we have
+λp(L{(L1, L2), d} + a(x)) ≤ λ∞ − 1,
+further,
+lim
+d→+∞ λp(L{(L1, L2), d} + a(x)) = λ∞ ≤ λ∞ − 1.
+We obtain the desired contradiction. Hence,
+lim
+d→+∞ λp(L{(L1, L2), d} + a(x)) = −∞.
+□
+Remark 3.5 Compared with [44], where the nonlocal operator is d
+�
+Ω J(x−y)(ψ(y)−ψ(x))dy,
+we consider the nonlocal operator d
+�
+Ω J(x − y)ϕ(y)dy − dϕ(x) and prove that when d → +∞,
+the limit of the principal eigenvalue is −∞ for some cases, which is different from the result
+in [44]; its limit is the average of a(x) over Ω.
+4
+Spreading-vanishing
+Since h(t) and −g(t) are monotonically increasing with t > 0, there exist h∞ and g∞ such that
+lim
+t→+∞ g(t) = g∞ ∈ [−∞, −h0) and lim
+t→+∞ h(t) = h∞ ∈ (h0, +∞]. Here we define that vanishing
+occurs if h∞ − g∞ < +∞ and
+lim
+t→+∞
+max
+x∈[g(t), h(t)] I(t, x) = 0; and spreading happens provided
+14
+
+that h∞ −g∞ = +∞ and lim sup
+t→+∞ ∥I(·, t)∥C([g(t), h(t)]) > 0. In this section, we always assume that
+(J) and (H) hold. The following proposition directly holds from Theorem 3.5 in [6].
+Proposition 4.1 Let (S, I; g, h) be the unique solution of (1.3). If h∞ − g∞ < +∞, then
+lim
+t→+∞ g′(t) = lim
+t→+∞ h′(t) = 0.
+Next, we discuss the asymptotic behavior of the solution of problem (1.3) when h∞ − g∞ <
++∞.
+Theorem 4.2 Let (S, I; g, h) be the unique solution of problem (1.3) with h∞ − g∞ < +∞,
+then lim
+t→+∞
+max
+x∈[g(t), h(t)] I(t, x) = 0, lim
+t→+∞ S(t, x) =
+σ
+µ1 and λp(L{(g∞, h∞), d} + a(x)) ≤ 0.
+Proof.
+Assume by contradiction that
+lim
+t→+∞
+max
+x∈[g(t), h(t)] I(t, x) > 0, there exists ǫ1 > 0 and
+sequence {(ti, xi)}∞
+i=1 with xi ∈ [g(t), h(t)] and ti → +∞ as i → +∞ such that I(ti, xi) ≥ ǫ1
+2
+for i ∈ N. Since g∞ < g(t) < xi < h(t) < h∞, there exists a subsequence {xij}∞
+j=1 such that
+xij → x0 ∈ (g∞, h∞) as j → +∞. For t ∈ (−ti, +∞) and x ∈ (g(t + ti), h(t + ti)), define
+Ii(t, x) = I(t + ti, x).
+Applying Theorem 2.2 gives that I and S are positive and bounded, and then Ii(t, x) satisfies
+Iit ≥ d
+� hi(t)
+gi(t)
+J(x − y)Ii(t, y)dy − dIi(t, x) − µ2Ii − γ(b, 0, x)Ii, t > −ti, x ∈ (gi(t), hi(t)).
+We next consider the following auxiliary problem
+�
+ut = d
+� hi(t)
+gi(t) J(x − y)u(t, y)dy − du(t, x) − µ2u − γ(b, 0, x)u,
+t > −ti, x ∈ (gi(t), hi(t)),
+u(0, x) = Ii(0, x),
+x ∈ (gi(t), hi(t)),
+it follows that u(t, x) → U(t, x) as i → +∞, and U(t, x) satisfies
+
+
+
+Ut(t, x) = d
+� h∞
+g∞ J(x − y)U(t, y)dy − dU(t, x) − µ2U − γ(b, 0, x)U,
+t ∈ R, x ∈ (g∞, h∞),
+U(0, x0) = lim
+i→+∞ Ii(0, xi) = lim
+i→+∞ I(ti, xi) ≥ ǫ1
+2 > 0,
+and then U(t, x) > 0 in R × (g∞, h∞) by the maximum principle ( [15]) for the nonlocal
+problem.
+On the other hand, considering h∞ − g∞ < +∞ and Proposition 4.1, we have lim
+t→+∞ g′(t) =
+lim
+t→+∞ h′(t) = 0 as t → +∞, which means
+0 = lim
+i→+∞ h′(t + ti)
+=
+k lim
+i→+∞
+� h(t+ti)
+g(t+ti)
+� +∞
+h(t+ti) J(x − y)Ii(t, x)dydx
+≥
+k
+� h(∞)
+g(∞)
+� +∞
+h(∞) J(x − y)U(t, x)dydx
+>
+0
+and
+0 = lim
+i→+∞ g′(t + ti)
+=
+−k lim
+i→+∞
+� h(t+ti)
+g(t+ti)
+� g(t+ti)
+−∞
+J(x − y)Ii(t, x)dydx
+≤
+−k
+� h(∞)
+g(∞)
+� g(∞)
+−∞ J(x − y)U(t, x)dydx
+<
+0.
+15
+
+It is a contradiction. Hence, lim
+t→+∞
+max
+x∈[g(t), h(t)] I(t, x) = 0.
+Next, we will prove that lim
+t→+∞ S(t, x) =
+σ
+µ1. Since lim
+t→+∞
+max
+x∈[g(t), h(t)] I(t, x) = 0, for any ǫ > 0,
+one can choose a T > 0 large such that
+0 < I(t, x) < ǫ
+for t > T, x ∈ (g(t), h(t)).
+Obviously, S(t, x) satisfies
+St ≥ dL1[S] + σ − µ1S − ǫβ(m, 0, x)S, t > T, x ∈ R,
+and then S(t, x) ≥ S(t), where S(t) is the solution to problem
+
+
+
+St = σ − (µ1 + ǫ sup
+x∈R
+β(m, 0, x))S,
+t > T,
+S(T) = inf
+x∈R S(T, x).
+It follows from Lemma 2.4 in [26] that lim
+t→+∞ S(t) =
+σ
+µ1+ǫ sup
+x∈R
+β(m,0,x), Therefore, lim inf
+t→+∞ S(t, x) ≥
+σ
+µ1+ǫ sup
+x∈R
+β(m,0,x). Letting ǫ → 0 yields
+lim inf
+t→+∞ S(t, x) ≥ σ
+µ1
+.
+(4.1)
+On the other hand, S(t, x) satisfies
+St ≤ dL1[S] + σ − µ1S + γ(b, ǫ, x)ǫ, t > T, x ∈ R.
+Let S(t) be the solution of
+
+
+
+St = σ − µ1S(t) + sup
+x∈R
+γ(b, ǫ, x)ǫ,
+t > T,
+S(T) = sup
+x∈R
+S(T, x).
+Apparently, S(t, x) ≤ S(t) by Lemma 2.4 in [26] and lim
+t→+∞ S(t) =
+1
+µ1[σ +sup
+x∈R
+γ(b, ǫ, x)ǫ]. There-
+fore,
+lim sup
+t→+∞ S(t, x) ≤ lim
+t→+∞ S(t) = 1
+µ1
+[σ + sup
+x∈R
+γ(b, ǫ, x)ǫ].
+Letting ǫ → 0 gives
+lim sup
+t→+∞ S(t, x) ≤ σ
+µ1
+,
+(4.2)
+which together with (4.1) yields lim
+t→+∞ S(t, x) =
+σ
+µ1 uniformly for x ∈ R.
+In what follows, we prove that λp(L{(g∞, h∞), d} + a(x)) ≤ 0.
+We argue by contradic-
+tion and suppose that λp(L{(g∞, h∞), d} + a(x)) > 0. Owing to the continuous dependence of
+λp(L{(g∞, h∞), d}+a(x)) on a(x) and (g∞, h∞), there exists a small ǫ such that λp(L{(g∞+ǫ, h∞−ǫ), d}+
+a(x) − β(m, 0, x)ǫ) > 0. Furthermore, in view of S(t, x) →
+σ
+µ1 as t → +∞ and h∞ − g∞ < +∞,
+there exists a T ∗ > 0 such that
+g(t) < g∞ + ǫ, h(t) > h∞ − ǫ, t > T ∗,
+S(t, x) > σ
+µ1
+− ǫ, t > T ∗, x ∈ R.
+16
+
+Then, for t > T ∗, x ∈ (g(t), h(t)),
+It(t, x)
+=
+d
+� h(t)
+g(t) J(x − y)I(t, y)dy − dI(t, x) − µ2I + β(m, I, x)SI − γ(b, I, x)I
+≥
+d
+� h∞−ǫ
+g∞+ǫ J(x − y)I(t, y)dy − dI(t, x) − µ2I + β(m, I, x)( σ
+µ1 − ǫ)I − γ(b, I, x)I
+≥
+d
+� h∞−ǫ
+g∞+ǫ J(x − y)I(t, y)dy − dI(t, x) − µ2I + β(m, 0, x)( σ
+µ1 − ǫ)I − γ(b, 0, x)I.
+Let φ(x) be the eigenfunction corresponding to λp(L{(g∞+ǫ, h∞−ǫ), d} + a(x) − β(m, 0, x)ǫ) and
+∥φ(x)∥L∞ = 1, and then φ(x) satisfies
+d
+� h∞−ǫ
+g∞+ǫ J(x − y)φ(y)dy − dφ − µ2φ + β(m, 0, x)( σ
+µ1 − ǫ)φ − γ(b, 0, x)φ
+=
+d
+� h∞−ǫ
+g∞+ǫ J(x − y)φ(y)dy − dφ + a(x)φ − β(m, 0, x)ǫφ
+=
+λp(L{(g∞+ǫ, h∞−ǫ), d} + a(x) − β(m, 0, x)ǫ)φ.
+If we choose δ sufficiently small such that δφ(x) ≤ I(T ∗, x) for x ∈ [g∞ + ǫ, h∞ − ǫ], then
+I(t, x) ≥ δφ(x) > 0 for t > T ∗, x ∈ [g∞ + ǫ, h∞ − ǫ]
+by the comparison principle in [15], which leads to a contradiction to the fact lim
+t→+∞
+max
+x∈[g(t), h(t)] I(t, x) =
+0.
+□
+Theorem 4.3 Suppose λp(L{(−h0, h0), d}+a(x)) < 0. Then h∞−g∞ < +∞, lim
+t→+∞
+max
+x∈[g(t), h(t)] I(t, x) =
+0 and lim
+t→+∞ S(t, x) =
+σ
+µ1 if ∥S0(x)∥L∞(R) + ∥I0(x)∥C([−h0, h0]) is sufficiently small.
+Proof.
+Applying Lemma 2.1 yields S(t, x) ≤
+σ
+µ1 for t > 0, x ∈ R if ∥S0(x)∥L∞(R) +
+∥I0(x)∥C([−h0, h0]) ≤
+σ
+µ1. In view of λp(L{(−h0, h0), d} + a(x)) < 0, there exists a ǫ > 0 small
+such that λp(L{(−hǫ, hǫ), d} + a(x)) < 0 with hǫ = h0 + ǫ by Theorem 3.3 (i). Let φ(x) be the
+eigenfunction of the principal eigenvalue λp(L{(−hǫ, hǫ), d} + a(x)), which satisfies
+d
+� hǫ
+−hǫ
+J(x − y)φ(y)dy − dφ(x) + a(x)φ(x) = λp(L{(−hǫ, hǫ), d} + a(x))φ(x), x ∈ (−hǫ, hǫ).
+Denote
+M = δC(
+� hǫ
+−hǫ
+φ(x)dx)−1, C = hǫ − h0
+k
+,
+h(t) = h0 + kC[1 − e−δt], g(t) = −h(t),
+h
+′(t) = kCδe−δt, I(t, x) = Me−δtφ(x).
+By direct calculations, we have
+k
+� h
+g
+� ∞
+h J(x − y)I(t, x)dydx
+≤
+k
+� h
+g I(t, x)dx
+=
+kδCe−δt = h
+′(t),
+t > 0.
+17
+
+In a similar way, one can deduce −k
+� h
+g
+� g
+−∞ J(x − y)I(t, x)dydx ≥ g′(t) for t > 0. Clearly,
+I(t, g(t)) > 0 and I(t, h(t)) > 0 for t > 0. For t > 0 and x ∈ (g, h), we obtain
+It(t, x) − d
+� h(t)
+g(t) J(x − y)I(t, y)dy + dI(t, x) + µ2I − β(m, I, x)SI + γ(b, I, x)I
+≥
+−[d
+� h(t)
+g(t) J(x − y)Me−δtφ(y)dy − dMe−δtφ(x) − µ2Me−δtφ(x)]
+−δI + γ(b, I, x)I − β(m, I, x)SI
+≥
+−δI − (λp(L(−hǫ, hǫ) + a(x)) + γ(b, 0, x) − β(m, 0, x) σ
+µ1)I + γ(b, I, x)I − β(m, I, x) σ
+µ1I
+=
+I(−δ − λp(L(−hǫ, hǫ) + a(x)) − γ(b, 0, x) + γ(b, I, x) + σ
+µ1(β(m, 0, x) − β(m, I, x))).
+Recalling that γ(b, I, x) → γ(b, 0, x) and β(m, I, x) → β(m, 0, x) as δ → 0, we can choose δ
+small enough so that
+It(t, x) − d
+� h(t)
+g(t)
+J(x − y)I(t, y)dy + dI(t, x) + µ2I − β(x)SI + γ(x, b, I)I ≥ 0.
+Moreover, if ∥S0(x)∥L∞(R)+∥I0(x)∥C([−h0, h0]) sufficiently small, then I0 ≤ Mφ(x), x ∈ [−h0, h0].
+Applying the comparison principle gives
+g(t) ≤ g(t), h(t) ≤ h(t) and I(t, x) ≤ I(t, x)
+for t > 0 and g(t) < x < h(t). Therefore,
+lim
+t→+∞ I(t, x) ≤ lim
+t→+∞ I(t, x) = 0
+(4.3)
+and
+h∞ − g∞ ≤ 2hǫ < +∞,
+which gives that lim
+t→+∞ S(t, x) =
+σ
+µ1 uniformly for x ∈ R by Theorem 4.2.
+□
+Remark 4.4 It follows from the proof of Theorem 4.3 that M → +∞ as k → 0. Therefore,
+there exists a k∗ > 0 such that (4.3) holds for all k ∈ (0, k∗) for any given initial function pair
+(S0(x), I0(x)).
+Theorem 4.5 If −g∞ = h∞ = +∞ and sup
+x∈R
+a(x) > 0, then lim sup
+t→+∞ ∥I(·, t)∥C([g(t), h(t)]) > 0.
+Proof.
+Conversely, suppose that
+lim
+t→+∞ ∥I(t, ·)∥C([g(t), h(t)]) = 0. It follows from Theorem 4.2
+that
+lim
+t→+∞ S(t, x) = σ
+µ1
+uniformly for x ∈ R.
+(4.4)
+Since −g∞ = h∞ = +∞, from (iii) of Theorem 3.3, there exists a T ∗ > 0 large enough such
+that
+λp(L{(g(T ∗), h(T ∗)), d} + a(x)) > 0
+for
+t ≥ T ∗.
+Now, we consider the eigenvalue problem
+d
+� h(T ∗)
+g(T ∗)
+J(x − y)ψ(y)dy − dψ(x) + a(x)ψ(x) = λp(L{(g(T ∗), h(T ∗)), d} + a(x))ψ(x), x ∈ (g(T ∗), h(T ∗)),
+18
+
+and positive function ψ(x) with ∥ψ∥L∞((g(T ∗), h(T ∗))) = 1 is its eigenfunction to the principal eigenvalue
+λp(L{(g(T ∗), h(T ∗)), d} + a(x)).
+In view of (4.4), for any given 0 < ǫ < min{ σ
+µ1 , λp(L{(g(T ∗), h(T ∗)), d} +a(x))(sup
+x∈R
+β)−1}, there exists
+a T ∗∗ > T ∗ such that
+S(t, x) > σ
+µ1
+− ǫ,
+t ≥ T ∗∗,
+x ∈ [g(T ∗), h(T ∗)],
+and then I(t, x) satisfies
+
+
+
+
+
+
+
+It ≥ d
+� h(T ∗)
+g(T ∗) J(x − y)I(t, y)dy − dI(t, x) − µ2I
++β(m, I, x)( σ
+µ1 − ǫ)I − γ(b, I, x)I,
+t > T ∗∗, x ∈ (g(T ∗), h(T ∗)),
+I(T ∗∗, x) > 0,
+g(T ∗) ≤ x ≤ h(T ∗).
+We now construct a suitable lower solution for the following auxiliary problem
+
+
+
+
+
+
+
+Wt = d
+� h(T ∗)
+g(T ∗) J(x − y)W(t, y)dy − dW(t, x)
+−µ2W + β(m, W, x)( σ
+µ1 − ǫ)W − γ(b, W, x)W,
+t > T ∗∗, x ∈ (g(T ∗), h(T ∗)),
+W(T ∗∗, x) = I(T ∗∗, x),
+g(T ∗) ≤ x ≤ h(T ∗).
+(4.5)
+Choose
+W(t, x) = δψ(x),
+t > T ∗∗,
+g(T ∗) ≤ x ≤ h(T ∗),
+where δ > 0 is small enough such that δψ(x) ≤ I(T ∗∗, x) for x ∈ [g(T ∗), h(T ∗)].
+For t > T ∗∗ and g(T ∗) < x < h(T ∗), direct computation yields
+W t − d
+� h(T ∗)
+g(T ∗) J(x − y)W(y)dy + dW + µ2W − β(m, W, x)( σ
+µ1 − ǫ)W + γ(b, W, x)W
+=
+−δ(d
+� h(T ∗)
+g(T ∗) J(x − y)ψ(t, y)dy − dψ(x) − µ2ψ + β(m, δψ, x)( σ
+µ1 − ǫ)ψ − γ(b, δψ, x)ψ)
+=
+−δψ(λp + γ(b, 0, x) − β(m, 0, x) σ
+µ1 + β(m, δψ, x)( σ
+µ1 − ǫ) − γ(b, δψ, x))
+≤
+δψ(−λp(L{(g(T ∗),h(T ∗)), d} + a(x)) + β(m, δψ, x)ǫ)
+<
+0.
+Note that the boundaries g(T ∗) and h(T ∗) are fixed, so there is no need to compare the boundary
+values of W(t, x) and I(t, x) by Lemma 3.1 in [15]. Applying the comparison principle in [15] gives
+I(t, x) ≥ W(t, x) ≥ W(t, x) = δψ(x) in [T ∗∗, +∞) × (g(T ∗), h(T ∗)).
+Therefore, lim inf
+t→+∞ I(t, x) ≥ lim inf
+t→+∞ W(t, x) ≥ δψ(0) > 0, which is a contradiction.
+□
+Remark 4.6 Suppose that a(x) is a positive constant. If h∞−g∞ = +∞, then lim sup
+t→+∞
+∥I(·, t)∥C([g(t), h(t)]) >
+0. In the special case, similar to Theorem 3.10 in [5], a spreading-vanishing dichotomy holds.
+Suppose that
+a(0) ≥ d,
+we can know that λp(L{(L1, L2), d} + a(x)) > 0 for any interval (L1, L2) by Theorem 3.3 and (i) of
+Proposition 3.1, which yields the following conclusion by Theorem 4.2.
+Theorem 4.7 If a(0) ≥ d holds, then spreading always occurs for (1.3).
+Next we consider the case 0 < a(0) < d. According to Theorem 3.3 and (i) of Proposition 3.1,
+there exists L∗ > 0 such that
+λp(L{(L1, L2), d} + a(x))
+
+
+
+
+
+
+
+< 0,
+(L1, L2) ⊂ (−L∗, L∗),
+= 0,
+−L1 = L2 = L∗,
+> 0,
+(−L∗, L∗) ⊂ (L1, L2).
+19
+
+Theorem 4.8 Assume 0 < a(0) < d holds, then
+(i) if h0 ≥ L∗, then spreading always occurs for (1.3).
+(ii) if h0 < L∗, then there exists a positive constant k∗ such that h∞ − g∞ = ∞ when k > k∗.
+Proof. (i) holds from Theorem 4.2 since
+λp(L{(g∞, h∞), d} + a(x)) > λp(L{(−L∗, L∗), d} + a(x)) = 0.
+In what follows, we prove (ii). Notice that
+−µ2 + β(m(x), I, x)S − γ(b(x), I, x) > −µ2 − γ(b(x), I, x) > −C
+for some C > 0. Clearly I(t, x) satisfies
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+It ≥ d
+� h(t)
+g(t) J(x − y)I(t, y)dy − dI(t, x) − CI(t, x),
+t > 0, x ∈ (g(t), h(t)),
+I(t, x) = 0,
+t ≥ 0, x ∈ R\(g(t), h(t)),
+h′(t) = k
+� h(t)
+g(t)
+� +∞
+h(t) J(x − y)I(t, x)dydx,
+t > 0,
+g′(t) = −k
+� h(t)
+g(t)
+� g(t)
+−∞ J(x − y)I(t, x)dydx,
+t > 0,
+g(0) = −h0, h(0) = h0,
+x ∈ R,
+I(0, x) = I0(x),
+x ∈ (−h0, h0),
+thereby, for any given constant M, there exists k∗ > 0 such that h∞ − g∞ > M provided that k > k∗
+by Lemma 3.9 in [16]. Then, h∞ − g∞ = +∞ by the arbitrariness of M.
+□
+Noting that the comparison principle for problem (1.3) is not valid, we cannot obtain that the
+monotonicity of the solution for (1.3) with k and thus cannot take k as a sharp criterion for the
+spreading-vanishing dichotomy as in [5]. However, recalling that Remark 4.4 and (ii) in Theorem 4.8,
+we have the following result:
+Theorem 4.9 Suppose that 0 < a(0) < d and h0 < L∗. For problem (1.3), there exists 0 < k∗ ≤ k∗
+such that vanishing occurs if 0 < k < k∗ and spreading happens provided that k > k∗.
+Finally, we will discuss the impact of the diffusion coefficient on the vanishing and spreading of
+infectious disease.
+Assume that
+� −2h0
+−∞
+J(z)dz > 0 (or
+� +∞
+2h0 J(z)dz > 0) holds. Using Theorem 3.4 with L1 = −h0
+and L2 = h0, if
+max
+x∈[−h0, h0] a(x) < 0, for any d > 0, we have λp(L{(−h0, h0), d} + a(x)) < 0; while if
+max
+x∈[−h0, h0] a(x) > 0, there exists a d∗ > 0 such that
+λp(L{(−h0, h0), d} + a(x))
+
+
+
+
+
+
+
+< 0,
+d > d∗,
+= 0,
+d = d∗,
+> 0,
+d < d∗.
+Inspired by the above analysis, combined with Theorem 4.2, we can obtain the following result.
+Theorem 4.10 Suppose
+� −2h0
+−∞
+J(z)dz > 0 (or
+� +∞
+2h0 J(z)dz > 0) holds. Then,
+(i) if
+max
+x∈[−h0, h0] a(x) > 0, there exists a d∗ > 0 such that for d < d∗, then spreading occurs;
+conversely, if d > d∗, then vanishing occurs provided that ∥S0(x)∥L∞(R) + ∥I0(x)∥C([−h0, h0]) is small
+enough;
+(ii) if
+max
+x∈[−h0, h0] a(x) ≤ 0, for any d > 0, then vanishing always occurs as long as ∥S0(x)∥L∞(R) +
+∥I0(x)∥C([−h0, h0]) is adequately small.
+20
+
+5
+Discussion
+In this paper, we study a free boundary problem (1.3) with media coverage and hospital bed numbers,
+which describes a nonlocal diffusive SIS epidemic model. The free boundary describes the moving
+front of the infected individuals, and the nonlocal diffusion operator characterizes the long-distance
+spatial movement of individuals.
+For the SIS model with nonlocal diffusion and free boundaries (1.3), the existence and uniqueness
+of the global solution are given by using two fixed point theorems (see Theorem 2.2). Then, we define
+the principal eigenvalue of the integral operator, and analyze impacts of media coverage and hospital
+bed number (Theorem 3.2), interval length (Theorem 3.3), and diffusion coefficient (Theorem 3.4) on
+the principal eigenvalue. In addition, sufficient conditions for disease spreading and vanishing (see
+Theorems 4.2, 4.3 and 4.5) are given. Finally, we discuss the impact of the principal eigenvalue on
+the spreading or vanishing of the infectious diseases. If a(0) ≥ d, then λp(L{(L1, L2), d} + a(x)) > 0
+for any L1 < L2, the disease always spreading (see Theorem 4.7). If 0 < a(0) < d, there exists a
+L∗ > 0, then spreading always appears for h0 ≥ L∗ (see Theorem 4.8); and when h0 < L∗, the impact
+of expanding capability k on the spreading or vanishing of disease is discussed. That is, there exists
+0 < k∗ ≤ k∗ such that vanishing occurs if 0 < k < k∗, and spreading happens provided that k > k∗
+(see Theorem 4.9). If maxx∈[−h0, h0] a(x) > 0, there exists a d∗ > 0 such that for d < d∗, then the
+disease spreads; if d > d∗ and ∥S0(x)∥L∞(R) + ∥I0(x)∥C([−h0, h0]) is small enough, then the disease
+vanishes; if maxx∈[−h0, h0] a(x) ≤ 0, then d∗ = 0, that is, then vanishing always appears provided that
+∥S0(x)∥L∞(R) + ∥I0(x)∥C([−h0, h0]) is adequately small (see Theorem 4.10).
+Finally, we may conclude that the differences between nonlocal diffusion in (1.3) and local diffusion
+in (1.2) in our mathematical analysis are as follows: first of all, the existence and uniqueness of the
+global solution for (1.2) are obtained by straightening the boundary and the first-order fixed point
+theorem. However, owing to lack of compactness, the existence and uniqueness of global solutions for
+(1.3) are given by using two fixed point theorems. Secondly, for (1.2), the corresponding principal
+eigenvalue always exists. However, for the nonlocal diffusion problem, the principal eigenvalue may
+not exist. In this paper, for the nonlocal diffusion model (1.3), we first define generalized principal
+eigenvalue λp(L{(L1, L2), d}+a(x)) of the integral operator, and then show that the generalized principal
+eigenvalue is the principal eigenvalue under the condition (H).
+Thirdly, different from the local
+diffusion whose principal eigenvalue is clear, the nonlocal operator leads more possibilities because of
+the choice of the kernel function.
+It is worth mentioning that the model (1.3) incorporates media coverage and hospital bed numbers.
+Based on the monotonicity of the generalized principal eigenvalue on media coverage and hospital bed
+numbers, we study the influence of the principal eigenvalue on infectious diseases, which implies that
+large media coverage and hospital bed numbers are beneficial to the prevention and control of disease.
+References
+[1] L. J. S. Allen, B. M. Bolker, Y. Lou, A. L. Nevai, Asymptotic profiles of the steady states for an
+SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20.
+[2]
+L. Basnarkov, SEAIR epidemic spreading model of COVID-19, Chaos Solitons Fractals, 142
+(2021), Paper No. 110394, 15 pp.
+[3]
+L. Basnarkov, I. Tomovski, T. Sandev, L. Kocarev,
+Non-Markovian SIR epidemic spreading
+model of COVID-19, Chaos Solitons Fractals, 160 (2022), Paper No. 112286, 8 pp.
+[4] H. Berestycki, J. Coville, H. Vo, On the difinition and the propperties of the principal eigeneigen
+of some nonlocal operators, J. Funct. Anal, 271 (2016), 2701-2751.
+[5] J. F. Cao, Y. H. Du, F. Li, W. T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model
+with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.
+21
+
+[6] J. F. Cao, W. T. Li, J. Wang, A Lotka-Volterra competition model with nonlocal diffusion and
+free boundaries, arXiv:1905.09584v1.
+[7] J. F. Cao, W. T. Li, J. Wang, F. Y. Yang, A free boundary problem of a diffusive SIRS model
+with nonlinear incidence, Z. Angew. Math. Phys., 68 (2017), 1-16.
+[8] J. F. Cao, W. T. Li, F. Y. Yang, Dynamics of a nonlocal SIS epidemic model with free boundary,
+Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 247-266.
+[9] T. Y. Chang, Y. H. Du, Long-time dynamics of an epidemic model with nonlocal diffusion and
+free boundaries, Electron. Res. Arch., 30 (2022), 289-313.
+[10] X. F. Chen, A. Friedman, A free boundary problem arising in a model of wound healing, SIAM
+J. Math. Anal., 32 (2000), 778-800.
+[11] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal
+operators, J. Differential Equations, 249 (2010), 2921-2953.
+[12] J. Cui, X. Tao, H. P. Zhu, An SIS infection on model in corporating media coverage, Rocky Mt.
+J. Math., 38 (2008), 1323-1334.
+[13] X. Dong, J. P. Wang, M. X. Wang, Free boundary problems with local-nonlocal diffusions and
+different free boundaries II: spreading-vanishing and long-time behavior, Nonlinear Anal. Real
+World Appl., 62 (2022), 23 pp.
+[14] Y. H. Du, Z. G. Lin, Spreading-Vanishing dichotomy in the diffusive logistic model with a free
+boundary, SIAM J. Math. Anal., 42 (2010), 377-405.
+[15]
+Y. H. Du, W. J. Ni,
+Analysis of a West Nile virus model with nonlocal diffusion and free
+boundaries, Nonlinearity, 33 (2020), 4407-4448.
+[16] Y. H. Du, M. X. Wang, M. Zhao, Two species nonlocal diffusion systems with free boundaries,
+Discrete Contin. Dyn. Syst., 42 (2022), 1127-1162.
+[17]
+Y. X. Feng, W. T. Li, S. G. Ruan, F. Y. Fei, Dynamics and asymptotic profiles of a nonlo-
+cal dispersal SIS epidemic model with bilinear incidence and Neumann boundary conditions, J.
+Differential Equations, 335 (2022), 294-346.
+[18]
+Y. X. Feng, W. T. Li, F. Y. Yang,
+Traveling waves in a nonlocal dispersal SIR model with
+non-monotone incidence, Commun. Nonlinear Sci. Numer. Simul., 95 (2021), 21 pp.
+[19] J. Garc´ıa-Meli´an, J. D. Rossi, Maximum and antimaximum principles for some nonlocal diffusion
+operators, Nonlinea Anal, 71 (2009), 6116-6121.
+[20] J. Garc´ıa-Meli´an, J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems,
+J. Differential Equations, 246 (2009), 21-38.
+[21] J. Garc´ıa-Meli´an, J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun.
+Pure Appl. Anal., 8 (2009), 2037-2053.
+[22] J. Ge, K. I. Kim, Z. G. Lin, H. P. Zhu, A SIS reaction-diffusion-advection model in a low-risk
+and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.
+[23] H. M. Huang, M. X. Wang, The reaction-diffusion system for an SIR epidemic model with a free
+boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050.
+[24] W. Z. Huang, M. A. Han, K. Y. Liu, Dynamics of an SIS reaction-diffusion epidemic model for
+disease transmission, Math. Biosci. Eng., 7 (2010), 51-66.
+22
+
+[25] D. Lacitignola, F. Diele, Using awareness to Z-control a SEIR model with overexposure: insights
+on Covid-19 pandemic, Chaos Solitons Fractals, 150 (2021), 14 pp.
+[26] L. Li, W. J. Sheng, M. X. Wang, Systems with nonlocal vs. local diffusions and free boundaries,
+J. Math. Anal. Appl., 483 (2020), 27 pp.
+[27] M. Li, Z. G. Lin, The spreading fronts in a mutualistic model with advection, Discrete Contin.
+Dyn. Syst. Ser. B, 20 (2015), 2089-2105.
+[28] X. Li, C. T. Wang, H. Li, A. L. Bertozzi, martingale formulation for stochastic compartmental
+susceptible-infected-recovered (SIR) models to analyze finite size effects in COVID-19 case studies,
+Netw. Heterog. Media, 17 (2022), 311-331.
+[29] L. L. Liu; R. Xu, Z. Jin, Global dynamics of a spatial heterogeneous viral infection model with
+intracellular delay and nonlocal diffusion, Appl. Math. Model., 82 (2020), 150-167.
+[30] A. Meiksin, Using the SEIR model to constrain the role of contaminated fomites in spreading an
+epidemic: an application to COVID-19 in the UK, Math. Biosci. Eng., 19 (2022), 3564-3590.
+[31] A. K. Misra, J. Maurya, Modeling the importance of temporary hospital beds on the dynamics of
+emerged infectious disease, Chaos, 31 (2021), 22 pp.
+[32]
+J. D. Murray, Mathematical Biology, II, Spatial Models and Biomedical Applications,
+Third
+edition. Interdisciplinary Applied Mathematical, 18. Springer-Verlag, New York, 2003.
+[33] R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion
+model. I, J. Differential Equations, 247 (2009), 1096-1119.
+[34] R. Peng, S. Q. Liu, Global stability of the steady states of an SIS epidemic reaction- diffusion
+model, Nonliner Anal., 71 (2009), 239-247.
+[35] R. Peng, X. Q, Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment,
+Nonlinerity, 25 (2012), 1451-1471.
+[36] S. P. Rajasekar, M. Pitchaimani, Q. X. Zhu, Higher order stochastically perturbed SIRS epidemic
+model with relapse and media impact, Math. Methods Appl. Sci., 45 (2022), 843-863.
+[37] C. H. Shan, H. P. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of
+the number of hospital beds, J. Differential Equations, 257 (2014), 1662-1688.
+[38]
+J. O. Takhirov, A. Norov, On a predator-prey model with free boundary, Uzbek Math. J., 4
+(2019), 162-168.
+[39]
+R. Wang, Y. H. Du, Long-time dynamics of a nonlocal epidemic model with free boundaries:
+spreading-vanishing dichotomy, J. Differential Equations, 327 (2022), 322-381.
+[40]
+Y. Z. Wang, S. J. Guo,
+A SIS reaction-diffusion model with a free boundary condition and
+nonhomogeneous coefficients, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1627-1652.
+[41] P. Wu, X. N. Wang, H. Wang, Threshold dynamics of a nonlocal dispersal HIV/AIDS epidemic
+model with spatial heterogeneity and antiretroviral therapy,
+Commun. Nonlinear Sci. Numer.
+Simul., 115 (2022), Paper No. 106728.
+[42] Y. X. Wu, X. F. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with
+mass action infection mechanism, J. Differential Equations., 261 (2016), 4424-4447.
+[43] W. B. Xu, W. T. Li, S. G. Ruan, Spatial propagation in an epidemic model with nonlocal diffusion:
+the influences of initial data and dispersals, Sci. China Math., 63 (2020), 2177-2206.
+23
+
+[44] F. Y. Yang, W. T. Li, S. G. Ruan, Dynamics of a nonlocal dispersal SIS epidemic model with
+Neumann boundary conditions, J. Differential Equations, 267 (2019), 2011-2051.
+[45] M. Zhao, W. T. Li, J. F. Cao, Dynamics for an SIR epidemic model with nonlocal diffusion and
+free boundaries, Acta Math. Sci. Ser. B, 41 (2021), 1081-1106.
+[46] Z. J. Zhao, Y. P. Yang, Threshold dynamics of an SEAIR epidemic model with application to
+COVID-19, J. Nonlinear Sci. Appl., 15 (2022), 136-151.
+[47] M. Zhu, X. F. Guo, Z. G. Lin, The risk index for an SIR epidemic model and spatial spreading
+of the infectious disease, Math. Biosci. Eng., 14 (2017), 1565-1583.
+24
+
diff --git a/ydFJT4oBgHgl3EQfhyyK/content/tmp_files/load_file.txt b/ydFJT4oBgHgl3EQfhyyK/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..07e0536d3abd0679a6a97d9bad783477da28a532
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@@ -0,0 +1,1025 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf,len=1024
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='11567v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='AP]  27 Jan 2023  Threshold dynamics of a nonlocal dispersal SIS epidemic  model with free boundaries ∗  Yachun Tonga, Inkyung Ahnb and Zhigui Lina†  a School of Mathematical Science, Yangzhou University, Yangzhou 225002, China  b Department of Mathematics, Korea University, Sejong 339-700, South Korea  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  To study the influence of the moving front of the infected interval and the  spatial movement of individuals on the spreading or vanishing of infectious disease, we  consider a nonlocal SIS (susceptible-infected-susceptible) reaction-diffusion model with  media coverage, hospital bed numbers and free boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' The principal eigenvalue of  the integral operator is defined, and the impacts of the diffusion rate of infected individuals  and interval length on the principal eigenvalue are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Furthermore, the sufficient  conditions for spreading and vanishing of the disease are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Our results show that  large media coverage and hospital bed numbers are beneficial to the prevention and control  of disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' The difference between the model with nonlocal diffusion and that with local  diffusion is also discussed and nonlocal diffusion leads more possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  MSC: 35K57, 92D30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' secondary: 35R35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Keywords: SIS model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Free boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Nonlocal diffusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Spreading and vanishing  1  Introduction  With the emergence and outbreak of COVID-19 [3,30] in recent years, infectious disease models  have become one of the most popular research topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' To study the spread and dynamics of  COVID-19, most scholars use the SIR (susceptible-infected-recovered) [3,28], SEIR (susceptible-  exposed-infected-recovered) [25,30] and SEAIR (susceptible-exposed-asymptomatic-infectious-  removed) [2, 46] models to describe the spread of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Meanwhile, the classical SIS  model has received great attention in mathematical epidemiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Considering the impact of the spatial heterogeneity of the environment and the movement  of individuals on infectious diseases, Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' in [1] proposed and discussed an SIS reaction-  diffusion system  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  St − dS∆S = −β(x)SI  S+I + γ(x)I,  t > 0, x ∈ Ω,  It − dI∆I = β(x)SI  S+I − γ(x)I,  t > 0, x ∈ Ω,  ∂S  ∂η = ∂I  ∂η = 0,  t > 0, x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1)  ∗The first author is supported by the Postgraduate Research & Practice Innovation Program of Jiangsu  Province (KYCX21-3188), the second author is supported under the framework of international cooperation  program managed by the National Research Foundation of Korea (NRF-2019K2A9A2A06025237) and the third  author is supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' 12271470).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Email: zglin@yzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='cn (Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Lin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  1  Here, Ω ⊂ Rn (n ≥ 1) is a bounded domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' S(t, x) and I(t, x) indicate the density of susceptible  and infected individuals at location x and time t, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' dS and dI are positive constants  that account for the diffusion rate of susceptible and infected individuals, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' and  the positive bounded H¨older continuous functions β(x) and γ(x) can be interpreted as rates of  disease transmission and recovery for x ∈ Ω, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' The authors in [1] mainly discussed  the existence, uniqueness and stability of DFE (disease-free equilibrium) and EE (endemic  equilibrium) and used the basic reproduction number R0 to characterize the risk of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Afterwards, Peng and Liu [34] confirmed the conjecture proposed by Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' in [1] that  a unique EE is globally asymptotically stable in some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Further results that the  effect of individual movement (large or small) on the existence and disappearance of disease were  obtained in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' For more results of the SIS reaction-diffusion model, one can see [24, 35, 42]  and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  It is easy to find that the above articles are devoted to the study of SIS models on a fixed  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In real life, the movement of species leads to changes in biological habitats, and in  mathematics, the free boundary can be used to described this phenomenon, such as the healing  of wounds [10] and the expansion of new species or invasive species [6,14,27,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Free boundary  problems can also be used to describe the transmission of disease, such as the SIRS model [7],  SIS model [22], SIR model [23,47] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  To explore the moving front of the infected individual,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang and Guo [40] introduced the  free boundary and studied the dynamics of the following SIS reaction-diffusion model:  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  St − d∆S = σ − µS − β(x)SI + γ(x)I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  It − d∆I = β(x)SI − µI − γ(x)I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x ∈ (g(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  I(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x ∈ R\\(g(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  g′(t) = −kIx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' g(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' g(0) = −h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  h′(t) = −kIx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h(0) = h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  S(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) = S0(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' I(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) = I0(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2)  The basic reproduction number was given, and the spreading-vanishing dichotomy was estab-  lished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Some conditions for disease spreading or vanishing were presented by investigating the  effect of the diffusion rate (d), initial value (I0) and expanding capability (k) on the asymptotic  behavior of the infected individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  It is widely known that random dispersal or local diffusion describes the local behavior  of the movements of organisms between adjacent spatial locations [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Briefly, the classical  Laplace diffusion operator is used to describe that the movement of the infectious agent and  infected population only occurs between adjacent spatial positions [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' However, Murray [32]  noted that a local or short-range diffusive flux proportional to the gradient is not suitable to  characterize some biological phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In the real world, the movements and interactions of  some organisms occur at nonadjacent spatial positions, and such dispersal are called nonlocal  diffusion [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Recently, nonlocal diffusion equations have attracted extensive attention and have been used  to characterize long-range dispersal in population ecology [6,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In addition, scholars have also  extensively investigated infectious disease models with nonlocal diffusion, such as the West Nile  virus model [15], SIS epidemic model [17, 44], and SIR reaction-diffusion model [18, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' For  other epidemic models with nonlocal diffusion, see references [9,39,41] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  In addition, there are many factors that affect the spread of infectious disease, such as the  contact transmission rate and the recovery rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Educating the public about the disease through  mass media (such as television, radio, newspapers, billboards, internet, magazines etc), is one  2  of the important precautions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Therefore, media coverage can indirectly reduce the contact rate  between people and infectious diseases, thus reducing the contact transmission rate of infectious  diseases [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In general, the main factor impacting the recovery rate is the availability of health  care (such as the number of physicians, nurses, hospital beds and isolation places).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In fact,  health and medical institutions use the hospital bed-population ratio (HBPR) (the number of  hospital beds per 10000 people) as a method of reckoning available resources to the public [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Taking into account of nonlocal diffusion, media coverage and hospital bed numbers, we  consider the following nonlocal dispersal SIS epidemic model with a free boundary  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  St = dL1[S] + σ − µ1S − β(m(x), I, x)SI + γ(b(x), I, x)I,  t > 0, x ∈ R,  It = dL2[I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' g, h] − µ2I + β(m(x), I, x)SI − γ(b(x), I, x)I,  t > 0, x ∈ (g(t), h(t)),  I(t, x) = 0,  t ≥ 0, x ∈ R\\(g(t), h(t)),  h′(t) = k  � h(t)  g(t)  � +∞  h(t) J(x − y)I(t, x)dydx,  t > 0,  g′(t) = −k  � h(t)  g(t)  � g(t)  −∞ J(x − y)I(t, x)dydx,  t > 0,  S(0, x) = S0(x), g(0) = −h0, h(0) = h0,  x ∈ R,  I(0, x) = I0(x),  x ∈ (−h0, h0),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3)  where  L1[S] =  �  R  J(x − y)S(t, y)dy − S(t, x),  L2[I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' g, h] =  � h(t)  g(t)  J(x − y)I(t, y)dy − I(t, x),  and d, S(t, x) and I(t, x) have the same epidemiological interpretation as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' The con-  stants σ, µ1 and µ2 are positive, where σ accounts for the environment carrying capability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' the  natural mortality rate of the susceptible individuals is expressed by µ1, and µ2 denotes the sum  of the natural mortality and disease-caused death rates of the infected individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' The func-  tions β(m(x), I, x), γ(b(x), I, x), m(x), b(x) are nonnegative, where m(x) represents the media  coverage, and b(x) stands for the number of hospital beds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In this paper, we assume that  (1) the contact infectious rate β(m(x), I, x) is Lipschitz continuous and monotonically de-  creasing in m(x) and increasing in I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (2) the recovery rate γ(b(x), I, x) is Lipschitz continuous and increasing in b(x) and mono-  tonically decreasing in I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (3) βI(m(x), I, x) and γI(b(x), I, x) are continuous and bounded for m(x) ∈ [0, ∞), I ∈  [0, ∞) and x ∈ (−∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  For instance, Cui and Zhu [12] used the function β(I) = βemI to model the impact of media  coverage on the transmission rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' and Shan and Zhu [37] used the function γ(b, I, x) = γ0 +  (γ1 − γ0)  b  b+I to describe the hospital resource impact factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Recalling that S(t, x) denotes the density at point x and time t, the kernel function J(x−y)  is regarded as the probability distribution of jumping from place y to place x, then the integral  operator  �  R J(x − y)S(t, y)dy accounts for the rate at which the individuals are gathering at  point x from all other places, and −S(t, x) is the rate at which the individuals are leaving  at point x to other places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In addition, the infected individuals stay in the infected interval  (g(t), h(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' We further suppose that the initial function S0(x) satisfies  S0(x) ∈ C(R) ∩ L∞(R) and S0(x) > 0 in R,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4)  and I0(x) satisfies  I0(x) ∈ C([−h0, h0]), I0(±h0) = 0, I0(x) > 0 in (−h0, h0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='5)  3  For system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3), we assume that the kernel function J : R → R is continuous and nonneg-  ative, and has the properties  (J) : J ∈ C(R) ∩ L∞(R) is symmetric, J(0) > 0,  �  R  J(x)dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  The free boundary conditions h′(t) = k  � h(t)  g(t)  � +∞  h(t) J(x−y)I(t, x)dydx and g′(t) = −k  � h(t)  g(t)  � g(t)  −∞ J(x−  y)I(t, x)dydx in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) imply that the expanding rate of the interval (g(t), h(t)) is determined  by the infected individuals and is proportional to the outward flux of the infected individuals  across the interval (g(t), h(t)) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  It is worth mentioning that there are links and differences between local diffusion and  nonlocal diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Local diffusion, expressed by the Laplace operator ∆u (the Laplace in Rn,  n ≥ 2) or uxx (in one-dimensional space), is used to describe the influence between adjacent  positions, and nonlocal diffusion, expressed by the integral operator (is given by  �  R J(x −  y)u(t, y)dy−u(t, x)), is used to describe long-distance dispersal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' However, the Laplace operator  can be regarded as a local approximation of a nonlocal diffusion operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In fact, when J(·) is  symmetric and has compact supports, such as J(x) = (1/ǫ)K(x/ǫ) with 0 < ǫ ≪ 1 and K(x) is  a general mollification function with support x ∈ [−1, 1], we can transform nonlocal operators  into local operators by using the Taylor formula [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  This article is organized as follows: the existence and uniqueness of the global solution  are given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Section 3 is devoted to defining and studying the properties of the  principal eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Section 4 gives some sufficient conditions for the disease to spread or  vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Finally, a brief discussion is presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  2  Global existence and uniqueness  In this section, we assume that h0 > 0, S0(x) and I0(x) satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' For any given  T > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' we first introduce the notations as follows:  HT := {h ∈ C([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T]) : h(0) = h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  inf  0≤t1<t2≤T  h(t2)−h(t1)  t2−t1  > 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  GT := {g ∈ C([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T]) : −g ∈ HT},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Dg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='h  T  := {(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) ∈ R2 : 0 < t ≤ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' g(t) < x < h(t)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Dh0  T := {(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) ∈ R2 : 0 < t ≤ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' −h0 < x < h0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  D∞  T := {(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) ∈ R2 : 0 < t ≤ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x ∈ R},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  XS0  T := {φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) ∈ C(D∞  T ) ∩ L∞(D∞  T ) : φ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) = S0(x) in R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) ≥ 0 in D∞  T },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  XI0  T := {ψ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) ∈ C(D∞  T ) : ψ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) = I0(x) in [−h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h0],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' ψ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) ≥ 0 in Dg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='h  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  ψ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) = 0 for t ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x ∈ R\\(g(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h(t))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  To prove the existence and uniqueness of the global solution of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3), we first give  the following result for problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) without a free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  4  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 For any given T > 0 and (g, h) ∈ HT × GT , the problem  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  St = dL1[S] + σ − µ1S − β(m(x), I, x)SI + γ(b(x), I, x)I,  0 < t ≤ T, x ∈ R,  It = dL2[I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' g, h] − µ2I + β(m(x), I, x)SI − γ(b(x), I, x)I,  0 < t ≤ T, x ∈ (g(t), h(t)),  I(t, x) = 0,  0 ≤ t ≤ T, x ∈ R\\(g(t), h(t)),  S(0, x) = S0(x),  x ∈ R,  I(0, x) = I0(x),  x ∈ (−h0, h0)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1)  admits a unique solution (Sg,h, Ig,h) ∈ C(D  ∞  T ) × C(D  g,h  T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Moreover,  0 < Sg,h(t, x) ≤ A  for any (t, x) ∈ D∞  T ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2)  0 < Ig,h(t, x) ≤ A  for any (t, x) ∈ Dg,h  T ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3)  where A = max{ σ  µ1, ∥S0∥∞ + ∥I0∥∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' The main idea of this proof comes from [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' We divide the proof into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' The parameterized ODE problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  For any given x ∈ R, s ∈ (0, T], denote  tx =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  tg  x,  x ∈ (g(s), −h0) and x = g(tg  x),  0,  x ∈ [−h0, h0],  th  x,  x ∈ (h0, h(s)) and x = h(th  x),  s,  x ∈ R\\(g(s), h(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Clearly, tx > 0 for x ∈ R\\[−h0, h0], tx < s for x ∈ (g(s), h(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' For any given (φ, ψ) ∈ XS0  s ×XI0  s ,  define  A1 = max{A, σ + ∥ψ∥∞ sup γ  µ1  , ∥φ∥∞},  A2 = max{A, (d + A1 sup β)∥ψ∥∞  d + µ2  }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  We discuss it in the following two cases:  Case 1: x ∈ R\\[−h0, h0], t ∈ [0, tx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Clearly, I(t, x) = 0 for (t, x) ∈ [0, tx] × R\\[−h0, h0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Consider the ODE problem  �  St = d  �  R J(x − y)φ(t, y)dy − dS + σ − µ1S,  0 < t ≤ tx,  S(0, x) = S0(x),  x ∈ R\\[−h0, h0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4)  For any S1, S2 ∈ [0, A1],  |d  �  R J(x − y)φ(t, y)dy − dS1 + σ − µ1S1 − d  �  R J(x − y)φ(t, y)dy + dS2 − σ + µ1S2|  =  (d + µ1)|S1 − S2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Therefore, F := d  �  R J(x−y)φ(t, y)dy−dS+σ−µ1S is Lipschitz continuous in S for S ∈ [0, A1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  By the fundamental theory of ODEs, problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4) has a unique solution Sφ(t, x) defined in  t ∈ [0, �tx), and Sφ(t, x) is continuous in both t and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' To see that t → S(·, x) can be uniquely  extended to [0, tx], we need to prove that if Sφ(t, x) is uniquely defined for t ∈ [0, �tx] with  �tx ∈ (0, tx], then  0 ≤ Sφ(t, x) ≤ A1, for t ∈ [0, �tx] and x ∈ R\\[−h0, h0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  5  Obviously,  d  �  R J(x − y)φ(t, y)dy − dA1 + σ − µ1A1  ≤  d∥φ∥∞ − dA1 + σ − µ1A1  ≤  0,  and ∥S0∥∞ ≤ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Thanks to the direct comparison argument, one can derive Sφ(t, x) ≤ A1  for t ∈ [0, �tx] and x ∈ R\\[−h0, h0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' We use similar method to prove that Sφ(t, x) ≥ 0 for  t ∈ [0, �tx], x ∈ R\\[−h0, h0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Case 2: x ∈ (g(s), h(s)), t ∈ [tx, s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Define  �Sφ(x) =  �  S0(x),  x ∈ [−h0, h0]  Sφ(tx, x),  x /∈ [−h0, h0]  and  �I(x) =  �  I0(x),  x ∈ [−h0, h0]  0,  x /∈ [−h0, h0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Consider the ODE problem  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  St = F1(t, x, S, I),  tx < t ≤ s,  It = F2(t, x, S, I),  tx < t ≤ s,  S(tx, x) = �Sφ(x), I(tx, x) = �I(x),  x ∈ (g(s), h(s))  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='5)  with  F1 = d  �  R  J(x − y)φ(t, y)dy − dS + σ − µ1S + γ(b, I, x)ψ − β(m, I, x)SI,  F2 = d  � h(t)  g(t)  J(x − y)ψ(t, y)dy − dI − µ2I − γ(b, I, x)I + β(m, I, x)Sψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  For any (Si, Ii) ∈ [0, A1] × [0, A2](i = 1, 2), obviously, Fi(t, x, S, I) is Lipschitz continuous in  (S, I) for (Si, Ii) ∈ [0, A1] × [0, A2] by the continuity and monotonicity of β(m(x), I, x) and  γ(b(x), I, x), and it is uniformly continuous for x ∈ (g(s), h(s)) and t ∈ [tx, s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In addition,  Fi(t, x, S, I) is continuous in all its variables in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='5) has a unique solution  (Sφ,ψ(t, x), Iφ,ψ(t, x)) for t ∈ [tx, sx), and (Sφ,ψ(t, x), Iφ,ψ(t, x)) is continuous in both t and x by  the fundamental theorem of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  To show that (Sφ,ψ(t, x), Iφ,ψ(t, x)) can be uniquely extended to [tx, s], it suffices to prove  that if (Sφ,ψ(t, x), Iφ,ψ(t, x)) is uniquely defined for t ∈ [tx, �t] with �t ∈ (tx, s], then  0 ≤ Sφ,ψ(t, x) ≤ A1, 0 ≤ Iφ,ψ(t, x) ≤ A2 for t ∈ [tx, �t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='6)  In fact, it is easy to see that  F1(t, x, A1, A2)  =  d  �  R J(x − y)φ(t, y)dy − dA1 + σ − µ1A1 + γ(b, A2, x)ψ − β(m, A2, x)A1A2  ≤  d∥φ∥∞ − dA1 + σ − µ1A1 + γ(b, A2, x)∥ψ∥ − β(m, A2, x)A1A2  <  d∥φ∥∞ − dA1 + σ − µ1A1 + ∥ψ∥∞ sup γ  ≤  0  and  F2(t, x, A1, A2)  =  d  � h(t)  g(t) J(x − y)ψ(t, y)dy − dA2 − µ2A2 − γ(b, A2, x)A2 + β(m, A2, x)A1ψ  ≤  d  � h(t)  g(t) J(x − y)ψ(t, y)dy − dA2 − µ2A2 + A1∥ψ∥∞ sup β  ≤  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  6  Since A1 ≥ ∥S0∥∞, A2 ≥ ∥I0∥∞, we have Sφ,ψ(t, x) ≤ A1 and Iφ,ψ(t, x) ≤ A2 in t ∈ [tx, �t] by the  comparison argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' The left part of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='6) can be obtained similarly by using Fi(t, x, 0, 0) ≥ 0  (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' A fixed point theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  For any s ∈ (0, T), we note  XS0  s  := {φ|D  ∞  s : φ ∈ XS0  T }, XI0  s := {ψ|D  g,h  s  : ψ ∈ XI0  T }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Denote  (�S(t, x), �I(t, x)) =  �  (Sφ(t,x), 0),  x ∈ R\\[−h0, h0], t = [0, tx],  (Sφ,ψ(t, x), Iφ,ψ(t, x)),  x ∈ (g(s), h(s)), t ∈ [tx, s],  where Sφ(t, x), Sφ,ψ(t, x) and Iφ,ψ(t, x)) are given in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' By Step 1, for any (φ, ψ), we have  a unique solution (�S, �I) for t ∈ [0, s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' It is easy to check that �S(t, x) is continuous in D  ∞  s ,  and �I(t, x) is continuous in D  g,h  s  due to the continuous dependence of the ODE solution on the  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Therefore, (�S, �I) ∈ XS0  s ×XI0  s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Note that XS0  s  and XI0  s are complete metric spaces,  respectively, with the norms  d1(φ1, φ2) = ∥φ1 − φ2∥C(D  ∞  s ), d2(ψ1, ψ2) = ∥ψ1 − ψ2∥C(Dg,h  s  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Hence, we find a mapping Γ : XS0  s × XI0  s → XS0  s × XI0  s by Γ(φ, ψ) = (�S, �I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Setting  M1 = max{A, 4∥S0∥∞, 4σ  µ1  , 4(σ + M2)  µ1 + d  }, M2 = max{A, 2∥I0∥∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Define  XM1  s  = {φ| φ ∈ XS0  s , ∥φ∥C(D∞  s ) ≤ M1},  XM2  s  = {ψ| ψ ∈ XI0  s , ∥ψ∥C(Dg,h  s  ) ≤ M2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Using the same arguments as Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 in [45], we can deduce that Γ is a contraction map and  has a unique fixed point (S∗, I∗) ∈ XM1  s  × XM2  s  for any s ∈ (0, �s] by the contraction mapping  theorem, where �s relies on d, M1, β, γ and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  To prove that (S∗, I∗) is the unique solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1) for t ∈ [0, s] with s ∈ (0, �s], it suffices to  discuss that any nonnegative solution (S, I) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1) for t ∈ [0, s] belongs to XM1  s  × XM2  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  We claim that  S + I ≤ A for t ∈ [0, s] and x ∈ R,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='7)  which implies that  0 ≤ S(t, x) ≤ A,  (t, x) ∈ [0, s] × R,  0 ≤ I(t, x) ≤ A,  (t, x) ∈ [0, s] × [g(t), h(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Consequently, we obtain that for any s ∈ (0, �s], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1) admits a unique solution for t ∈ [0, s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' To  complete the proof, it only needs to prove that the claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='7) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Let N = S + I, then for t ∈ [0, s] and x ∈ (g(t), h(t)),  Nt  ≤  d  �  R J(x − y)N(t, y)dy − dN(t, x) − d(  � g(t)  −∞ +  � ∞  h(t))J(x − y)I(t, y)dy + σ − µ1N  ≤  d  �  R J(x − y)N(t, y)dy − dN(t, x) − µ1N + σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Since ∥S0∥∞ + ∥I0∥∞ ≤ A, we have N(t, x) ≤ A for t ∈ [0, s] and x ∈ (g(t), h(t)) by using the  comparison principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  7  While t ∈ [0, s] and x ∈ R\\(g(t), h(t)), then I(t, x) = 0, which implies  Nt = d  �  R  J(x − y)N(t, y)dy − dN(t, x) + σ − µ1N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  It is clear that N ≤ A for t ∈ [0, s] and x ∈ R\\(g(t), h(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Next, we prove that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='7) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' We  argue by contradiction and suppose that  max  (t,x)∈[0, s]×R N(t, x) > A, there exists a point (t0, x0) ∈  [0, s] × R such that max N = N(t0, x0) > A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' According to the above analysis, we can obtain  that x0 = g(t0) or x0 = h(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Without loss of generality, we assume that x0 = g(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Since  I(t0, g(t0)) = 0, S(t0, x0) satisfies  St(t0, x0) = d  �  R  J(x0 − y)S(t0, y)dy − dS(t0, x0) + σ − µ1S(t0, x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Obviously, St(t0, x0) ≥ 0 and S(t0, x0) ≤ A, which contradicts the assumption that  max  (t,x)∈[0, s]×R N(t, x) >  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Extension of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  We now prove that the unique solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1) for 0 < t ≤ s can be extended to 0 < t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  In Step 2, �s depends only on d, γ, β, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' With the help of the iterative method, we obtain  that problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1) has a unique solution for t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' We omit it here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' see Step 3 of the proof  in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 in [45] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  □  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2 Assume that (J) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  For any S0 satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4) and I0 satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='5),  problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) admits a unique positive solution (S(t, x), I(t, x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' g(t), h(t)) defined for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  We will prove this result by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 and the fixed point theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' For any  given T > 0 and (g∗, h∗) ∈ HT × GT , we know that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1) with (g, h) = (g∗, h∗) has a unique  solution (S∗, I∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Define  \uf8f1  \uf8f2  \uf8f3  �g = −h0 − k  � t  0  � h∗(τ)  g∗(τ)  � g∗(τ)  −∞  J(x − y)I∗(τ, x)dydxdτ,  �h = h0 + k  � t  0  � h∗(τ)  g∗(τ)  � +∞  h∗(τ) J(x − y)I∗(τ, x)dydxdτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='8)  In view of (J) and J(0) > 0, there exist constants ǫ0 ∈ (0, h0/4) and δ0 > 0 such that  J(x) ≥ δ0 if |x| ≤ ǫ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  By virtue of the above inequality and proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 in [5], there exists  T0 = T0(k, A, h0, ǫ0, I0, J) > 0,  such that, for any T ∈ (0, T0],  sup  0≤t1<t2≤T  �g(t2) − �g(t1)  t2 − t1  ≤ −kη1,  inf  0≤t1<t2≤T  �h(t2) − �h(t1)  t2 − t1  ≥ kη2,  �h(t) − �g(t) ≤ 2h0 + ǫ0/4 for t ∈ [0, T],  where  η1 = 1  4ǫ0δ0e−(d+µ2+sup γ)T0  � −h0+ ǫ0  4  −h0  I0(x)dx, η2 = 1  4ǫ0δ0e−(d+µ2+sup γ)T0  � h0  h0− ǫ0  4  I0(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  8  Let  ΣT := {(g, h) ∈ Hh0  T × Gh0  T :  sup  0≤t1<t2≤T  g(t2)−g(t1)  t2−t1  ≤ −kη1,  inf  0≤t1<t2≤T  h(t2)−h(t1)  t2−t1  ≥ kη2, h(t) − g(t) ≤ 2h0 + ǫ0  4 for t ∈ [0, T]},  and define the mapping F(g∗, h∗) = (�g,�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Clearly, the above analysis implies that  F(ΣT) ⊂ ΣT for T ∈ (0, T0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  In the following, similar to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 in [45], we first prove that F is a contraction  mapping, and then F admits a unique fixed point in ΣT by the contraction mapping theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Next, we can derive that (g, h) ∈ ΣT holds for any solution (S, I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' g, h) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) for t ∈ [0, T],  that is, (S, I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' g, h) is a unique solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) for t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Finally, we can show that the  solution (S, I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' g, h) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) is uniquely extended to t ∈ (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Here, we omit the proof here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  see Step 3 in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 in [45] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  □  3  The eigenvalue problem  For any −∞ < L1 < L2 < +∞ and d > 0, denote  L{(L1, L2), d}[φ](x) = d  � L2  L1  J(x − y)φ(y)dy − dφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Now we introduce some results on the principal eigenvalue of the linear operator L{(L1, L2), d} +  a(x) : C([L1, L2]) �→ C([L1, L2]) defined by  (L{(L1, L2), d} + a(x))[φ](x) = d  � L2  L1  J(x − y)φ(y)dy − dφ(x) + a(x)φ(x),  where a(x) = σβ(m(x),0,x)  µ1  − µ2 − γ(b(x), 0, x) ∈ C([L1, L2]) and J satisfies (J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Furthermore, we  assume  (H) : a(x) is Lipschitz continuous and achieves its maximum in [L1, L2] at some point x0 ∈ (L1, L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Define the generalized principal eigenvalue as  λp(L{(L1, L2), d} + a(x))  := inf{λ ∈ R | ∃φ ∈ C([L1, L2]), φ > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' (L{(L1, L2), d} + a(x))[φ] ≤ λφ in (L1, L2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1)  Furthermore, we call it a principal eigenvalue if λp(L{(L1, L2), d} + a(x)) is an eigenvalue of the  operator L{(L1, L2), d} + a(x) with a continuous and positive eigenfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Recalling that a(x)  is Lipschitz continuous and achieves a global maximum in (L1, L2), thus a(x) automatically  satisfies the condition  1  (supx∈(L1, L2) a(x))−a(x) ̸∈ L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Therefore, it follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 or  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2 in [11] that the generalized principal eigenvalue λp(L{(L1, L2), d}+a(x)) is a principal  eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  In this section, we are interested in the properties of the generalized principal eigenvalue  λp(L{(L1, L2), d}+a(x)) with media coverage m(x) and hospital bed number b(x), and asymptotic  behavior of the principal eigenvalue in large and small interval lengths (L1, L2) or diffusion rate  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Before stating our primary result, we recall a useful proposition from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  9  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 ( [11]) The following assertions hold:  (i) Assume (L1, L2) ⊂ (L3, L4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Then,  λp(L{(L1, L2), d} + a(x)) ≤ λp(L{(L3, L4), d} + a(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (ii) Fix L1, L2 and suppose that a1(x) ≤ a2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Then,  λp(L{(L1, L2), d} + a1(x)) ≤ λp(L{(L1, L2), d} + a2(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Moreover, if a1(x) + δ < a2(x) for some δ > 0, then  λp(L{(L1, L2), d} + a1(x)) < λp(L{(L1, L2), d} + a2(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (iii) λp(L{(L1, L2), d} + a(x)) is Lipschitz continuous in a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' More precisely,  |λp(L{(L1, L2), d} + a1(x)) − λp(L{(L1, L2), d} + a2(x))| ≤ ∥a1(x) − a2(x)∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Let us now analyze the impact of media coverage m(x) and hospital bed number b(x) on  the generalized principal eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Obviously, the following result holds by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1  (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2 Suppose that (J) and (H) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Then,  (i) λp(L{(L1, L2), d} + a(x)) is a strictly monotone decreasing in m(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (ii) λp(L{(L1, L2), d} + a(x)) is a strictly monotone decreasing in b(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  From now on, we discuss the effect of interval length on the principal eigenvalue λp(L{(L1,L2), d}+  a(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3 Suppose that (J) and (H) hold, then the following three conclusions hold:  (i) λp(L{(L1, L2), d} + a(x)) is continuous for L1, L2 ∈ (−∞, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (ii)  lim  L1, L2→0 λp(L{(L1, L2), d} + a(x)) = a(0) − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (iii)  lim  −L1,L2→+∞ λp(L{(L1, L2), d} + a(x)) = sup  x∈R  a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' The proof of (i) is similar to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4 in [5], and we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (ii) Due to the continuity of a(x), for any given ǫ > 0, there exists h > 0 small enough such  that  |a(x) − a(0)| < ǫ, x ∈ [−h, h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Since λp(L{(−h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' d} + a(x)) is a principal eigenvalue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' there exists a positive function φ(x) ∈  C([−h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h]) such that  d  � h  −h  J(x − y)φ(y)dy − dφ(x) + a(x)φ(x) = λp(L{(−h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' d} + a(x))φ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x ∈ [−h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  which gives by integrating,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  |λp(L{(−h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' d} + a(x)) − a(0) + d|  =  |  d  � h  −h  � h  −h J(x−y)φ(y)φ(x)dydx  � h  −h φ2(x)dx  +  � h  −h a(x)φ2(x)dx  � h  −h φ2(x)dx  − a(0)|  =  |  d  � h  −h  � h  −h J(x−y)φ(y)φ(x)dydx  � h  −h φ2(x)dx  +  � h  −h(a(x)−a(0))φ2(x)dx  � h  −h φ2(x)dx  |  ≤  d∥J∥∞(� h  −h φ(x)dx)2  � h  −h φ2(x)dx  + |  � h  −h(a(x)−a(0))φ2(x)dx  � h  −h φ2(x)dx  |  ≤  2d∥J∥∞h + ǫ → ǫ as h → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  10  From the arbitrariness of ǫ, we have  |λp(L{(−h, h), d} + a(x)) − a(0) + d| → 0 as h → 0+,  which together with the continuity of λp(L{(L1, L2), d} + a(x)) about L1, L2 give  lim  L1, L2→0 λp(L{(L1, L2), d} + a(x)) = a(0) − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (iii) According to the monotonicity of λp(L{(L1, L2), d}+a(x)) with respect to interval (L1, L2)  and function a(x) yields  λp(L{(L1, L2), d} + a(x)) ≤ λp(L{(L1, L2), d} + sup  x∈R  a(x)) ≤ λp(L{(−∞, +∞), d} + sup  x∈R  a(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Consider the following eigenvalue problem  d  � ∞  −∞  J(x − y)φ(y)dy − dφ(x) + φ(x) sup  x∈R  a(x) = λp(L{(−∞, +∞), d} + sup  x∈R  a(x))φ, x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2)  It is easily seen that λp(L{(−∞, +∞), d} + sup  x∈R  a(x)) = sup  x∈R  a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  So the principal eigenvalue  λp(L{(L1, L2), d} + a(x)) ≤ sup  x∈R  a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Therefore,  lim sup  −L1, L2→+∞  λp(L{(L1, L2), d} + a(x)) ≤ sup  x∈R  a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  To prove (iii), it suffices to prove that  lim inf  −L1, L2→+∞ λp(L{(L1, L2), d} + a(x)) ≥ sup  x∈R  a(x) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  In fact, by the continuity of a(x) and the definition of sup, for given ǫ > 0, there exists some  x0 ∈ R such that  sup  x∈R  a(x) − ǫ ≤ a(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  By (J), for given ǫ > 0, there exist L1 < x0 − 1 and L2 > x0 + 1 such that  � L2  L1  J(z)dz > 1 − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Now take  δn(x − x0) =  �  k1e−1/(1−n2(x−x0)2) > 0,  x ∈ (x0 − 1/n, x0 + 1/n),  = 0,  x /∈ (x0 − 1/n, x0 + 1/n),  where k1 is positive and satisfies  �  R δn(x − x0)dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  It is easy to check that the se-  quence {δn(x − x0)} weakly converges to some δ(x − x0) in L1((L1, L2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' By the definition  of λp(L{(L1, L2), d} + a(x)), one can easily obtain  d  � L2  L1 J(x − y)δn(y − x0)dy − dδn(x − x0) + a(x)δn(x − x0)  ≤  λp(L{(L1, L2), d} + a(x))δn(x − x0), x ∈ (L1, L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3)  Integrating the equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) over (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' L2) yields  λp(L{(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' L2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' d} + a(x))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='≥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 J(x−y)δn(y−x0)dydx−d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 δn(x−x0)dx+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 a(x)δn(x−x0)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 δn(x−x0)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='≥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� x0+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='x0−1 J(x−y)δn(y−x0)dydx−d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 δn(x−x0)dx+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 a(x)δn(x−x0)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 δn(x−x0)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='≥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='d � x0+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='x0−1 δn(y−x0)[� L2−x0−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1−x0+1 J(z)dz]dy−d � L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 δn(x−x0)dx+� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 a(x)δn(x−x0)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 δn(x−x0)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='≥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='(d(1−ǫ)−d) � L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 δn(x−x0)dx+� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 a(x)δn(x−x0)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 δn(x−x0)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='−dǫ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 a(x)δn(x−x0)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='� L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='L1 δn(x−x0)dx ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  11  where we have used that  � x0+1  x0−1 δn(x−x0)dx =  � L2  L1 δn(x−x0)dx for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Therefore,  by taking n → +∞,  lim inf  −L1, L2→+∞ λp(L{(L1,L2), d} + a(x))  ≥  −dǫ +  � +∞  −∞ a(x)δ(x − x0)dx  =  −dǫ + a(x0)  ≥  −dǫ + sup  x∈R  a(x) − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  It follows from the arbitrarily of ǫ that  lim inf  −L1,L2→+∞ λp(L{(L1, L2), d} + a(x)) ≥ sup  x∈R  a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  □  In the following, we discuss the monotonicity of the principal eigenvalue with respect to d  and its limiting behaviors as d → 0 or d → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4 Suppose that (J) and (H) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Then, the following statements hold:  (i) λp(L{(L1, L2), d} + a(x)) is a strictly monotone decreasing function of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (ii) lim  d→0 λp(L{(L1, L2), d} + a(x)) =  max  x∈[L1, L2] a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (iii) If  � L1−L2  −∞  J(z)dz > 0(symmetrically,  � +∞  L2−L1 J(z)dz > 0) holds, then lim  d→+∞ λp(L{(L1, L2), d}+  a(x)) = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (i) Assume that λp(d1) := λp(L{(L1,L2), d} + a(x)) is the principal eigenvalue and  φ(x) is the corresponding positive eigenfunction with ∥φ∥L2 = 1, we have  λp(d1)φ(x) = d1  � L2  L1  J(x − y)φ(y)dy − d1φ(x) + a(x)φ(x), x ∈ (L1, L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Suppose that d < d1, then  λp(d1)  =  d1  � L2  L1  � L2  L1 J(x − y)φ(y)φ(x)dydx − d1 +  � L2  L1 a(x)φ2(x)dx  =  d  � L2  L1  � L2  L1 J(x − y)φ(y)φ(x)dydx − d1 +  � L2  L1 a(x)φ2(x)dx  +(d1 − d)  � L2  L1  � L2  L1 J(x − y)φ(y)φ(x)dydx  <  d  � L2  L1  � L2  L1 J(x − y)φ(y)φ(x)dydx − d1 +  � L2  L1 a(x)φ2(x)dx + d1 − d  =  d  � L2  L1  � L2  L1 J(x − y)φ(y)φ(x)dydx − d +  � L2  L1 a(x)φ2(x)dx  ≤  λp(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Therefore λp(d1) < λp(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (ii) The idea of this proof is from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='8 in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' For the eigenvalue problem  d  � L2  L1  J(x − y)ϕ(y)dy − dϕ(x) + ( max  x∈[L1, L2] a(x)) ϕ(x) = λ∗  pϕ(x), x ∈ (L1, L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4)  It follows from [19] that λ∗  p ≤  max  x∈[L1, L2] a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  So we have λp(L{(L1, L2), d} + a(x)) ≤ λ∗  p ≤  max  x∈[L1, L2] a(x) by (ii) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  12  Next, we prove that lim inf  d→0  λp(L{(L1, L2), d} + a(x)) ≥  max  x∈[L1, L2] a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Assume for the contrary  that lim inf  d→0  λp(L{(L1, L2), d} + a(x)) ≤  max  x∈[L1, L2] a(x) − ǫ for some ǫ > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' By the definition of  lim inf, there exists some �d > 0 such that if d ≤ �d, then  λp(L{(L1, L2), d} + a(x)) ≤  max  x∈[L1, L2] a(x) − ǫ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  On the other hand, by the continuity of a(x), there exist x0 ∈ (L1, L2) and r > 0 such that  max  x∈[L1, L2] a(x) ≤ a(x) + ǫ  4, x ∈ Ur(x0) ⊂ (L1, L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Therefore,  λp(L{(L1, L2), d} + a(x)) ≤ a(x) − ǫ  4  for 0 < d < �d and x ∈ Ur(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Let (λp(L{(L1, L2), d} + a(x)), ψ(x)) be the eigenpair of the  following eigenvalue problem:  d  � L2  L1  J(x − y)ψ(y)dy − dψ(x) + a(x)ψ(x) = λp(L{(L1, L2), d} + a(x))ψ(x), x ∈ (L1, L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Then,  � L2  L1  J(x − y)ψ(y)dy − ψ(x) = λp(L{(L1, L2), d} + a(x)) − a(x)  d  ψ(x) ≤ − ǫ  4dψ(x) in Ur(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Let �λ be the principal eigenvalue of the linear problem  � �  R J(x − y)u(y)dy − u(x) = λu(x)  in Ur(x0),  u(x) = 0,  in R\\Ur(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='5)  It is well known that −1 < �λ < 0 by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Let Ψ(x) be the eigenfunction  corresponding to �λ and ∥Ψ(x)∥L∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Take  ψ(x) =  ψ(x)  infUr(x0) ψ(x), ψ(x) = Ψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  We consider the following problem  � �  R J(x − y)u(y)dy − u(x) = − ǫ  4du(x)  in Ur(x0),  u(x) = 0,  in R\\Ur(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='6)  Direct calculation yields  �  Ur(x0) J(x − y)ψ(y)dy − ψ(x) +  ǫ  4dψ(x)  =  1  inf ψ[  �  Ur(x0) J(x − y)ψ(y)dy − ψ(x)] +  ǫ  4dψ(x)  ≤  1  inf ψ(− ǫ  4dψ(x) +  ǫ  4dψ(x))  =  0,  13  and  �  Ur(x0) J(x − y)ψ(y)dy − ψ(x) +  ǫ  4dψ(x)  =  �  Ur(x0) J(x − y)Ψ(x)dy − Ψ(x)] +  ǫ  4dΨ(x)  =  �λΨ(x) +  ǫ  4dΨ(x)  ≥  0  provided d < min{�d, − ǫ  4�λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Hence, by the super-sub solution method in [21], one can yield (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='6)  has a positive solution between ψ(x) and ψ , which implies that �λ = − ǫ  4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' This contradicts to  the independence of �λ about d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Therefore, lim  d→0 λp(L{(L1, L2), d} + a(x)) =  max  x∈[L1, L2] a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (iii) We claim that if  � L1−L2  −∞  J(z)dz > 0 (or  � +∞  L2−L1 J(z)dz > 0), then lim  d→+∞ λp(L{(L1, L2), d} +  a(x)) := λ∞ = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  We argue by contradiction and suppose that λ∞ > −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Now let  ϕ(x) = C1 (positive constant), without loss of generality, taking C1 = 1, then for x ∈ (L1, L2),  d  � L2  L1 J(x − y)ϕ(y)dy − dϕ(x) + a(x)ϕ(x)  <  d  � L2  L1 J(x − y)dy − d +  max  x∈[L1,L2] a(x)  =  −d  �  R\\[L1,L2] J(x − y)dy +  max  x∈[L1,L2] a(x)  =  −d(  � x−L2  −∞  +  � +∞  x−L1)J(z)dz +  max  x∈[L1,L2] a(x)  ≤  −d(  � L1−L2  −∞  +  � +∞  L2−L1)J(z)dz +  max  x∈[L1,L2] a(x)  as  �  R J(x)dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Owing to the assumption that  � L1−L2  −∞  J(z)dz > 0 (or  � +∞  L2−L1 J(z)dz > 0),  there exists a d adequately large such that  d  � L2  L1  J(x − y)ϕ(y)dy − dϕ(x) + a(x)ϕ(x) ≤ (λ∞ − 1)ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Thus, by definition of λp(L{(L1, L2), d} + a(x)), we have  λp(L{(L1, L2), d} + a(x)) ≤ λ∞ − 1,  further,  lim  d→+∞ λp(L{(L1, L2), d} + a(x)) = λ∞ ≤ λ∞ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  We obtain the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Hence,  lim  d→+∞ λp(L{(L1, L2), d} + a(x)) = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='5 Compared with [44], where the nonlocal operator is d  �  Ω J(x−y)(ψ(y)−ψ(x))dy,  we consider the nonlocal operator d  �  Ω J(x − y)ϕ(y)dy − dϕ(x) and prove that when d → +∞,  the limit of the principal eigenvalue is −∞ for some cases, which is different from the result  in [44];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' its limit is the average of a(x) over Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  4  Spreading-vanishing  Since h(t) and −g(t) are monotonically increasing with t > 0, there exist h∞ and g∞ such that  lim  t→+∞ g(t) = g∞ ∈ [−∞, −h0) and lim  t→+∞ h(t) = h∞ ∈ (h0, +∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Here we define that vanishing  occurs if h∞ − g∞ < +∞ and  lim  t→+∞  max  x∈[g(t), h(t)] I(t, x) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' and spreading happens provided  14  that h∞ −g∞ = +∞ and lim sup  t→+∞ ∥I(·, t)∥C([g(t), h(t)]) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In this section, we always assume that  (J) and (H) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' The following proposition directly holds from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='5 in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 Let (S, I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' g, h) be the unique solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' If h∞ − g∞ < +∞, then  lim  t→+∞ g′(t) = lim  t→+∞ h′(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Next, we discuss the asymptotic behavior of the solution of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) when h∞ − g∞ <  +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2 Let (S, I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' g, h) be the unique solution of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) with h∞ − g∞ < +∞,  then lim  t→+∞  max  x∈[g(t), h(t)] I(t, x) = 0, lim  t→+∞ S(t, x) =  σ  µ1 and λp(L{(g∞, h∞), d} + a(x)) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Assume by contradiction that  lim  t→+∞  max  x∈[g(t), h(t)] I(t, x) > 0, there exists ǫ1 > 0 and  sequence {(ti, xi)}∞  i=1 with xi ∈ [g(t), h(t)] and ti → +∞ as i → +∞ such that I(ti, xi) ≥ ǫ1  2  for i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Since g∞ < g(t) < xi < h(t) < h∞, there exists a subsequence {xij}∞  j=1 such that  xij → x0 ∈ (g∞, h∞) as j → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' For t ∈ (−ti, +∞) and x ∈ (g(t + ti), h(t + ti)), define  Ii(t, x) = I(t + ti, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2 gives that I and S are positive and bounded, and then Ii(t, x) satisfies  Iit ≥ d  � hi(t)  gi(t)  J(x − y)Ii(t, y)dy − dIi(t, x) − µ2Ii − γ(b, 0, x)Ii, t > −ti, x ∈ (gi(t), hi(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  We next consider the following auxiliary problem  �  ut = d  � hi(t)  gi(t) J(x − y)u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' y)dy − du(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) − µ2u − γ(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x)u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  t > −ti,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x ∈ (gi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' hi(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  u(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) = Ii(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  x ∈ (gi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' hi(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  it follows that u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) → U(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) as i → +∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' and U(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) satisfies  \uf8f1  \uf8f2  \uf8f3  Ut(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) = d  � h∞  g∞ J(x − y)U(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' y)dy − dU(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) − µ2U − γ(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x)U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  t ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x ∈ (g∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  U(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x0) = lim  i→+∞ Ii(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' xi) = lim  i→+∞ I(ti,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' xi) ≥ ǫ1  2 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  and then U(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) > 0 in R × (g∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h∞) by the maximum principle ( [15]) for the nonlocal  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  On the other hand, considering h∞ − g∞ < +∞ and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1, we have lim  t→+∞ g′(t) =  lim  t→+∞ h′(t) = 0 as t → +∞, which means  0 = lim  i→+∞ h′(t + ti)  =  k lim  i→+∞  � h(t+ti)  g(t+ti)  � +∞  h(t+ti) J(x − y)Ii(t, x)dydx  ≥  k  � h(∞)  g(∞)  � +∞  h(∞) J(x − y)U(t, x)dydx  >  0  and  0 = lim  i→+∞ g′(t + ti)  =  −k lim  i→+∞  � h(t+ti)  g(t+ti)  � g(t+ti)  −∞  J(x − y)Ii(t, x)dydx  ≤  −k  � h(∞)  g(∞)  � g(∞)  −∞ J(x − y)U(t, x)dydx  <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  15  It is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Hence, lim  t→+∞  max  x∈[g(t), h(t)] I(t, x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Next, we will prove that lim  t→+∞ S(t, x) =  σ  µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Since lim  t→+∞  max  x∈[g(t), h(t)] I(t, x) = 0, for any ǫ > 0,  one can choose a T > 0 large such that  0 < I(t, x) < ǫ  for t > T, x ∈ (g(t), h(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Obviously, S(t, x) satisfies  St ≥ dL1[S] + σ − µ1S − ǫβ(m, 0, x)S, t > T, x ∈ R,  and then S(t, x) ≥ S(t), where S(t) is the solution to problem  \uf8f1  \uf8f2  \uf8f3  St = σ − (µ1 + ǫ sup  x∈R  β(m, 0, x))S,  t > T,  S(T) = inf  x∈R S(T, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4 in [26] that lim  t→+∞ S(t) =  σ  µ1+ǫ sup  x∈R  β(m,0,x), Therefore, lim inf  t→+∞ S(t, x) ≥  σ  µ1+ǫ sup  x∈R  β(m,0,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Letting ǫ → 0 yields  lim inf  t→+∞ S(t, x) ≥ σ  µ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1)  On the other hand, S(t, x) satisfies  St ≤ dL1[S] + σ − µ1S + γ(b, ǫ, x)ǫ, t > T, x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Let S(t) be the solution of  \uf8f1  \uf8f2  \uf8f3  St = σ − µ1S(t) + sup  x∈R  γ(b, ǫ, x)ǫ,  t > T,  S(T) = sup  x∈R  S(T, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Apparently, S(t, x) ≤ S(t) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4 in [26] and lim  t→+∞ S(t) =  1  µ1[σ +sup  x∈R  γ(b, ǫ, x)ǫ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' There-  fore,  lim sup  t→+∞ S(t, x) ≤ lim  t→+∞ S(t) = 1  µ1  [σ + sup  x∈R  γ(b, ǫ, x)ǫ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Letting ǫ → 0 gives  lim sup  t→+∞ S(t, x) ≤ σ  µ1  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2)  which together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1) yields lim  t→+∞ S(t, x) =  σ  µ1 uniformly for x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  In what follows, we prove that λp(L{(g∞, h∞), d} + a(x)) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  We argue by contradic-  tion and suppose that λp(L{(g∞, h∞), d} + a(x)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Owing to the continuous dependence of  λp(L{(g∞, h∞), d}+a(x)) on a(x) and (g∞, h∞), there exists a small ǫ such that λp(L{(g∞+ǫ, h∞−ǫ), d}+  a(x) − β(m, 0, x)ǫ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Furthermore, in view of S(t, x) →  σ  µ1 as t → +∞ and h∞ − g∞ < +∞,  there exists a T ∗ > 0 such that  g(t) < g∞ + ǫ, h(t) > h∞ − ǫ, t > T ∗,  S(t, x) > σ  µ1  − ǫ, t > T ∗, x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  16  Then, for t > T ∗, x ∈ (g(t), h(t)),  It(t, x)  =  d  � h(t)  g(t) J(x − y)I(t, y)dy − dI(t, x) − µ2I + β(m, I, x)SI − γ(b, I, x)I  ≥  d  � h∞−ǫ  g∞+ǫ J(x − y)I(t, y)dy − dI(t, x) − µ2I + β(m, I, x)( σ  µ1 − ǫ)I − γ(b, I, x)I  ≥  d  � h∞−ǫ  g∞+ǫ J(x − y)I(t, y)dy − dI(t, x) − µ2I + β(m, 0, x)( σ  µ1 − ǫ)I − γ(b, 0, x)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Let φ(x) be the eigenfunction corresponding to λp(L{(g∞+ǫ, h∞−ǫ), d} + a(x) − β(m, 0, x)ǫ) and  ∥φ(x)∥L∞ = 1, and then φ(x) satisfies  d  � h∞−ǫ  g∞+ǫ J(x − y)φ(y)dy − dφ − µ2φ + β(m, 0, x)( σ  µ1 − ǫ)φ − γ(b, 0, x)φ  =  d  � h∞−ǫ  g∞+ǫ J(x − y)φ(y)dy − dφ + a(x)φ − β(m, 0, x)ǫφ  =  λp(L{(g∞+ǫ, h∞−ǫ), d} + a(x) − β(m, 0, x)ǫ)φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  If we choose δ sufficiently small such that δφ(x) ≤ I(T ∗, x) for x ∈ [g∞ + ǫ, h∞ − ǫ], then  I(t, x) ≥ δφ(x) > 0 for t > T ∗, x ∈ [g∞ + ǫ, h∞ − ǫ]  by the comparison principle in [15], which leads to a contradiction to the fact lim  t→+∞  max  x∈[g(t), h(t)] I(t, x) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  □  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3 Suppose λp(L{(−h0, h0), d}+a(x)) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Then h∞−g∞ < +∞, lim  t→+∞  max  x∈[g(t), h(t)] I(t, x) =  0 and lim  t→+∞ S(t, x) =  σ  µ1 if ∥S0(x)∥L∞(R) + ∥I0(x)∥C([−h0, h0]) is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 yields S(t, x) ≤  σ  µ1 for t > 0, x ∈ R if ∥S0(x)∥L∞(R) +  ∥I0(x)∥C([−h0, h0]) ≤  σ  µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In view of λp(L{(−h0, h0), d} + a(x)) < 0, there exists a ǫ > 0 small  such that λp(L{(−hǫ, hǫ), d} + a(x)) < 0 with hǫ = h0 + ǫ by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Let φ(x) be the  eigenfunction of the principal eigenvalue λp(L{(−hǫ, hǫ), d} + a(x)), which satisfies  d  � hǫ  −hǫ  J(x − y)φ(y)dy − dφ(x) + a(x)φ(x) = λp(L{(−hǫ, hǫ), d} + a(x))φ(x), x ∈ (−hǫ, hǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Denote  M = δC(  � hǫ  −hǫ  φ(x)dx)−1, C = hǫ − h0  k  ,  h(t) = h0 + kC[1 − e−δt], g(t) = −h(t),  h  ′(t) = kCδe−δt, I(t, x) = Me−δtφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  By direct calculations, we have  k  � h  g  � ∞  h J(x − y)I(t, x)dydx  ≤  k  � h  g I(t, x)dx  =  kδCe−δt = h  ′(t),  t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  17  In a similar way, one can deduce −k  � h  g  � g  −∞ J(x − y)I(t, x)dydx ≥ g′(t) for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Clearly,  I(t, g(t)) > 0 and I(t, h(t)) > 0 for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' For t > 0 and x ∈ (g, h), we obtain  It(t, x) − d  � h(t)  g(t) J(x − y)I(t, y)dy + dI(t, x) + µ2I − β(m, I, x)SI + γ(b, I, x)I  ≥  −[d  � h(t)  g(t) J(x − y)Me−δtφ(y)dy − dMe−δtφ(x) − µ2Me−δtφ(x)]  −δI + γ(b, I, x)I − β(m, I, x)SI  ≥  −δI − (λp(L(−hǫ, hǫ) + a(x)) + γ(b, 0, x) − β(m, 0, x) σ  µ1)I + γ(b, I, x)I − β(m, I, x) σ  µ1I  =  I(−δ − λp(L(−hǫ, hǫ) + a(x)) − γ(b, 0, x) + γ(b, I, x) + σ  µ1(β(m, 0, x) − β(m, I, x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Recalling that γ(b, I, x) → γ(b, 0, x) and β(m, I, x) → β(m, 0, x) as δ → 0, we can choose δ  small enough so that  It(t, x) − d  � h(t)  g(t)  J(x − y)I(t, y)dy + dI(t, x) + µ2I − β(x)SI + γ(x, b, I)I ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Moreover, if ∥S0(x)∥L∞(R)+∥I0(x)∥C([−h0, h0]) sufficiently small, then I0 ≤ Mφ(x), x ∈ [−h0, h0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Applying the comparison principle gives  g(t) ≤ g(t), h(t) ≤ h(t) and I(t, x) ≤ I(t, x)  for t > 0 and g(t) < x < h(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Therefore,  lim  t→+∞ I(t, x) ≤ lim  t→+∞ I(t, x) = 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3)  and  h∞ − g∞ ≤ 2hǫ < +∞,  which gives that lim  t→+∞ S(t, x) =  σ  µ1 uniformly for x ∈ R by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4 It follows from the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3 that M → +∞ as k → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Therefore,  there exists a k∗ > 0 such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) holds for all k ∈ (0, k∗) for any given initial function pair  (S0(x), I0(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='5 If −g∞ = h∞ = +∞ and sup  x∈R  a(x) > 0, then lim sup  t→+∞ ∥I(·, t)∥C([g(t), h(t)]) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Conversely, suppose that  lim  t→+∞ ∥I(t, ·)∥C([g(t), h(t)]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' It follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2  that  lim  t→+∞ S(t, x) = σ  µ1  uniformly for x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4)  Since −g∞ = h∞ = +∞, from (iii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3, there exists a T ∗ > 0 large enough such  that  λp(L{(g(T ∗), h(T ∗)), d} + a(x)) > 0  for  t ≥ T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Now, we consider the eigenvalue problem  d  � h(T ∗)  g(T ∗)  J(x − y)ψ(y)dy − dψ(x) + a(x)ψ(x) = λp(L{(g(T ∗), h(T ∗)), d} + a(x))ψ(x), x ∈ (g(T ∗), h(T ∗)),  18  and positive function ψ(x) with ∥ψ∥L∞((g(T ∗), h(T ∗))) = 1 is its eigenfunction to the principal eigenvalue  λp(L{(g(T ∗), h(T ∗)), d} + a(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4), for any given 0 < ǫ < min{ σ  µ1 , λp(L{(g(T ∗), h(T ∗)), d} +a(x))(sup  x∈R  β)−1}, there exists  a T ∗∗ > T ∗ such that  S(t, x) > σ  µ1  − ǫ,  t ≥ T ∗∗,  x ∈ [g(T ∗), h(T ∗)],  and then I(t, x) satisfies  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  It ≥ d  � h(T ∗)  g(T ∗) J(x − y)I(t, y)dy − dI(t, x) − µ2I  +β(m, I, x)( σ  µ1 − ǫ)I − γ(b, I, x)I,  t > T ∗∗, x ∈ (g(T ∗), h(T ∗)),  I(T ∗∗, x) > 0,  g(T ∗) ≤ x ≤ h(T ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  We now construct a suitable lower solution for the following auxiliary problem  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  Wt = d  � h(T ∗)  g(T ∗) J(x − y)W(t, y)dy − dW(t, x)  −µ2W + β(m, W, x)( σ  µ1 − ǫ)W − γ(b, W, x)W,  t > T ∗∗, x ∈ (g(T ∗), h(T ∗)),  W(T ∗∗, x) = I(T ∗∗, x),  g(T ∗) ≤ x ≤ h(T ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='5)  Choose  W(t, x) = δψ(x),  t > T ∗∗,  g(T ∗) ≤ x ≤ h(T ∗),  where δ > 0 is small enough such that δψ(x) ≤ I(T ∗∗, x) for x ∈ [g(T ∗), h(T ∗)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  For t > T ∗∗ and g(T ∗) < x < h(T ∗), direct computation yields  W t − d  � h(T ∗)  g(T ∗) J(x − y)W(y)dy + dW + µ2W − β(m, W, x)( σ  µ1 − ǫ)W + γ(b, W, x)W  =  −δ(d  � h(T ∗)  g(T ∗) J(x − y)ψ(t, y)dy − dψ(x) − µ2ψ + β(m, δψ, x)( σ  µ1 − ǫ)ψ − γ(b, δψ, x)ψ)  =  −δψ(λp + γ(b, 0, x) − β(m, 0, x) σ  µ1 + β(m, δψ, x)( σ  µ1 − ǫ) − γ(b, δψ, x))  ≤  δψ(−λp(L{(g(T ∗),h(T ∗)), d} + a(x)) + β(m, δψ, x)ǫ)  <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Note that the boundaries g(T ∗) and h(T ∗) are fixed, so there is no need to compare the boundary  values of W(t, x) and I(t, x) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1 in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Applying the comparison principle in [15] gives  I(t, x) ≥ W(t, x) ≥ W(t, x) = δψ(x) in [T ∗∗, +∞) × (g(T ∗), h(T ∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Therefore, lim inf  t→+∞ I(t, x) ≥ lim inf  t→+∞ W(t, x) ≥ δψ(0) > 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='6 Suppose that a(x) is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' If h∞−g∞ = +∞, then lim sup  t→+∞  ∥I(·, t)∥C([g(t), h(t)]) >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In the special case, similar to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='10 in [5], a spreading-vanishing dichotomy holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Suppose that  a(0) ≥ d,  we can know that λp(L{(L1, L2), d} + a(x)) > 0 for any interval (L1, L2) by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3 and (i) of  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1, which yields the following conclusion by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='7 If a(0) ≥ d holds, then spreading always occurs for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Next we consider the case 0 < a(0) < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' According to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3 and (i) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='1,  there exists L∗ > 0 such that  λp(L{(L1, L2), d} + a(x))  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  < 0,  (L1, L2) ⊂ (−L∗, L∗),  = 0,  −L1 = L2 = L∗,  > 0,  (−L∗, L∗) ⊂ (L1, L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  19  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='8 Assume 0 < a(0) < d holds, then  (i) if h0 ≥ L∗, then spreading always occurs for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (ii) if h0 < L∗, then there exists a positive constant k∗ such that h∞ − g∞ = ∞ when k > k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' (i) holds from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2 since  λp(L{(g∞, h∞), d} + a(x)) > λp(L{(−L∗, L∗), d} + a(x)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  In what follows, we prove (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Notice that  −µ2 + β(m(x), I, x)S − γ(b(x), I, x) > −µ2 − γ(b(x), I, x) > −C  for some C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Clearly I(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) satisfies  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  It ≥ d  � h(t)  g(t) J(x − y)I(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' y)dy − dI(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) − CI(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x ∈ (g(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  I(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x ∈ R\\(g(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  h′(t) = k  � h(t)  g(t)  � +∞  h(t) J(x − y)I(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x)dydx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  g′(t) = −k  � h(t)  g(t)  � g(t)  −∞ J(x − y)I(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x)dydx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  g(0) = −h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h(0) = h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  x ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  I(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' x) = I0(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  x ∈ (−h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' h0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  thereby,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' for any given constant M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' there exists k∗ > 0 such that h∞ − g∞ > M provided that k > k∗  by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='9 in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Then, h∞ − g∞ = +∞ by the arbitrariness of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  □  Noting that the comparison principle for problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) is not valid, we cannot obtain that the  monotonicity of the solution for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) with k and thus cannot take k as a sharp criterion for the  spreading-vanishing dichotomy as in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' However, recalling that Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4 and (ii) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='8,  we have the following result:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='9 Suppose that 0 < a(0) < d and h0 < L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' For problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3), there exists 0 < k∗ ≤ k∗  such that vanishing occurs if 0 < k < k∗ and spreading happens provided that k > k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Finally, we will discuss the impact of the diffusion coefficient on the vanishing and spreading of  infectious disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Assume that  � −2h0  −∞  J(z)dz > 0 (or  � +∞  2h0 J(z)dz > 0) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4 with L1 = −h0  and L2 = h0, if  max  x∈[−h0, h0] a(x) < 0, for any d > 0, we have λp(L{(−h0, h0), d} + a(x)) < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' while if  max  x∈[−h0, h0] a(x) > 0, there exists a d∗ > 0 such that  λp(L{(−h0, h0), d} + a(x))  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  < 0,  d > d∗,  = 0,  d = d∗,  > 0,  d < d∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Inspired by the above analysis, combined with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2, we can obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='10 Suppose  � −2h0  −∞  J(z)dz > 0 (or  � +∞  2h0 J(z)dz > 0) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Then,  (i) if  max  x∈[−h0, h0] a(x) > 0, there exists a d∗ > 0 such that for d < d∗, then spreading occurs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  conversely, if d > d∗, then vanishing occurs provided that ∥S0(x)∥L∞(R) + ∥I0(x)∥C([−h0, h0]) is small  enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  (ii) if  max  x∈[−h0, h0] a(x) ≤ 0, for any d > 0, then vanishing always occurs as long as ∥S0(x)∥L∞(R) +  ∥I0(x)∥C([−h0, h0]) is adequately small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  20  5  Discussion  In this paper, we study a free boundary problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) with media coverage and hospital bed numbers,  which describes a nonlocal diffusive SIS epidemic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' The free boundary describes the moving  front of the infected individuals, and the nonlocal diffusion operator characterizes the long-distance  spatial movement of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  For the SIS model with nonlocal diffusion and free boundaries (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3), the existence and uniqueness  of the global solution are given by using two fixed point theorems (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Then, we define  the principal eigenvalue of the integral operator, and analyze impacts of media coverage and hospital  bed number (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2), interval length (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3), and diffusion coefficient (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='4) on  the principal eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In addition, sufficient conditions for disease spreading and vanishing (see  Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='5) are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Finally, we discuss the impact of the principal eigenvalue on  the spreading or vanishing of the infectious diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' If a(0) ≥ d, then λp(L{(L1, L2), d} + a(x)) > 0  for any L1 < L2, the disease always spreading (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' If 0 < a(0) < d, there exists a  L∗ > 0, then spreading always appears for h0 ≥ L∗ (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' and when h0 < L∗, the impact  of expanding capability k on the spreading or vanishing of disease is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' That is, there exists  0 < k∗ ≤ k∗ such that vanishing occurs if 0 < k < k∗, and spreading happens provided that k > k∗  (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' If maxx∈[−h0, h0] a(x) > 0, there exists a d∗ > 0 such that for d < d∗, then the  disease spreads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' if d > d∗ and ∥S0(x)∥L∞(R) + ∥I0(x)∥C([−h0, h0]) is small enough, then the disease  vanishes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' if maxx∈[−h0, h0] a(x) ≤ 0, then d∗ = 0, that is, then vanishing always appears provided that  ∥S0(x)∥L∞(R) + ∥I0(x)∥C([−h0, h0]) is adequately small (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Finally, we may conclude that the differences between nonlocal diffusion in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) and local diffusion  in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2) in our mathematical analysis are as follows: first of all, the existence and uniqueness of the  global solution for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2) are obtained by straightening the boundary and the first-order fixed point  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' However, owing to lack of compactness, the existence and uniqueness of global solutions for  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) are given by using two fixed point theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Secondly, for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='2), the corresponding principal  eigenvalue always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' However, for the nonlocal diffusion problem, the principal eigenvalue may  not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' In this paper, for the nonlocal diffusion model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3), we first define generalized principal  eigenvalue λp(L{(L1, L2), d}+a(x)) of the integral operator, and then show that the generalized principal  eigenvalue is the principal eigenvalue under the condition (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Thirdly, different from the local  diffusion whose principal eigenvalue is clear, the nonlocal operator leads more possibilities because of  the choice of the kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  It is worth mentioning that the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='3) incorporates media coverage and hospital bed numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Based on the monotonicity of the generalized principal eigenvalue on media coverage and hospital bed  numbers, we study the influence of the principal eigenvalue on infectious diseases, which implies that  large media coverage and hospital bed numbers are beneficial to the prevention and control of disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  References  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Allen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Bolker, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Lou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Nevai, Asymptotic profiles of the steady states for an  SIS epidemic reaction-diffusion model, Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 21 (2008), 1-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [2]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Basnarkov, SEAIR epidemic spreading model of COVID-19, Chaos Solitons Fractals, 142  (2021), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' 110394, 15 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [3]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Basnarkov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Tomovski, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Sandev, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Kocarev,  Non-Markovian SIR epidemic spreading  model of COVID-19, Chaos Solitons Fractals, 160 (2022), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' 112286, 8 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [4] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Berestycki, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Coville, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Vo, On the difinition and the propperties of the principal eigeneigen  of some nonlocal operators, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Anal, 271 (2016), 2701-2751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Du, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, The dynamics of a Fisher-KPP nonlocal diffusion model  with free boundaries, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 277 (2019), 2772-2814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  21  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Cao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang, A Lotka-Volterra competition model with nonlocal diffusion and  free boundaries, arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='09584v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Cao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Yang, A free boundary problem of a diffusive SIRS model  with nonlinear incidence, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 68 (2017), 1-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Cao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Yang, Dynamics of a nonlocal SIS epidemic model with free boundary,  Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' B, 22 (2017), 247-266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Chang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Du, Long-time dynamics of an epidemic model with nonlocal diffusion and  free boundaries, Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 30 (2022), 289-313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [10] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Friedman, A free boundary problem arising in a model of wound healing, SIAM  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 32 (2000), 778-800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal  operators, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Differential Equations, 249 (2010), 2921-2953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Cui, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Tao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Zhu, An SIS infection on model in corporating media coverage, Rocky Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 38 (2008), 1323-1334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [13] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Dong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang, Free boundary problems with local-nonlocal diffusions and  different free boundaries II: spreading-vanishing and long-time behavior, Nonlinear Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Real  World Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 62 (2022), 23 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [14] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Du, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Lin, Spreading-Vanishing dichotomy in the diffusive logistic model with a free  boundary, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 42 (2010), 377-405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [15]  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Du, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Ni,  Analysis of a West Nile virus model with nonlocal diffusion and free  boundaries, Nonlinearity, 33 (2020), 4407-4448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [16] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Du, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Zhao, Two species nonlocal diffusion systems with free boundaries,  Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 42 (2022), 1127-1162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [17]  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Feng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Ruan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Fei, Dynamics and asymptotic profiles of a nonlo-  cal dispersal SIS epidemic model with bilinear incidence and Neumann boundary conditions, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Differential Equations, 335 (2022), 294-346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [18]  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Feng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Yang,  Traveling waves in a nonlocal dispersal SIR model with  non-monotone incidence, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Simul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 95 (2021), 21 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Garc´ıa-Meli´an, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Rossi, Maximum and antimaximum principles for some nonlocal diffusion  operators, Nonlinea Anal, 71 (2009), 6116-6121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Garc´ıa-Meli´an, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Rossi, On the principal eigenvalue of some nonlocal diffusion problems,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Differential Equations, 246 (2009), 21-38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Garc´ıa-Meli´an, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Rossi, A logistic equation with refuge and nonlocal diffusion, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 8 (2009), 2037-2053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Ge, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Kim, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Lin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Zhu, A SIS reaction-diffusion-advection model in a low-risk  and high-risk domain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Differential Equations, 259 (2015), 5486-5509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang, The reaction-diffusion system for an SIR epidemic model with a free  boundary, Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' B, 20 (2015), 2039-2050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [24] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Han, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Liu, Dynamics of an SIS reaction-diffusion epidemic model for  disease transmission, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Biosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 7 (2010), 51-66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  22  [25] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Lacitignola, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Diele, Using awareness to Z-control a SEIR model with overexposure: insights  on Covid-19 pandemic, Chaos Solitons Fractals, 150 (2021), 14 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [26] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Sheng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang, Systems with nonlocal vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' local diffusions and free boundaries,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 483 (2020), 27 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Lin, The spreading fronts in a mutualistic model with advection, Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' B, 20 (2015), 2089-2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [28] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Bertozzi, martingale formulation for stochastic compartmental  susceptible-infected-recovered (SIR) models to analyze finite size effects in COVID-19 case studies,  Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Heterog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Media, 17 (2022), 311-331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [29] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Liu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Xu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Jin, Global dynamics of a spatial heterogeneous viral infection model with  intracellular delay and nonlocal diffusion, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 82 (2020), 150-167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Meiksin, Using the SEIR model to constrain the role of contaminated fomites in spreading an  epidemic: an application to COVID-19 in the UK, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Biosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 19 (2022), 3564-3590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Misra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Maurya, Modeling the importance of temporary hospital beds on the dynamics of  emerged infectious disease, Chaos, 31 (2021), 22 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [32]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Murray, Mathematical Biology, II, Spatial Models and Biomedical Applications,  Third  edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Interdisciplinary Applied Mathematical, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Springer-Verlag, New York, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [33] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' I, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Differential Equations, 247 (2009), 1096-1119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [34] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Liu, Global stability of the steady states of an SIS epidemic reaction- diffusion  model, Nonliner Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 71 (2009), 239-247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Peng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Q, Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment,  Nonlinerity, 25 (2012), 1451-1471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [36] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Rajasekar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Pitchaimani, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Zhu, Higher order stochastically perturbed SIRS epidemic  model with relapse and media impact, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 45 (2022), 843-863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [37] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Shan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Zhu, Bifurcations and complex dynamics of an SIR model with the impact of  the number of hospital beds, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Differential Equations, 257 (2014), 1662-1688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [38]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Takhirov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Norov, On a predator-prey model with free boundary, Uzbek Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 4  (2019), 162-168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [39]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Du, Long-time dynamics of a nonlocal epidemic model with free boundaries:  spreading-vanishing dichotomy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Differential Equations, 327 (2022), 322-381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [40]  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Guo,  A SIS reaction-diffusion model with a free boundary condition and  nonhomogeneous coefficients, Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' B, 24 (2019), 1627-1652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [41] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wang, Threshold dynamics of a nonlocal dispersal HIV/AIDS epidemic  model with spatial heterogeneity and antiretroviral therapy,  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  Simul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 115 (2022), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' 106728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [42] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with  mass action infection mechanism, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Differential Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 261 (2016), 4424-4447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [43] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Ruan, Spatial propagation in an epidemic model with nonlocal diffusion:  the influences of initial data and dispersals, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' China Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 63 (2020), 2177-2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  23  [44] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Yang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Ruan, Dynamics of a nonlocal dispersal SIS epidemic model with  Neumann boundary conditions, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Differential Equations, 267 (2019), 2011-2051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [45] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Zhao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Cao, Dynamics for an SIR epidemic model with nonlocal diffusion and  free boundaries, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' B, 41 (2021), 1081-1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [46] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Yang, Threshold dynamics of an SEAIR epidemic model with application to  COVID-19, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 15 (2022), 136-151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  [47] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Guo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Lin, The risk index for an SIR epidemic model and spatial spreading  of the infectious disease, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Biosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content=', 14 (2017), 1565-1583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
+page_content='  24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFJT4oBgHgl3EQfhyyK/content/2301.11567v1.pdf'}
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+Draft version January 12, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+Observed polarized scattered light phase functions of planet-forming disks
+Christian Ginski
+,1, 2 Ryo Tazaki
+,2, 3, 4 Carsten Dominik
+,2 and Tomas Stolker
+1
+1Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands
+2Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands
+3Institute of Planetology and Astrophysics, Université Grenoble Alpes, 38000 Grenoble, France
+4Astronomical Institute, Graduate School of Science, Tohoku University, 6-3 Aramaki, Aoba-ku, Sendai 980-8578, Japan
+(Received January 11, 2023; Revised –; Accepted –)
+Submitted to ApJ
+ABSTRACT
+Dust particles are the building blocks from which planetary bodies are made.
+A major goal of
+the studies of planet-forming disks is to constrain the properties of dust particles and aggregates in
+order to trace their origin, structure, and the associated growth and mixing processes in the disk.
+Observations of scattering and/or emission of dust in a location of the disk often lead to degenerate
+information about the kind of particles, such as size, porosity, or fractal dimension of aggregates.
+Progress can be made by deriving the full (polarizing) scattering phase function of such particles at
+multiple wavelengths. This has now become possible by careful extraction from scattered light images.
+Such an extraction requires knowledge about the shape of the scattering surface in the disk and we
+discuss how to obtain such knowledge as well as the associated uncertainties. We use a sample of
+disk images from observations with VLT/SPHERE to, for the first time, extract the phase functions
+of a whole sample of disks with broad phase angle coverage. We find that polarized phase functions
+come in two categories. Comparing the extracted functions with theoretical predictions from rigorous
+T-Matrix computations of aggregates, we show that one category can be linked back to fractal, porous
+aggregates, while the other is consistent with more compact, less porous aggregates. We speculate that
+the more compact particles become visible in disks where embedded planets trigger enhanced vertical
+mixing.
+Keywords: Exoplanet formation(492) — Circumstellar disks(235) — Direct imaging(387) — Polarime-
+try(1278)
+1. INTRODUCTION
+Gas- and dust-rich circumstellar disks are the sites
+of ongoing planet formation. However, the early stage
+of planet formation, such as the formation of planetes-
+imals, remains a matter of discussion, as it comes with
+a number of barriers inhibiting grain growth and form-
+ing planetesimals (e.g., Brauer et al. 2008; Zsom et al.
+2010; Birnstiel et al. 2012). The sizes and structures of
+growing dust aggregates are of crucial relevance to the
+barriers because these quantities influence the sticking
+and aerodynamic properties (Ormel et al. 2007; Zsom
+Corresponding author: Christian Ginski
+ginski@strw.leidenuniv.nl
+et al. 2010; Okuzumi et al. 2012; Kataoka et al. 2013;
+Krijt et al. 2016; Lorek et al. 2018; Garcia & Gonzalez
+2020; Kobayashi & Tanaka 2021; Estrada et al. 2022)
+Thus by determining these properties by disk observa-
+tions, one can conclude the early planet formation and
+transport processes including vertical mixing.
+The past decade was revolutionary for resolved ob-
+servations of young planet-forming disks in the near-
+infrared. Driven by advances in instrumentation several
+large surveys have been conducted or are still ongoing
+such as SEEDS (Tamura 2016), DARTTS-S (Avenhaus
+et al. 2018; Garufi et al. 2020), Gemini-LIGHTS (Rich
+et al. 2022) and SPHERE-DESTINYS (Ginski et al.
+2021). A summary of the field was recently presented
+by Benisty et al. (2022). Due to these ongoing observa-
+arXiv:2301.04617v1  [astro-ph.EP]  11 Jan 2023
+
+ID2
+Ginski et al.
+tional programs more than 150 systems have now been
+observed in (polarized) near-infrared scattered light. A
+large diversity of sub-structures has been discovered,
+such as rings, gaps and spirals, which are typically asso-
+ciated with ongoing planet formation (see e.g. Benisty
+et al. 2022).
+The analysis of the scattered light data
+often focused on either the basic disk geometry, tracing
+illumination and shadowing (e.g. Garufi et al. 2022) as
+a function of disk aspect ratio and disk symmetry, or on
+the disk morphology (e.g. Garufi et al. 2018).
+However, the appearance of these objects in (polar-
+ized) scattered light is also strongly dependent on the
+properties of the dust grains or aggregates in the upper
+disk atmosphere (see e.g.
+Min et al. 2005).
+In par-
+ticular the scattering angle dependent amount of flux
+that we receive from the different regions of the disk,
+the so-called (polarized) scattering phase function, en-
+codes dust grain and aggregate properties (Tazaki et al.
+2019). While the extraction of polarized scattered light
+phase functions has been done for geometrically flat de-
+bris disks (e.g. Milli et al. 2017, 2019; Olofsson et al.
+2020; Engler et al. 2022), it is less common for young
+gas rich disks, due to their more complex geometry. So
+far this has only been done for the disks around the Her-
+big stars HD 97048 (Ginski et al. 2016) and HD 100546
+(Quanz et al. 2011; Stolker et al. 2016) in polarized light
+and the T Tauri star multiple system GG Tau (McCabe
+et al. 2002) in total intensity.
+In this study we gathered a sample of 10 disks for
+which the surface height profile has been determined
+in the literature from near-infrared scattered light ob-
+servations. The sample is comprised of Herbig and T
+Tauri stars with the earliest spectral type being the B9.5
+star HD 34282 and the latest spectral type the M0 star
+IM Lup. The systems cover a range of ages from 1.1 Myr
+(IM Lup, Avenhaus et al. 2018) to 12.7 Myr (MY Lup,
+Avenhaus et al. 2018).
+All systems were also already
+observed at (sub)mm-wavelengths which allowed to es-
+timate dust masses based on their continuum flux. Our
+sample spans roughly an order of magnitude between
+the lowest mass disk around PDS 70 (13 M⊕) and the
+highest mass disk in the RXJ 1615.3-3255 system with
+(140 M⊕). A summary of all included systems with the
+appropriate references is given in table 1. Due to the
+ring-shaped sub-structure in most of these disks, planet
+formation is thought to be ongoing. In particular our
+study includes the PDS 70 system, in which two young
+planets have been detected inside the disk cavity (Kep-
+pler et al. 2018; Haffert et al. 2019), and the HD 163296
+system for which the presence of two wide separation
+planets has been inferred from ALMA gas kinematic ob-
+servations (Pinte et al. 2018; Teague et al. 2018). In the
+following section we briefly describe the observational
+data. In section 3 we discuss how the phase functions
+were extracted, while we discuss their interpretation in
+light of dust aggregate models in section 4. We summa-
+rize our results in section 5.
+System
+SpType
+Mdust (M⊕)
+Ref.
+RXJ 1615.3-3255
+K5
+140
+(1),(12)
+HD 163296
+A1
+75
+(2),(3)
+IM Lup
+M0
+54
+(1),(3)
+LkCa 15
+K5.5
+33
+(4),(3)
+PDS 66
+K1.5
+15
+(5),(3)
+PDS 70
+K7
+12
+(1),(3)
+2MASSJ18521730-3700119
+K4
+13
+(6),(3)
+V 4046 Sgr
+K4
+48
+(7),(10)
+HD 34282
+B9.5
+87
+(8),(11)
+MY Lup
+K0
+53
+(1),(9)
+Table 1. (1) Luhman (2022) (2) Sartori et al. (2003) (3) van
+der Marel & Mulders (2021) (4) Krolikowski et al. (2021)
+(5) Pecaut & Mamajek (2016) (6) Herczeg & Hillenbrand
+(2014) (7) Pecaut & Mamajek (2013) (8) Kharchenko (2001)
+(9) Mulders et al. (2017) (10) Martinez-Brunner et al. (2022)
+(11) Stapper et al. (2022) (12) van der Marel et al. (2019)
+2. OBSERVATIONS AND DATA REDUCTION
+All datasets used in our study have been obtained with
+VLT/SPHERE (Beuzit et al. 2019) and its near infrared
+camera IRDIS (Dohlen et al. 2008).
+The instrument
+was operated in dual-beam polarization imaging mode
+(DPI, de Boer et al. 2020a; van Holstein et al. 2020), in
+either J or H broad band filters, to obtain (linear) po-
+larized scattered light images of the circumstellar disks
+in each system. An overview of the observation setup
+and conditions is given in table 3 in appendix A. In
+all cases the innermost 92.5 mas around the stellar po-
+sition where covered by the standard YJH_S apodized
+Lyot coronagraph (Carbillet et al. 2011) and are there-
+fore inaccessible for the analysis. All data sets that are
+included in our study have been previously discussed in
+the literature.
+We give the relevant references in ta-
+ble 2. The data reduction has in all cases been carried
+out with the public IRDAP (IRDIS Data reduction for
+Accurate Polarimetry, van Holstein et al. 2020) pipeline
+with default settings1. This includes a full model based
+determination and removal of instrumental polarization,
+as well as the measurement and subsequent subtraction
+of astrophysical stellar polarization.
+1 https://irdap.readthedocs.io
+
+Phase functions of planet-forming disks
+3
+3. PHASE FUNCTION EXTRACTION
+3.1. Surface height profiles
+The key challenge in extracting the scattered light
+phase function of young gas-rich disks is the uncertainty
+of the vertical structure. Here we are particularly in-
+terested at which height above the disk mid-plane the
+optical depth τ becomes equal to 1, i.e. the surface layer
+from which the majority of the disk scattered light orig-
+inates. For ease of use within the further discussion we
+will refer to this as the surface height of the disk within
+this study.2
+While the surface height may generally be inferred from
+detailed radiative transfer modeling, there exists a sub-
+class of disks for which it can be directly determined
+from the data itself. As shown by de Boer et al. (2016)
+and Ginski et al. (2016) the disk surface height can be
+computed for disks with radial sub-structures such as
+multiple rings. This is done by measuring the inclina-
+tion of the ring and its offset along the minor axis from
+the central star position in the image. This directly gives
+the surface height at the ring location. If there are multi-
+ple rings then the radial dependence of the disk surface
+height can be directly traced. Both of the mentioned
+studies found that the surface height profile for the two
+studied target systems (RXJ 1615 and HD97048) can be
+described reasonably well with a single power law profile
+of the form:
+H(r) = Href/(rref/1au)α ∗ (r/1au)α,
+(1)
+wherein H(r) is the radial dependent surface height,
+Href is a reference height at reference separation rref
+(all in au) and α is the flaring exponent. Avenhaus et al.
+(2018) found that the surface height profile for five disks
+in their study could all be described by the same power-
+law profile with a flaring exponent of α = 1.22 ± 0.03.
+This indicates that for single-ringed disks, i.e. when only
+a surface height at a single radial separation is known,
+we may still infer a reasonable guess of the radial de-
+pendent surface height profile by using the height mea-
+surement as reference height in the power-law and by
+assuming a standard flaring exponent of α = 1.22.
+For our sample we draw the surface profile parame-
+ters from the literature. For the RXJ 1615, IM Lup and
+V 4046 Sgr multi-ring systems we use the specific power-
+law profiles fitted by Avenhaus et al. (2018). For the
+HD 34282 multi-ring system we likewise use the power-
+2 We note that this is not identical to the pressure scale height of
+the disk, which is typically a factor 3-4 smaller than the scattered
+light surface height (Chiang et al. 2001).
+law profile given by de Boer et al. (2020a). For the re-
+maining single-ringed disks we use the literature values
+for the surface height of the rings as reference height and
+the standard flaring exponent of α = 1.22 by Avenhaus
+et al. (2018).
+For the RXJ1852 and PDS 70 systems
+a different flaring exponent was infered from radiative
+transfer modelling by Villenave et al. (2019) and Kep-
+pler et al. (2018), respectively, and we make use of their
+fitted values.
+In order to capture the uncertainty of the extracted
+phase functions due to the uncertainty of the surface
+height we always consider three scenarios for each sys-
+tem:
+(1) the nominal surface height profile, (2) a
+strongly "flared" profile and a (3) "flat" profile.
+The
+flared and flat profile consider the uncertainty of the
+reference height, the reference separation and the flar-
+ing exponent. For the "flared" profile we use the upper
+bound on the reference height, the lower bound on the
+reference separation and the upper bound on the flaring
+exponent, while we switch lower and upper bounds for
+each parameter for the "flat" profile. We summarize all
+profile parameters for each system in table 2 and illus-
+trate all three profiles (nominal, flared and flat) for the
+RXJ 1615 system in figure 1, top panel. In the bottom
+panel of figure 1 we show the corresponding deviations in
+scattering angle between the "flared" and "flat" surface
+profile extremes. While we see deviations of up to ∼5◦
+in the inner disk region, these are smaller in the outer
+disk. We note that the intermediate region between in-
+ner disk and outer edge, which shows deviations of less
+than 1◦ is close to the reference separation for which
+the reference height was directly measured by Avenhaus
+et al. (2018). Thus the deviation in this region is dom-
+inated by the (small) uncertainties of these quantities.
+We show similar figures for the maximum deviation of
+the scattering angle for our complete sample in figure 7
+in the appendix. Unsurprisingly the largest deviations
+are found close to the outer disk edge for the systems
+with large uncertainties in their flaring exponent, i.e.
+IM Lup and HD 34282 in particular. In general we are
+avoiding these outer disk regions for phase function ex-
+traction and typically consider regions for which the un-
+certainty in scattering angle is less than ∼10◦, with the
+main exception being the two aforementioned systems.
+However, as we explain in the following section we do
+for all systems incorporate the resulting deviations of
+the extracted phase functions between all three surface
+height profiles in their uncertainties.
+3.2. Extraction with diskmap
+For the extraction of the phase functions we use the
+publicly available diskmap python package by Stolker
+
+4
+Ginski et al.
+System
+d
+i
+PA
+h/rα
+ref
+rref
+α
+Ref.
+(pc)
+(◦)
+(◦)
+(au)
+RXJ 1615.3-3255
+nominal
+155.6
+47.2
+145.0
+0.091
+161.8
+1.12
+(3)
+min
+0.140
+162.3
+1.02
+max
+0.059
+161.3
+1.22
+HD 163296
+nominal
+101.0
+46.0
+134.8
+0.086
+63.6
+1.22
+(1),(2),(3)
+min
+0.081
+68.7
+1.19
+max
+0.090
+58.6
+1.25
+IM Lup
+nominal
+155.8
+55.0
+325.0
+0.046
+149.6
+1.27
+(3)
+min
+0.095
+154.3
+1.07
+max
+0.022
+144.9
+1.47
+LkCa 15
+nominal
+157.2
+50
+60
+0.074
+57.7
+1.22
+(3),(4),(5)
+min
+0.059
+60.1
+1.19
+max
+0.089
+53.7
+1.25
+PDS 66
+nominal
+97.9
+30.3
+189.2
+0.052
+84.2
+1.22
+(3)
+min
+0.055
+84.7
+1.19
+max
+0.050
+83.5
+1.25
+PDS 70
+nominal
+112.4
+49.7
+158.6
+0.041
+99.1
+1.25
+(6),(7)
+min
+0.044
+105.1
+1.22
+max
+0.039
+93.2
+1.28
+2MASSJ18521730-3700119
+nominal
+147.1
+30
+124
+0.046
+43.4
+1.10
+(8)
+min
+0.065
+45.2
+1.00
+max
+0.034
+41.6
+1.20
+V 4046 Sgr
+nominal
+71.5
+32.2
+74.7
+0.017
+26.7
+1.61
+(3)
+min
+0.027
+26.8
+1.47
+max
+0.012
+26.6
+1.74
+HD 34282
+nominal
+308.6
+57.0
+118.0
+0.064
+86.4
+1.35
+(9)
+min
+0.071
+89.5
+1.27
+max
+0.057
+83.3
+1.43
+MY Lup
+nominal
+157.2
+77.0
+239.0
+0.073
+121.0
+1.22
+(3)
+min
+0.073
+125.8
+1.19
+max
+0.072
+116.3
+1.25
+Table 2. Values for the surface height geometry of the studied disks. We give the nominal values used as well as the values
+for the minimal and maximal surface height that we considered. References are: (1) Muro-Arena et al. (2018) (2) Isella et al.
+(2016) (3) Avenhaus et al. (2018) (4) Thalmann et al. (2014) (5) Thalmann et al. (2016) (6) Keppler et al. (2018) (7) Hashimoto
+et al. (2012) (8) Villenave et al. (2019) (9) de Boer et al. (2020b)
+
+Phase functions of planet-forming disks
+5
+1
+0
+1
+∆RA (arcsec)
+1
+0
+1
+∆Dec (arcsec)
+RXJ1615
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+scattering angle deviation (deg)
+0
+50
+100
+150
+200
+250
+300
+r (au)
+0
+10
+20
+30
+40
+50
+60
+70
+H (au)
+RXJ1615 surface profile
+100
+101
+10-1
+100
+101
+inner region
+Figure 1.
+Top:
+Exemplary τ=1 surface profile for the
+RXJ 1615 system, as used in our phase function extraction.
+The solid black curve indicates the nominal surface profile,
+while the blue-dotted and red-dashed curves indicate the
+flared and flat extremes, respectively.
+We consider theses
+as the boundaries for the region of uncertainty. The inset
+shows the region inside 50 au on a log scale for clarity. Bot-
+tom: Maximum deviation of the scattering angles between
+the flared and flat disk profiles for each position within the
+image of the RXJ 1615 system. We find the largest devia-
+tion of up to 5◦ close to the star and significantly smaller
+deviations further out. Note that the dark region with the
+smallest deviations was used for normalization of the surface
+power-law profile.
+et al. (2016)3. While the package is described in detail
+in Stolker et al. (2016), we give a brief summary here
+on the extraction steps. As described in the previous
+section the τ=1 surface for all our system is described
+3 https://diskmap.readthedocs.io
+by a single power-law profile. Given this profile and the
+inclination of the disk diskmap calculates for each pixel
+in the image the distance from the central star and sub-
+sequently the angle under which scattered light is re-
+ceived from this part of the disk. We show this for the
+RXJ 1615 system in figure 2 (middle panel). The flux
+is corrected for the square-distance dependent illumina-
+tion drop-off and is then extracted for each pixel, giving
+a single data point for the polarized scattering phase
+function. These data points are then placed in bins of
+scattering angles with a width of 5◦. For each bin the
+median scattering angle of all included pixels and the
+median flux is computed, giving the final data point for
+this angular bin. The standard deviation within each
+bin is used as a measure for the uncertainty of the phase
+function and captures effects due to the width of the bin
+as well as photometric uncertainties.
+To include the uncertainty of the surface height profile
+we repeat the extraction for the "flat" and the "flared"
+extreme cases and calculate for each angle bin the flux
+difference. We consider this difference as the uncertainty
+of the phase function introduced by the uncertainty of
+the surface height profile. We quadratically combine this
+uncertainty with the standard deviation in each bin for
+the nominal extraction and consider the result the total
+uncertainty of the extracted phase function at each an-
+gle.
+For each system we selected extraction regions centered
+on known bright ring features. Since the surface height
+profile is directly measured at the ring locations, this
+minimizes the introduced uncertainty while simultane-
+ously selecting the region of the disk with the highest
+signal-to-noise ratio. If multiple rings are present then
+we separately extracted the phase function for each ring.
+In figure 2 we show the two selected extraction regions
+for the RXJ 1615 system. The extraction and individual
+phase functions for all systems are shown in appendix B.
+In figure 3 we show the final extracted phase function for
+all systems and photometric bands. We note that in fig-
+ure 3 we show the average phase function for the IM Lup
+and V 4046 Sgr systems instead of extractions for indi-
+vidual rings. For the HD 163296 and RXJ 1615 systems
+we show phase functions after correction for azimuthal
+shadowing as discussed in the following section.
+3.3. Effects of azimuthal shadowing
+An aspect that may complicate the extraction of the
+phase function in some systems is azimuthal shadowing
+since it changes the brightness of disk regions due to re-
+duced illumination, an effect that needs to be separated
+from the phase function. There are now a number of
+class II objects known for which shadows are observed
+
+6
+Ginski et al.
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+∆RA (arcsec)
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+∆Dec (arcsec)
+RXJ1615
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+∆RA (arcsec)
+0
+24
+48
+72
+96
+120
+144
+168
+scattering angle (deg)
+20
+40
+60
+80
+100
+120
+140
+scattering angle (deg)
+0.0
+0.5
+1.0
+1.5
+2.0
+polarized scattering phase function
+28..59 au
+146..181 au
+146..181 au, symmetric
+Figure 2. Extracted polarized scattered phase functions and extraction regions for the H-band data of the RXJ 1615 system.
+Left: H-band data of RXJ 1615 scaled with the square of the distance from the central star. The blue and red transparent
+overlays highlight the two regions used for phase function extraction, i.e. the inner disk region close to the coronagraph and
+the bright outer ring. Middle: Position dependent scattering angles calculated with the nominal surface height profile of the
+system. Right: Extracted phase functions for the inner disk region (blue-dashed), the outer ring (red-dashed), and an azimuthal
+region of 180◦ centered on the north-western ansae of the outher ring (black-solid), which should be less affected by azimuthal
+shadowing.
+in scattered light (see Benisty et al. 2022 for an overview
+and detailed discussion). One of the most iconic systems
+is probably HD 142527 (Canovas et al. 2013; Avenhaus
+et al. 2014) for which inner and outer disk are strongly
+misaligned (70◦, Marino et al. 2015), leading to two nar-
+row shadows projected on the outer disk. Such narrow
+and well defined shadows are not of major concern for
+the extraction of the overall phase function as the small
+region affected by the shadows can simply be excluded.
+However, there as was shown for the HD 139614 sys-
+tem by Muro-Arena et al. (2020), small misalignments
+or warps in the disk can lead to very broad and some-
+what diffuse shadowing, which covers in extreme cases
+an azimuthal range of more than 180◦. For such systems
+the problem is two fold. On the one hand the exclusion
+of large azimuthal regions of the disk from the extrac-
+tion may severely limit the range of scattering angles
+for which the phase function can be extracted. On the
+other hand, these shadows are less well defined making it
+more difficult to decide which regions of the disk might
+be trusted for phase function extraction.
+To estimate if the disks in our sample may be affected
+by broad shadowing effects we performed a simple anal-
+ysis. If there is no shadowing present and the brightness
+distribution of disk structures is solely due to the dust
+scattering phase function4, then we would expect the
+disk surface brightness to be axis-symmetric relative to
+4 We imply here the assumptions that there are no azimuthal vari-
+ations in the dust grain size distribution or composition, that the
+disks are not eccentric and there are no or only small azimuthal
+variations in surface density.
+the disk minor axis. Thus for all our systems we flipped
+the disk images around the minor axis and then divided
+the original image by the flip image. This indicates the
+brightness ratio between the two "mirrored" sides of the
+disk. We show the axis-symmetric brightness ratio for all
+systems in the appendix in figure 17. For most systems
+the deviation in brightness is smaller than a factor ∼2,
+with the notable exceptions of the RXJ 1615 system (up
+to factor ∼4), the HD 163296 system (up to factor ∼5)
+and the HD 34282 system (up to factor ∼6). We thus
+consider that the RXJ 1615 and the HD 163296 systems
+maybe be affected by broad shadowing. This was in fact
+discussed for both systems in the literature by de Boer
+et al. (2016) and Muro-Arena et al. (2018), respectively.
+In the case of HD 34282 de Boer et al. (2020b) discuss a
+possible spiral within the disk, thus in this case we may
+rather trace a genuine azimuthal asymmetry in the dust
+surface density rather than shadowing effects.
+To give an indication how strongly these shadowing ef-
+fects may affect the extracted phase function we per-
+formed two separate extractions for the RXJ 1615 sys-
+tem choosing the bright ring between deprojected radii
+of 146 au and 181 au. One extraction only considered the
+brighter north-west side of the disk, while the second ex-
+traction only considered the fainter south-east side, i.e.
+the side of the ring more strongly affected by shadowing.
+The results are shown in figure 4 (both phase functions
+were normalized at scattering angles of 90◦). While the
+profiles somewhat match between 60◦ and 90◦ they de-
+viate significantly at larger and smaller scattering an-
+gles.
+In particular the phase function extracted from
+the south-east half of the disk shows more relative flux
+
+Phase functions of planet-forming disks
+7
+20
+40
+60
+80
+100
+120
+140
+160
+scattering angle (deg)
+1
+2
+3
+4
+5
+6
+7
+8
+normalized polarized flux
+H-band
+HD34282
+MY Lup
+IM Lup
+LkCa 15
+V4046 Sgr
+PDS 66
+RXJ1852
+RXJ1615
+HD163296
+PDS 70
+20
+40
+60
+80
+100
+120
+140
+160
+scattering angle (deg)
+1
+2
+3
+4
+5
+6
+7
+8
+normalized polarized flux
+J-band
+CategoryI
+CategoryII
+Figure 3. Final extracted polarized scattered light phase functions for all targets in our sample. We show the H-band data on
+the left and the J-band data on the right. All phase functions were normalized at scattering angles of 90◦ and had an arbitrary
+offset applied for better visibility. Colors indicate the same systems in both panels and the order of the phase functions from
+top to bottom is the same as indicated in the legend. Note that not for all targets both H and J-band were available.
+in both ranges. If we consider that the shadow is cen-
+tered on the south-east ansae, this might be explained
+by the stronger shadowing close to 90◦ scattering angles,
+or put in a different way, the disk may rise out of the
+shadow at regions seen under small and large scattering
+angles.
+We note that there is no geometric reason why the disk
+should specifically be warped or misaligned around the
+minor axis. In practice there will in fact most likely be
+a deviation between the disk minor axis (defined by our
+arbitrary viewing geometry), and the warp or misalign-
+ment axis around which the disk tilts as a function of
+radial separation from the central star. However, if the
+axis of misalignment were closer to the disk major axis,
+then one would not expect a strong brightness asymme-
+try between the two disk ansae as seen in figure 17. de
+Boer et al. (2016) also noted that the brightness asym-
+metry seemed to switch sides between subsequent ring
+structures in the disk, i.e.
+in the next outer (overall
+fainter) ring the south-east ansae is brighter than the
+north-west ansae indicating a continuous warp in the
+disk. If the disk near or far side were strongly affected
+by this effect then one may expect strong changes in
+brightness asymmetries between disk near and far side
+
+8
+Ginski et al.
+seen in subsequent rings. This does however not seem
+to be the case. We can thus conclude that the broad
+shadowing effect is likely centered at least near the disk
+ansae. This is further supported by the fact that the
+phase function extracted from the south-east side of the
+disk and shown in figure 4 increases the relative flux
+level toward both the forward and back-scattering sides
+of the disk. If the shadow were not centered close to
+the south-east ansae, then we would naively expect ei-
+ther a steeper phase function relative to the north-east
+at larger scattering angles or a flatter phase function at
+small scattering angles.
+Given our analysis we thus consider the phase function
+extracted from the north west side of the disk in the
+RXJ 1615 system to be minimally or in any case less af-
+fected by shadowing and use it for further analysis and
+comparison to other systems.
+This phase function is
+shown as black solid line in figure 2 (named "symmet-
+ric" in the legend as this is the phase function of the
+point symmetric version of this disk relative to the mi-
+nor axis). For the HD 163296 system we then follow the
+same strategy and choose the north-west side of the disk
+for phase function extraction as it appears less affected
+by shadowing compared to the south-east.
+For the HD 34282 system the situation is different, as
+the detected asymmetry likely traces a genuine asym-
+metry in dust surface density profile, possibly due to
+spiral density waves in the disk gas. It is then not clear
+which regions may be best suited to extract an unbiased
+scattering phase function. We thus include in this case
+the full azimuthal range in the extraction and caution
+that a more detailed analysis of the system with dedi-
+cated radiative transfer modelling should be performed
+in the future to revise our preliminary results.
+3.4. Limb brightening
+In addition to shadowing effects, the shape of the
+phase function may be influenced by the viewing ge-
+ometry. In recent studies of optically thin debris disks
+Olofsson et al. (2020) and Engler et al. (2022) found that
+the relative flux extracted at ∼90◦ scattering angles, i.e.
+close to the disk ansae, may be enhanced compared to
+the intrinsic phase function of the present dust particles
+due to a higher column density along the line of sight.
+This effect is sometimes referred to as "limb brighten-
+ing" (Engler et al. 2022). The systems in our sample are
+all at an earlier evolutionary stage before the gas disper-
+sal in the disk and it is generally assumed that they are
+optically thick at near infrared wavelengths (Chiang &
+Goldreich 1997). Thus we do not expect that the col-
+umn density along the line of sight will change signifi-
+cantly for different scattering angles. However, due to
+40
+60
+80
+100
+120
+scattering angle (deg)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+polarized scattering phase function
+RXJ1615 phase function asymmetry
+South-East
+North-West
+Figure 4.
+Polarized scattered phase functions of the
+RX 1615 H-band data extracted from the brightest ring in
+the data set between projected separations of 146 and 181 au.
+The blue-dashed phase function was extracted from an az-
+imuthal region of 180◦ centered on the north-west ansae of
+the disk, while the red solid curve was extracted from a re-
+gion centered on the south-east ansae. Both phase functions
+were normalized at scattering angles of 90◦. The differences
+may be explained by azimuthal shadowing of the extraction
+region due to an inner disk component, which strongly af-
+fected the south-western region.
+the flaring surface height profiles of these systems we
+may instead expect an artificial increase in the flux ra-
+tio between the disk forward and back scattering sides,
+i.e. between small and large scattering angles. At small
+scattering angles (on the disk near side) the line of sight
+encompasses more of the illuminated disk surface than is
+the case for large scattering angles (on the disk far side).
+We discuss this effect in detail for the IM Lup system in
+Tazaki, Ginski & Dominik (submitted). Using radiative
+transfer models we found in this study that the effect de-
+pends on the inclination, the local aspect ratio and flar-
+ing exponent of the disk surface height profile. For the
+specific case of the IM Lup system this may introduce
+a ∼25% brightening for scattering angles smaller than
+∼50◦. However, we caution that this strongly depends
+on the dust composition, i.e. for dust with high scat-
+tering albedos limb brightening may be much smaller
+or even insignificant because multiple scattering reduces
+the polarized flux at smaller scattering angles and starts
+to (at least partly) counteract this effect. Dedicated ra-
+diative transfer modelling is required to determine the
+influence of this effect for individual systems.
+In a more general sense this limb brightening effect will
+
+Phase functions of planet-forming disks
+9
+typically not strongly alter the shape of the phase func-
+tions for optically thick disks, such as the ones discussed
+in our current study.
+Rather it will slightly increase
+the slope of the overall phase functions. We note that
+there may be one exception to this, which is the MY Lup
+system, which is seen under a particularly high inclina-
+tion of 77◦. The extracted phase function in figure 3 is
+strongly peaked toward small scattering angles. Based
+on the results in Tazaki, Ginski & Dominik (submitted),
+we find it likely that the intrinsic phase function of dust
+particles in MY Lup has a significantly smaller slope,
+possibly with no up turn at small scattering angles. As
+we summarize in table 2 the remaining systems in our
+study have either a comparable inclination to IM Lup
+(this is the case for HD 34282) or are seen under signifi-
+cantly smaller inclination. Thus, while the effect should
+be considered for future detailed modelling of individual
+systems, it will not strongly affect the general popula-
+tion level trends that we recover.
+4. RESULTS AND DISCUSSION
+4.1. Qualitative inference of dust properties
+Figure 3 shows that the extracted polarization phase
+function is diverse in terms of their shapes. The shapes
+can be roughly divided into two categories: those that
+are monotonically increasing in polarized flux with de-
+creasing scattering angle (category I: e.g., HD 34282,
+MY Lup, IM Lup, LkCa 15, V4046 Sgr) and those that
+turn around at a scattering angle of 60◦–80◦, go through
+a local minimum, and then increase again at the small-
+est scattering angles (category II: e.g., HD 163296, PDS
+70).
+To illustrate the origins of these variations, we per-
+form T-matrix light-scattering calculations for various
+dust aggregates (Tazaki & Dominik 2022) (see also Sec-
+tion C). The scattering matrix element (−S12 in Bohren
+& Huffman 1983) obtained by the simulations are sum-
+marized in figure 5. There is a caveat when comparing
+the scattering matrix element and the observed phase
+function: a planet-forming disk is optically thick in the
+near infrared, and the scattering angle dependence of
+the observed polarization flux might have been affected
+by radiative transfer effects, such as multiple scatter-
+ing within the disk surface and limb brightening (see a
+detailed discussion of these effects in Tazaki, Ginski &
+Dominik, submitted).
+To distinguish it from the one
+extracted from an observed image, we will refer to the
+computed scattering matrix element as the intrinsic po-
+larization phase function.
+While a three-dimensional
+radiative transfer calculation is necessary to determine
+the dust parameters more accurately from the extracted
+polarization phase function, it is possible to capture the
+general trend from the intrinsic phase function and infer
+the origins of the variations in shape shown in figure 3.
+Figure 5 (a) illustrates how the intrinsic polarization
+phase function of aggregates varies with their size. The
+scattering angle dependency of the polarized intensity is
+the result of the dependence of the two quantities over-
+lapping: total-intensity phase function and the degree of
+linear polarization. When the aggregates are small com-
+pared to the wavelength, the scattered light distribution
+will be close to isotropic in total intensity, but the degree
+of polarization approaches 0 in the forward scattering
+direction. As a result, the intrinsic polarization phase
+function turns around at an angle of scattering around
+90◦ (for the case of pure Rayleigh scattering, the intrin-
+sic polarization function is proportional to 1−cos2 θ, see
+Bohren & Huffman (1983)). As the aggregates become
+larger, forward scattering in total intensity develops and
+compensates for the decrease in the degree of polariza-
+tion. Consequently, the turn-around position shifts to
+the small-angle side.
+Except for RXJ1852, none of the observed phase
+functions presented in figure 3 exhibits the Rayleigh-
+scattering-like profile. This indicates that the aggregates
+have grown to at least micron sizes in those disks. For
+RXJ1852, the function appears to have a peak in polar-
+ized flux around a scattering angle of 80◦, and the profile
+may be consistent with the presence of small, submicron-
+sized aggregates. Since the presence of such small parti-
+cles makes the disk scattered light bluish (Tazaki et al.
+2019), future multi-color observations would be useful
+to draw a conclusion.
+The shape diversity in polarization phase functions
+could be resulting from a diversity of the structure and
+porosity of micron-sized aggregates. First of all, we fo-
+cus on category I (the top six curves in the J band); all
+show monotonically increasing polarizing flux with de-
+creasing scattering angle. In particular, the curves for
+V4046 Sgr, LkCa 15, and IM Lup seem to have approx-
+imately constant slopes, while the slope of the curves
+for MY Lup and HD 34282 become steeper at scatter-
+ing angles below 80◦. Figure 5 (b) demonstrates that
+aggregates with different fractal dimensions can explain
+these differences. Aggregates with a low fractal dimen-
+sion (around 1.9) exhibit nearly constant slopes except
+for scattering angles below 30◦, whereas aggregates with
+a high fractal dimension (around 3.0) show a similar
+slope to fractal aggregates in the large scattering an-
+gle region, but the slope becomes steeper in the small
+scattering angle region. Therefore, the approximately
+constant slope observed in V4046 Sgr, LkCa 15, and IM
+Lup may be explained by fractal aggregates. In fact, the
+detailed radiative transfer calculations by Tazaki, Gin-
+
+10
+Ginski et al.
+0
+20
+40
+60
+80 100 120 140 160 180
+Scattering angle (degrees)
+0
+1
+2
+3
+4
+Intrinsic polarization phase function
+(a) Aggregate size
+amax =0.2 m, ac, max =0.359 m
+amax =0.4 m, ac, max =1.05 m
+amax =0.8 m, ac, max =3.24 m
+amax =1.6 m, ac, max =10.0 m
+0
+20
+40
+60
+80 100 120 140 160 180
+Scattering angle (degrees)
+0
+1
+2
+3
+4
+Intrinsic polarization phase function
+amax =1.6 m
+(b) Fractal dimension
+ac, max =3.14 m, Df =3.0
+ac, max =10.0 m, Df =1.9
+0
+20
+40
+60
+80 100 120 140 160 180
+Scattering angle (degrees)
+0
+1
+2
+3
+4
+Intrinsic polarization phase function
+amax =1.6 m
+(c) Porosity
+ac, max =3.12 m,
+max =86.5%
+ac, max =2.47 m,
+max =72.8%
+ac, max =2.09 m,
+max =54.9%
+Figure 5. Effect of aggregate size, fractal dimension, and porosity on the intrinsic polarization phase function. The phase
+functions are normalized to a scattering angle of 90◦. (a) The intrinsic functions for BCCA aggregates for various radii. The
+monomer radius is set as amon = 0.1 µm. (b) The blue and violet lines represent the results for BPCA (ac,max = 3.14 µm, Df =
+3.0) and BCCA (ac,max = 10.0 µm, Df = 1.9) aggregates, respectively. In these computations we used amax = 1.6 µm and
+amon = 0.1 µm. (c) The blue, orange, and brown lines represent the results for BPCA (ac,max = 3.12 µm, Pmax = 86.5%),
+BAM1 (ac,max = 2.47 µm, Pmax = 72.8%), and BAM2 (ac,max = 2.09 µm, Pmax = 54.9%) aggregates, respectively. In these
+computations we used amax = 1.6 µm and amon = 0.4 µm.
+ski and Dominik (submitted) showed that fractal aggre-
+gates best explains the observed phase function of the
+IM Lup disk. On the other hand, the phase functions for
+MY Lup and HD 34282 point to the presence of aggre-
+gates with a high fractal dimension, although the poros-
+ity is still high (∼ 87%), unless the limb brightening is
+responsible for the steepening of the slope. Therefore,
+these disks might contain highly porous aggregates, al-
+though a full radiative transfer modeling is needed to
+determine the fractal dimension.
+Category II exhibits peculiar phase functions (e.g.,
+RXJ 1615, HD 163296, PDS 70 in figure 3).
+Firstly,
+there is a turn-around at scattering angles of 60◦–80◦,
+and secondly the polarized flux increases again at the
+smallest scattering angles. The similar trend has also
+been reported in the disk around HD 100546 (Stolker
+et al. 2016). The effects of radiative transfer within the
+disk surface would not account for this trends as long
+as the disk structure is axisymmetric and is uniformly
+illuminated by the central star. For example, multiple
+scattering tends to decrease the polarization flux on the
+forward scattering side, but its dependency is monotonic
+and would not explain the re-rise at the smallest scat-
+tering angles. If this is the case, the tendency has to be
+attributed to the intrinsic properties of dust particles.
+However, the high porosity aggregates that are thought
+to explain category I do not show such a tendency, as
+already shown in figure 5 (a) and (b). This means that
+
+Phase functions of planet-forming disks
+11
+the dust particles in these system are different from the
+ones in category I.
+One possibility is low-porosity aggregates. Figure 5(c)
+shows that a turn-around at scattering angles around
+80◦ and a re-rise in polarization flux at small scattering
+angles appear simultaneously when the porosity is as
+low as ∼ 55%. Since this feature is not prominent for a
+higher porosity, it seems to be triggered by an increased
+contribution of monomer-monomer electromagnetic in-
+teraction as the monomers get packed closely for lower
+porosity. Therefore, the phase functions for the disks
+around HD 163296, PDS 70, and perhaps RXJ1615,
+might be explained by low porosity aggregates.
+Low
+porosity aggregates tend to show a reddish polarized
+intensity color (Tazaki et al. 2019).
+For RXJ 1615
+Avenhaus et al. (2018) found that J/H=0.78±0.42 thus
+indeed the disk appears red, i.e.
+is brighter in the
+H-band relative to the central star than in the J-
+band.
+We repeated an analog measurement to that
+described in Avenhaus et al. (2018) for the HD 163296
+and the PDS 70 systems.
+We found J/H=0.75±0.11
+and J/H=0.94±0.21 for HD 163296 and PDS 70, respec-
+tively. Thus the polarized intensity colors for all three
+systems are consistent with the presence of low porosity
+aggregates in the surface layers of the disk (although we
+caution that in the cases of RXJ 1615 and PDS 70 the
+error bars are large on the color measurement).
+Low porosity aggregates have relative small area-to-
+mass ratios (i.e.g, weak dynamical coupling with gas),
+making them prone to settling on the disk midplane. Ef-
+ficient vertical mixing would be necessary to keep these
+aggregates in the disk surface layers (e.g., Mulders et al.
+2013; Tazaki et al. 2021). Interestingly, the disks around
+HD 163296, PDS 70, and HD 100546 (studied in Stolker
+et al. 2016) have been suggested to host a planet in the
+disk (Teague et al. 2018; Keppler et al. 2018; Haffert
+et al. 2019; Quanz et al. 2013; Currie et al. 2014; Casas-
+sus & Pérez 2019). The presence of planets has been
+suggested to influence vertical mixing of dust particles
+in disks by meridional circulation(Bi et al. 2021; Binkert
+et al. 2021; Szulágyi et al. 2022). Therefore, we specu-
+late that the presence of planets might indirectly affect
+the dust properties at the disk surface, which might in
+turn explain the distinct phase functions from the oth-
+ers.
+4.2. Trends with system properties
+As discussed in section 4.1 we find an empirical di-
+chotomy in the shape of the phase functions indicating
+different dust aggregate porosities. In order to find pos-
+sible trends with basic system parameters we plot the
+phase functions of all systems in order of stellar spec-
+tral type, system age, disk dust mass and disk inclina-
+tion in figure 6. For the stellar spectral type and the
+disk dust mass we do not see any correlation between
+the two different categories of phase functions. For the
+system age we notice that the three disks with lower
+dust porosities are among the younger sources in our
+sample, with ages of 5.4 Myr for PDS 70 (Müller et al.
+2018), 5.1 Myr for HD163296 (Alecian et al. 2013) and
+1.4 Myr for RXJ 1615 (Wahhaj et al. 2010).
+However
+the presumably youngest source in our sample, IM Lup
+(1.1 Myr, Avenhaus et al. 2018) is again part of the
+higher porosity category. We stress that our sample is
+small and that individual system ages are inherently un-
+certain, thus an expanded study with a large number of
+systems will be needed to confirm if such a trend indeed
+exists.
+In the rightmost panel of figure 6 we order the phase
+functions by disk inclination. Here we find that the two
+phase functions with a monotonous slope and strong
+forward scattering peak were extracted from the two
+systems with the highest inclination in our study, i.e.
+MY Lup (77◦) and HD 34282 (57◦). Conversely the two
+phase functions for the systems with the lowest inclina-
+tion show a monotonous and shallow slope with no in-
+dication for a similar forward scattering peak (although
+we note that in these cases the smallest scattering angles
+could not be sampled). This trend with inclination may
+well correspond to the limb brightening effect discussed
+in section 3.4.
+4.3. Multi-ringed systems
+For three systems with multiple ring-like features we
+were able to extract the polarized phase function at mul-
+tiple separations: RXJ 1615, IM Lup and V 4046 Sgr.
+For RXJ 1615 we extracted the phase function from the
+innermost resolved ring between 28 au and 59 au as well
+as the brightest full ring at a radial separation between
+146 au and 181 au. The resulting extractions in the H-
+band are shown in figure 2. For the comparison we con-
+sider the extraction of the outer ring which was corrected
+for azimuthal shadowing as discussed in section 3.3 (the
+black, solid curve in figure 2).
+The two phase func-
+tions show a very different shape. While the outer disk
+shows the previously mentioned peak between 60◦ and
+80◦, the phase function of the inner disk zone is well
+described by a single slope for angles between 40◦ and
+120◦, but shows strong peaks at larger and smaller an-
+gles. Both of them seem to favor compact aggregates
+that have relatively high fractal dimensions, e.g., the
+BPCA aggregates shown in figure 5 (b). In order to ex-
+plain the dip, the porosity needs to be relatively low,
+e.g., Pmax ∼ 55 %, at least for the outer region (figure 5
+
+12
+Ginski et al.
+20
+40
+60
+80
+100 120 140 160
+scattering angle (deg)
+1
+2
+3
+4
+5
+6
+7
+8
+spectral type
+early
+late
+B9. 5
+M0
+20
+40
+60
+80
+100 120 140 160
+scattering angle (deg)
+1
+2
+3
+4
+5
+6
+7
+8
+age
+old
+young
+12. 7Myr
+1. 1Myr
+20
+40
+60
+80
+100 120 140 160
+scattering angle (deg)
+1
+2
+3
+4
+5
+6
+7
+8
+dust mass
+high
+low
+140M ⊕
+12M ⊕
+20
+40
+60
+80
+100 120 140 160
+scattering angle (deg)
+1
+2
+3
+4
+5
+6
+7
+8
+normalized polarized flux
+inclination
+high
+low
+i = 77. 0 ◦
+i = 30. 0 ◦
+Figure 6. Phase functions for all 10 target systems ordered by various system parameters. We show the J-band data for
+all systems but one since it is more complete. For the RXJ 1852 system we show the H-band data as there are no J-band
+observations of this system. The color code is the same as in figure 3 for all systems. We indicate for each panel on the left
+y-axis the system parameter by which the phase functions were sorted and indicate the extreme values of these parameters in
+the plot.
+c). Since the phase functions are very different from each
+other, the inner and outer disk surfaces are likely domi-
+nated by different types of compact dust aggregates. As
+we already discuss in the previous section, the presence
+of low porosity aggregates in the upper disk atmosphere
+may indicate the presence of a perturber that leads to
+more efficient vertical mixing. This fits also well with
+our discussion in section 3.3 which indicates that there
+is a warp present in the outer disk of the RXJ 1615 sys-
+tem, consistent with previous results by de Boer et al.
+(2016).
+We discuss the IM Lup system in great detail in Tazaki,
+Ginski & Dominik (submitted). As a brief summary we
+note that the two innermost disk zones between 70 au
+and 110 au and between 130 au and 170 au are well con-
+sistent with each other (see figure 10). The outermost
+extraction zone between 217 au and 257 au shows a much
+shallower slope toward small scattering angles (but also
+has intrinsically much larger uncertainties due to the
+less well defined surface height profile in the outer disk,
+see figure 7).
+As we argue in Tazaki, Ginski & Do-
+minik (submitted), this may simply be an indication
+that the outer disk region is not well described anymore
+by the same power-law profile as the inner regions due
+to decreasing surface density and thus decreasing optical
+depth.
+The V 4046 Sgr system has arguably the most well de-
+fined dual ring structure of all disks in this study and
+shows no indication of significant azimuthal shadowing.
+We extracted the phase function from the inner region
+between 11 au and 19 au as well as from the outer ring
+between 23 au and 34 au (see figure 12). We find that
+both phase functions are well consistent with each other
+for angles larger than ∼80◦, while the inner disk shows
+a slightly (but significantly) smaller slope for smaller
+scattering angles. It is unclear if this slight difference
+is due to the intrinsic phase functions in the inner and
+outer ring (and thus indicates slightly different aggre-
+gate properties) or if it can be attributed to observa-
+tional effects such as limb brightening. Indeed, due to
+the steeper slope in the outer ring of the flared disk
+surface we would expect a slightly stronger limb bright-
+ening effect for the outer ring, which may then explain
+the small deviations between the two observed phase
+functions.
+5. SUMMARY
+We measure for the first time the polarized scattering
+phase functions of 10 young planet forming disks ob-
+served in the near infrared. We find that even though
+the geometry of these disks is complex, phase functions
+with meaningful uncertainties can be extracted. While
+detailed radiative transfer models for individual systems
+are required to disentangle observational effects such as
+limb brightening or azimuthal shadowing from the in-
+trinsic phase function of the dust particles, we can still
+infer some general trends from the extracted phase func-
+tions.
+1. We find empirically two distinct categories of
+phase functions.
+Category I has a monotonous
+slope, while category II displays a local maximum
+between 60◦ to 80◦.
+2. Disks in category I have phase functions consis-
+tent with micron-sized, high porosity aggregates,
+while disks in category II require micron-sized, low
+porosity aggregates to explain their phase func-
+tion.
+3. Category II disks appear consistent with red polar-
+ized intensity colors between the J/H band, as pre-
+dicted for micron-sized, low porosity aggregates.
+
+Phase functions of planet-forming disks
+13
+4. While we do not find general correlations with ba-
+sic system parameters for the two phase function
+categories we do note that category II disks include
+the HD 163296 and the PDS 70 systems, both of
+which host embedded planets.
+Furthermore the
+literature data of the HD 100546 system, which is
+also suggested to host planets is consistent with
+phase function category II.
+5. If the presence of low porosity aggregates is an
+indication for the presence of embedded planets,
+then this may indicate that the RXJ 1615 system,
+which also belongs to the category II disks, hosts
+an embedded planet similar to the cases of PDS 70
+and HD 163296
+As further near infrared observations of young disks
+become available it will be most interesting to repeat
+the extraction performed for the small sample in this
+study to investigate if the two tentative categories that
+we identify are indeed present in the larger population.
+ACKNOWLEDGMENTS
+C.G. dedicates this work to the memory of Gisela
+Else Mäder.
+Her many years of selfless support en-
+abled his past and present scientific work.
+C.G. ac-
+knowledges funding from the Netherlands Organisation
+for Scientific Research (NWO) TOP-1 grant as part
+of the research program “Herbig Ae/Be stars, Rosetta
+stones for understanding the formation of planetary sys-
+tems”, project number 614.001.552. R.T. acknowledges
+the JSPS overseas research fellowship. R.T. also thank
+Daniel Mackowski for making the MSTM codes pub-
+licly available.
+R.T. awould also like to thank Bruce
+Draine for the availability of particle data of BA, BAM1,
+and BAM2 and Yasuhiko Okada for providing a genera-
+tion code for BCCA. SPHERE is an instrument designed
+and built by a consortium consisting of IPAG (Greno-
+ble, France), MPIA (Heidelberg, Germany), LAM (Mar-
+seille, France), LESIA (Paris, France), Laboratoire La-
+grange (Nice, France), INAF - Osservatorio di Padova
+(Italy), Observatoire de Genève (Switzerland), ETH
+Zurich (Switzerland), NOVA (Netherlands), ONERA
+(France), and ASTRON (The Netherlands) in collabora-
+tion with ESO. SPHERE was funded by ESO, with addi-
+tional contributions from CNRS (France), MPIA (Ger-
+many), INAF (Italy), FINES (Switzerland), and NOVA
+(The Netherlands). SPHERE also received funding from
+the European Commission Sixth and Seventh Frame-
+work Programmes as part of the Optical Infrared Co-
+ordination Network for Astronomy (OPTICON) under
+grant number RII3-Ct2004-001566 for FP6 (2004-2008),
+grant number 226604 for FP7 (2009-2012), and grant
+number 312430 for FP7 (2013-2016). This research has
+used the SIMBAD database, operated at CDS, Stras-
+bourg, France (Wenger et al. 2000). We used the Python
+programming language5, especially the SciPy (Virta-
+nen et al. 2020), NumPy (Oliphant 2006), Matplotlib
+(Hunter 2007) and astropy (Astropy Collaboration et al.
+2013, 2018) packages.
+We thank the writers of these
+software packages for making their work available to the
+astronomical community.
+
+14
+Ginski et al.
+Facilities: VLT(SPHERE)
+Software:
+MSTM v3.0 (Mackowski & Mishchenko
+2011)
+REFERENCES
+Alecian, E., Wade, G. A., Catala, C., et al. 2013, MNRAS,
+429, 1027, doi: 10.1093/mnras/sts384
+Astropy Collaboration, Robitaille, T. P., Tollerud, E. J.,
+et al. 2013, A&A, 558, A33,
+doi: 10.1051/0004-6361/201322068
+Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M.,
+et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
+Avenhaus, H., Quanz, S. P., Schmid, H. M., et al. 2014,
+ApJ, 781, 87, doi: 10.1088/0004-637X/781/2/87
+Avenhaus, H., Quanz, S. P., Garufi, A., et al. 2018, ApJ,
+863, 44, doi: 10.3847/1538-4357/aab846
+Benisty, M., Dominik, C., Follette, K., et al. 2022, arXiv
+e-prints, arXiv:2203.09991.
+https://arxiv.org/abs/2203.09991
+Beuzit, J. L., Vigan, A., Mouillet, D., et al. 2019, A&A,
+631, A155, doi: 10.1051/0004-6361/201935251
+Bi, J., Lin, M.-K., & Dong, R. 2021, ApJ, 912, 107,
+doi: 10.3847/1538-4357/abef6b
+Binkert, F., Szulágyi, J., & Birnstiel, T. 2021, MNRAS,
+506, 5969, doi: 10.1093/mnras/stab2075
+Birnstiel, T., Andrews, S. M., & Ercolano, B. 2012, A&A,
+544, A79, doi: 10.1051/0004-6361/201219262
+Birnstiel, T., Dullemond, C. P., Zhu, Z., et al. 2018, ApJL,
+869, L45, doi: 10.3847/2041-8213/aaf743
+Bohren, C. F., & Huffman, D. R. 1983, Absorption and
+scattering of light by small particles (New York: Wiley)
+Brauer, F., Dullemond, C. P., & Henning, T. 2008, A&A,
+480, 859, doi: 10.1051/0004-6361:20077759
+Canovas, H., Ménard, F., Hales, A., et al. 2013, A&A, 556,
+A123, doi: 10.1051/0004-6361/201321924
+Carbillet, M., Bendjoya, P., Abe, L., et al. 2011,
+Experimental Astronomy, 30, 39,
+doi: 10.1007/s10686-011-9219-4
+Casassus, S., & Pérez, S. 2019, ApJL, 883, L41,
+doi: 10.3847/2041-8213/ab4425
+Chiang, E. I., & Goldreich, P. 1997, ApJ, 490, 368,
+doi: 10.1086/304869
+Chiang, E. I., Joung, M. K., Creech-Eakman, M. J., et al.
+2001, ApJ, 547, 1077, doi: 10.1086/318427
+Currie, T., Muto, T., Kudo, T., et al. 2014, ApJL, 796,
+L30, doi: 10.1088/2041-8205/796/2/L30
+de Boer, J., Salter, G., Benisty, M., et al. 2016, A&A, 595,
+A114, doi: 10.1051/0004-6361/201629267
+de Boer, J., Langlois, M., van Holstein, R. G., et al. 2020a,
+A&A, 633, A63, doi: 10.1051/0004-6361/201834989
+de Boer, J., Ginski, C., Chauvin, G., et al. 2020b, arXiv
+e-prints, arXiv:2010.12202.
+https://arxiv.org/abs/2010.12202
+Dohlen, K., Langlois, M., Saisse, M., et al. 2008, in Society
+of Photo-Optical Instrumentation Engineers (SPIE)
+Conference Series, Vol. 7014, Ground-based and Airborne
+Instrumentation for Astronomy II, ed. I. S. McLean &
+M. M. Casali, 70143L, doi: 10.1117/12.789786
+Dorschner, J., Begemann, B., Henning, T., Jaeger, C., &
+Mutschke, H. 1995, A&A, 300, 503
+Engler, N., Milli, J., Gratton, R., et al. 2022, arXiv e-prints,
+arXiv:2211.11767. https://arxiv.org/abs/2211.11767
+Estrada, P. R., Cuzzi, J. N., & Umurhan, O. M. 2022, ApJ,
+936, 42, doi: 10.3847/1538-4357/ac7ffd
+Garcia, A. J. L., & Gonzalez, J.-F. 2020, MNRAS, 493,
+1788, doi: 10.1093/mnras/staa382
+Garufi, A., Benisty, M., Pinilla, P., et al. 2018, A&A, 620,
+A94, doi: 10.1051/0004-6361/201833872
+Garufi, A., Avenhaus, H., Pérez, S., et al. 2020, A&A, 633,
+A82, doi: 10.1051/0004-6361/201936946
+Garufi, A., Dominik, C., Ginski, C., et al. 2022, A&A, 658,
+A137, doi: 10.1051/0004-6361/202141692
+Ginski, C., Stolker, T., Pinilla, P., et al. 2016, A&A, 595,
+A112, doi: 10.1051/0004-6361/201629265
+Ginski, C., Facchini, S., Huang, J., et al. 2021, ApJL, 908,
+L25, doi: 10.3847/2041-8213/abdf57
+Haffert, S. Y., Bohn, A. J., de Boer, J., et al. 2019, Nature
+Astronomy, 3, 749, doi: 10.1038/s41550-019-0780-5
+Hashimoto, J., Dong, R., Kudo, T., et al. 2012, ApJL, 758,
+L19, doi: 10.1088/2041-8205/758/1/L19
+Henning, T., & Stognienko, R. 1996, A&A, 311, 291
+Herczeg, G. J., & Hillenbrand, L. A. 2014, ApJ, 786, 97,
+doi: 10.1088/0004-637X/786/2/97
+Hunter, J. D. 2007, Computing in Science and Engineering,
+9, 90, doi: 10.1109/MCSE.2007.55
+Isella, A., Guidi, G., Testi, L., et al. 2016, PhRvL, 117,
+251101, doi: 10.1103/PhysRevLett.117.251101
+Kataoka, A., Tanaka, H., Okuzumi, S., & Wada, K. 2013,
+A&A, 557, L4, doi: 10.1051/0004-6361/201322151
+Keppler, M., Benisty, M., Müller, A., et al. 2018, A&A,
+617, A44, doi: 10.1051/0004-6361/201832957
+Kharchenko, N. V. 2001, Kinematika i Fizika Nebesnykh
+Tel, 17, 409
+Kobayashi, H., & Tanaka, H. 2021, ApJ, 922, 16,
+doi: 10.3847/1538-4357/ac289c
+
+Phase functions of planet-forming disks
+15
+Krijt, S., Ormel, C. W., Dominik, C., & Tielens,
+A. G. G. M. 2016, A&A, 586, A20,
+doi: 10.1051/0004-6361/201527533
+Krolikowski, D. M., Kraus, A. L., & Rizzuto, A. C. 2021,
+AJ, 162, 110, doi: 10.3847/1538-3881/ac0632
+Lorek, S., Lacerda, P., & Blum, J. 2018, A&A, 611, A18,
+doi: 10.1051/0004-6361/201630175
+Luhman, K. L. 2022, AJ, 163, 24,
+doi: 10.3847/1538-3881/ac35e2
+Mackowski, D. W., & Mishchenko, M. I. 2011, JQSRT, 112,
+2182, doi: 10.1016/j.jqsrt.2011.02.019
+Marino, S., Perez, S., & Casassus, S. 2015, ApJL, 798, L44,
+doi: 10.1088/2041-8205/798/2/L44
+Martinez-Brunner, R., Casassus, S., Pérez, S., et al. 2022,
+MNRAS, 510, 1248, doi: 10.1093/mnras/stab3440
+McCabe, C., Duchêne, G., & Ghez, A. M. 2002, ApJ, 575,
+974, doi: 10.1086/341479
+Milli, J., Vigan, A., Mouillet, D., et al. 2017, A&A, 599,
+A108, doi: 10.1051/0004-6361/201527838
+Milli, J., Engler, N., Schmid, H. M., et al. 2019, A&A, 626,
+A54, doi: 10.1051/0004-6361/201935363
+Min, M., Hovenier, J. W., & de Koter, A. 2005, A&A, 432,
+909, doi: 10.1051/0004-6361:20041920
+Mukai, T., Ishimoto, H., Kozasa, T., Blum, J., &
+Greenberg, J. M. 1992, A&A, 262, 315
+Mulders, G. D., Min, M., Dominik, C., Debes, J. H., &
+Schneider, G. 2013, A&A, 549, A112,
+doi: 10.1051/0004-6361/201219522
+Mulders, G. D., Pascucci, I., Manara, C. F., et al. 2017,
+ApJ, 847, 31, doi: 10.3847/1538-4357/aa8906
+Müller, A., Keppler, M., Henning, T., et al. 2018, A&A,
+617, L2, doi: 10.1051/0004-6361/201833584
+Muro-Arena, G. A., Dominik, C., Waters, L. B. F. M., et al.
+2018, A&A, 614, A24, doi: 10.1051/0004-6361/201732299
+Muro-Arena, G. A., Benisty, M., Ginski, C., et al. 2020,
+A&A, 635, A121, doi: 10.1051/0004-6361/201936509
+Okuzumi, S., Tanaka, H., Kobayashi, H., & Wada, K. 2012,
+ApJ, 752, 106, doi: 10.1088/0004-637X/752/2/106
+Oliphant, T. E. 2006, A guide to NumPy, Vol. 1 (Trelgol
+Publishing USA)
+Olofsson, J., Milli, J., Bayo, A., Henning, T., & Engler, N.
+2020, A&A, 640, A12, doi: 10.1051/0004-6361/202038237
+Ormel, C. W., Spaans, M., & Tielens, A. G. G. M. 2007,
+A&A, 461, 215, doi: 10.1051/0004-6361:20065949
+Pecaut, M. J., & Mamajek, E. E. 2013, ApJS, 208, 9,
+doi: 10.1088/0067-0049/208/1/9
+—. 2016, MNRAS, 461, 794, doi: 10.1093/mnras/stw1300
+Pinte, C., Price, D. J., Ménard, F., et al. 2018, ApJL, 860,
+L13, doi: 10.3847/2041-8213/aac6dc
+Quanz, S. P., Amara, A., Meyer, M. R., et al. 2013, ApJL,
+766, L1, doi: 10.1088/2041-8205/766/1/L1
+Quanz, S. P., Schmid, H. M., Geissler, K., et al. 2011, ApJ,
+738, 23, doi: 10.1088/0004-637X/738/1/23
+Rich, E. A., Monnier, J. D., Aarnio, A., et al. 2022, AJ,
+164, 109, doi: 10.3847/1538-3881/ac7be4
+Sartori, M. J., Lépine, J. R. D., & Dias, W. S. 2003, A&A,
+404, 913, doi: 10.1051/0004-6361:20030581
+Shen, Y., Draine, B. T., & Johnson, E. T. 2008, ApJ, 689,
+260, doi: 10.1086/592765
+Stapper, L. M., Hogerheijde, M. R., van Dishoeck, E. F., &
+Mentel, R. 2022, A&A, 658, A112,
+doi: 10.1051/0004-6361/202142164
+Stolker, T., Dominik, C., Avenhaus, H., et al. 2016, A&A,
+595, A113, doi: 10.1051/0004-6361/201528039
+Szulágyi, J., Binkert, F., & Surville, C. 2022, ApJ, 924, 1,
+doi: 10.3847/1538-4357/ac32d1
+Tamura, M. 2016, Proceedings of the Japan Academy,
+Series B, 92, 45, doi: 10.2183/pjab.92.45
+Tazaki, R., & Dominik, C. 2022, A&A, 663, A57,
+doi: 10.1051/0004-6361/202243485
+Tazaki, R., Murakawa, K., Muto, T., Honda, M., & Inoue,
+A. K. 2021, ApJ, 921, 173, doi: 10.3847/1538-4357/ac1f8c
+Tazaki, R., Tanaka, H., Muto, T., Kataoka, A., & Okuzumi,
+S. 2019, MNRAS, 485, 4951, doi: 10.1093/mnras/stz662
+Teague, R., Bae, J., Bergin, E. A., Birnstiel, T., &
+Foreman-Mackey, D. 2018, ApJL, 860, L12,
+doi: 10.3847/2041-8213/aac6d7
+Thalmann, C., Mulders, G. D., Hodapp, K., et al. 2014,
+A&A, 566, A51, doi: 10.1051/0004-6361/201322915
+Thalmann, C., Janson, M., Garufi, A., et al. 2016, ApJL,
+828, L17, doi: 10.3847/2041-8205/828/2/L17
+van der Marel, N., Dong, R., di Francesco, J., Williams,
+J. P., & Tobin, J. 2019, ApJ, 872, 112,
+doi: 10.3847/1538-4357/aafd31
+van der Marel, N., & Mulders, G. D. 2021, AJ, 162, 28,
+doi: 10.3847/1538-3881/ac0255
+van Holstein, R. G., Girard, J. H., de Boer, J., et al. 2020,
+A&A, 633, A64, doi: 10.1051/0004-6361/201834996
+Villenave, M., Benisty, M., Dent, W. R. F., et al. 2019,
+A&A, 624, A7, doi: 10.1051/0004-6361/201834800
+Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020,
+Nature Methods, 17, 261,
+doi: https://doi.org/10.1038/s41592-019-0686-2
+Wahhaj, Z., Cieza, L., Koerner, D. W., et al. 2010, ApJ,
+724, 835, doi: 10.1088/0004-637X/724/2/835
+Warren, S. G., & Brandt, R. E. 2008, Journal of
+Geophysical Research (Atmospheres), 113, D14220,
+doi: 10.1029/2007JD009744
+
+16
+Ginski et al.
+Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, A&AS,
+143, 9, doi: 10.1051/aas:2000332
+Zsom, A., Ormel, C. W., Güttler, C., Blum, J., &
+Dullemond, C. P. 2010, A&A, 513, A57,
+doi: 10.1051/0004-6361/200912976
+Zubko, V. G., Mennella, V., Colangeli, L., & Bussoletti, E.
+1996, MNRAS, 282, 1321, doi: 10.1093/mnras/282.4.1321
+
+Phase functions of planet-forming disks
+17
+Table 3. Observing dates, instrument setup and weather conditions for all systems in our study.
+Target
+Date
+Filter
+DIT [s]
+# frames
+Seeing [arcsec]
+τ0 [ms]
+ESO ID
+RXJ 1615.3-3255
+14-03-2016
+BB_H
+64
+80
+1.1
+2.6
+096.C-0523(A)
+14-03-2016
+BB_J
+64
+48
+1.1
+2.2
+096.C-0523(A)
+HD 163296
+26-05-2016
+BB_H
+8 (16)
+64 (64)
+1.0
+2.8
+097.C-0523(A)
+26-05-2016
+BB_J
+16
+200
+0.8
+2.0
+097.C-0523(A)
+IM Lup
+13-03-2016
+BB_H
+64
+48
+1.3
+3.4
+096.C-0523(A)
+11-03-2016
+BB_J
+64
+56
+0.9
+1.5
+096.C-0523(A)
+LkCa 15
+19-12-2015
+BB_J
+32
+120
+0.7
+2.4
+096.C-0248(A)
+PDS 66
+15-03-2016
+BB_H
+64
+56
+0.9
+2.8
+096.C-0523(A)
+14-03-2016
+BB_J
+64
+48
+1.1
+2.5
+096.C-0523(A)
+PDS 70
+09-08-2019
+BB_H
+64
+36
+1.5
+2.1
+0102.C-0916(B)
+26-03-2016
+BB_J
+64
+52
+1.9
+1.3
+096.C-0333(A)
+2MASSJ18521730-3700119
+15-05-2017
+BB_H
+64
+12
+1.2
+-
+099.C-0147(B)
+V 4046 Sgr
+13-03-2016
+BB_H
+64
+48
+1.0
+2.3
+096.C-0523(A)
+12-03-2016
+BB_J
+64
+48
+1.5
+1.7
+096.C-0523(A)
+HD 34282
+19-12-2015
+BB_J
+64
+88
+0.6
+3.0
+096.C-0248(A)
+MY Lup
+15-03-2016
+BB_H
+64
+40
+0.7
+3.7
+096.C-0523(A)
+15-03-2016
+BB_J
+64
+35
+0.9
+2.4
+096.C-0523(A)
+APPENDIX
+A. SUMMARY OF OBSERVATION SETUPS AND CONDITIONS
+B. PHASE FUNCTION EXTRACTION OF ALL SYSTEMS
+C. T-MATRIX CALCULATIONS
+We considered four different types of dust aggregation: Ballistic Cluster-Cluster Aggregation (BCCA); Ballistic
+Particle-Cluster Aggregation (BPCA), and two modified versions of BPCA, known as BAM1 and BAM2 (Shen et al.
+2008). BCCA has a fractal dimension of 1.9 and therefore has a highly open structure, whereas the other three have a
+fractal dimension close to three. The main difference between BPCA, BAM1, and BAM2 are their porosities; the lowest
+for BAM2. We assume a spherical and single-sized monomer for the computational convenience. Each monomer has a
+dust composition with a mixture of water ice (Warren & Brandt 2008), pyroxene silicate (Mg0.7Fe0.3SiO3) (Dorschner
+et al. 1995), amorphous carbon (Zubko et al. 1996), and troilite (Henning & Stognienko 1996) with the mass abundance
+ratios similar to the DSHARP model (Birnstiel et al. 2018). We calculated the effective refractive index (m) by using
+the Bruggeman mixing rule and found m = 1.92 + 0.404i at a wavelength of 1.63 µm.
+Given dust geometry and composition, we calculate the scattering matrix elements of dust aggregates using by
+Multiple Sphere T-Matrix Method (Mackowski & Mishchenko 2011). In the simulations, we assume that aggregates
+are randomly orientated, i.e.., ignoring grain alignment, and their optical properties were averaged over all possible
+orientation with equal probability by using the analytical orientation averaging technique of the T-matrix method.
+The results were also averaged over four realizations of each aggregate model.
+Once we obtain the optical properties for each aggregate, we then averaged the optical properties by considering
+aggregate-size distribution:
+n(a)da ∝ a−3.5da (amin ≤ a ≤ amax),
+(C1)
+where a is the volume-equivalent radius of an aggregate, defined by a = amonN 1/3; amon being the radius of the
+monomer and N being the number of monomers. The minimum aggregate radius is fixed to amin = 2amon, and the
+maximum aggregate radius amax is a parameter. We consider two different monomer radii: amon = 0.1 µm and 0.4 µm.
+The largest aggregates we investigated have amax = 1.6 µm. Since the volume-equivalent radius does not necessarily
+represents the apparent size of an aggregate, we introduce the characteristic radius ac (Mukai et al. 1992), which better
+
+18
+Ginski et al.
+2
+1
+0
+1
+2
+RXJ1615
+HD163296
+30
+20
+10
+8
+6
+4
+2
+0
+scattering angle deviation (deg)
+2
+1
+0
+1
+2
+IMLup
+LkCa15
+2
+1
+0
+1
+2
+∆Dec (arcsec)
+PDS66
+PDS70
+2
+1
+0
+1
+2
+RXJ1852
+V4046Sgr
+2
+1
+0
+1
+2
+∆RA (arcsec)
+2
+1
+0
+1
+2
+HD34282
+2
+1
+0
+1
+2
+MYLup
+Figure 7. Same as the bottom panel of figure 1 but for all disks in our sample. Shown are the maximum deviations in scattering
+angle based on the flared and the flat surface height profiles.
+
+Phase functions of planet-forming disks
+19
+1.0
+0.5
+0.0
+0.5
+1.0
+∆RA (arcsec)
+1.0
+0.5
+0.0
+0.5
+1.0
+∆Dec (arcsec)
+HD34282
+1.0
+0.5
+0.0
+0.5
+1.0
+∆RA (arcsec)
+0
+24
+48
+72
+96
+120
+144
+168
+scattering angle (deg)
+50
+60
+70
+80
+90
+100
+110
+120
+130
+scattering angle (deg)
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+polarized scattering phase function
+40..240 au
+Figure 8. Extracted polarized scattered phase functions and extraction regions for the J-band data of the HD 34282 system.
+Panels are analog to figure 2.
+0.5
+0.0
+0.5
+∆RA (arcsec)
+0.5
+0.0
+0.5
+∆Dec (arcsec)
+0.5
+0.0
+0.5
+∆RA (arcsec)
+0
+24
+48
+72
+96
+120
+144
+168
+scattering angle (deg)
+20
+40
+60
+80
+100
+120
+140
+160
+scattering angle (deg)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+polarized scattering phase function
+79..140 au
+Figure 9. Extracted polarized scattered phase functions and extraction regions for the H-band data of the MY Lup system.
+Panels are analog to figure 2.
+3
+2
+1
+0
+1
+2
+3
+∆RA (arcsec)
+3
+2
+1
+0
+1
+2
+3
+∆Dec (arcsec)
+IMLup
+3
+2
+1
+0
+1
+2
+3
+∆RA (arcsec)
+0
+24
+48
+72
+96
+120
+144
+168
+scattering angle (deg)
+20
+40
+60
+80
+100
+120
+140
+scattering angle (deg)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+polarized scattering phase function
+70..110 au
+130..170 au
+217..257 au
+Figure 10. Extracted polarized scattered phase functions and extraction regions for the H-band data of the IM Lup system.
+Panels are analog to figure 2. We note that this figure is identical to figure 1 in Tazaki et al. (submitted).
+
+20
+Ginski et al.
+0.5
+0.0
+0.5
+∆RA (arcsec)
+0.5
+0.0
+0.5
+∆Dec (arcsec)
+LkCa15
+0.5
+0.0
+0.5
+∆RA (arcsec)
+0
+24
+48
+72
+96
+120
+144
+168
+scattering angle (deg)
+20
+40
+60
+80
+100
+120
+scattering angle (deg)
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+polarized scattering phase function
+58..94 au
+Figure 11. Extracted polarized scattered phase functions and extraction regions for the J-band data of the LkCa 15 system.
+Panels are analog to figure 2.
+0.5
+0.0
+0.5
+∆RA (arcsec)
+0.5
+0.0
+0.5
+∆Dec (arcsec)
+V4046Sgr
+0.5
+0.0
+0.5
+∆RA (arcsec)
+0
+24
+48
+72
+96
+120
+144
+168
+scattering angle (deg)
+40
+50
+60
+70
+80
+90
+100
+110
+120
+scattering angle (deg)
+0.0
+0.5
+1.0
+1.5
+2.0
+polarized scattering phase function
+11..19 au
+23..34 au
+Figure 12. Extracted polarized scattered phase functions and extraction regions for the H-band data of the V 4046 Sgr system.
+Panels are analog to figure 2.
+1.0
+0.5
+0.0
+0.5
+1.0
+∆RA (arcsec)
+1.0
+0.5
+0.0
+0.5
+1.0
+∆Dec (arcsec)
+PDS66
+1.0
+0.5
+0.0
+0.5
+1.0
+∆RA (arcsec)
+0
+24
+48
+72
+96
+120
+144
+168
+scattering angle (deg)
+40
+50
+60
+70
+80
+90
+100
+110
+120
+scattering angle (deg)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+polarized scattering phase function
+64..104 au
+Figure 13. Extracted polarized scattered phase functions and extraction regions for the H-band data of the PDS 66 system.
+Panels are analog to figure 2.
+
+Phase functions of planet-forming disks
+21
+0.5
+0.0
+0.5
+∆RA (arcsec)
+0.5
+0.0
+0.5
+∆Dec (arcsec)
+2MASS1852
+0.5
+0.0
+0.5
+∆RA (arcsec)
+0
+24
+48
+72
+96
+120
+144
+168
+scattering angle (deg)
+40
+50
+60
+70
+80
+90
+100
+110
+120
+scattering angle (deg)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+polarized scattering phase function
+36..68 au
+Figure 14. Extracted polarized scattered phase functions and extraction regions for the H-band data of the RXJ 1852 system.
+Panels are analog to figure 2.
+0.5
+0.0
+0.5
+∆RA (arcsec)
+0.5
+0.0
+0.5
+∆Dec (arcsec)
+HD163296
+0.5
+0.0
+0.5
+∆RA (arcsec)
+0
+24
+48
+72
+96
+120
+144
+168
+scattering angle (deg)
+20
+40
+60
+80
+100
+120
+140
+scattering angle (deg)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+polarized scattering phase function
+55..75 au
+54..74 au, symmetric
+Figure 15. Extracted polarized scattered phase functions and extraction regions for the H-band data of the HD 163296 system.
+Panels are analog to figure 2.
+0.5
+0.0
+0.5
+∆RA (arcsec)
+0.5
+0.0
+0.5
+∆Dec (arcsec)
+PDS70
+0.5
+0.0
+0.5
+∆RA (arcsec)
+0
+24
+48
+72
+96
+120
+144
+168
+scattering angle (deg)
+20
+40
+60
+80
+100
+120
+140
+scattering angle (deg)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+polarized scattering phase function
+42..73 au
+Figure 16. Extracted polarized scattered phase functions and extraction regions for the H-band data of the PDS 70 system.
+Panels are analog to figure 2.
+
+22
+Ginski et al.
+2
+1
+0
+1
+2
+RXJ1615
+H-band
+HD163296
+H-band
+6.00
+4.00
+2.00
+1.00
+0.50
+0.25
+flux ratio
+2
+1
+0
+1
+2
+IMLup
+H-band
+LkCa15
+J-band
+2
+1
+0
+1
+2
+∆Dec (arcsec)
+PDS66
+H-band
+PDS70
+H-band
+2
+1
+0
+1
+2
+RXJ1852
+H-band
+V4046Sgr
+H-band
+2
+1
+0
+1
+2
+∆RA (arcsec)
+2
+1
+0
+1
+2
+HD34282
+J-band
+2
+1
+0
+1
+2
+MYLup
+H-band
+Figure 17. Axis symmetry of all disks in our sample relative to the disk minor axis. H-band data is shown when available
+otherwise J-band data. The disk images were flipped around the minor axis and then the original images was divided by the
+flipped image. Thus flux ratios>1 indicate the factor by which the disk region is brighter than the corresponding axis-symmetric
+region.
+
+Phase functions of planet-forming disks
+23
+describes the apparent size. We measure the porosity by P = 1 − (a/ac)3. The characteristic radius and porosity of
+the maximum aggregate in the size distribution will be denoted by ac,max and Pmax, respectively.
+
diff --git a/ztE3T4oBgHgl3EQfmgqk/content/tmp_files/load_file.txt b/ztE3T4oBgHgl3EQfmgqk/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..bb2f0bc0eded80f2338ceca21a3380dfa4e9505a
--- /dev/null
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf,len=1659
+page_content='Draft version January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2023  Typeset using LATEX twocolumn style in AASTeX63  Observed polarized scattered light phase functions of planet-forming disks  Christian Ginski  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2 Ryo Tazaki  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 4 Carsten Dominik  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2 and Tomas Stolker  1  1Leiden Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Leiden University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' PO Box 9513,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2300 RA Leiden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The Netherlands  2Anton Pannekoek Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' University of Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Science Park 904,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 1098XH Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The Netherlands  3Institute of Planetology and Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Université Grenoble Alpes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 38000 Grenoble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' France  4Astronomical Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Graduate School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 6-3 Aramaki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Aoba-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Sendai 980-8578,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Japan  (Received January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Revised –;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Accepted –)  Submitted to ApJ  ABSTRACT  Dust particles are the building blocks from which planetary bodies are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  A major goal of  the studies of planet-forming disks is to constrain the properties of dust particles and aggregates in  order to trace their origin, structure, and the associated growth and mixing processes in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Observations of scattering and/or emission of dust in a location of the disk often lead to degenerate  information about the kind of particles, such as size, porosity, or fractal dimension of aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Progress can be made by deriving the full (polarizing) scattering phase function of such particles at  multiple wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This has now become possible by careful extraction from scattered light images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Such an extraction requires knowledge about the shape of the scattering surface in the disk and we  discuss how to obtain such knowledge as well as the associated uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We use a sample of  disk images from observations with VLT/SPHERE to, for the first time, extract the phase functions  of a whole sample of disks with broad phase angle coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We find that polarized phase functions  come in two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Comparing the extracted functions with theoretical predictions from rigorous  T-Matrix computations of aggregates, we show that one category can be linked back to fractal, porous  aggregates, while the other is consistent with more compact, less porous aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We speculate that  the more compact particles become visible in disks where embedded planets trigger enhanced vertical  mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Keywords: Exoplanet formation(492) — Circumstellar disks(235) — Direct imaging(387) — Polarime-  try(1278)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' INTRODUCTION  Gas- and dust-rich circumstellar disks are the sites  of ongoing planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' However, the early stage  of planet formation, such as the formation of planetes-  imals, remains a matter of discussion, as it comes with  a number of barriers inhibiting grain growth and form-  ing planetesimals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Brauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Zsom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Birnstiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The sizes and structures of  growing dust aggregates are of crucial relevance to the  barriers because these quantities influence the sticking  and aerodynamic properties (Ormel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Zsom  Corresponding author: Christian Ginski  ginski@strw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='leidenuniv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='nl  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Okuzumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Kataoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Krijt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Lorek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Garcia & Gonzalez  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Kobayashi & Tanaka 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Estrada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022)  Thus by determining these properties by disk observa-  tions, one can conclude the early planet formation and  transport processes including vertical mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  The past decade was revolutionary for resolved ob-  servations of young planet-forming disks in the near-  infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Driven by advances in instrumentation several  large surveys have been conducted or are still ongoing  such as SEEDS (Tamura 2016), DARTTS-S (Avenhaus  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2020), Gemini-LIGHTS (Rich  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022) and SPHERE-DESTINYS (Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' A summary of the field was recently presented  by Benisty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Due to these ongoing observa-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='04617v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='EP]  11 Jan 2023  ID2  Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  tional programs more than 150 systems have now been  observed in (polarized) near-infrared scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' A  large diversity of sub-structures has been discovered,  such as rings, gaps and spirals, which are typically asso-  ciated with ongoing planet formation (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Benisty  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  The analysis of the scattered light data  often focused on either the basic disk geometry, tracing  illumination and shadowing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022) as  a function of disk aspect ratio and disk symmetry, or on  the disk morphology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  However, the appearance of these objects in (polar-  ized) scattered light is also strongly dependent on the  properties of the dust grains or aggregates in the upper  disk atmosphere (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  In par-  ticular the scattering angle dependent amount of flux  that we receive from the different regions of the disk,  the so-called (polarized) scattering phase function, en-  codes dust grain and aggregate properties (Tazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' While the extraction of polarized scattered light  phase functions has been done for geometrically flat de-  bris disks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Milli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2017, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Olofsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Engler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022), it is less common for young  gas rich disks, due to their more complex geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' So  far this has only been done for the disks around the Her-  big stars HD 97048 (Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016) and HD 100546  (Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Stolker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016) in polarized light  and the T Tauri star multiple system GG Tau (McCabe  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2002) in total intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  In this study we gathered a sample of 10 disks for  which the surface height profile has been determined  in the literature from near-infrared scattered light ob-  servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The sample is comprised of Herbig and T  Tauri stars with the earliest spectral type being the B9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  star HD 34282 and the latest spectral type the M0 star  IM Lup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The systems cover a range of ages from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1 Myr  (IM Lup, Avenhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018) to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='7 Myr (MY Lup,  Avenhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  All systems were also already  observed at (sub)mm-wavelengths which allowed to es-  timate dust masses based on their continuum flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Our  sample spans roughly an order of magnitude between  the lowest mass disk around PDS 70 (13 M⊕) and the  highest mass disk in the RXJ 1615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3-3255 system with  (140 M⊕).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' A summary of all included systems with the  appropriate references is given in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Due to the  ring-shaped sub-structure in most of these disks, planet  formation is thought to be ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In particular our  study includes the PDS 70 system, in which two young  planets have been detected inside the disk cavity (Kep-  pler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Haffert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2019), and the HD 163296  system for which the presence of two wide separation  planets has been inferred from ALMA gas kinematic ob-  servations (Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Teague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In the  following section we briefly describe the observational  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In section 3 we discuss how the phase functions  were extracted, while we discuss their interpretation in  light of dust aggregate models in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We summa-  rize our results in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  System  SpType  Mdust (M⊕)  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  RXJ 1615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3-3255  K5  140  (1),(12)  HD 163296  A1  75  (2),(3)  IM Lup  M0  54  (1),(3)  LkCa 15  K5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  33  (4),(3)  PDS 66  K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  15  (5),(3)  PDS 70  K7  12  (1),(3)  2MASSJ18521730-3700119  K4  13  (6),(3)  V 4046 Sgr  K4  48  (7),(10)  HD 34282  B9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  87  (8),(11)  MY Lup  K0  53  (1),(9)  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (1) Luhman (2022) (2) Sartori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2003) (3) van  der Marel & Mulders (2021) (4) Krolikowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2021)  (5) Pecaut & Mamajek (2016) (6) Herczeg & Hillenbrand  (2014) (7) Pecaut & Mamajek (2013) (8) Kharchenko (2001)  (9) Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2017) (10) Martinez-Brunner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2022)  (11) Stapper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2022) (12) van der Marel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2019)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' OBSERVATIONS AND DATA REDUCTION  All datasets used in our study have been obtained with  VLT/SPHERE (Beuzit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2019) and its near infrared  camera IRDIS (Dohlen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  The instrument  was operated in dual-beam polarization imaging mode  (DPI, de Boer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' van Holstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2020), in  either J or H broad band filters, to obtain (linear) po-  larized scattered light images of the circumstellar disks  in each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' An overview of the observation setup  and conditions is given in table 3 in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In  all cases the innermost 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5 mas around the stellar po-  sition where covered by the standard YJH_S apodized  Lyot coronagraph (Carbillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2011) and are there-  fore inaccessible for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' All data sets that are  included in our study have been previously discussed in  the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  We give the relevant references in ta-  ble 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The data reduction has in all cases been carried  out with the public IRDAP (IRDIS Data reduction for  Accurate Polarimetry, van Holstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2020) pipeline  with default settings1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This includes a full model based  determination and removal of instrumental polarization,  as well as the measurement and subsequent subtraction  of astrophysical stellar polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  1 https://irdap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='io  Phase functions of planet-forming disks  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' PHASE FUNCTION EXTRACTION  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Surface height profiles  The key challenge in extracting the scattered light  phase function of young gas-rich disks is the uncertainty  of the vertical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Here we are particularly in-  terested at which height above the disk mid-plane the  optical depth τ becomes equal to 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' the surface layer  from which the majority of the disk scattered light orig-  inates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For ease of use within the further discussion we  will refer to this as the surface height of the disk within  this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2  While the surface height may generally be inferred from  detailed radiative transfer modeling, there exists a sub-  class of disks for which it can be directly determined  from the data itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' As shown by de Boer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2016)  and Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2016) the disk surface height can be  computed for disks with radial sub-structures such as  multiple rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This is done by measuring the inclina-  tion of the ring and its offset along the minor axis from  the central star position in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This directly gives  the surface height at the ring location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' If there are multi-  ple rings then the radial dependence of the disk surface  height can be directly traced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Both of the mentioned  studies found that the surface height profile for the two  studied target systems (RXJ 1615 and HD97048) can be  described reasonably well with a single power law profile  of the form:  H(r) = Href/(rref/1au)α ∗ (r/1au)α,  (1)  wherein H(r) is the radial dependent surface height,  Href is a reference height at reference separation rref  (all in au) and α is the flaring exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Avenhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  (2018) found that the surface height profile for five disks  in their study could all be described by the same power-  law profile with a flaring exponent of α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  This indicates that for single-ringed disks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' when only  a surface height at a single radial separation is known,  we may still infer a reasonable guess of the radial de-  pendent surface height profile by using the height mea-  surement as reference height in the power-law and by  assuming a standard flaring exponent of α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  For our sample we draw the surface profile parame-  ters from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For the RXJ 1615, IM Lup and  V 4046 Sgr multi-ring systems we use the specific power-  law profiles fitted by Avenhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For the  HD 34282 multi-ring system we likewise use the power-  2 We note that this is not identical to the pressure scale height of  the disk, which is typically a factor 3-4 smaller than the scattered  light surface height (Chiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  law profile given by de Boer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For the re-  maining single-ringed disks we use the literature values  for the surface height of the rings as reference height and  the standard flaring exponent of α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='22 by Avenhaus  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  For the RXJ1852 and PDS 70 systems  a different flaring exponent was infered from radiative  transfer modelling by Villenave et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2019) and Kep-  pler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2018), respectively, and we make use of their  fitted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  In order to capture the uncertainty of the extracted  phase functions due to the uncertainty of the surface  height we always consider three scenarios for each sys-  tem:  (1) the nominal surface height profile, (2) a  strongly "flared" profile and a (3) "flat" profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  The  flared and flat profile consider the uncertainty of the  reference height, the reference separation and the flar-  ing exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For the "flared" profile we use the upper  bound on the reference height, the lower bound on the  reference separation and the upper bound on the flaring  exponent, while we switch lower and upper bounds for  each parameter for the "flat" profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We summarize all  profile parameters for each system in table 2 and illus-  trate all three profiles (nominal, flared and flat) for the  RXJ 1615 system in figure 1, top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In the bottom  panel of figure 1 we show the corresponding deviations in  scattering angle between the "flared" and "flat" surface  profile extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' While we see deviations of up to ∼5◦  in the inner disk region, these are smaller in the outer  disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We note that the intermediate region between in-  ner disk and outer edge, which shows deviations of less  than 1◦ is close to the reference separation for which  the reference height was directly measured by Avenhaus  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Thus the deviation in this region is dom-  inated by the (small) uncertainties of these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  We show similar figures for the maximum deviation of  the scattering angle for our complete sample in figure 7  in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Unsurprisingly the largest deviations  are found close to the outer disk edge for the systems  with large uncertainties in their flaring exponent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  IM Lup and HD 34282 in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In general we are  avoiding these outer disk regions for phase function ex-  traction and typically consider regions for which the un-  certainty in scattering angle is less than ∼10◦, with the  main exception being the two aforementioned systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  However, as we explain in the following section we do  for all systems incorporate the resulting deviations of  the extracted phase functions between all three surface  height profiles in their uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Extraction with diskmap  For the extraction of the phase functions we use the  publicly available diskmap python package by Stolker  4  Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  System  d  i  PA  h/rα  ref  rref  α  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='25  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Values for the surface height geometry of the studied disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We give the nominal values used as well as the values  for the minimal and maximal surface height that we considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' References are: (1) Muro-Arena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2018) (2) Isella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  (2016) (3) Avenhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2018) (4) Thalmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2014) (5) Thalmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2016) (6) Keppler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2018) (7) Hashimoto  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2012) (8) Villenave et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2019) (9) de Boer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2020b)  Phase functions of planet-forming disks  5  1  0  1  ∆RA (arcsec)  1  0  1  ∆Dec (arcsec)  RXJ1615  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='0  scattering angle deviation (deg)  0  50  100  150  200  250  300  r (au)  0  10  20  30  40  50  60  70  H (au)  RXJ1615 surface profile  100  101  10-1  100  101  inner region  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Top:  Exemplary τ=1 surface profile for the  RXJ 1615 system, as used in our phase function extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  The solid black curve indicates the nominal surface profile,  while the blue-dotted and red-dashed curves indicate the  flared and flat extremes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  We consider theses  as the boundaries for the region of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The inset  shows the region inside 50 au on a log scale for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Bot-  tom: Maximum deviation of the scattering angles between  the flared and flat disk profiles for each position within the  image of the RXJ 1615 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We find the largest devia-  tion of up to 5◦ close to the star and significantly smaller  deviations further out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Note that the dark region with the  smallest deviations was used for normalization of the surface  power-law profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2016)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' While the package is described in detail  in Stolker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2016), we give a brief summary here  on the extraction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' As described in the previous  section the τ=1 surface for all our system is described  3 https://diskmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='io  by a single power-law profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Given this profile and the  inclination of the disk diskmap calculates for each pixel  in the image the distance from the central star and sub-  sequently the angle under which scattered light is re-  ceived from this part of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We show this for the  RXJ 1615 system in figure 2 (middle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The flux  is corrected for the square-distance dependent illumina-  tion drop-off and is then extracted for each pixel, giving  a single data point for the polarized scattering phase  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' These data points are then placed in bins of  scattering angles with a width of 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For each bin the  median scattering angle of all included pixels and the  median flux is computed, giving the final data point for  this angular bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The standard deviation within each  bin is used as a measure for the uncertainty of the phase  function and captures effects due to the width of the bin  as well as photometric uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  To include the uncertainty of the surface height profile  we repeat the extraction for the "flat" and the "flared"  extreme cases and calculate for each angle bin the flux  difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We consider this difference as the uncertainty  of the phase function introduced by the uncertainty of  the surface height profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We quadratically combine this  uncertainty with the standard deviation in each bin for  the nominal extraction and consider the result the total  uncertainty of the extracted phase function at each an-  gle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  For each system we selected extraction regions centered  on known bright ring features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Since the surface height  profile is directly measured at the ring locations, this  minimizes the introduced uncertainty while simultane-  ously selecting the region of the disk with the highest  signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' If multiple rings are present then  we separately extracted the phase function for each ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  In figure 2 we show the two selected extraction regions  for the RXJ 1615 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The extraction and individual  phase functions for all systems are shown in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  In figure 3 we show the final extracted phase function for  all systems and photometric bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We note that in fig-  ure 3 we show the average phase function for the IM Lup  and V 4046 Sgr systems instead of extractions for indi-  vidual rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For the HD 163296 and RXJ 1615 systems  we show phase functions after correction for azimuthal  shadowing as discussed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Effects of azimuthal shadowing  An aspect that may complicate the extraction of the  phase function in some systems is azimuthal shadowing  since it changes the brightness of disk regions due to re-  duced illumination, an effect that needs to be separated  from the phase function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' There are now a number of  class II objects known for which shadows are observed  6  Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='5  ∆RA (arcsec)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  ∆RA (arcsec)  0  24  48  72  96  120  144  168  scattering angle (deg)  20  40  60  80  100  120  140  scattering angle (deg)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  polarized scattering phase function  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='.59 au  146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='.181 au  146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='.181 au, symmetric  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Extracted polarized scattered phase functions and extraction regions for the H-band data of the RXJ 1615 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Left: H-band data of RXJ 1615 scaled with the square of the distance from the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The blue and red transparent  overlays highlight the two regions used for phase function extraction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' the inner disk region close to the coronagraph and  the bright outer ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Middle: Position dependent scattering angles calculated with the nominal surface height profile of the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Right: Extracted phase functions for the inner disk region (blue-dashed), the outer ring (red-dashed), and an azimuthal  region of 180◦ centered on the north-western ansae of the outher ring (black-solid), which should be less affected by azimuthal  shadowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  in scattered light (see Benisty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022 for an overview  and detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' One of the most iconic systems  is probably HD 142527 (Canovas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Avenhaus  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2014) for which inner and outer disk are strongly  misaligned (70◦, Marino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2015), leading to two nar-  row shadows projected on the outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Such narrow  and well defined shadows are not of major concern for  the extraction of the overall phase function as the small  region affected by the shadows can simply be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  However, there as was shown for the HD 139614 sys-  tem by Muro-Arena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2020), small misalignments  or warps in the disk can lead to very broad and some-  what diffuse shadowing, which covers in extreme cases  an azimuthal range of more than 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For such systems  the problem is two fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' On the one hand the exclusion  of large azimuthal regions of the disk from the extrac-  tion may severely limit the range of scattering angles  for which the phase function can be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' On the  other hand, these shadows are less well defined making it  more difficult to decide which regions of the disk might  be trusted for phase function extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  To estimate if the disks in our sample may be affected  by broad shadowing effects we performed a simple anal-  ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' If there is no shadowing present and the brightness  distribution of disk structures is solely due to the dust  scattering phase function4, then we would expect the  disk surface brightness to be axis-symmetric relative to  4 We imply here the assumptions that there are no azimuthal vari-  ations in the dust grain size distribution or composition, that the  disks are not eccentric and there are no or only small azimuthal  variations in surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  the disk minor axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Thus for all our systems we flipped  the disk images around the minor axis and then divided  the original image by the flip image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This indicates the  brightness ratio between the two "mirrored" sides of the  disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We show the axis-symmetric brightness ratio for all  systems in the appendix in figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For most systems  the deviation in brightness is smaller than a factor ∼2,  with the notable exceptions of the RXJ 1615 system (up  to factor ∼4), the HD 163296 system (up to factor ∼5)  and the HD 34282 system (up to factor ∼6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We thus  consider that the RXJ 1615 and the HD 163296 systems  maybe be affected by broad shadowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This was in fact  discussed for both systems in the literature by de Boer  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2016) and Muro-Arena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2018), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  In the case of HD 34282 de Boer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2020b) discuss a  possible spiral within the disk, thus in this case we may  rather trace a genuine azimuthal asymmetry in the dust  surface density rather than shadowing effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  To give an indication how strongly these shadowing ef-  fects may affect the extracted phase function we per-  formed two separate extractions for the RXJ 1615 sys-  tem choosing the bright ring between deprojected radii  of 146 au and 181 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' One extraction only considered the  brighter north-west side of the disk, while the second ex-  traction only considered the fainter south-east side, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  the side of the ring more strongly affected by shadowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  The results are shown in figure 4 (both phase functions  were normalized at scattering angles of 90◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' While the  profiles somewhat match between 60◦ and 90◦ they de-  viate significantly at larger and smaller scattering an-  gles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='In particular the phase function extracted from  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='the south-east half of the disk shows more relative flux  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='Phase functions of planet-forming disks  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='120  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='140  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='160  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='scattering angle (deg)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='normalized polarized flux  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='H-band  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='HD34282  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='MY Lup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='IM Lup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='LkCa 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='V4046 Sgr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='PDS 66  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='RXJ1852  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='RXJ1615  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='HD163296  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='PDS 70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='120  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='140  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='160  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='scattering angle (deg)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='normalized polarized flux  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='J-band  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='CategoryI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='CategoryII  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Final extracted polarized scattered light phase functions for all targets in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We show the H-band data on  the left and the J-band data on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' All phase functions were normalized at scattering angles of 90◦ and had an arbitrary  offset applied for better visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Colors indicate the same systems in both panels and the order of the phase functions from  top to bottom is the same as indicated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Note that not for all targets both H and J-band were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  in both ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' If we consider that the shadow is cen-  tered on the south-east ansae, this might be explained  by the stronger shadowing close to 90◦ scattering angles,  or put in a different way, the disk may rise out of the  shadow at regions seen under small and large scattering  angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  We note that there is no geometric reason why the disk  should specifically be warped or misaligned around the  minor axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In practice there will in fact most likely be  a deviation between the disk minor axis (defined by our  arbitrary viewing geometry), and the warp or misalign-  ment axis around which the disk tilts as a function of  radial separation from the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' However, if the  axis of misalignment were closer to the disk major axis,  then one would not expect a strong brightness asymme-  try between the two disk ansae as seen in figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' de  Boer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2016) also noted that the brightness asym-  metry seemed to switch sides between subsequent ring  structures in the disk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  in the next outer (overall  fainter) ring the south-east ansae is brighter than the  north-west ansae indicating a continuous warp in the  disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' If the disk near or far side were strongly affected  by this effect then one may expect strong changes in  brightness asymmetries between disk near and far side  8  Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  seen in subsequent rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This does however not seem  to be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We can thus conclude that the broad  shadowing effect is likely centered at least near the disk  ansae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This is further supported by the fact that the  phase function extracted from the south-east side of the  disk and shown in figure 4 increases the relative flux  level toward both the forward and back-scattering sides  of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' If the shadow were not centered close to  the south-east ansae, then we would naively expect ei-  ther a steeper phase function relative to the north-east  at larger scattering angles or a flatter phase function at  small scattering angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Given our analysis we thus consider the phase function  extracted from the north west side of the disk in the  RXJ 1615 system to be minimally or in any case less af-  fected by shadowing and use it for further analysis and  comparison to other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  This phase function is  shown as black solid line in figure 2 (named "symmet-  ric" in the legend as this is the phase function of the  point symmetric version of this disk relative to the mi-  nor axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For the HD 163296 system we then follow the  same strategy and choose the north-west side of the disk  for phase function extraction as it appears less affected  by shadowing compared to the south-east.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  For the HD 34282 system the situation is different, as  the detected asymmetry likely traces a genuine asym-  metry in dust surface density profile, possibly due to  spiral density waves in the disk gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' It is then not clear  which regions may be best suited to extract an unbiased  scattering phase function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We thus include in this case  the full azimuthal range in the extraction and caution  that a more detailed analysis of the system with dedi-  cated radiative transfer modelling should be performed  in the future to revise our preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Limb brightening  In addition to shadowing effects, the shape of the  phase function may be influenced by the viewing ge-  ometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In recent studies of optically thin debris disks  Olofsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2020) and Engler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2022) found that  the relative flux extracted at ∼90◦ scattering angles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  close to the disk ansae, may be enhanced compared to  the intrinsic phase function of the present dust particles  due to a higher column density along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  This effect is sometimes referred to as "limb brighten-  ing" (Engler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The systems in our sample are  all at an earlier evolutionary stage before the gas disper-  sal in the disk and it is generally assumed that they are  optically thick at near infrared wavelengths (Chiang &  Goldreich 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Thus we do not expect that the col-  umn density along the line of sight will change signifi-  cantly for different scattering angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' However, due to  40  60  80  100  120  scattering angle (deg)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  polarized scattering phase function  RXJ1615 phase function asymmetry  South-East  North-West  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Polarized scattered phase functions of the  RX 1615 H-band data extracted from the brightest ring in  the data set between projected separations of 146 and 181 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  The blue-dashed phase function was extracted from an az-  imuthal region of 180◦ centered on the north-west ansae of  the disk, while the red solid curve was extracted from a re-  gion centered on the south-east ansae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Both phase functions  were normalized at scattering angles of 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The differences  may be explained by azimuthal shadowing of the extraction  region due to an inner disk component, which strongly af-  fected the south-western region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  the flaring surface height profiles of these systems we  may instead expect an artificial increase in the flux ra-  tio between the disk forward and back scattering sides,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' between small and large scattering angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' At small  scattering angles (on the disk near side) the line of sight  encompasses more of the illuminated disk surface than is  the case for large scattering angles (on the disk far side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  We discuss this effect in detail for the IM Lup system in  Tazaki, Ginski & Dominik (submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Using radiative  transfer models we found in this study that the effect de-  pends on the inclination, the local aspect ratio and flar-  ing exponent of the disk surface height profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For the  specific case of the IM Lup system this may introduce  a ∼25% brightening for scattering angles smaller than  ∼50◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' However, we caution that this strongly depends  on the dust composition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' for dust with high scat-  tering albedos limb brightening may be much smaller  or even insignificant because multiple scattering reduces  the polarized flux at smaller scattering angles and starts  to (at least partly) counteract this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Dedicated ra-  diative transfer modelling is required to determine the  influence of this effect for individual systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  In a more general sense this limb brightening effect will  Phase functions of planet-forming disks  9  typically not strongly alter the shape of the phase func-  tions for optically thick disks, such as the ones discussed  in our current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Rather it will slightly increase  the slope of the overall phase functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We note that  there may be one exception to this, which is the MY Lup  system, which is seen under a particularly high inclina-  tion of 77◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The extracted phase function in figure 3 is  strongly peaked toward small scattering angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Based  on the results in Tazaki, Ginski & Dominik (submitted),  we find it likely that the intrinsic phase function of dust  particles in MY Lup has a significantly smaller slope,  possibly with no up turn at small scattering angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' As  we summarize in table 2 the remaining systems in our  study have either a comparable inclination to IM Lup  (this is the case for HD 34282) or are seen under signifi-  cantly smaller inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Thus, while the effect should  be considered for future detailed modelling of individual  systems, it will not strongly affect the general popula-  tion level trends that we recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' RESULTS AND DISCUSSION  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Qualitative inference of dust properties  Figure 3 shows that the extracted polarization phase  function is diverse in terms of their shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The shapes  can be roughly divided into two categories: those that  are monotonically increasing in polarized flux with de-  creasing scattering angle (category I: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', HD 34282,  MY Lup, IM Lup, LkCa 15, V4046 Sgr) and those that  turn around at a scattering angle of 60◦–80◦, go through  a local minimum, and then increase again at the small-  est scattering angles (category II: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', HD 163296, PDS  70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  To illustrate the origins of these variations, we per-  form T-matrix light-scattering calculations for various  dust aggregates (Tazaki & Dominik 2022) (see also Sec-  tion C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The scattering matrix element (−S12 in Bohren  & Huffman 1983) obtained by the simulations are sum-  marized in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' There is a caveat when comparing  the scattering matrix element and the observed phase  function: a planet-forming disk is optically thick in the  near infrared, and the scattering angle dependence of  the observed polarization flux might have been affected  by radiative transfer effects, such as multiple scatter-  ing within the disk surface and limb brightening (see a  detailed discussion of these effects in Tazaki, Ginski &  Dominik, submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  To distinguish it from the one  extracted from an observed image, we will refer to the  computed scattering matrix element as the intrinsic po-  larization phase function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  While a three-dimensional  radiative transfer calculation is necessary to determine  the dust parameters more accurately from the extracted  polarization phase function, it is possible to capture the  general trend from the intrinsic phase function and infer  the origins of the variations in shape shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Figure 5 (a) illustrates how the intrinsic polarization  phase function of aggregates varies with their size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The  scattering angle dependency of the polarized intensity is  the result of the dependence of the two quantities over-  lapping: total-intensity phase function and the degree of  linear polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' When the aggregates are small com-  pared to the wavelength, the scattered light distribution  will be close to isotropic in total intensity, but the degree  of polarization approaches 0 in the forward scattering  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' As a result, the intrinsic polarization phase  function turns around at an angle of scattering around  90◦ (for the case of pure Rayleigh scattering, the intrin-  sic polarization function is proportional to 1−cos2 θ, see  Bohren & Huffman (1983)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' As the aggregates become  larger, forward scattering in total intensity develops and  compensates for the decrease in the degree of polariza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Consequently, the turn-around position shifts to  the small-angle side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Except for RXJ1852, none of the observed phase  functions presented in figure 3 exhibits the Rayleigh-  scattering-like profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This indicates that the aggregates  have grown to at least micron sizes in those disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For  RXJ1852, the function appears to have a peak in polar-  ized flux around a scattering angle of 80◦, and the profile  may be consistent with the presence of small, submicron-  sized aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Since the presence of such small parti-  cles makes the disk scattered light bluish (Tazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2019), future multi-color observations would be useful  to draw a conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  The shape diversity in polarization phase functions  could be resulting from a diversity of the structure and  porosity of micron-sized aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' First of all, we fo-  cus on category I (the top six curves in the J band);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' all  show monotonically increasing polarizing flux with de-  creasing scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In particular, the curves for  V4046 Sgr, LkCa 15, and IM Lup seem to have approx-  imately constant slopes, while the slope of the curves  for MY Lup and HD 34282 become steeper at scatter-  ing angles below 80◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Figure 5 (b) demonstrates that  aggregates with different fractal dimensions can explain  these differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Aggregates with a low fractal dimen-  sion (around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='9) exhibit nearly constant slopes except  for scattering angles below 30◦, whereas aggregates with  a high fractal dimension (around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0) show a similar  slope to fractal aggregates in the large scattering an-  gle region, but the slope becomes steeper in the small  scattering angle region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Therefore, the approximately  constant slope observed in V4046 Sgr, LkCa 15, and IM  Lup may be explained by fractal aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In fact, the  detailed radiative transfer calculations by Tazaki, Gin-  10  Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  0  20  40  60  80 100 120 140 160 180  Scattering angle (degrees)  0  1  2  3  4  Intrinsic polarization phase function  (a) Aggregate size  amax =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2 m, ac, max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='359 m  amax =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='4 m, ac, max =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='05 m  amax =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='8 m, ac, max =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='24 m  amax =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='6 m, ac, max =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0 m  0  20  40  60  80 100 120 140 160 180  Scattering angle (degrees)  0  1  2  3  4  Intrinsic polarization phase function  amax =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='6 m  (b) Fractal dimension  ac, max =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='14 m, Df =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  ac, max =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0 m, Df =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='9  0  20  40  60  80 100 120 140 160 180  Scattering angle (degrees)  0  1  2  3  4  Intrinsic polarization phase function  amax =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='6 m  (c) Porosity  ac, max =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='12 m,  max =86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5%  ac, max =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='47 m,  max =72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='8%  ac, max =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='09 m,  max =54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='9%  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Effect of aggregate size, fractal dimension, and porosity on the intrinsic polarization phase function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The phase  functions are normalized to a scattering angle of 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (a) The intrinsic functions for BCCA aggregates for various radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The  monomer radius is set as amon = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (b) The blue and violet lines represent the results for BPCA (ac,max = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='14 µm, Df =  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0) and BCCA (ac,max = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0 µm, Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='9) aggregates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In these computations we used amax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='6 µm and  amon = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (c) The blue, orange, and brown lines represent the results for BPCA (ac,max = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='12 µm, Pmax = 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5%),  BAM1 (ac,max = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='47 µm, Pmax = 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='8%), and BAM2 (ac,max = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='09 µm, Pmax = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='9%) aggregates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In these  computations we used amax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='6 µm and amon = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  ski and Dominik (submitted) showed that fractal aggre-  gates best explains the observed phase function of the  IM Lup disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' On the other hand, the phase functions for  MY Lup and HD 34282 point to the presence of aggre-  gates with a high fractal dimension, although the poros-  ity is still high (∼ 87%), unless the limb brightening is  responsible for the steepening of the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Therefore,  these disks might contain highly porous aggregates, al-  though a full radiative transfer modeling is needed to  determine the fractal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Category II exhibits peculiar phase functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=',  RXJ 1615, HD 163296, PDS 70 in figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Firstly,  there is a turn-around at scattering angles of 60◦–80◦,  and secondly the polarized flux increases again at the  smallest scattering angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The similar trend has also  been reported in the disk around HD 100546 (Stolker  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The effects of radiative transfer within the  disk surface would not account for this trends as long  as the disk structure is axisymmetric and is uniformly  illuminated by the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For example, multiple  scattering tends to decrease the polarization flux on the  forward scattering side, but its dependency is monotonic  and would not explain the re-rise at the smallest scat-  tering angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' If this is the case, the tendency has to be  attributed to the intrinsic properties of dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  However, the high porosity aggregates that are thought  to explain category I do not show such a tendency, as  already shown in figure 5 (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This means that  Phase functions of planet-forming disks  11  the dust particles in these system are different from the  ones in category I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  One possibility is low-porosity aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Figure 5(c)  shows that a turn-around at scattering angles around  80◦ and a re-rise in polarization flux at small scattering  angles appear simultaneously when the porosity is as  low as ∼ 55%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Since this feature is not prominent for a  higher porosity, it seems to be triggered by an increased  contribution of monomer-monomer electromagnetic in-  teraction as the monomers get packed closely for lower  porosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Therefore, the phase functions for the disks  around HD 163296, PDS 70, and perhaps RXJ1615,  might be explained by low porosity aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Low  porosity aggregates tend to show a reddish polarized  intensity color (Tazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  For RXJ 1615  Avenhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2018) found that J/H=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='42 thus  indeed the disk appears red, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  is brighter in the  H-band relative to the central star than in the J-  band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  We repeated an analog measurement to that  described in Avenhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (2018) for the HD 163296  and the PDS 70 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  We found J/H=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='11  and J/H=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='21 for HD 163296 and PDS 70, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Thus the polarized intensity colors for all three  systems are consistent with the presence of low porosity  aggregates in the surface layers of the disk (although we  caution that in the cases of RXJ 1615 and PDS 70 the  error bars are large on the color measurement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Low porosity aggregates have relative small area-to-  mass ratios (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g, weak dynamical coupling with gas),  making them prone to settling on the disk midplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Ef-  ficient vertical mixing would be necessary to keep these  aggregates in the disk surface layers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Tazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Interestingly, the disks around  HD 163296, PDS 70, and HD 100546 (studied in Stolker  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016) have been suggested to host a planet in the  disk (Teague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Keppler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Haffert  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Casas-  sus & Pérez 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The presence of planets has been  suggested to influence vertical mixing of dust particles  in disks by meridional circulation(Bi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Binkert  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Szulágyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Therefore, we specu-  late that the presence of planets might indirectly affect  the dust properties at the disk surface, which might in  turn explain the distinct phase functions from the oth-  ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Trends with system properties  As discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1 we find an empirical di-  chotomy in the shape of the phase functions indicating  different dust aggregate porosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In order to find pos-  sible trends with basic system parameters we plot the  phase functions of all systems in order of stellar spec-  tral type, system age, disk dust mass and disk inclina-  tion in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For the stellar spectral type and the  disk dust mass we do not see any correlation between  the two different categories of phase functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For the  system age we notice that the three disks with lower  dust porosities are among the younger sources in our  sample, with ages of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='4 Myr for PDS 70 (Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2018), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1 Myr for HD163296 (Alecian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2013) and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='4 Myr for RXJ 1615 (Wahhaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  However  the presumably youngest source in our sample, IM Lup  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1 Myr, Avenhaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018) is again part of the  higher porosity category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We stress that our sample is  small and that individual system ages are inherently un-  certain, thus an expanded study with a large number of  systems will be needed to confirm if such a trend indeed  exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  In the rightmost panel of figure 6 we order the phase  functions by disk inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Here we find that the two  phase functions with a monotonous slope and strong  forward scattering peak were extracted from the two  systems with the highest inclination in our study, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  MY Lup (77◦) and HD 34282 (57◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Conversely the two  phase functions for the systems with the lowest inclina-  tion show a monotonous and shallow slope with no in-  dication for a similar forward scattering peak (although  we note that in these cases the smallest scattering angles  could not be sampled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This trend with inclination may  well correspond to the limb brightening effect discussed  in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Multi-ringed systems  For three systems with multiple ring-like features we  were able to extract the polarized phase function at mul-  tiple separations: RXJ 1615, IM Lup and V 4046 Sgr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  For RXJ 1615 we extracted the phase function from the  innermost resolved ring between 28 au and 59 au as well  as the brightest full ring at a radial separation between  146 au and 181 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The resulting extractions in the H-  band are shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For the comparison we con-  sider the extraction of the outer ring which was corrected  for azimuthal shadowing as discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3 (the  black, solid curve in figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  The two phase func-  tions show a very different shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' While the outer disk  shows the previously mentioned peak between 60◦ and  80◦, the phase function of the inner disk zone is well  described by a single slope for angles between 40◦ and  120◦, but shows strong peaks at larger and smaller an-  gles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Both of them seem to favor compact aggregates  that have relatively high fractal dimensions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', the  BPCA aggregates shown in figure 5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In order to ex-  plain the dip, the porosity needs to be relatively low,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Pmax ∼ 55 %, at least for the outer region (figure 5  12  Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  20  40  60  80  100 120 140 160  scattering angle (deg)  1  2  3  4  5  6  7  8  spectral type  early  late  B9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 5  M0  20  40  60  80  100 120 140 160  scattering angle (deg)  1  2  3  4  5  6  7  8  age  old  young  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 7Myr  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 1Myr  20  40  60  80  100 120 140 160  scattering angle (deg)  1  2  3  4  5  6  7  8  dust mass  high  low  140M ⊕  12M ⊕  20  40  60  80  100 120 140 160  scattering angle (deg)  1  2  3  4  5  6  7  8  normalized polarized flux  inclination  high  low  i = 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 0 ◦  i = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 0 ◦  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Phase functions for all 10 target systems ordered by various system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We show the J-band data for  all systems but one since it is more complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' For the RXJ 1852 system we show the H-band data as there are no J-band  observations of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The color code is the same as in figure 3 for all systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We indicate for each panel on the left  y-axis the system parameter by which the phase functions were sorted and indicate the extreme values of these parameters in  the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Since the phase functions are very different from each  other, the inner and outer disk surfaces are likely domi-  nated by different types of compact dust aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' As  we already discuss in the previous section, the presence  of low porosity aggregates in the upper disk atmosphere  may indicate the presence of a perturber that leads to  more efficient vertical mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This fits also well with  our discussion in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3 which indicates that there  is a warp present in the outer disk of the RXJ 1615 sys-  tem, consistent with previous results by de Boer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  We discuss the IM Lup system in great detail in Tazaki,  Ginski & Dominik (submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' As a brief summary we  note that the two innermost disk zones between 70 au  and 110 au and between 130 au and 170 au are well con-  sistent with each other (see figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The outermost  extraction zone between 217 au and 257 au shows a much  shallower slope toward small scattering angles (but also  has intrinsically much larger uncertainties due to the  less well defined surface height profile in the outer disk,  see figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  As we argue in Tazaki, Ginski & Do-  minik (submitted), this may simply be an indication  that the outer disk region is not well described anymore  by the same power-law profile as the inner regions due  to decreasing surface density and thus decreasing optical  depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  The V 4046 Sgr system has arguably the most well de-  fined dual ring structure of all disks in this study and  shows no indication of significant azimuthal shadowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  We extracted the phase function from the inner region  between 11 au and 19 au as well as from the outer ring  between 23 au and 34 au (see figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We find that  both phase functions are well consistent with each other  for angles larger than ∼80◦, while the inner disk shows  a slightly (but significantly) smaller slope for smaller  scattering angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' It is unclear if this slight difference  is due to the intrinsic phase functions in the inner and  outer ring (and thus indicates slightly different aggre-  gate properties) or if it can be attributed to observa-  tional effects such as limb brightening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Indeed, due to  the steeper slope in the outer ring of the flared disk  surface we would expect a slightly stronger limb bright-  ening effect for the outer ring, which may then explain  the small deviations between the two observed phase  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' SUMMARY  We measure for the first time the polarized scattering  phase functions of 10 young planet forming disks ob-  served in the near infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We find that even though  the geometry of these disks is complex, phase functions  with meaningful uncertainties can be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' While  detailed radiative transfer models for individual systems  are required to disentangle observational effects such as  limb brightening or azimuthal shadowing from the in-  trinsic phase function of the dust particles, we can still  infer some general trends from the extracted phase func-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We find empirically two distinct categories of  phase functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Category I has a monotonous  slope, while category II displays a local maximum  between 60◦ to 80◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Disks in category I have phase functions consis-  tent with micron-sized, high porosity aggregates,  while disks in category II require micron-sized, low  porosity aggregates to explain their phase func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Category II disks appear consistent with red polar-  ized intensity colors between the J/H band, as pre-  dicted for micron-sized, low porosity aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Phase functions of planet-forming disks  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' While we do not find general correlations with ba-  sic system parameters for the two phase function  categories we do note that category II disks include  the HD 163296 and the PDS 70 systems, both of  which host embedded planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Furthermore the  literature data of the HD 100546 system, which is  also suggested to host planets is consistent with  phase function category II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' If the presence of low porosity aggregates is an  indication for the presence of embedded planets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  then this may indicate that the RXJ 1615 system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  which also belongs to the category II disks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' hosts  an embedded planet similar to the cases of PDS 70  and HD 163296  As further near infrared observations of young disks  become available it will be most interesting to repeat  the extraction performed for the small sample in this  study to investigate if the two tentative categories that  we identify are indeed present in the larger population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' dedicates this work to the memory of Gisela  Else Mäder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Her many years of selfless support en-  abled his past and present scientific work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' ac-  knowledges funding from the Netherlands Organisation  for Scientific Research (NWO) TOP-1 grant as part  of the research program “Herbig Ae/Be stars, Rosetta  stones for understanding the formation of planetary sys-  tems”, project number 614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' acknowledges  the JSPS overseas research fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' also thank  Daniel Mackowski for making the MSTM codes pub-  licly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' awould also like to thank Bruce  Draine for the availability of particle data of BA, BAM1,  and BAM2 and Yasuhiko Okada for providing a genera-  tion code for BCCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' SPHERE is an instrument designed  and built by a consortium consisting of IPAG (Greno-  ble, France), MPIA (Heidelberg, Germany), LAM (Mar-  seille, France), LESIA (Paris, France), Laboratoire La-  grange (Nice, France), INAF - Osservatorio di Padova  (Italy), Observatoire de Genève (Switzerland), ETH  Zurich (Switzerland), NOVA (Netherlands), ONERA  (France), and ASTRON (The Netherlands) in collabora-  tion with ESO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' SPHERE was funded by ESO, with addi-  tional contributions from CNRS (France), MPIA (Ger-  many), INAF (Italy), FINES (Switzerland), and NOVA  (The Netherlands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' SPHERE also received funding from  the European Commission Sixth and Seventh Frame-  work Programmes as part of the Optical Infrared Co-  ordination Network for Astronomy (OPTICON) under  grant number RII3-Ct2004-001566 for FP6 (2004-2008),  grant number 226604 for FP7 (2009-2012), and grant  number 312430 for FP7 (2013-2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' This research has  used the SIMBAD database, operated at CDS, Stras-  bourg, France (Wenger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We used the Python  programming language5, especially the SciPy (Virta-  nen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2020), NumPy (Oliphant 2006), Matplotlib  (Hunter 2007) and astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2013, 2018) packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  We thank the writers of these  software packages for making their work available to the  astronomical community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  14  Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Facilities: VLT(SPHERE)  Software:  MSTM v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0 (Mackowski & Mishchenko  2011)  REFERENCES  Alecian, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Wade, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Catala, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2013, MNRAS,  429, 1027, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1093/mnras/sts384  Astropy Collaboration, Robitaille, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Tollerud, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2013, A&A, 558, A33,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201322068  Astropy Collaboration, Price-Whelan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Sipőcz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018, AJ, 156, 123, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-3881/aabc4f  Avenhaus, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Quanz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2014,  ApJ, 781, 87, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1088/0004-637X/781/2/87  Avenhaus, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Quanz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Garufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018, ApJ,  863, 44, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-4357/aab846  Benisty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Dominik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Follette, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022, arXiv  e-prints, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='09991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='org/abs/2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='09991  Beuzit, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Vigan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Mouillet, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2019, A&A,  631, A155, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201935251  Bi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Lin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Dong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2021, ApJ, 912, 107,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-4357/abef6b  Binkert, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Szulágyi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2021, MNRAS,  506, 5969, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1093/mnras/stab2075  Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2012, A&A,  544, A79, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201219262  Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018, ApJL,  869, L45, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/2041-8213/aaf743  Bohren, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Huffman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 1983, Absorption and  scattering of light by small particles (New York: Wiley)  Brauer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Henning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2008, A&A,  480, 859, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361:20077759  Canovas, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Ménard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Hales, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2013, A&A, 556,  A123, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201321924  Carbillet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Bendjoya, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Abe, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2011,  Experimental Astronomy, 30, 39,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1007/s10686-011-9219-4  Casassus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Pérez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2019, ApJL, 883, L41,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/2041-8213/ab4425  Chiang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Goldreich, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 1997, ApJ, 490, 368,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1086/304869  Chiang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Joung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Creech-Eakman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2001, ApJ, 547, 1077, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1086/318427  Currie, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Muto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Kudo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2014, ApJL, 796,  L30, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1088/2041-8205/796/2/L30  de Boer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Salter, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Benisty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016, A&A, 595,  A114, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201629267  de Boer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Langlois, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', van Holstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2020a,  A&A, 633, A63, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201834989  de Boer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Ginski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Chauvin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2020b, arXiv  e-prints, arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='12202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='org/abs/2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='12202  Dohlen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Langlois, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Saisse, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2008, in Society  of Photo-Optical Instrumentation Engineers (SPIE)  Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 7014, Ground-based and Airborne  Instrumentation for Astronomy II, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' McLean &  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Casali, 70143L, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='789786  Dorschner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Begemann, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Henning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Jaeger, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', &  Mutschke, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 1995, A&A, 300, 503  Engler, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Milli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Gratton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022, arXiv e-prints,  arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='11767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='11767  Estrada, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Cuzzi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Umurhan, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022, ApJ,  936, 42, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-4357/ac7ffd  Garcia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Gonzalez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2020, MNRAS, 493,  1788, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1093/mnras/staa382  Garufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Benisty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Pinilla, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018, A&A, 620,  A94, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201833872  Garufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Avenhaus, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Pérez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2020, A&A, 633,  A82, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201936946  Garufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Dominik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Ginski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022, A&A, 658,  A137, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/202141692  Ginski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Stolker, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Pinilla, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016, A&A, 595,  A112, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201629265  Ginski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Facchini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2021, ApJL, 908,  L25, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/2041-8213/abdf57  Haffert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Bohn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', de Boer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2019, Nature  Astronomy, 3, 749, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1038/s41550-019-0780-5  Hashimoto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Dong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Kudo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2012, ApJL, 758,  L19, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1088/2041-8205/758/1/L19  Henning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Stognienko, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 1996, A&A, 311, 291  Herczeg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Hillenbrand, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2014, ApJ, 786, 97,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1088/0004-637X/786/2/97  Hunter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2007, Computing in Science and Engineering,  9, 90, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1109/MCSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='55  Isella, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Guidi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Testi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016, PhRvL, 117,  251101, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='251101  Kataoka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Tanaka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Okuzumi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Wada, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2013,  A&A, 557, L4, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201322151  Keppler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Benisty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Müller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018, A&A,  617, A44, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201832957  Kharchenko, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2001, Kinematika i Fizika Nebesnykh  Tel, 17, 409  Kobayashi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Tanaka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2021, ApJ, 922, 16,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-4357/ac289c  Phase functions of planet-forming disks  15  Krijt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Ormel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Dominik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Tielens,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016, A&A, 586, A20,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201527533  Krolikowski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Kraus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Rizzuto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2021,  AJ, 162, 110, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-3881/ac0632  Lorek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Lacerda, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Blum, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018, A&A, 611, A18,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201630175  Luhman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022, AJ, 163, 24,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-3881/ac35e2  Mackowski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Mishchenko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2011, JQSRT, 112,  2182, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='jqsrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='019  Marino, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Perez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Casassus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2015, ApJL, 798, L44,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1088/2041-8205/798/2/L44  Martinez-Brunner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Casassus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Pérez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022,  MNRAS, 510, 1248, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1093/mnras/stab3440  McCabe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Duchêne, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Ghez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2002, ApJ, 575,  974, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1086/341479  Milli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Vigan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Mouillet, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2017, A&A, 599,  A108, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201527838  Milli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Engler, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2019, A&A, 626,  A54, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201935363  Min, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Hovenier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & de Koter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2005, A&A, 432,  909, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361:20041920  Mukai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Ishimoto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Kozasa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Blum, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', &  Greenberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 1992, A&A, 262, 315  Mulders, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Min, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Dominik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Debes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', &  Schneider, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2013, A&A, 549, A112,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201219522  Mulders, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Pascucci, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2017,  ApJ, 847, 31, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-4357/aa8906  Müller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Keppler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Henning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018, A&A,  617, L2, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201833584  Muro-Arena, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Dominik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Waters, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2018, A&A, 614, A24, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201732299  Muro-Arena, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Benisty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Ginski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2020,  A&A, 635, A121, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201936509  Okuzumi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Tanaka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Kobayashi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Wada, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2012,  ApJ, 752, 106, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1088/0004-637X/752/2/106  Oliphant, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2006, A guide to NumPy, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 1 (Trelgol  Publishing USA)  Olofsson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Milli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Bayo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Henning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Engler, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2020, A&A, 640, A12, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/202038237  Ormel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Spaans, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Tielens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2007,  A&A, 461, 215, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361:20065949  Pecaut, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Mamajek, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2013, ApJS, 208, 9,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1088/0067-0049/208/1/9  —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016, MNRAS, 461, 794, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1093/mnras/stw1300  Pinte, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Price, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Ménard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018, ApJL, 860,  L13, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/2041-8213/aac6dc  Quanz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Amara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Meyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2013, ApJL,  766, L1, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1088/2041-8205/766/1/L1  Quanz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Geissler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2011, ApJ,  738, 23, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1088/0004-637X/738/1/23  Rich, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Monnier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Aarnio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022, AJ,  164, 109, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-3881/ac7be4  Sartori, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Lépine, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Dias, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2003, A&A,  404, 913, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361:20030581  Shen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Draine, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Johnson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2008, ApJ, 689,  260, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1086/592765  Stapper, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Hogerheijde, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', van Dishoeck, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', &  Mentel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022, A&A, 658, A112,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/202142164  Stolker, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Dominik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Avenhaus, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016, A&A,  595, A113, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201528039  Szulágyi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Binkert, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Surville, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022, ApJ, 924, 1,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-4357/ac32d1  Tamura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016, Proceedings of the Japan Academy,  Series B, 92, 45, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2183/pjab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='45  Tazaki, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Dominik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2022, A&A, 663, A57,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/202243485  Tazaki, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Murakawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Muto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Honda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Inoue,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2021, ApJ, 921, 173, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-4357/ac1f8c  Tazaki, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Tanaka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Muto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Kataoka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Okuzumi,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2019, MNRAS, 485, 4951, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1093/mnras/stz662  Teague, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Bae, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Bergin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', &  Foreman-Mackey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018, ApJL, 860, L12,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/2041-8213/aac6d7  Thalmann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Mulders, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Hodapp, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2014,  A&A, 566, A51, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201322915  Thalmann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Janson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Garufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2016, ApJL,  828, L17, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/2041-8205/828/2/L17  van der Marel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Dong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', di Francesco, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Williams,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Tobin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2019, ApJ, 872, 112,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-4357/aafd31  van der Marel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Mulders, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2021, AJ, 162, 28,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3847/1538-3881/ac0255  van Holstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Girard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', de Boer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2020,  A&A, 633, A64, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201834996  Villenave, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Benisty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Dent, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2019,  A&A, 624, A7, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1051/0004-6361/201834800  Virtanen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Gommers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Oliphant, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2020,  Nature Methods, 17, 261,  doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1038/s41592-019-0686-2  Wahhaj, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Cieza, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Koerner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2010, ApJ,  724, 835, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1088/0004-637X/724/2/835  Warren, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Brandt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2008, Journal of  Geophysical Research (Atmospheres), 113, D14220,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1029/2007JD009744  16  Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Wenger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Ochsenbein, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Egret, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2000, A&AS,  143, 9, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content=', Ormel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Güttler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Blum, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', &  Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2010, A&A, 513, A57,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Mennella, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', Colangeli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=', & Bussoletti, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  1996, MNRAS, 282, 1321, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='C-0147(B)  V 4046 Sgr  13-03-2016  BB_H  64  48  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='C-0523(A)  15-03-2016  BB_J  64  35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='4  096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='C-0523(A)  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' SUMMARY OF OBSERVATION SETUPS AND CONDITIONS  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' PHASE FUNCTION EXTRACTION OF ALL SYSTEMS  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' T-MATRIX CALCULATIONS  We considered four different types of dust aggregation: Ballistic Cluster-Cluster Aggregation (BCCA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Ballistic  Particle-Cluster Aggregation (BPCA), and two modified versions of BPCA, known as BAM1 and BAM2 (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' BCCA has a fractal dimension of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='9 and therefore has a highly open structure, whereas the other three have a  fractal dimension close to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The main difference between BPCA, BAM1, and BAM2 are their porosities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' the lowest  for BAM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We assume a spherical and single-sized monomer for the computational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Each monomer has a  dust composition with a mixture of water ice (Warren & Brandt 2008), pyroxene silicate (Mg0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='7Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='3SiO3) (Dorschner  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 1995), amorphous carbon (Zubko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 1996), and troilite (Henning & Stognienko 1996) with the mass abundance  ratios similar to the DSHARP model (Birnstiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We calculated the effective refractive index (m) by using  the Bruggeman mixing rule and found m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='92 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='404i at a wavelength of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='63 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Given dust geometry and composition, we calculate the scattering matrix elements of dust aggregates using by  Multiple Sphere T-Matrix Method (Mackowski & Mishchenko 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' In the simulations, we assume that aggregates  are randomly orientated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='., ignoring grain alignment, and their optical properties were averaged over all possible  orientation with equal probability by using the analytical orientation averaging technique of the T-matrix method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  The results were also averaged over four realizations of each aggregate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Once we obtain the optical properties for each aggregate, we then averaged the optical properties by considering  aggregate-size distribution:  n(a)da ∝ a−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5da (amin ≤ a ≤ amax),  (C1)  where a is the volume-equivalent radius of an aggregate, defined by a = amonN 1/3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' amon being the radius of the  monomer and N being the number of monomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The minimum aggregate radius is fixed to amin = 2amon, and the  maximum aggregate radius amax is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We consider two different monomer radii: amon = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='1 µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  The largest aggregates we investigated have amax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Since the volume-equivalent radius does not necessarily  represents the apparent size of an aggregate, we introduce the characteristic radius ac (Mukai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' 1992), which better  18  Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2  1  0  1  2  RXJ1615  HD163296  30  20  10  8  6  4  2  0  scattering angle deviation (deg)  2  1  0  1  2  IMLup  LkCa15  2  1  0  1  2  ∆Dec (arcsec)  PDS66  PDS70  2  1  0  1  2  RXJ1852  V4046Sgr  2  1  0  1  2  ∆RA (arcsec)  2  1  0  1  2  HD34282  2  1  0  1  2  MYLup  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Same as the bottom panel of figure 1 but for all disks in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Shown are the maximum deviations in scattering  angle based on the flared and the flat surface height profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Phase functions of planet-forming disks  19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  ∆RA (arcsec)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  ∆Dec (arcsec)  HD34282  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  ∆RA (arcsec)  0  24  48  72  96  120  144  168  scattering angle (deg)  50  60  70  80  90  100  110  120  130  scattering angle (deg)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  polarized scattering phase function  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='.240 au  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Extracted polarized scattered phase functions and extraction regions for the J-band data of the HD 34282 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Panels are analog to figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='5  ∆RA (arcsec)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='5  ∆Dec (arcsec)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='5  ∆RA (arcsec)  0  24  48  72  96  120  144  168  scattering angle (deg)  20  40  60  80  100  120  140  160  scattering angle (deg)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='25  polarized scattering phase function  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='.140 au  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Extracted polarized scattered phase functions and extraction regions for the H-band data of the MY Lup system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Panels are analog to figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  3  2  1  0  1  2  3  ∆RA (arcsec)  3  2  1  0  1  2  3  ∆Dec (arcsec)  IMLup  3  2  1  0  1  2  3  ∆RA (arcsec)  0  24  48  72  96  120  144  168  scattering angle (deg)  20  40  60  80  100  120  140  scattering angle (deg)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='5  polarized scattering phase function  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='.110 au  130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='.170 au  217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='.257 au  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Extracted polarized scattered phase functions and extraction regions for the H-band data of the IM Lup system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Panels are analog to figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We note that this figure is identical to figure 1 in Tazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' (submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  20  Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='0  polarized scattering phase function  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content=' Extracted polarized scattered phase functions and extraction regions for the J-band data of the LkCa 15 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Panels are analog to figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content=' Extracted polarized scattered phase functions and extraction regions for the H-band data of the V 4046 Sgr system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Panels are analog to figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content=' Extracted polarized scattered phase functions and extraction regions for the H-band data of the PDS 66 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Panels are analog to figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Phase functions of planet-forming disks  21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='8  polarized scattering phase function  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='.75 au  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='.74 au, symmetric  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Extracted polarized scattered phase functions and extraction regions for the H-band data of the HD 163296 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Panels are analog to figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  ∆RA (arcsec)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  ∆Dec (arcsec)  PDS70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='5  ∆RA (arcsec)  0  24  48  72  96  120  144  168  scattering angle (deg)  20  40  60  80  100  120  140  scattering angle (deg)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='0  polarized scattering phase function  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='.73 au  Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Extracted polarized scattered phase functions and extraction regions for the H-band data of the PDS 70 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Panels are analog to figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  22  Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  2  1  0  1  2  RXJ1615  H-band  HD163296  H-band  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='00  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='25  flux ratio  2  1  0  1  2  IMLup  H-band  LkCa15  J-band  2  1  0  1  2  ∆Dec (arcsec)  PDS66  H-band  PDS70  H-band  2  1  0  1  2  RXJ1852  H-band  V4046Sgr  H-band  2  1  0  1  2  ∆RA (arcsec)  2  1  0  1  2  HD34282  J-band  2  1  0  1  2  MYLup  H-band  Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Axis symmetry of all disks in our sample relative to the disk minor axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' H-band data is shown when available  otherwise J-band data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The disk images were flipped around the minor axis and then the original images was divided by the  flipped image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' Thus flux ratios>1 indicate the factor by which the disk region is brighter than the corresponding axis-symmetric  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content='  Phase functions of planet-forming disks  23  describes the apparent size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' We measure the porosity by P = 1 − (a/ac)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}
+page_content=' The characteristic radius and porosity of  the maximum aggregate in the size distribution will be denoted by ac,max and Pmax, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE3T4oBgHgl3EQfmgqk/content/2301.04617v1.pdf'}